M. Blundevile his exercises containing sixe treatises, the titles wherof are set down in the next printed page: which treatises are verie necessarie to be read and learned of all yoong gentlemen that haue not bene exercised in such disciplines, and yet are desirous to haue knowledge as well in cosmographie, astronomie, and geographie, as also in the arte of navigation ... To the furtherance of which arte of navigation, the said M. Blundevile speciallie wrote the said treatises and of meere good will doth dedicate the same to all the young gentlemen of this realme. Blundeville, Thomas, fl. 1561. This text is an enriched version of the TCP digital transcription A16221 of text S102547 in the English Short Title Catalog (STC 3146). Textual changes and metadata enrichments aim at making the text more computationally tractable, easier to read, and suitable for network-based collaborative curation by amateur and professional end users from many walks of life. The text has been tokenized and linguistically annotated with MorphAdorner. The annotation includes standard spellings that support the display of a text in a standardized format that preserves archaic forms ('loveth', 'seekest'). Textual changes aim at restoring the text the author or stationer meant to publish. This text has not been fully proofread Approx. 1934 KB of XML-encoded text transcribed from 366 1-bit group-IV TIFF page images. EarlyPrint Project Evanston,IL, Notre Dame, IN, St. Louis, MO 2018 A16221 STC 3146 ESTC S102547 99838324 99838324 2698 2240947165

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Early English books online. (EEBO-TCP ; phase 1, no. A16221) Transcribed from: (Early English Books Online ; image set 2698) Images scanned from microfilm: (Early English books, 1475-1640 ; 280:07) M. Blundevile his exercises containing sixe treatises, the titles wherof are set down in the next printed page: which treatises are verie necessarie to be read and learned of all yoong gentlemen that haue not bene exercised in such disciplines, and yet are desirous to haue knowledge as well in cosmographie, astronomie, and geographie, as also in the arte of navigation ... To the furtherance of which arte of navigation, the said M. Blundevile speciallie wrote the said treatises and of meere good will doth dedicate the same to all the young gentlemen of this realme. Blundeville, Thomas, fl. 1561. [8], 350 leaves, 3 folded plates : ill. Printed by Iohn Windet, dwelling at the signe of the crosse Keies, neere Paules wharffe, and are there to be solde, London : 1594. The first leaf is blank except for large signature-mark "A". With volvelles on leaves 149 [i.e. 151]r, 338v, 350r. "A briefe description of the tables of the three speciall right lines belonging to a circle, called sines, lines tangent, and lines secant" and "A plaine treatise of the first principles of cosmographie" each have separate dated title page; "A plaine description of Mercator his two globes", "A plaine and full description of Petrus Plancius his vniversall map", "A very brief and most plaine description of Maister Blagraue his astrolabe", and "A nevv and necessarie treatise of nauigation" each have separate title page; foliation and register are continuous. The sexagenarie table has first line of heading ending "in". Variant 1: first line ends "Division,". Variant 2: with cancel errata pasted over the original. Reproduction of the original in the Henry E. Huntington Library and Art Gallery. eng Mercator, Gerhard, 1512-1594. Plancius, Petrus, 1552-1622. Blagrave, John, d. 1611. Astronomy -- Early works to 1800. Arithmetic -- Early works to 1900. Trigonometry -- Early works to 1800. Early maps -- Early works to 1800. eebo M. Blundevile his exercises, containing sixe treatises, the titles wherof are set down in the next printed page: which treatises are verie necessarie Blundeville, Thomas, 1594 694 253838 178 0 29 0 0 0 8.15 B The rate of 8.15 defects per 10,000 words puts this text in the B category of texts with fewer than 10 defects per 10,000 words. 2005-10 Assigned for keying and markup 2005-11 Keyed and coded from ProQuest page images 2005-12 Sampled and proofread 2006-06 Rekeyed and resubmitted 2006-07 Sampled and proofread 2006-07 Text and markup reviewed and edited 2006-09 Batch review (QC) and XML conversion

M. BLVNDEVILE His Exercises , containing sixe Treatises , the titles wherof are set down in the next printed page : which Treatises are verie necessarie to be read and learned of all yoong Gentlemen that haue not bene exercised in such disciplines , and yet are desirous to haue knowledge as well in Cosmographie , Astronomie , and Geographie , as also in the Arte of Navigation , in which Arte it is impossible to profite without the helpe of these , or such like instructions . To the furtherance of which Arte of Navigation , the said M. Blundevile speciallie wrote the said Treatises and of meere good will doth dedicate the same to all the young Gentlemen of this Realme .

THOV SHALT LABOR FOR PEACE PLENTIE

LONDON . Printed by John Windet , dwelling at the signe of the crosse Keies , neere Paules wharffe , and are there to be solde . 1594.

The Titles of the Treatises contayned in this booke .

FIrst , a verie easie Arithmeticke so plainlie written as any man of a mean capacitie may easilie learn the same without the helpe of any teacher .

Item the first principles of Cosmographie , and especially a plaine treatise of the Spheare , representing the shape of the whole world , together with the chiefest and most necessarie vses of the said Spheare .

Item a plaine and full description of both the Globes , aswell Terrestriall as Celestiall , and all the chiefest and most necessary vses of the same , in the end whereof are set downe the chiefest vses of the Ephemerides of Iohannes Stadius , and of certaine necessarie Tables therein contained for the better finding out of the true place of the Sunne and Moone , and of all the rest of the Planets vpon the Celestiall Globe .

Item a plaine and full description of Petrus Plancius his vniversall Mappe , lately set forth in the yeare of our Lord 1592. contayning more places newly found , aswell in the East and West Indies , as also towards the North Pole , which no other Map made heretofore hath , whereunto is also added how to find out the true distance betwixt anie two places on the land or sea , their longitudes and latitudes being first knowne , and thereby you may correct the skales or Tronkes that be not trulie set downe in anie Map or Carde .

Item , A briefe and plaine description of M. Blagraue his Astrolabe , otherwise called the Mathematicall Iewel , shewing the most necessary vses thereof , and meetest for sea men to know .

Item the first & chiefest principles of Navigation more plainlie and more orderly taught than they haue bene heretofore by some that haue written thereof , lately collected out of the best modern writers , and treaters of that Arte.

And moreover , I haue thought good to adde vnto mine Arithmeticke , as an appendix depending thereon , the vse of the Tables of the three right lines belonging to a circle , which lines are called Sines , lines tangent , and lines secant , whereby many profitable and necessarie conclusions aswell of Astronomie , as of Geometrie are to be wrought only by the help of Arithmeticke , which Tables are set downe by Clauius the Iesuite , a most excellent Mathematician , in his booke of demonstrations made vpon the Spherickes of Theodosius , more trulie printed than those of Monte Regio , which booke whilest I read at mine owne house , together with a loving friend of mine , I took such delight therein , as I mind ( God willing ) if God giue me life , to translate all those propositions , which Clauius himselfe hath set downe of his owne , touching the quantitie of Angles , and of their sides , as well in right line triangles , as in Sphericall triangles : of which matter , a Monte Regio wrote diffusedlie and at large , so Copernicus wrote of the same brieflie , but therewith somewhat obscurelie , as Clauius saith . Moreover , in reading the Geometrie of Albertus Durcrus , that excellent painter , and finding manie of his conclusions verie obscurelie interpreted by his Latine interpreter ( for he himselfe wrote in high Dutch ) I requested a friend of mine , whome I knewe to haue spent some time in the studie of the Mathematicals , not onelie plainelie to translate the foresaide Durerus into English , but also to adde thereunto manie necessary propositions of his owne , which my request he hath ( I thanke him ) verie well perfourmed , not onely to my satisfaction , but also to the great commoditie and profite of all those that desire to bee perfect in Architecture , in the Arte of Painting , in free Masons craft , in Ioyners craft , in Carvers craft , or anie such like Arte commodious and serviceable in any common Wealth , and I hope that he will put the same in print ere it be long , his name I conceale at his owne earnest intreatie , although much against my will , but I hope that he will make himselfe known in the publishing of his Arithmeticke , and the great Arte of Algebra , the one being almost finished , and the other to bee vndertaken at his best leasure , as also in the printing of Durerus , vnto whom he hath added many necessary Geometrical conclusions , not heard of heretofore , together with divers other of his workes as wel in Geometrie as as in other of the Mathematicall sciences , if he be not called away from these his studies by other affaires . In the mean time I pray al young Gentlemen and seamen to take these my labours already ended in good part , whereby I seeke neither praise nor glorie , but onely to profite my countrey .

To the Reader .

I Greatlie reioyce to see so manie of our English Gentlemen , both of the Court and Countrie in these dayes so earnestlie giuen to trauell aswell by sea as land , into straunge and vnknowne countries , and speciallie into the East and West Indies , following therein the good example of diuers worthy knightes and Gentlemen that haue ventured their liues to discouer strange countries to the great honour of their countrie , and to their owne immortall fame . And because that to trauell by sea requireth skil in the Art of Nauigation , in which it is vnpossible for any man to be perfect vnles he first haue his Arithmetick , and also some knowledge in the principles of Cosmographie , and specially to haue the vse of the Spheare , of the two Globes , of the Astrolabe , and crosse staffe , and such like instruments belonging to the Arte of Navigation , I thought good therefore to write the Treatises before mentioned , to serue as an introduction for such yoong Gentlemen as haue not bene exercised in such kind of studies , which Treatises if they shall vouchsafe to read with attentiue minde , and in such order as they are before set downe , I doubt not but that it will cause them hereafter to seeke for further knowledge therein . And in any wise I wish them to begin with my Arithmeticke , the contents whereof are declared in the next chapter following . In the meane time I do earnestly request all yoong Gentlemen to take these my simple pamphlets no lesse thankfullie than they haue done my horse booke , and in so doing I shall haue iust cause to thinke my labour well bestowed .

What cause first mooved the Author to write this Arithmeticke , and with what order it is here taught , which order the contents of the chapters therof hereafter following doe plainly shew .

I Began this Arithmeticke more than seuen yeares since for a vertuous Gentlewoman , and my verie deare frend M. Elizabeth Bacon , the daughter of Sir Nicholas Bacon Knight , a man of most excellent wit , and of most deepe iudgment , and sometime Lord Keeper of the great Seale of England , and latelie ( as shee hath bene manie yeares past ) the most loving and faithfull wife of my worshipfull friend M. Iustice Wyndham , not long since deceased , who for his integritie of life , and for his wisedome and iustice daylie shewed in gouernement , and also for his good hospitalitie deserued great commendation . And though at her request I had made this Arithmeticke so plaine and easie as was possible ( to my seeming ) yet her continuall sicknesse would not suffer her to exercise her selfe therein . And because that diuerse having seene it , and liking my plaine order of teaching therein , were desirous to haue copies thereof , I thought good therefore to print the same , and to augment it with many necessarie rules meet for those that are desirous to studie any part of Cosmographie , Astronomie , or Geographie , and speciallie the Arte of Navigation , in which without Arithmeticke , as I haue said before , they shall hardly profit . But nowe to returne to my matter , the contents of this Arithmeticke are these here following . First , hauing defined what Arithmeticke is , and what numeration or numbring is , and of what parts it consisteth , and how to number any great summe written in many figures , I deale with the foure speciall kindes of Arithmeticke , that is , Addition , Subtraction , multiplication , and division . Then I shewe the order of working in whole numbers , called in Latine Integra , by the rule of proportion , otherwise called the golden rule , or the rule of three . Then I treat of broken numbers called Fractions , setting downe seuen necessarie rules belonging thereunto , by the helpe whereof you shall be the better able to adde , subtract , to multiplie and divide the same . That done , I shew how to vse the rule of three aswell in dealing with sole fractions as with fractions annexed to integrums , which rule of three is threefold , that is , the common rule , the rule reverse , and the double rule , the order of all which three kindes I doe plainlie teach by examples , shewing wherein , how , and when , they are to be vsed . Next to that I set downe the rule of fellowship , a necessarie rule for those that haue to traffique in anie trade of merchandize , giving divers examples thereof . Next to that I treat of Arithmeticall , and Geometricall progression , and also of proportion , and of the three kindes thereof , that is , of proportion Arithmeticall , Geometricall , and musicall . Then I shew how to finde out the square roote of anie number , and also the vse thereof in setting of Battels , and also how to finde out the cubique roote of any number . And last of all I treat of Astronomicall fractions , shewing how to adde , to subtract , to multiply and divide the same . And also to take the square root thereof , without the knowledge of which fractions , you can neuer calculate any thing trulie out of the Astronomicall Tables .

Some perhaps do look here that I should speak somewhat of the rules of Algebra , whereby all subtill and intricate questions of Arithmeticke are to be vnfolded , wherewith I leaue indeede deale , partlie , for that I haue not of long time exercised my selfe therein , and partlie because I knowe that one hath begun to write thereof , whose booke being once ended , I doubt not but that hee will shortlie after print the same to the great profite and furtherance of al those that delight in such good exercises . But in the mean time I haue thought good to adde ( as I haue saide before ) vnto this Arithmeticke a plaine description , together with the vse of the Tables of Sines , of lines tangent and secant , which Tables will pleasure many that would gladlie know how and wherein to vse them , and specially such seamen as haue some taste of Arithmeticke , without the which no good almost is to bee done in anie science .

This Treatise of Arithmeticke containeth 26. Chapters as followeth .

WHat Arithmetique is , what numeration is , and of what partes it consisteth , and what signification euerie digit hath according to his place , and how to expresse or tell a great number written in manie figures . Chapter . 1.

Of the foure speciall kindes of Arithmetique , and first of Addition , with examples thereof . Chap. 2.

Of Subtraction , with examples thereof , and how to trie the same . Chap. 3.

Of Multiplication and certaine Tables belonging thereunto , together with the vse thereof , and what is to be obserued therein , with examples and triall thereof . Chap. 4.

Of Diuision , and what is to be obserued therein , with examples and triall thereof , and of halfing any number . Chap. 5.

Of the rule of three , called the Golden rule , and what order is to bee obserued in working thereby , and of the three kindes thereof . Chap. 6.

Of Fractions what they be , with a Demonstration thereof , together with seuen necessarie rules belonging to the same , and what euerie rule teacheth . Chap. 7.

Of Addition , Subtraction , Multiplication , and Diuision of Fractions . Chap. 8.

Of the common rule of three belonging to Fractions with examples . Chap. 9.

Of the rule reuerse , called in Latine Regula eversa , and the order of working thereby with examples . Chap. 10.

Of the double rule called in Latine Regula duplex , and the order of working thereby with examples , and in working thereby how to know when you haue to vse Regula eversa , or the common rule of three . Chap. 11.

Of the rule of Fellowship , and the order of working thereby , with divers examples thereof . Chap. 12.

Of Progression , what it is , and of the two kindes thereof , that is , Arithmeticall and Geometricall . Chap. 13.

Of Addition belonging to progression Arithmeticall , and the order thereof , with examples . Chap. 14.

Of Addition belonging to progression Geometricall , and the order thereof , with examples . Chap. 15.

Of proportion , what it is , and of the three kindes thereof , that is , Arithmeticall , Geometricall and Musicall . Chap. 16.

Of Arithmeticall proportion , what it is , and how it is diuided . Chap. 17.

Of proportion Geometricall , what it is , and how it is divided . chap. 18.

Of the chiefe and speciall kinds of Geometrical proportion , that is , of equalitie and inequalitie both greater and lesser . Chap. 19.

Of proportion of the greater inequalitie , and of the 2. kinds thereof , that is , Simplex , and Multiplex , and of their diuers kindes , with certaine Tables belonging thereunto Chap. 21.

Of proportion of the lesser inequality . Chap. 21.

Of Musicall proportion , what it is , and of the two kindes thereof , that is , Simple and compound , of which compound there be also 2. kyndes that is proper and vnproper . Chap. 22.

How proportions are to bee set downe in writing , and how they are to be added , subtracted , multiplyed , and divided euen like to fractions in all respectes . Chap. 23.

How to finde out the square roote of any number with examples . Chap. 24.

The vse of the square roote in setting of Battels , which according to the Italian vse are to be set foure manner of wayes , the order whereof is heere set downe with examples . Chap. 25.

How to finde the cubique roote of any number , and the order therof with examples . Chap. 26.

Of Astronomicall Fractions , whereto they serue , and how to adde , subtract , multiplie , and to diuide the same . Chap. 27.

How to diuide such fractions when the diuisor is greater than the dividend . Chap. 28.

How to take the square roote of Astronomicall Fractions . Chap. 29.

And immediately after these Chapters doe followe the vse of the Tables of Sines , lines tangent , and lines secant before mentioned , together with the Tables themselues .

To the Reader touching the order of finding and correcting the faults escaped in printing .

FIrst , I pray you vouchsafe with your penne to correct the numbers of the leaues of these bookes , which numbers are not in some leaues trulie set downe , that done , for the rest of the faults resort to the Table following , which consisteth of fiue collums , whereof the first on the left hand sheweth the true number of the leafe , the seconde collum sheweth the page both first and second : the third sheweth the line : in counting whereof you must begin at the head of everie page , and so proceede downward , leaving no printed line vntold , though it contayneth but one word , omitting notwithstanding such lines as doe containe the title of any chapter , so shall you bee sure to finde any fault mentioned in the sayde Table , and the correction thereof hard by it on the right hand : for the fourth collum of this Table containeth the faults , and the fift collum the correction thereof : by helpe of which Table you may correct your owne booke , where need is , in setting downe the correction in the margent right against the fault found .

Errata. Leafe Page Line The faultes . The Correction . 1 2 27 for , 4270570. read 43 20 , 70. 2 1 15 for , what , read . Why. 3 2 5 for , calling , read Culling . 5 2 9 for , squarewise , read squirewise . 10 1 32 for , limediate , read immediate . 24 2 33 for , the former 4. read the former doubled 4. 29 1 29 for , 2650. read 2550. 32 1 4 for , the productes , read the particular productes . 32 1 27 for , containe manie , read containe so manie . 32 2 31 for , ouer 1. read over 1‴ . 32 2 32 for , appointing to his , read appointing to the product his 34 1 1 for , the number must be 24. r. the number 24. must be 37 1 18 for , 53″ . read 50″ . 42 1 4 for , 10. read 16. 42 2 23 for , 20. read 19. 53 1 16 for , 890. read 899. 56 2 13 for , 23. degr. 38. read 23. degr. 28′ 131 1 30 for , precession , read precession . 137 2 13 for , euer the heauen , read everie heaven , 139 2 3 for Anges , read Auges . 169 1 6 for returning , read returneth . 173 1 16 for , or , read and. 181 1 32 for , 2160. miles , read 2160 b. miles . 193 2 6 for , Iland , read Island . 199 2 5 for , Aphiscii , read Amphiscii . 210 1 31 for , before fastned , read before taken . 215 2 2 for , the hour of the north , r. the houre of the night . 216 2 2 for , dawning twilight , read dawning and twilight 217 1 28 for , the Celestial globe , read the terrestriall Globe . 227 2 20 for , meere lately , read more lately . 229 2 7 for , in a great sph are , reade in a right spheare . 233 1 32 for , the 7. of Scorpio , read the 8. of Scorpio . 236 2 16 for , 30. deg. 47′ . of , read the 3. deg. 47′ . of 249 1 22 for , Equinoctionall , read Equinoctiall . 300 1 In the figure , for 1. ♉ . read ♑ . 1.

Of Arithmetike . Cap. 1.

WHat is Arithmetike ?

It is the art of counting or numbring by figures .

What is to number by figures ?

It is to expresse the value of any number in his proper Characters and figures , which is called by a Latine name Numeration .

What belongeth to Numeration ?

Two things , to know the shapes of the figures , and the signification of their places .

How many figures are there ?

These ten . 1. 2. 3. 4. 5. 6. 7. 8. 9. 0. Whereof the tenth made like an. 0. as you sée here , is called a Cypher , which is no number of it selfe , but serueth onely to fill vp a number .

What is number ?

Number is a collection or somme of many ones added together .

How is number deuided ?

Into thrée kinds , that is Digit , Article and compound .

Which be they ?

The Digit is any of the first 9. figures before set downe . Article is any number ending in ten , as ten is one Article , twentie is two Articles , thirtie is thrée Articles , &c. Compound is that which is compounded of Article & Digit , as 13. 14. 17. 24. &c.

Shew what signification euery Digit hath according to his place , and in what order such place is to be considered in expressing any number ?

The order as touching the place , is to beginne at the right hand , and so to procéede towardes the left hand . For any of the 9. Digits whatsoeuer , standing in the first place , which is on the right hand , signifieth the value of himselfe onely , in the second place ten times himselfe , in the third place Hundreth times himselfe , in the fourth place a Thousand times himselfe , in the fift place ten Thousand times himself , in the sixt place a hundreth Thousand times himselfe , in the seuenth place a Myllion , in the eight place ten Myllions , in the ninth place a hundreth Myllions in the tenth place a thousand Myllions , &c.

Doth the Cypher signifie nothing ?

Yes it maketh a place wheresoeuer it standeth , so as it be not the outermost on the left hand : for there it hath no place at all , as here you may sée in this number , 04500. whereof the first Cypher on the right hand , signifieth the first place , the second Cypher the second place , but the last and outermost Cypher on the left hand , signifieth no place at all , because it hath no Digit standing before it towards the left , hand , and therefore though in this number there be 5. figures , yet it signifieth no more but foure Thousand and fiue Hundreth .

By what meanes may a great number written in many figures , be readily expressed or tolde ?

By diuiding the same into diuerse parts with stréekes , or prickes made at the ende of euery third figure , beginning to tell from the right hand towards the left , as in this number , 4/270/570. In which , beginning with the Cypher on the right hand , I tell one , two , and thrée , and there make a Stréeke , and so procéeding forth still towards the left hand , I make a Stréeke at the ende of euery thirde figure , by which Stréekes or partitions I make them now seuerst numbers , and euery Stréeke must bée named by this word Thousand . Notwithstanding , in expressing this number being thus deuided , or any other such like , you must begin at the left hand , and say thus foure Thousand thousand , thrée Hundreth twentie Thousand , fiue Hundreth and seuentie : for by reason of this Diuision , the figure 4. standeth here alone , and in the first place , and in déed signifieth 4. Myllions , and by that means , you may more fitly expresse the said number in saying thus , foure Myllions , thrée Hundreth and twentie Thousand , fiue Hundreth and seuentie : And the better to discerne the Myllions from the rest in great numbers , it shal not be a misse to set an M. signifiing Myllions , right ouer the head of the Stréek which is drawne betwixt the sixth and seuenth figures , as in this example containing a leauen figures , 34/545 ᵐ / 678/694. which is to be vttered thus , thirtie foure Thousand , fiue Hundreth fortie fiue Myllions , sixe Hundreth seuentie eight Thousand , sixe Hundreth nintie and foure Crownes , or Pounds , or whatsoeuer other denomination or name , it shall please you to giue them .

Of the foure speciall kinds or partes of Arithmetike . Cap. 2.

WHich are those foure kindes ?

These , Addition , Subtraction , Multiplication and Deuision .

What , is not Numeration also counted as a parte ?

Because Numeration together with the figures , and places whereof it consisteth , are counted rather as first Elements , and principles of Arithmetike , then as partes or speciall kindes thereof .

Of Addition .

WHat is Addition ?

It is that which teacheth to bring many seuerall Sommes into one somme .

How is that done ?

First by placing euery seuerall number one right vnder another , vnder which you must drawe a line , that done , you must adde together the numbers of the first rancke , beginning on the right hand with the lowest figure of the same rancke , and so going vpward to the highest figure of the same rancke , and so from rancke to rancke , til you come to the last , and if the Somme of any rancke doe not excéede the number of any of the foresaide 9. Digits , then set downe that Digit which comprehendeth that number right vnder his proper rancke , beneath the line , but if the Somme of that rancke , excéedeth the number of any one Digit by reason that it consisteth of Articles and Digits , then set downe the Digit and kéepe the Article or Articles in your minde , to be added to the first figure of the next rancke on the left hand , but if the Somme be an euen Article or Articles : then set downe a Cypher , kéeping the number of Article or Articles in your minde , be it , one , two , or thrée , to be added to the next rancke , all which things you shall better vnderstand by this example here following . As for example , I spent in one yeare 125. l. in another yeare 234. l , and in another yeare 240. l. Now to knowe the totall Somme of all this , I place these seueral Sommes one right vnder another . l 125. 234. 240. 599. and then I drawe a line vnder them as here you sée .

Then beginning on the right hand with the lowest figure of the first rancke aboue the line , I say that a Cypher and 4. is but 4. Againe 4. and 5. maketh 9. which I set down vnder the line , then procéeding to the second rancke towards the left hand , I say that 4. and 3. maketh 7. and 7. and 2. maketh 9. which I also set downe , then remoouing to the thirde rancke , I say that 2. and 2. maketh 4. and 4. and 1. maketh 5. which I also set downe as you sée in the former example , so as the totall Somme vnder the line is . 599. l.

Another example hauing Cyphers mixt with Digits .

3047. 4508. 3049. 10/604.

Here I say that 9. and 8. maketh 17. and 17. and 7 maketh 24. wherefore I set downe the Digit 4. and kéepe 2. Articles in minde , which being added to the lowest figure of the second rancke , which is 14. maketh 6. then 6. and 4. maketh 10. here I set downe a Cypher , kéeping one Article in minde , which being added to the figure 5. of the third ranck maketh 6. which I also set downe , then I say that 3. and 4. maketh 7. and 7. and 3. maketh 10. for the which I set down first a Cypher , and then because there is no more to be added . I set downe on the left hand the one Article which I had in minde , so as the whole Somme commeth to 10/604. as in the former example .

How are pounds , shillings , pence , halfe pence , and farthings and all other numbers of diuerse Denominations to be added ?

You must deuide euery seuerall name into diuerse Collums or Spaces by themselues , and then beginning with the first on the right hand , you must adde euery Collum by it selfe , bringing farthings to halfe pence , and halfe pence to pence , pence to shillings , and shillings to poundes , setting the Somme of euery Collum vnder the nether line as you sée in this example following .

l s d ob. que 345 13 1 0 1 234 11 0 1 1 45 14 9 1 0 320 6 8 1 1 946 5 8 0 1

Here first beginning with the Collum of farthings , I finde therein 3. farthings which is one halfe peny and one farthing . Wherefore I set downe the od farthing as you sée , and kéepe the halfe peny in minde : then adding the halfe pennie in minde to the lowest halfe pennie of the second Collum , I say that 1. in minde and 1. maketh 2. and 2. and 1. maketh 3. then 3. and 1. maketh 4. which 4. halfe pence because they make iust two pence , I set downe a Cypher kéeping the two pence in minde , which two pence being added to 8. maketh 10. then 10. and 9. maketh 19. and 19. and 1. maketh 20 : How because that 20. d. maketh one shilling and 8. d. I set down the 8. d. kéeping the shilling in minde , which on shilling being added to the 6. of the next Collum maketh 7. then 7. and 4. maketh 11. and 11. and 1. maketh 12. then 12. & 3. maketh 15. wherefore I set downe 5. kéeping the Article in minde , which being added to one of the next Collum maketh 2. and 2. and 1. maketh 3. and 3. and one maketh 4. Articles , which 4. Articles maketh 40. s which is two pound which I kéepe in minde , and therefore I adde the 2. l. to the Collum of pounds , saying that 2. and 5. maketh 7. and 7. and 4. maketh a 11. and 11. and 5. maketh 16. wherefore I set downe 6. kéeping the one Article in mind , which being added to 2. of the next Collum , maketh 3. then 3. and 4. maketh 7. and 7. and 3. maketh 10. then 10. and 4. maketh 14. wherefore I set downe 4. kéeping one Article in mind , which being added to 3. of the next Collum maketh 4. then 4. and 2. maketh 6. and 6. and 3. maketh 9. which I also set downe , so as the totall Somme amounteth to 946. l. 5. s. 8. d. no halfe pennie , one farthing as you sée in the former example .

How shall I know whether these seuerall Sommes be truely added or not ?

Some doe teach it to be done by calling out all the nines , which way is more tedious then sure : for the surest tryall indeéde is to be done by Subtracting the seuerall Sommes out of the totall Somme , of which Subtraction we come now to speake , for all the foure speciall kindes are tryed one by an other .

Of Subtraction . Cap. 3.

WHat doth Subtraction teach ?

It teacheth to take a lesser number out of a greater and to sée what remayneth .

What is to be obserued in this kind ?

First , you must set downe your greater number aboue , and then the lesser number right vnder the same . As for example , I haue lent to one 564. l. and hée hath paide me thereof 57. l. Here to knowe what remaineth , I first set downe the number lent , and vnder that the number paid , and then drawe a line as you sée in this manner .

Lent . 564. l. Paide . 57. l.

Here beginning on the right hand , I first say , take 7. out of 4. that cannot bée : wherfore I take one Article of the next figure or place of the lent number , which Article being added to 4. maketh 14. then I say take 7. out of 14. and there remaineth 7. which I set downe vnder the 4. then I adde that one Article which I borrowed , to the second figure of the paide number which is 5. saying that 5. and 1. in minde maketh 6. then take 6. out of 6. and there remaineth nothing , wherefore I set downe a Cypher , vnder the 5. of the paide number , then I procéede to the third figure of the lent number , which is 5. and because I finde nothing written vnder it , nor haue nothing in minde to take out of it , I say , take nothing out of 5. and their remaineth still 5. so as the remainder is 507. l as you sée in this example following .

Lent . 564. l. Paid . 57. l. Remaine . 507

How shall I knowe whether this bee right or not ?

By adding the remainder and the number paide together , the Somme whereof ( if you haue done well ) wil be all one with the number lent , as in the former example , I first adde 7. and 7. together and that maketh 14. wherefore according to the precepts of Addition before taught , I set downe 4. kéeping the Article in minde , then I say , one in mind and 5. maketh 6. which I also set downe , then I say nothing and 5. is 5. which I set downe in the third place , which in all maketh 564. a number equall to the number lent , as you sée here following .

Lent , 564. l. Paid . 57 : Remaine . 507. Proofe . 564.

You may perceiue by this , that if any figure of the paid number be greater then the figure ouer him , out of the which it is to bée Subtracted , you must alwaies borrow one Article of his next fellow , to be added againe to him in his proper place . But you haue to note , that hauing to deale with numbers of diuerse Denominations , them in borrowing any number , you must alwaies haue respect to the Denomination or name of the thing , from whence you borrow , as in borrowing from shillings you borrow 12. and not 10. from pounds you borrow not one Article but 2. Articles which doe make 20. s but when the whole number is altogether of one selfe Denomination , then you must alwaies borrow one Article which is 10. to make vp your number that wanteth . As you shall more plainely perceiue by this example containing numbers of diuerse Denominations , as of pounds , shillinges and pence , halfe pence , and farthings . Suppose therefore that you haue lent to one 467. l 13. s. 4. d. ob . q. and hee hath paide you againe thereof , 89. l. 16. s. 9. d. ob . q. Here hauing set downe the Somme lent in seuerall Collums , according to their diuerse names , and then the Somme paide , right vnder the same , drawe a line as you sée in this example following .

l s d ob que Lent . 467 13 4 1 1 Paide . 89 16 9 1 0 Remaine . 377 16 7 0 1

Here beginning with the first Collum on the right hand , I say take nothing out of one , and one stil remaineth , which I set down , then procéeding to the next , I say take one out of one and nothing remaineth , wherefore I set downe a Cypher , then procéeding to the next Collum , I say take 9. out of 4. that cannot bée , wherefore I borrow a shilling of the next Collum that is 12. d. which being added to 4. maketh 16. pence , then I say take 9. out of 16. and there remaineth 7. which I set downe , then procéeding to the next , I adde the one shilling which I borrowed , to the 16. which maketh 17. s. then I say take 17. s. out of 13. s. that cannot be . wherefore I borrow one pound which is 2. Articles of the next rancke , that is 20. s. which being added to the 13. s. maketh 33. s. then I say take 17. out of 33. s. and there remaineth 16. s. which I set downe , then I adde the one pound which I borrowed , to 9. and that maketh 10. then I say take 10. out of 7. that cannot be , wherfore I borrow one Article out of the next 6. which being added to the 7. maketh 17. then I say take 10. out of 17. and there remayneth 7. which I set downe , then the one , which I borrowed , I adde to the 8. of the next rancke , and that maketh 9. Againe I say take 9. out of 6. that cannot be : wherfore I borrow one Article of the next 4. which being added to 6. maketh 16. then I say take 9. out of 16 and there remaineth 7. which I set downe , then I take the one which I borrowed out of the 4. & there remaineth 3. so as the remainder is as you sée in the former example . 377. l. 16. s. 7. d. 0. q.

How shall I trie whether this be true or not ?

By adding the remainder and the Somme paid together : as in the former example , and of that Addition , will rise if you haue done truely a Somme like in euery condition to the Somme lent : In making which proofe or triall you cannot lightly erre , if you remember to reduce pence to shillings , and shillings to poundes , and therefore in the Collum of pence , no particular Somme can be aboue 11. d , nor in the Collum of shillings no particular somme can be aboue 19 , s for if it be 20 , s , then it is a pound and must bée brought to the Collum of poundes ,

Of Multiplication . Cap. 4.

WHat is Multiplication ?

It is the producing or bringing foorth of a third number , by Multiplying two other numbers the one into the other : And it consisteth of thrée numbers , that is the multiplycand , the multiplyer , and the product .

What signifie those names ?

The Multiplicand is that number which is to be multiplyed , and the multiplyer is that whereby the same is multiplied , & the product is the Somme of such . Multiplication : As for example , if I would multiply 4. by 3. as in saying 3. times 4. maketh 12. here the number of 4. is the Multiplycand , and the number 3. is the multiplyer , and the number 12. is the product of that Multiplycation .

What order is to be obserued in multiplying , and how are those numbers to be set ?

Before I teach you the true order of multiplying , I thinke it good to set your downe a table of Multiplycation , which vnlesse you learne perfectly by hart , you shal neuer multiply readily nor quicklie .

TheTableofMultiplycation. 2 2 4 2 3 6 2 4 8 2 5 10 2 6 12 2 7 14 2 8 16 2 9 18 2 10 20 3 3 9 3 4 12 3 5 15 3 6 18 3 7 21 3 8 24 3 9 27 3 10 30 4 4 16 4 5 20 4 6 24 4 7 28 4 8 32 4 9 36 4 10 40 5 5 25 5 6 30 5 7 35 5 8 40 5 9 45 5 10 50 6 6 36 6 7 42 6 8 48 6 9 54 6 10 60 7 7 49 7 8 56 7 9 63 7 10 70 8 8 64 8 9 72 8 10 80 9 9 81 9 10 90 10 10 100

How is this Table to be read ?

In this manner , 2. times 2. maketh . 4. and 2. times 3. maketh 6. and 2. times 4. maketh 8. and so foorth : multiplying still one Digit by another , vntill you come to a 100. for this Table serueth onely for Digits , which may bee made to extend so farre as you will , and vntil you haue learned the foresaid Table without booke , you may helpe your selfe with this other Table of Digits made squire-wise as you sée here .

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1 4 6 8 10 12 14 16 18 2 9 12 15 18 21 24 27 3 16 20 24 28 32 36 4 25 30 35 40 45 5 36 42 48 54 6 49 56 63 7 64 72 8 81 9

In the front of which Table are set down the 9. Digits beginning on the left hand , and so procéeding to the right hand , from 1. to 9. Againe on the right side of the said Table are set downe the foresaid 9. Digits , beginning aboue , and so procéeding right downe from 1. to 9. the vse whereof is thus , first séeke the Digit to be Multiplyed in the front , and séek the other wherby you haue to multiply the same on the right hand , and the square Angle aunswering to these . 2. Digits will shew the product of such Multiplycation . As for example hauing to multiply 7. by 6. I first séeke the greater Digit which is 7. in the front , and the lesser which is 6. on the right hand , the product whereof I finde in the square Angle answering both to 7. and to 6. to be 42. & the like is to be obserued in any 2. Digits that are of like value as 7. times 7. the product whereof is 49. But the first Table being perfectly learned without booke , whensoeuer you haue to multiply one number by another , you must obserue these rules héere following .

First that you set downe the first figure of your multiplyer right vnder the first figure of the number that is to be multiplyed , on the right hand , and then orderly to place the rest of the figures of your multiplyer , be they few or many , towards your left hand , directly vnder the rest of the figures of the number that is to beé multiplyed , for if the figures stand not in order one right ouer another it will bréede a confusion in your working . Secondly you must not forget to multiply all the figures of the number that is to be multiplyed , by the first figure of your multiplyer before you deale with the next multiplyer , beginning alwaies on your right hand , and so to procéed from one to another , wherby you shal make as many seuerall products as there be figures in your multiplyer . Thirdly you must remember to set downe the first figure of euery seuerall product , right vnder the figure of that multiplyer , whereby you doe multiply , and hauing ended your Multiplication , draw a line vnder the seuerall products : that done , ad the seueral products together according to the rules of Addition , & the Somme therof shal be the total or general product of that Multiplycation : al which rules you shall the better vnderstand by working this example following . Suppose then that you would know how many houres there are in a yeare , knowing first that a yeare consisteth of 365. daies , here because that euery naturall day containeth 24. houres comprehending both day & night , you haue to multiply 365. by 24. and therefore you must first set down 365. because it is the greater number , and is the multiplycand , which must alwaies stand aboue , & right vnder that , your multiplyer which is 24. is to be set downe in his due place , according to the Rules before taught thus as you sée here . 365. the multiplicand 24. the multiplier . Then say thus 4. times 5. maketh 20. & hauing set down a Cypher right vnder the 4. kéep the 2. Articles in mind , them say 4. times 6. is 24. & 2. in mind is 26. here set down 6. & kéepe 2. in mind , then say 4. times 3. is 12. which with the 2. in minde maketh 14. here first set downe the 4. vnder the 3. and because you can procéede no further : you must therefore set downe hard by the 4. on the left hand the one Article which you had in minde , then hauing cancelled the first figure of the multiplyer , by making a Dash through it with your pen , as you sée in the example following : Procéede with the other figure of the multiplyer , saying that 2. times 5. maketh 10. wherefore set downe a Cypher right vnder the said 2. kéeping the one Article in minde , then say 2. times 6. is 12. and one in minde maketh 13. wherfore set downe 3. and kéepe one in minde , then say 2. times 3. is 6. which with one in minde maketh 7. the which you must set downe , and because you haue made an end of your Multiplycation , cancell the 2. and draw another line vnder the 2. seuerall products , that done , ad together whatsoeuer is contained betwixt the two lines , and you shall finde the generall product to be 8/760. houres .

365. 24. 1460. 730. 8760.

How shall I knowe whether the last Multiplycation be right or not ?

By diuiding the generall product by the Multiplyer , for in so doing your quotient will bee like vnto the first Somme that was Multiplyed , which you cannot do , vntill such time as you haue learned to deuide , and therefore hauing first shewed certaine compendious waies of Multiplycation , I will then procéede to Diuision .

Certaine compendious waies of Multiplycation .

WHen is any such way to be vsed ?

When the Multiplyer beginning on the right hand with one Cypher or with many , endeth on the left hand with the digit 1. as these numbers following , 10/100 100/1000 / &c.

Why what is then to be done ?

Yf you haue to Multiply by 10. then you haue no more to doe , but to set downe on the right hand of the number that is to bée Multiplyed , one Cypher , if by a 100. then 2. Cyphers , if by a 1000. then 3. Cyphers , as for example : if you would Multiply 365. by 10. then by setting downe on the right hand one Cypher as hath béene said , the product will be 3/650. if you set downe 2. Cyphers the product wil be 36/500. if you set down 3. Cyphers , then the product will be 365/000.

What if the number do end on the left hand with any other Digit , as 2. 3. or more , as 200. 300. 400 ?

Then you must Multiply the number of the Multiplycand first by that Digit , and then adde to the end of the product on the right hand all the Cyphers annexed to the said Digit , as if you woulde Multiply 365. by 200. First say 2. times 5. maketh 10. and 2. 6. and one in minde is 13. and so to procéed in Multiplying euery figure of the Multiplycand by 2. and you shall finde the product to be 730. whereunto if you adde on the right hand 2. Cyphers the whole product wil be 73/000.

Of Diuision . Cap. 5.

WHat is Diuision ?

Diuision is that whereby any number is diuided into as many parts as you will .

How many numbers are incident to Diuision and how are they called ?

These 4. that is to say , the number which is to be diuided which is called the diuidend , the second number whereby you doe diuide which is called the Diuisor , the third number is called the quotient , which sheweth how many times the Diuisor is comprehended in the diuidend , and the fourth number is called the remainder , if any be .

What order is to be obserued in Diuision ?

This here following , first set downe your diuidend and directly vnder that , beginning on the left hand , set downe your Diuisor , that is to say the first figure of your Diuisor right vnder the first figure of your diuidend on the left hand , and so consequently one after another procéeding towardes the right hand , which is alwaies to be done , so often as your Diuisor doth not excéed in quantitie , the figure standing right ouer his head , for if it doe , then you must remoue your Diuisor one figure further towardes the right hand . As for example if you would diuide 487. by 53. you must not set the first figure of your Diuisor which is 5. vnder the first figure of your diuidend , which is but 4. but vnder the second figure of your diuidend which is 8. for you cannot take 53. out of 48. and therefore you must set the first figure of your Diuisor , vnder the second figure of the diuidend , and so follow on with the rest and then draw a line as you sée here in this example following .

What is them to be done ? The diuidend . 487. The diuisor . 53. ( 9. the quotient

Then you must aske how many times 5. is comprehended in 48. and you shall finde that 5. is comprehended in 48. 9. times which 9. being the quotient , must be placed on your right hand , behind a crooked line made like a halfe Moone as you sée in the example aboue , then Multiply the first figure of the Diuisor by the quotient 9. and the product thereof shal be 45. which being taken out of 48. there will remaine 3. which you must set downe ouer the head of 8. and stréeke out the 48. and also the first figure of your Diuisor which is 5. that done Multiply the second figure of your diuisor which is 3. by the foresaid quotient which is 9. & the product therof wil be 27. which being taken out of 37. ther remaineth 10. which you must set ouer the head of 37. and cancell the 37. and also the 3. beneath , as you sée in this example following .

Wherein you sée that the diuidend is 487. the Diuisor 53 the quotient 9. and the remainder 10. which if you will apply to any vse you may imagine that there is 487. l. to be diuided amongst 53. soldiours , and by working as before , you shal find that euery Souldiour must haue 9. l. and there shal be remaining 10. l. to be diuided amongst the foresaid 53. Souldiours , which 10. l. being reduced to shillings , may bee diuided amongst the Souldiours as well as the pounds : As for example if you Multiply 10. l. by 20. s. it will make 200. s. which being diuided by 53. the first Diuisor , the quotient will be 3. s. and the remainder 41. s. and that being reduced into pence , which is done by Multiplying 41. by 12. the quotient , will be 9. d. and the remainder 15. d. which if you Multiply by 4. you shall reduce them to farthings , the product whereof will be 60. which product being diuided by 53. the first Diuisor you shall find in the quotient 1. farthing and the remainder to be 7/53 of a farthing , the value of which Fraction how to find out is taught hereafter when wee come to speake of Fractions . So as you sée by this meanes that euery Souldiour shall haue for his share 9. l. 3. s. 9. d. q. and some what more a thing of no moment .

How many things are to be remembred in Diuision ?

These fiue Rules here following . First that you put no number at one time in the quotient aboue 9. Secondly to moderate your quotient in such sorte , as hauing Multiplyed the first figure of the Deuisor by the quotient , there may remaine sufficient for the next figure of the Diuisor being Multiplyed by the quotient to be deducted out of that number which standeth right ouer his head . Thirdly that you Multiply euery figure contained in the Diuisor by the quotient . Fourthly that if at any time in working it happeneth so as your Diuisor is not comprehended in the number ouer his head , then to put a Cypher in the quotient and to remoue your Diuisor one figure further towards the right hand , and that done to worke as before . Fiftly to sée that the last remainder , if there be any left , doe not excéede in quantitie the Diuisor , all which things you shall better vnderstand by this one example following : Suppose then that you haue to diuide 819096. by 92. here hauing set downe your diuidend and Diuisor in such order as is before taught , and as you sée here in this example .

First aske howe many times 9. is in 81. and you shal find the quotient to be 8. which being Multiplyed into 9. maketh 72. which if you take out of 81. there shall remaine 9. which must be set downe ouer the head of the figure 1. and cancel the said 81. together with the first figure of the Diuisor , which is 9. then say 2. times 8. is 16. which being Subtracted out of 99. there remaineth 83. which you must set ouer the head of the 99. and cancell the 99. that is aboue , and also the 2. beneath , as you sée in the former example . That done remoue your Diuisor towardes the right hand , that is to say by setting the last figure of your Diuisor which is 2. vnder the next Cypher on the right hand , and place the first figure of your Diuisor which is 9. next to that towardes the left hand : then aske how many times 9. is comprehended in 83. and you shall find 9. times , which 9. must be set downe in the quotient next to the 8. then say 9. times 9. is 81. which being taken out of 83. there remaineth 2. then hauing set downe the 2. ouer the head of 3. cancel the 83. aboue & also the 9. beneath , then Multiply the last figure of the Diuisor which is 2. by the 9. which is in the quotient , and the product thereof is 18. which being taken out of 20. there remaineth 2. which 2. must be set downe ouer the Cypher , and the Cypher cancelled and also the 2. beneath , as you sée in this example following .

That doone remooue your Diuisor one figure further towards the right hand by setting the 2. vnder the 9. which is the last figure saue one of the diuidend , and place the first figure of your Diuisor which is 9. next to that on the left hand , then aske how many times 9. is conteined in 2. and you shall finde none , wherefore you must set downe a Cypher in the quotient and cancell the Diuisor as you sée in this example

That done remoue your Diuisor againe by setting the 2. vnder the last figure of the diuidend and place the 9. next to that on the left hand , then aske how many times 9. is contained in 29. and you shall find 3. times , which 3. you must set downe in the quotient as you sée in the example following , then Multiply the first figure of your Diuisor which is 9. by 3. and the product thereof will be 27. which being taken out of 29. there will remaine 2. which 2. you must set ouer the head of 9 and cancell the 9. aboue , together with the 9. beneath , then Multiply the 3. which is in the quotient by the last figure of your Diuisor which is 2. and the product thereof will be 6. which being taken out of 26. there remaineth 20. wherefore you must set downe a Cypher ouer the 6. which is the last figure of the diuidend , and cancell the said 6. aboue , and also the 2. beneath and then you haue done rightly , and in such order as all the former examples and this also next following doe plainely shew .

For heere you sée that the diuidende is 819/096. the Diuisor 92. the quotient 8903. and the remainder 20. which remainder must be set on the right hand of the quotient ouer the Diuisor , hauing a line drawne betwixt them as you sée in the last example .

How shall I know whether I haue diuided truely or not ?

By Multiplying the quotient by your Diuisor & by adding to the product thereof the remainder , for so shall you haue a number like to the first diuidend if you haue wrought well , if not , then your Somme will be either more or lesse then the diuidend .

Certaine compendious waies of Diuision .

IS there no shorter way of Diuision ?

No , vnlesse the Diuisor hath one Cypher or more on the right hand , for then you may vse a bréefer way by cutting off the last figure , or figures of the diuidend : As for example if you had to diuide 3708. by 10. heere cut off the last figure 8. with a dash of your penne in this wise 370/8. and the quotient will be 370. and the remainder shall be 8. Againe if you had to diuide the foresaide number by 100. then cut off the 2. last figures of the diuidend , that is the 8. & the cypher standing next it , and then the quotient wil be but 37. and the remainder 8. as before . but if you haue to diuide any number by 1000. then you must cut off 3. figures of the diuidend , and so forth , remembring alwaies for euery Cypher in the Diuisor , to cut off one figure or Cypher of the diuidend .

Of halfing any number .

HOw is that done ?

By diuiding the number by 2. As for example , if you would knowe the halfe of 3708. diuide the same by 2. and you shall finde the quotient to be 1854. which is the iust one halfe of the diuidend .

Of the Rule of three otherwise called the golden Rule whereof there be three kinds , that is , the common Rule , the rule Reuerse , and the Double rule . Cap. 6.

WHerefore is it called the Rule of three ?

Because that by 3. knowne numbers , it teacheth you to finde out a fourth number vnknowne .

How are such 3. numbers to be placed ?

The first is to be placed on the left hand , the third on the right hand , and the second in the middest betwixt both .

How shall I know in what place each one is to be set ?

By marking to what number the question is annexed , for that must alwaies be the third number , and hauing the third number , you shall quickely haue the first , because the first and third must alwaies be of one selfe Denomination or name , betwixt which two , the second number is to be placed : As for example let this bee the question . If I pay 35. s. for 13. yards of Linnen cloth , how much shall I pay for 3. yards , here because the question is annexed to the number 3. yards , that must bee the third number and is to be placed on the right hand , then because that the number 13. yards is of the selfe same Denomination , that must bee the first number , and is to be placed on the left hand , & the number 35. s. ( which is the price ) is to be placed in the middest betwixt the other two , as you sée here following . yards , s. yards : 13. 35. 3.

Now must you make your question in this manner , if 13. yards did cost me 35. s. what shall 3. yards cost me , and so the first and last shal be of one Denomination or name , that is to sayyards .

What order is to be obserued in working by this rule ?

You must Multiply the third and second numbers the one into the other , and diuide the product ther of by the first , and the quotient will shew you what the fourth number shoulde bee , which fourth number is alwaies of like Denomination to the second or middle number , as in the former example , first Multiply 35. by 3. and the product will be 105. which if you diuide by 13. the quotient will be 8. s. and one third of a shilling which is 4. d. and that is the price of 3. yards which is the number that you séek to know . And this is the common kind of working by the Rule of 3. whereof it is called the common Rule of thrée .

What if there be diuerse Denominations either of Coyne as of pounds , shillings , and pence , or parts of yards , as quarters , and halfe quarters , and such like ?

Then you must reduce them all to the smallest Denomination which belongeth to Fractions , whereof we come now to treate of .

Of Fractions . Cap. 7.

WHat are Fractions ?

They are broken partes of some whole thing called in Latine Integrum .

What is Integrum ?

Any thing that is whole , and not broken , or diuided into parts : As one whole yard , a pound , a shilling .

Of how many numbers doth a Fraction consist ?

Of two , that is the Numerator , and the denominator , whereof the Numerator is alwaies set aboue , and the denominator beneath , hauing a little line drawne betwixt them thus ½ which signifieth one second or one halfe , againe two thirds and thrée fourths are written thus , 23/34 whereof the first signifieth two thirds , and the other thrée fourths and such like .

What is ment by these two words Numerator , and Denominator , and whereto serue they ?

The denominator is as much to say as the namer of the parts , which sheweth into how many parts the Integrum is to be diuided , and the Numerator is as much to say as the Numberer , and sheweth how many of those parts are to be taken : As for example ⅔ of a shilling , here the nethermost called the denominator , sheweth that the shilling is to be diuided into thrée parts , as into thrée groats , and the vpper number called the Numerator , sheweth that you must take 2. out of those 3. parts , so as ⅔ of a shilling is as much to say as 2. groats or 8. d. Againe Fractions may be diuided into two kinds , that is , into simple Fractions , and into Fractions of Fractions .

Define those two kinds ?

Simple Fractions are such as I speake of before which are the first & limediate parts of any Integrum that is diuided into parts , but if those Fractions bee diuided into more Fractions , then are they called Fractions of Fractions , as when I say thrée fourthes of two thirds , of six seuenthes and such like .

How are such Fractions to be set downe in writing ?

In manner of simple Fractions thus ¾ of ⅔ of 6/7.

Make some demonstration of this example that I may the better vnderstand it ?

Imagine that there is a whole peece of Golde of vii . s. as our Angel in times past hath bene , which vii . s. be the vii . partes of that péece of Gold , for tryall whereof laie downe before you vii . twelue pennie péeces of Siluer , or in stéede thereof vii counters , to signifie those vii . partes , of which vii . parts you must first take away vi . reiecting or laying a side the od seuenth part , then diuide those 6. parts into 3. parts , and euery such part will be iust 2. s. of which 3. parts you must take away 2. partes that is 4. s. reiecting the other third part which remaineth , that done , diuide those 2. parts which you haue taken away into 4. parts , which is 4. s. and take 3. of them reiecting the fourth part , so shall you finde that ¾ of ⅔ of 6/7 of the foresaid péece of Golde is iust 3. s. and there remaineth still of the parts reiected 4. s. which being added to 3. s. that was taken away , doe make vp againe the whole Integrum of 7. s. Notwithstanding Fractions of Fractions doe séeldome chaunce in the Diuision of any number , but if they doe , you must reduce them into simple Fractions , before you can deale with them any manner of way , and because there are diuerse rules belonging to Fractions , without the knowledge whereof you can nether adde , nor subtract them , nor yet ioyne them with any Integrum , I wil briefly set downe here seauen necessarie rules for the same .

Seauen necessarie Rules belonging to Fractions .

WHat doth the first rule teach ?

To bring Fractions of Fractions into simple Fractions .

How is that done ?

Thus , Multiply the first Numerator into the second , and if there be any more Fractions , then Multiply also the said product of the first two Numerators into the third Numerator , and that shall be the Numerator of the simple Fraction , then Multiply the denominators in like manner , and the Somme thereof shall be the denominator of the simple Fraction , as in the former example ¾ of ⅔ of 6/7 here I Multiply the first 2. Numerators together , in saying 2. times 3. maketh 6. now because I haue a third Numerator , I multiply the product of the first 2. Numerators , which is 6. into the third Numerator which is also 6. saying , 6. times 6. maketh 36. and that shall be the Numerator of the simple Fraction , which I set downe . Then by Multiplying the 3. denominators in like manner , I finde the denominator of the simple Fraction to be 84. the which I set downe vnder the 36. and draw a line betwixt them so as now I finde that ¾ of ⅔ of 6/7 do make 36/84 which indéed is no more but 3/7 as you shall learne hereafter by the sixt rule .

What doth the second rule teach ?

To bring Fractions being more in value then Integrums into integrums .

When are Fractions said to be more then integrums ?

When the Numerator is a greater number than the denominator , for if they be both of like value as 34/34 and such like , then such Fraction is an euen integrum .

How shall I know how many Integrums such Fractions as be more then Integrums doe containe ?

By diuiding the Numerator by the denominator , and if any thing remaine , write that aboue the denominator , as in this example 806/7 here if you diuide 806. by 7. which is the denominator , you shall finde the quotient to containe 115. Integrums , and the remainder to be 1. which is 1/7.

What doth the third rule teach ?

To bring Integrums into parts by Multiplying the number of the Integrums by the denominator of those partes , as if you would bring 64. yards into quarters , Multiply 64. by 4. and there will arise thereof 256. quarters .

What doth the fourth rule teach ?

It teacheth to bring Integrums hauing Fractions annexed to them , into one Fraction .

How is that done ?

Multiply the number of the Integrums by the denominator of the Fraction , and then adde to the product the Numerator of the saide Fraction , and that somme shall be the Numerator , vnder which write the denominator aforesaid , and so you shall finde that 23. Integrums hauing ⅔ annexed thereunto shall make ●1/3.

What doth the fift rule teach ?

It teacheth to expresse a Fraction written with many figures , in so fewe as may be .

How shall I doe that ?

Thus , finde out some number that may first diuide the Numerator , and then the denominator seuerally by themselues , without leauing in either of them any remainder , and the quotient of the first Diuision will shew the Numerator , and the quotient of the second Diuision will shew the denominator , but if you cannot readily finde out a number that will diuide them both without leauing some remainder , then leaue not to Subtract the lesser number out of the greater , vntill you finde two like numbers , by one of the which two like numbers diuide both the Numerator , and also the denominator seuerally as before , and the quotient will shew that which you séeke : but if such 2. like numbers cannot be found , then you may assure your selfe that the Fraction cannot be written in fewer figures then they alreadie be .

Giue examples of these two waies ?

For example of the first way , suppose that you would expresse 9/12 in lesser numbers , here séeke out some number that may diuide euenly both the Numerator , and also the denominator , without leauing in either of them any remainder , which by diuiding each of them by the number of 3. you may doe : For by asking how many times 3. is contained in 9. the quotient will be 3. and in asking how many times 3 is conteined in 12. the quotient shall be 4. so shall you find 9/12 to be no more then ¾ . Now for example of the second way , let 27/81 be the Fraction which you would set down in fewer figures , here because you cannot readily finde out a number that wil euenly diuide both the Numerator and the denominator , Subtract the lesser out of the greater , that is to say 27. out of 81. and there will still remaine 54. from which Subtract 27. and there will remaine 27. which 2. numbers because they are both like , diuide the Numerator and denominator of the foresaid Fraction , each of them by one of these numbers , that is to say , by 27. and the quotient of the first Diuision will shew you the Numerator which is 1. and the quotient of the second Diuision wil shew you the denominator which is 3. so as you shal find ⅓ to be as much in value as 27/81 and note that when there be Cyphers both in the Numerator and in the denominator standing in such sorte as they may bee euenly cut off , the remainder will shew the fewest figures , wher in the Fraction may be written as 200/500 here by cutting off the two Cyphers as well beneath as aboue the line , with a dash of your penne in this manner ⅖ 00/00 the remainder shall bee ⅖ which is as much in value as 200/500.

What doth the sixt rule teach ?

It teacheth to find out the value of the Fraction of any integrum .

How is that done ?

Thus , Multiply the Numerator by the knowne partes of the Integrum , and diuide the product thereof by the denominator , so shall you haue the value of the Fraction : As for example , if you would know the value of ¾ of an Angel , consider first what partes an Angel hath , and you shall find the parts thereof to be 10. s. or 30. groats , here if you Multiply 3. by 10. s. it will make 30. s. which being diuided by 4. you shall finde in the quotient 7. s. and the remainder to be 2/4 or one half of a shilling , which is 6. d. Againe if you Multiply 3. by 30. groates it will make 90. groats , which being diuided by 4. you shall finde in the quotient 22. groats , and the remainder to be 2/4 or one halfe of a groat which is 2. pence , making in all 7. s. 6. d. like vnto the first Somme , so as you sée that ¾ of an Angel is 7. s. 6. d. and thus you may deale with the Fractions of any other Integrum that hath knowne parts .

What doth the seauenth rule teach ?

It teacheth to bring Fractions of diuerse denominations to one selfe denomination , without the which nether Addition , nor Subtraction of Fractions can be made : As for example if you would adde ⅔ and ⅘ together , you must first bring them to one selfe denomination thus , Multiply the denominators the one into the other , and the product thereof shall be a common denominator to both the Fractions , wherefore say 3. times 5. doe make 15. which must be set downe in 2. seueral places by them selues thus , 15. and 15. then Multiply the Numerator of the first Fraction into the denominator of the second , as 2. into 5. maketh 10. which set downe ouer the first common denominator thus 10/15 then Multiply the Numerator of the second Fraction into the denominator of the first Fraction , as 4. into 3. which maketh 12. and that must bée likewise set downe ouer the second common denominator thus 12/15 so shall you find that ⅔ and ⅘ are all one in value with 10/15 and 12/15.

Addition of Fractions . Cap. 8.

HOw are Fractions to be added together ?

Fractions being first brought to one denomination are easily added , for then you haue no more to doe but to adde the Numerators together , and to write the common denominator vnder the Somme of such Addition , as in the former example , 10/15 being added to 12/15 maketh 22/15 but if the denominators be not like , then you must make them like by the seuenth rule before taught .

What if there be three Fractions of diuerse denominations to be added together as ⅔ ¾ ⅘ ?

Then hauing reduced the 2. first to one self denomination adde the 2. Numerators together and write vnder the Somme thereof the common denominator , that done , deale with that Fraction last found , and with the third as you did before with the 2. Fractions that had diuerse denominations , and so you shall find that ⅔ and ¾ being first brought to one denomination , and then added together doe make 17/12. whereto if you will adde ⅘ . then you must bring them againe to one selfe denomination , and so you shall finde that 17/12. and ⅘ doe make 85/60. and 48/60. which last 2. Numerators being added together doe make in all 133/60. which is 2. Integrums and 13/60.

Subtraction of Fractions .

HOw are Fractions to be Subtracted , one out of another , or out of Integrums ?

First make the denominators like as you did before in Addition , then take the lesser Numerator out of the greater , and vnder the remainder thereof write the common denominator , so shall you find that 3/7. being taken out of 6/7. there remaineth 3/7. but if you would Subtract Fractions out of Integrums , then you must take one of the Integrums and breake it into partes . As for example , to Subtract 3/7. out of 9. Integrums take one from 9. and breake it into parts making it 7/7. for that is one Integrum , for whensoeuer the Numerator is made like and equall to the denominator , it signifieth one Integrum , then take 3/7. out of 7/7. and there remaineth 4/7. and 8. integrums .

What is to be obserued in breaking the integrum ?

In breaking the Integrum , you are to be directed alwaies by the denominator of the Fraction which you haue to Subtract , for the Integrum which should be the Numerator , must be equall in value to the said denominator , as before is said .

Multiplycation of Fractions .

HOw are Fractions to be Multiplyed by Fractions , or Integrums by Fractions ?

As touching Fractions , Multiply the Numerators one into another , and the product thereof shall bee the Numerator , then Multiply the denominators in like manner , and the product therof shall be the denominator , so shall you find that 5/7. being Multiplyed by ¾ . doe make 15/28. But if you would Multiply integrums by Fractions , then Multiply the integrums by the Numerator of the Fraction , and vnder the product thereof , set downe the denominator of the same Fraction , drawing a line betwixt them , so shal you find that 20. integrums being Multiplyed by 5/12. doe make 100/12. that is to say 8. integrums and 4/12. which is ⅓ , &c.

Diuision of Fractions .

HOw are Fractions to be diuided by Fractions , or integrums by Fractions , or Fractions by integrums ?

The diuison of Fractions is done by Multiplycation thus , first set downe your diuidend on the left hand , and the Diuisor alwaies on the right hand , and then draw 2. crosse lines like a Saint Andrews crosse betwixt them , which shal direct you in your working : As for example , if you woulde diuide ⅔ . by ⅘ . set them downe diuidend thus ⅔ ⅘ diuisor worke as followeth , first Multiply the Numerator of the diuidend , by the denominator of the Diuisor , and the product there of shal be the Numerator , then Multiply the denominator of the dividend , by the Numerator of the Diuisor , and that shall be the denominator , so shall you find that ⅔ . being diuided by ⅘ . there remaineth 10/12. that is ⅚ . but if you would diuide Integrums by Fractions , or contrarywise Fractions by Integrums , then make of the Integrums a Fraction , by setting down 1. in the place of a denominator vnder the Integrums , and worke as before , so shall you find that 7/1. Integrums being diuided by ¾ . doe make 28/3. contrarily if you diuide ¾ . by 7/1. Integrums there will arise 3/28.

What if I haue to diuide Fractions annexed to integrums ?

Then you must first reduce each Integrum with his Fraction annexed , into one selfe Fraction by the fourth rule of Fractions before taught . As for example , if you would diuide 9. Integrums hauing ¾ . thereunto annexed , by 3. Integrums and ⅓ . heere the fourth rule of Fractions teacheth you first to Multiply the Integrum 9. by 4. which is the denominator of the Fraction annexed , the product whereof is 36. whereunto by adding the Numerator of the same Fraction which is 3. you shall make the Numerator of the Diuidend to be 39. vnder which you must set the denominator 4. thus 39/4. and that is your whole diuidend , then hauing in like manner brought the Diuisor which is 3. Integrums and ⅓ . into one selfe Fraction , worke as the former rule of Diuision teacheth you , and you shall produce 117/40. which is 2. Integrums and 37/40.

The rule of three belonging to Fractions . Cap. 9.

WHat order is to bee obserued in this rule hauing to deale with Fractions onely ?

The selfe same order that hath beene taught before touching Integrums : for in working with Fractions you must haue also 3. seuerall numbers , and you must sée that the first and third numbers be of one selfe denomination , and that number to bee placed alwaies in the third place whereunto the question is annexed , and then to Multiply the second by the third , and to diuide by the first , and so the fourth number which you séeke to know , shall appeare : As for example if ¾ . of an ell of fine Holland cost me ⅔ . of an English crowne , in value 15. groats , what shall ⅚ . of an ell cost me , here first you must set downe your 3. seuerall numbers in order thus , ¾ ell ⅔ ⅚ ell . so as the first and third may be of one selfe denomination , then Multiply the second and third Fraction the one into the other , which will make 10/18. and that being diuided by ¾ . which is the first Fraction will produce 40/54. of a crowne , the value of which Fraction if you séeke to knowe by helpe of the sixt rule of Fractions , teaching you to Multiply the Numerator of the Fraction , by the knowne parts of a crown which are 5 s or 15. groats , you shall finde the value of that Fraction to be 11. groats and 6/54. of a groat , which is one farthing and somewhat more , supposing the least knowen parts of a groat to be 16. farthings .

The golden Rule reuerse called in Latine regula euersa that is to say turned back-ward . Cap. 10.

WHat is the order of this rule ?

Multiply the first by the second , and diuide the product therof by the third , as if a pennie Loafe must waigh 2. l. Wheate being at 3. s. the bushell , what shall a pennie Loafe waigh when Wheate is at 2. s. the hushell , the question must be framed thus , if 3. s. require 2. l waight , what shall 2. s. haue , then by working according to this rule you shall finde that the pennie Loafe must waigh . 3. l.

Another example of the same rule .

I would know how many yards of Bayes bearing in breadth 7/4. wil suffice to line 7. yards of Silke bearing in breadth 3. quarters & a halfe . Here you must frame your question thus , if ¾ . and ½ require 7. yards , what shall 7/4. require , but because the first and third number of this question are not of one selfe denomination by reason of the Fraction annexed to the first number , you shall doe well to reduce the first and third number both into halfe quarters , and then to worke as though they were all Integrums , which is more easie then to make al the numbers Fractions , wherefore say thus : If 7. halfe quarters doe require 7. yards , what shall 14. halfe quarters require , and in working by the rule Reuerse , you shall finde in the quotient 3. yards of Bayes and a halfe .

The double rule called in Latine regula duplex . Cap. 11.

WHereto serues this rule and what order is to be obserued therein ?

This rule serueth to vnfold two questions wrapt in one , as thus . If I pay 4. d. for the carriage of 20. l. waight 30. miles , what shal I pay for the carriage of 50. l. waight 60. miles , here of this and such like demaunds , you must make 2. sundrie questions , and the fourth Somme of the first question being found , shall be the second or middle number of the second question : wherefore frame your first question thus , if 20. l. cost 4. d what shall 50. l cost , and shall find that it will cost you 10. d. then say if 30. miles cost 10. d. what shall 60. miles cost and you shall finde that it will cost 20. d. And note that each of these 2. questions is to bee wrought by the common rule of 3. that is to say by Multiplying the second into the third , and by diuiding the product thereof by the first , and the fourth found number of the first question must be the second , or middle number of the second question , as in the former example , you sée that 10. d. which was the fourth found number , is here the middle number of the second question .

Another example .

If 25. l. doe gaine me 8. l. in 4. yeares , how much shall a 100. l. win one in 10. yeares , both these questions are also to bee wrought by the common rule of 3. Wherefore set downe the first question thus , if 25. l. yéeldeth 8. l. what shall 100. l. yéelde , and you shall finde 32. l. then say 4. yeares yéeldeth 32. l. what shall 10. yeares yéelde , and you shall find 80. l. But note that these double questions , may be put in such sort as you must worke the first or second question , sometimes by the rule reuerse . As in this question here following , if 6. l. win 8. Crownes in 10. yeares , in how many yeares shall 3. l. win 12. Crownes , here frame your first question thus , if 6. l. require 10. yeares how many yeares shall 3. l. require : And in working this question by the rule Reuerse , you shall finde 20. yeares , then for the second question say thus , if 8. Crownes require 20. yeares , how many yeares shal 12. Crownes require : Here if you worke by the common rule of 3. you shall finde 30 yeares .

Another example .

If 7. horses doe eate 12. bushels of Dates in 20. daies , how many bushels shall 14. horses eate in 15. daies , here frame your first question thus , if 7. horses doe eate 12. bushels , what will 14 horses eate , and in working by the common rule of 3. you shall finde in the quotient 24. bushels , then frame your question thus , if 20. daies require 24. bushels , what will 15. daies require , here in working by the common rule of 3. you shall finde in the quotient 18. bushels .

Another example .

If ten reapers reape 15. Acres in 7. dayes , in how many daies shall 16. reapers reape 20. Acres : Here frame your first question thus , if 10. reapers require 7. daies , how many daies shall 16. reapers require , which question must be wrought by the rule Reuerse , and so you shall finde 4. daies and ⅜ . of a day which is 9. houres , then say , if 15. Acres require 4. daies and ⅜ . of a day , how many daies shall 20. Acres require , and in working this second question by the common rule of 3. belonging of Fractions as is before taught , you shall finde that 20. Acres wil require 5. daies and 10/12. or ⅚ . of a day which is 20 houres . But it were more easie in this second question to reduce the daies into houres by Multiplying the 4. daies by 24. houres , the product whereof wil be 96. houres , whereunto if you adde the od 9. houres it will make in all 105. houres , which being Multiplyed by the third number of this second question , which is 20. the product shall be 2100. houres , which diuided by the first number of the said question which is 15. you shall finde in the quotient 140. houres , which if you diuide againe by 24. you shall finde in the quotient 5. daies , and the remainder to be 20. houres , which agréeth in all pointes with the first manner of working by Fractions , and is the easier way of the two .

How shall I know hauing to worke by this Double rule , when to vse the rule reuerse ?

By considering whether the third number requireth more or lesse of time , or of any other measure or quantitie , as in the former example , of Bayes for lining , the more breadth it had , the lesse did serue for lining . Againe in the example of the gaine by yeares , of 6. l. and 3. l. you did sée that 3. l. require more yeares then 6. l. and therefore that first question was wrought by the rule reuerse : Also in this last example of the reapers , the more reapers , the lesse time they require , & therefore that question was wrought by the rule reuerse .

The rule of Fellowship Cap. 12.

WHat doth this rule teach ?

To knowe the gaine or losse of such as doe make a stocke and doe occupie together in the trade of Marchandize : As for example , foure Marchants did put their mony in lot in this manner . The first brought 30. Crownes , the second 50. the third 60 and the fourth 100. and with these portions they gained 3000. Crownes , the question is how much euery one shall haue to his share of that gaine according to the portion which he brought . To know this , you must first gather all the seuerall portions together by Addition into one Somme , and you shall finde the somme of the portions to be 240. Crownes , then say if 240. Crownes do gaine 3000. Crownes , what shall 30. gaine , and after this manner worke by the common rule of 3. with all the rest of the portions , and so you shall finde that he which brought 30. ought to haue 375. Crownes , and hee that brought 50. ought to haue 625. Crownes , and hee that brought 60. ought to haue 750. Crownes , and he that brought 100. ought to haue 1250. which maketh in all 3000. Crownes , for that is the very Somme of the gaine before set downe , the order of working whereof , this figure plainely sheweth .

240. — 3000. — 30. — 375. 50. — 625. 60. — 750. 100. — 1250. The common Diuisor which is the Somme of the particular portions added together . the generall gaine . the particuler portions . euery mans single share .

A like example of losse receiued by Shipwracke .

Thrée Marchants doe venture their goods in one Shippe , the goods of the first were worth 300. Crownes , the second 400. the third 500. there were as much goods cast out as was worth 100. Crownes , the question is how much euery one should loose according to his portion , here worke as before , and you shall finde that euery one shall loose so much as this figure following sheweth .

1200. — 100. — 300. — 25. 400. — 33. ⅓ which is 5. groats 500. — 41. ⅔ which is 10. groats The common diuisor which is the Som of the particular portions added together . the generall losse . the particular portions . euery mans seueral losse

Here to know whether the 3. seuerall losses doe make vp the generall losse , do thus , first adde the Integrums of the seuerall losses together , the Somme whereof will bee 99. whereunto if you adde the 2. Fractions which doe make one whole Crowne , the Somme will be 100. a number like vnto the generall losse .

Another example .

Thrée Marchants haue bought 1000. l. of Pepper for 300. crownes , the first taketh 200. l. the second 350. l. and the third 450. l. what shal euery man pay according to the portion which he hath receiued , then say if 1000. l. be worth 300. Crownes , what is 200. l. worth , here working by the common rule of 3. with euery mans seuerall portion receiued , you shall finde that the first must pay 60. Crownes , the second 105. and the third 135. Crownes , as this figure here following sheweth .

l. 1000. — 300. — 200. — 60. 350. — 105. 450. — 135. The common Diuisor . the generall price . the seueral portions receiued . euery mans particular paiment .

Another like example of diuerse distance of time .

Thrée Marchaunts occupying together did gaine 2345. Crownes , the first put in 40. Crownes for the space of 14. Moneths , the second put in 50. Crownes for the space of 8. Moneths , and the third put in 85. Crownes for the space of 6. Moneths , the question is , how much euery one shall haue after the rate of his mony , and according of the quantitie of time : This is to bée wrought according to the rule of Fellowship thus , Multiply euery mans mony by his time , either of which , that is to say , the mony and the time must be of one selfe denomination , then adde the Sommes of those seuerall portions together , and the total somme of such Addition shall be the first number , and the common gaine shall be the second number , and the third number shall bee euery mans mony Multiplyed by his time , then in working by the common rule of 3. you shall finde that euery man shall haue such share as this figure here following sheweth .

1470. — 2345. — 560. — 893. ⅓ 400. — 638. 2 / 21. 510. — 813. 4 / 7 The common Diuisor which is the totall Somme of the seuerall portions of mony added together . the general Somme of the gaine . the particular portions of the mony multiplyed by his time . euery mans single share .

Now if you would know whether the single shares in this example doe make vp the generall Somme of the gaine , then adde together the Integrums or whole Sommes of euery mans single share , and you shall finde the Somme to bee but 2344. which wanteth one whole Integrum of the second or middle number of the question , which you shal easily supply by adding the 3. seueral Fractions together according to the rule of Addition of Fractions before taught , for so shall you finde the said 3. Fractions to make in all one whole Integrum , which being added to 2344. will be aunswerable to the second number of the question , which as you sée in the example is 2345. And remember in adding together the said 3. Fractions , to set them in this order ⅓ / 2/21 21/4/7. and then to bring them to one selfe denomination , as the seuenth rule of Fractions teacheth you . But first because you shal find the 3. remainders after the first 3. Diuisor of the 3. seuerall portions , made by the first common Diuisor of the question to be written in many figures , you must set them downe in fewer figures according as the fift rule of Fractions teacheth you , so shall you finde the first remainder containing 490/1470. to be no more but ⅓ . and the second remainder containing 140/1470. to be 2/21. and the third remainder containing 840/1470. to be 4/7. which 3. Fractions being added together according to the seuenth rule of Fractions , wil make 441/441. which is one whole Integrum and must be added by the name of 1 to the middle or second number of the question , as I haue said before : you may also make up the foresaid number by séeking to knowe the value of euery Fraction annexed to the Integrums , according as the sixt rule of Fractions teacheth , for so shall you finde the value of the first Fraction to be fiue groats , & the value of the second Fraction to be one groat 6. farthings , and 18/21. of a farthing thing , & you shal find the value of the third Fraction to be 8. groats 9. farthings and ½ . of a farthing , & if you adde 18/2● & 1/7 of a farthing together , you shall find that it will make 147/147. which is one whole Integrum or one whole farthing . Now if you adde 6. 9. and 1. farthings together , it wil make in al 16. farthings , and that is one groat which being added to the 14 groats before found out , will make in al 15. groats which is one Crowne , and that being added to the Somme 2344. will make it 2345. Crownes , which is a number agreeable to the middle Somme of the question . Truely if you exercise your selfe in this and such like questions , it will make you perfect not onely in Addition , Subtraction , Multiplycation , and Diuision of whole numbers , but also of Fractions , and almost in al the other rules belonging as wel to Fractions , as to Integrums : Wherefore I would wish you often to vse your penne therein .

Hauing hitherto treated of the 4. speciall kinds of Arithmetike , that is of Addition , Subtraction , Multiplycation and Diuision , as well belonging to whole numbers , as to Fractions , and also shewed the vse of the Golden rule , otherwise called the rule of 3. and of all the thrée kindes thereof , that is the common rule , the rule Reuerse , and the Double rule , and giuen examples how and when euery one is to be vsed , together with the rule of Fellowship , necessarie for them that vse any trade of Marchandize , I thinke good now to speake somewhat of Arithmeticall and Geometricall progression , and also of Proportion , and of the thrée kinds thereof , that is Arithmeticall , Geometricall , and Musicall proportion , and then of the extraction of rootes both square and Cubicall , and last of all , of Astronomicall Fractions , shewing how they are to be added , Subtracted , Multiplyed & diuided .

Of Progression and how manifold it is . Cap. 13.

WHat is Progression ?

It is a certaine order of procéeding with diuerse numbers in such sorte as euery one may excéede each other , either by like difference of quantitie , or else by likenesse of Proportion , whereof springeth 2. kinds of Progression , the one called Arithemeticall , and the other Geometricall .

What is Arithmeticall Progression ?

It is that which procéedeth by like difference of quantitie , as thus , 3. 5. 7. 9. 11. 13. whereof euery one excéedeth his fellow by the difference of 2.

What is Progression Geometricall ?

It is that wherein euery number excéedeth his fellow by like proportion , for as 6. containeth 3. twise , so doth 12. containe 6. twise , &c.

Of Addition belonging to Progression Arithmeticall . Cap. 14.

HOwe are numbers being set in order according to Progression Arithmeticall , to be added together and to be reduced into one totall somme ?

Thus , first sée how many seuerall numbers there be in all , and note the Somme by it selfe , then adde the first number of the Progression to the last thereof , and note that Somme by it selfe , that done , Multiply the one of these 2. reserued Sommes by the one halfe of the other , and you shall haue the totall Somme . As for example let this be your Progression 6. 10. 14 / 18/22 22/26 26/30 30/34 / wherein euery number excéedeth his fellow by the difference of 4. here hauing tolde the numbers , I finde them to be 8. which I set downe by it selfe , then I adde 6. which is the first number to 34. which is the last , and the Somme thereof is 40. which I also set downe by it selfe , then in Multiplying 40. by 4. which is the one halfe of 8. before set downe a part by it selfe , I finde the totall Somme of the Progression to be 160.

Of Addition belonging to progression Geometricall . Cap. 15.

HOw are numbers being set in order according to progression Geometricall , to be added and to bee brought into one Somme ?

Thus , Multiply the last Somme of the progression by the number of the proportion whereby such progression procéedeth , and from the product of that Multiplycation , Subtract the first number of the progression , that done , diuide that which remaineth by a number which is lesse then the proportion by one , and the quotient of such Diuision shall shew you the totall Somme of the said progression . As for example , let this bee the progression 2. 6. 18. 54. 162. 486. 1458. the proportion of which progression is tryple , wherefore according to the rule , I Multiply the last number of the progression by 3. and the product of such Multiplycation amounteth to 4/374. out of which Somme I Subtract the first number which is 2. and there remaineth 4372. which I diuide by 2. ( for it is lesse then be proportion 3. by 1. ) and so I finde in the quotient 2186. which is the totall Somme of the progression Geometricall .

Is there no brieffer way of adding such kinds of progression ?

Yes indéede , but not so plaine as this way is , and therefore I thinke not good to trouble your memorie therewith .

Of Proportion . Cap. 16.

WHat is Proportion ?

Proportion is taken generally for the comparing of 2. diuerse quantities or numbers together , shewing what likenes is betwixt them . But before we deale with Proportion , and with the 3. kinds thereof , that is Proportion Arithmeticall , Geometricall , and Musicall : You haue first to vnderstand , that of numbers some are said to be abstract , and some concreate .

The abstract are such as are not tyed to any denomination , and such are twofold , that is absolute and relatiue .

The absolute , are simply pronounced without hauing any relation to any other number , measure , or quantitie , as 2. 3. 4. &c. and all numbers whatsoeuer , that are without denomination and are not attributed to any other thing .

The relatiue are those which haue relation one to another , which may be thrée manner of waies . First in respect of difference which is found by Subtraction , secondly in respect of the quotient found by Diuision , thirdly in respect of both . Of the first way springeth Arithmeticall proportion : Of the second way Geometricall proportion : And of the third Musicall proportion .

Cap. 17.

WHat is Arithmeticall proportion ?

Arithmeticall proportion vnproperly so called , ( because it is no proportion indéed ) is when many seuerall numbers haue one selfe and like difference , as 8. 12. 16. which doe onely differ one from another by 4. and this Proportion is twofold , that is continual and disiunct .

Continuall is when many numbers procéede with like difference , as hath bene said before when we spake of Arithmeticall Progression , as 8. 12. 16. 20. &c. whose difference betwixt euery 2. numbers is 4.

The Disiunct , is when many numbers being seuerally propounded , the difference of the first 2. numbers is not like to the difference that is betwixt the second and the third , and so forth as 5. 8. 4. 7. for 8. differeth from 5. by 3. and 4. from 8. by 4. and 7. from 4. by 3. Now of the second way of comparing which is done by Diuision , springeth as hath beene said before , Geometricall proportion .

Cap. 18.

WHat is Geometricall proportion ?

Geometricall proportion is that which sheweth what part or parts one number is of another number , as 3. is the halfe of 6. which proportion is found by Diuision , wherein you haue to note that if the Diuisor be greater then the diuidend , then it is to be made a Fraction , as in the former example , if you would diuide 3. by 6. then you must make it a Fraction thus , 3/6. or ½ , and this kinde of proportion which may be truely and properly called a proportion indéed or rather a proportionalitie , is said to be twofold , that is , direct , and reuerse , and the direct is either Coniunct , or Disiunct

Coniunct differeth not from Geometricall Progression before taught .

Disiunct proportion Geometricall , consisting most commonly of 4. numbers or of 3. at the least , is when there is not like proportion betwixt the second and the third , that is betwixt the first and the second , or betwixt the third and the fourth , as 3. 6. 4. 8. for here 6. containeth 4. once , and one halfe thereof , which is called Proportio Sesquialtera , and 6. containeth 3. twise , which is called Proportio dupla : and so is 8. to 4.

Cap. 19.

AGaine , proportion Reuerse differeth not from the rule of 3. called Regula euersa . But you haue to vnderstand that the two chéefe and speciall kinds of Geometricall proportion , are these , that is , proportion of equalitie , and proportion of inequalitie .

Proportion of equalitie , is when 2. numbers compared togethtr , are equall the one to the other , as 3. to 3. 4. to 4.

The proportion of inequalitie , is when 2. vnequal numbers are compared together , as 6. to 5 , 4. to 9. and of this there are two kinds , that is proportion of the greater inequalitie , and proportion of lesser inequalitie .

Cap. 20.

PRoportion of the greater inequalitie , is when the greater number is compared to the lesser numbers , as 6. to 5.

Proportion of the lesser inequality , is when the lesser is compared to the greater , as 5. to 6.

Of proportion of the greater inequalitie there be two kindes , Simplex and Multiplex , that is to say , simple and manifold .

SImplex , is when the Antecedent , that is to say the former number containeth the consequent , that is to say the latter number once and somewhat more , which ouerplus must alwaies bee lesse then the consequent it selfe , as 5. containeth 4. once and one parte thereof more , for if you diuide 5. by 4. the quotient will be 1. and ¼ ouer . Againe , this proportion is twofold , that is Superparticuler , and Superpartient : Superparticular , is when the Antecedent containeth the consequent once and some one part thereof , as 3. to 2. for 3. containeth 2. once and one halfe thereof , which is called Sesquialtera , or as 4. to 3. for 4. containeth 3. once and one third part thereof , and is called Sesquitertia , the like is to be said of Sesquiquarta , Sesquiquinta , Sesquisexta , and so forth infinitely as this Table sheweth .

Superparticular proportions are these & such like . Sesquialtera as 3. to 2. 6. to 4. 9. to 6. which is as much as . 1 ½ Sesquitertia as 4. to 3. 8. to 6. 12. to 9. 1 ⅓ Sesquiquarta as 5. to 4. 10. to 8. 15. to 12. 1 ¼ Sesquiquinta as 6. to 5. 12. to 10. 18. to 15. 1 ⅕ Sesquisexta as 7. to 8. 14. to 12. 21. to 18. 1 ⅙ Sesquiseptima as 8. to 7. 16. to 14. 24. to 21. 1 1/7 Sesquioctaua as 9. to 8. 18. to 16. 27. to 24. 1 ⅛ Sesquinona as 10. to 9. 20. to 18. 30. to 27. 1 1/9 Sesquidecima as 11. to 10. 22. to 20. 33. to 30 1 1/10 Sesquiundecima as 12. to 11. 24. to 22. 36. to 33 1 1/11 Sesqiuduodec . as 13. to 12. 26. to 24. 39. to 36 1 1/12

But superpartient is when the Antecedent containeth the consequent once and some parts thereof , that is to say more parts then one , as 5. to 3. for 5. containeth 3. once and 2. third parts thereof , which is called Superbipartiens tertias , of which kinds are these set downe in the Table following .

Proportions superpartient are these and such like . Superbipartiens . Tertias as 5. to 3. 10. to 6. 15. to 9. which is as much as . 1 ⅔ Quintas as 7. to 5. 14. to 10. 21. to 15. 2 ⅖ Septimas as 9. to 7. 18. to 14. 27. to 21. 1 2/7 Nonas as 11. to 9. 22. to 18. 33. to 27. 1 2/9 Vndecimas as 13. to 11. 26. to 22. 39. to 33. 1 2/11 Decimas tertias as 15. to 13. 30. to 26. 45. to 39. 1 2/13 Supertripartiens . Quartas as 7. to 4. 14. to 8. 21. to 12. which is as much as . 1 ¾ Quintas as 8. to 5. 16. to 10. 24. to 15. 1 ⅗ Septimas as 10. to 7. 20. to 14. 30. to 21. 1 3/7 Octauas as 11. to 8. 22. to 16. 33. to 24. 1 ⅜ Decimas as 13. to 10. 26 , to 20. 39. to 30. 1 3/10 Vndecimas as 14. to 11. 28. to 22. 42. to 33. 1 3/11 Superquadripartiens . Quintas as 9. to 5. 18. to 10. 27. to 15. which is as much as . 1 ⅘ Septimas as 11. to 7. 22. to 14. 33. to 21. 1 4/7 Nonas as 13. to 9. 26. to 18. 49. to 27. 1 4/9 Vndecimas as 15. to 11. 30. to 22. 45. to 33. 1 4/11 Decimas tertias as 17. to 13. 34. to 26. 51. to 39. 1 4/13 Decim quintas as 19. to 15. 38. to 30. 57. to 45. 1 4/15 Superquintupartiens . Sextas as 11. to 6. 22. to 12. 33. to 18. which is as much as . 1 ⅚ Septimas as 12. to 7. 24. to 14. 36. to 21. 1 5/7 Octauas as 13. to 8. 26. to 16. 39. to 24. 1 ⅝ Nonas as 14. to 9. 28. to 18. 42. to 27. 1 5/9 Vndecimas as 16. to 11. 32. to 22. 48. to 33. 1 5/11 Duodecimas as 17. to 12. 34. to 24. 51. to 36. 1 5/12 Supersextupartiens . Seqtimas as 13. to 7. 26. to 14. 39. to 21. which is as much as . 1 6/7 Vndecimas as 17. to 11. 34. to 22. 51. to 33. 1 6/11 Decimas tertias as 19. to 13. 38. to 26. 57. to 39. 1 6/13 Decimas septimas as 23. to 17. 46. to 34. 60. to 51. 1 6/17 Decimas nonas as 25. to 19. 50. to 38. 75. to 57. 1 6/19 Vicessimas tertias as 29. to 23. 58. to 46. 1 6/23

Hitherto of Simplex proportio . Now of Multiplex proportio .

Multiplex proportio is when the Antecedent containeth the consequent more then once , as 6. to 2. for 6. containeth 2. thrée tunes , which is called Tripla proportio , Also 12. to 5. for 12. comprehendeth 5. twise and ⅖ . And this Multiplex proportio is twofolde , that is either exact or not exact .

Multiplex exact , is when the Antecedent containeth the consequet more then once , and nothing remaineth , as 4. to 2. 6. to 3. &c. whereof are infinite kinds , as Dupla , Tripla , and so forth as this Table sheweth .

The kinds of Multiplex exact are these and such like . Dupla as 4. to 2. 6. to 3. 8. to 4. which is as much as . 2/1 Tripla as 6. to 2. 9. to 3. 12. to 4. 3/1 Quadrupla as 8. to 2. 12. to 3. 16. to 4. 4/1 Quintupla as 10. to 2. 15. to 3. 20. to 4. 5/1 Sextupla as 12. to 2. 18. to 3. 24. to 4. 6/1 Septupla as 14. to 2. 21. to 3. 56. to 8. 7/1 Octupla as 16. to 2. 24. to 3. 32. to 4. 8/1 Nondupla as 18. to 2. 27. to 3. 36. to 4. 9/1 Decupla as 20. to 2. 30. to 3. 40. to 4. 10/1 Vndecupla as 22. to 2. 33. to 3. 495. to 45. 11/1

But Multiplex not exact , is when the Antecedent containeth the consequent more then once and some thing remaineth ouer , as 5. to 2. for 5. containeth . 2. twise and one remayneth , and this is also twofolde , that is , Multiplex superparticularis , and Multiplex superpartiens

Multiplex superparticularis , is when the Antecedent containeth the consequent more then once & one only remaineth , as 7. to 3. for 7. containeth 3. twise and one only remaineth , wherof are diuers kinds , as Duplex sesquialtera , Duplex sesquitertia , Triplex sesquisexta , & so forth as the table hereafter following sheweth .

But Multiplex superpartiens , is when the Antecedent containeth the consequent more then once & the remainder is more then 1. as 8. to 3. for 8. containeth 3. twise and 2 thirds ouer , whereof there be many , kinds , as Dupla superbipartiens tertias , Dupla supertripartiens quartas , and so forth as this Table following sheweth , which comprehendeth both kinds , that is Multiplex superparticularis , and Multiplex superpartiens

Multiplex not exact , is either Multiplex Superparticularis or els . Duplex . Sesquialtera as 5. to 2. 15. to 6. Sesquitertia as 7. to 3. 21. to 9. Sesquiquinta as 11. to 5 , 22. to 10. Triplex . Sesquisexta as 19. to 6. 38. to 12. Sesquiseptima as 22. to 7. 44. to 14. Sesquioctaua as 25. to 8. 50. to 16. Quadruplex . Sesquinona as 37. to 9. 148. to 36. Sesquidecima as 41. to 10. 82. to 20. Sesquiundecima as 45. to 11. 90. to 22. which is . 2 ½ 2 ⅓ 2 ⅕ 3 ⅙ 3 1/7 3 ⅛ 4 1/9 4 1/10 4 1/11 Multiplex Superpartiens . Dupla . Superbipartiens . Tertias as 8. to 3. Quintas as 12. to 5. Supertripartiens . Quartas as 11. to 4. Quintas as 13. to 5. Superquadripartiens . Quintas as 14. to 5. Septimas as 18. to 7. Tripla . Superquintupartiens . Octauas as 29. to 8. Nonas as 34. to 9. Supersextupartiens . Vndecimas as 39. to 11. Decim , tertias as 45. to 13 Superseptupartiens . Octauas as 31. to 8. Nonas as 34. to 9. Quadrupla . superbipartiens . Tertias as 14. to 3. Quintas as 22. to 5. Superquadripartiens . Quintas as 24. to 5. Nonas as 40. to 9. Superquintupartiens . Sextas as 29. to 6. Septimas as 33. to 7. which is . 2 ⅔ 2 ⅖ 2 ¾ 2 ⅗ 2 ⅘ 2 4/7 3 ⅝ 3 5/9 3 6/11 3 6/13 3 ⅞ 3 7/9 4 ⅔ 4 ⅖ 4 ⅘ 4 4/9 4 ⅚ 4 5/7

Thus much of proportion of the greater inequalitie : Now wee will speake somewhat of proportion of the lesser inequalitie .

Cap. 21.

PRoportion of the lesser inequalitie , is when the Antecedent is lesse then the consequent , as 2. to 3. 5. to 7. for if you diuide 2. by 3. the quotient is ⅔ . if 5. by 7. then the quotient is 5/7 and this Proportion hath the same names which the Proportion of the greate inequalitie hath , sauing that you must adde to the beginning of euery name , this word Sub. as Subdupla , Subtripla , Subsesquialtera , &c. as 1 to 2 is Subdupla proportio , 3. to 9. is Subtripla proportio , &c.

Of Musicall proportion called in Latine Harmoniaca proportio . Cap. 22.

Musical proportion which requireth 3. numbers at the least , is when the first number hath the same proportion vnto the third , which the difference betwixt the first and the second , hath to the difference which is betwixt the second and the third , as 3. 4. and 6. for looke what proportion 3. hath to 6. which is Subdupla , the same hath the difference betwixt 3. and 4. which is 1. to the difference betwixt 4. and 6. which is 2. for 1. to 2. is Subdupla , and this is called Musicall proportion , because the numbers therein haue the same proportions one to another , which are found to be in Musicall consorts , as 6. 4. and 3. for the proportion which 6. hath to 4. is Sesquialtera , called of the Musitians Diapente , or a fift , and the proportion which 4. hath to 3. is Sesquitertia , called of the Musitians Diatesseron , or a fourth , and the proportion which 6. hath to 3. is Dupla , called a Diapason , or an Eight , and of this Musicall proportion there be two kinds , that is simple and compound .

It is said to be simple when it consisteth onely of 3. numbers .

And it is called compound when it consisteth of more then 3. numbers : And of compound there be 2. kinds , that is vnproper and proper .

The vnproper is , when to 3. numbers giuen , 2. other seueral numbers are ioyned , which doe containe the same proportions with the third , which the first 3. numbers haue one to another : As for example , let be giuen these 3. numbers , 3. 4. 6. vnto which if you ioyne 8. and 12. here 6. 8. 12. haue the same proportion one to another which 3. 4. and 6. haue amongst themselues , for like as here betwixt 6. and 8. the proportion is Subsesquitertia , and betwixt 6. and 12. it is Subdupla , and betwixt 8. and 12. it is Subsesquialtera , so betwixt 3. and 4. is Subsesquitertia , and betwixt 3. and 6. is Subdupla , and betwixt 4. and 6. is Subsesquialtera ,

Compound proper , is when diuerse numbers in Musicall proportion standing together , and the first being omitted , the 3. next doe continue still in Musicall proportion . Also when omitting the 2. first numbers the 3. next following are in Musicall proportion . And when omitting the first 3. the next 3. are in Musicall proportion , and so forth how many numbers so euer there be : As for example 10. 12. 15. 20. 30. these are numbers belonging to the proper kind of compound Musical proportion , for 10. 12. and 15. are in Musicall proportion , then omitting 10. which is the first number , 12. 15. and 20 are still in Musicall proportion , and if you omit the 2. first numbers , that is 10 and 12. then the next 3. that is 15. 20. and 30. are still in Musicall proportion , and so forth how many so euer there be of them , so as they be in Musicall proportion : But our Musitians doe make no more but 8. Musicall proportions in all , that is .

Dupla . Tripla . Quadrupla . Sesquialtera . Sesquitertia . Sesquiquarta . Dupla superbipartiens . Sesquioctaua . which are thus named Diapason . Diapason diapente . Bis diapason . Diapente . Diatesseron . Diatonus semitonus . Diapason diatesseron . Tonus .

The vse wherof is to be learned at the hands of the Musitians .

Cap. 23.

But now because it is not inough to knowe the foresaid proportions and their names , vnlesse you can both adde , Subtract , Multiply , and diuide them when néede is , I will therefore briefly here set downe the order thereof , and first shewe how to set them downe in writing , and then how they are to be added , Subtracted , Multiplyed , and diuided .

They are to be set downe in writing in like manner that Fractions are wont to be set downe , sauing that as in Fractions the vpper number is called Numerator , and the inferior number Denominator , so in proportions the vpper number is called Antecedent , and the inferior number Consequent , but whereas in Fractions , ther is wont to be drawn a litle line betwixt the Numerator and the Denominator , in proportions no line is drawne betwixt the Antecedent and the Consequent , but the Antecedent is set ouer the Consequent , without any line drawne betwixt them thus . 3. 4. 2. 5. 7. 1.

And looke as Fractions are to be written in as fewe figures as may be , so are proportions to be set downe in so small numbers as may be , which is to be done by the self same rule that Fractions are . Againe as you can neither adde nor Subtract Fractions , hauing diuerse denominations , before that you haue brought them to one selfe denomination : So can you neither adde nor subtract proportions hauing diuerse consequents , vntill you haue brought them to one selfe consequent by the same rule that Fractions are reduced to one selfe denomination : And looke what order is to be obserued in adding , subtracting , multiplying and diuiding of Fractions , the same is to be kept in adding , subtracting , multiplying and diuiding of proportions . Wherefore I wholly referre you to the rules of Fractions before most plainly taught . And thus I end with proportions , the knowledge wherof is very necessarie to such as haue to deale in matters either of Geometry or of Astronomie or with Musicke .

How to finde out the square roote of any number . Cap. 24.

WHat is a square roote ?

It is any Digit or any other number , which being multiplyed into it self bringeth forth a square number , as 4. being multiplyed in it selfe , in saying 4. times 4. maketh 16. which is a square number , the roote whereof is 4. And to finde out the square roote of any number be it square or not square , you must doe thus . First hauing set downe the number propounded which must consist of 3. figures at the least , set a pricke vnder the first Digit or Cypher of the said number on the right hand , that done , pricke euery other figure thereof towards the left hand , leauing alwaies one voide space betwixt euery 2. pricks as you sée here done in this number ⁴ .⁷ ⁵ .⁶ ⁴ .⁵ and looke how many prickes there be , so many figures shall you haue in the quotient , and this differeth nothing in a manner from Diuision , the order in working is thus , first séeke out a Digit , which being multiplyed in it selfe , may take away that which standeth ouer the last pricke on the left hand , or as much thereof as may be , and set the Digit in the quotient , that done , double the said quotient , then consider whether the double doe consist of one figure or of many , if of one onely , then place that in the next voide space on the right hand , but if the double doe consist of many figures , then take the first figure thereof , that is to say , that figure or cipher which standeth in the first place of the double , and place that in the foresaide next voide space towards the right hand , and next to that , place all the rest of the figures of the sayd double orderly towardes the left hand , that done , séeke another Digit , which being multiplyed in it selfe together with the doubled number , may take away all that which standeth right ouer it , or as much thereof as may be , which Digit you must not onely put in the quotient to his fellow , but also set downe the same right vnder the next pricke on the right hand , and then multiply that Digit together with the double whereto it is ioyned , by the said last found Digit set downe in the quotient , the product whereof you must subtract out of the vpper number , standing right ouer the foresaid pricke , and if there be any remainder , write it aboue , cancelling the other , and looke how many pricks ther be , so many times must you double the quotient , obseruing alwaies like order of working as before , which you shall more plainely vnderstand by these examples following , and for your better vnderstanding thereof , I will first giue you an example of 3. figures onely as thus , 464. here to finde out the square roote of this number , you must first pricke the said number in such order as is before taught , and then séeke out some one of the 9. Digits , which being multiplyed in it selfe , may take away the first 4. on the left hand , or as much thereof as may be , which you shall finde to be 2. which 2. being set in the quotient , multiply the same in it selfe , in saying 2. times 2. maketh 4. which 4. doth cleane take away the first 4. standing ouer the last pricke on the left hand , and therefore cancell the 4. that done , double the quotient which maketh 4. and set the same in the next voide space on the right hand vnder the figure 6. thus .

And then séeke out some Digit , which being multiplyed in it selfe together with the doubled , may take away the Somme of 64. that remaineth , which by asking how many times 4. is comprehended in 6. you shall finde to be but one , which you must not onely put in the quotient , but also set it downe vnder the first pricke on the right hand thus .

So as you shall make the lower Somme to be 41. which is to be multiplyed by the last found quotient which is 1. wherefore you must say that 1. times 41. is 41 which being taken out of 64. there remaineth 23. so as the roote is 21. and the remainder 23. as this example sheweth .

And to know whether you haue done well or not , multiply the quotient in it selfe , and if there be any remainder left , adde that vnto the product of the multiplyed quotient , and if you finde the summe thereof to be like to the first , then you haue done well , if it be not like , then you haue erred , the product of this quotient being multiplyed in it selfe amounteth vnto 441. whereto by adding the remainder which is 23. you shall finde the summe to be like vnto the first number that is 464. whereof you sought the square roote . But if such number do consist of many figures , then in working you must double the quotient once , twice or thrice , according as the number doth require , as you shall more plainely perceiue by this example following : as let 50/467. be the number whereof you would know the square roote , here hauing pricked this number in such order as before , first séeke out some Digit which being multiplyed in it selfe may take away the Digit or number standing right ouer the last pricke on the left hand , which you shall finde to be 2. and hauing set downe the saide 2. in the quotient , saye 2. times 2. maketh 4. which being subtracted from 5. there remaineth 1. which you must set ouer 5. and cancell the sayde 5. as this example sheweth .

That done , double the quotient which maketh 4. and set downe the 4. in the next space towardes the right hand , vnder the cipher as you sée in the former example , then séeke out a Digit , which being multiplyed in it selfe together with the double quotient , may take away all that which standeth ouer the second pricke towards the left hand , or as much thereof as may be , which Digit by asking how many times 4. is comprehended in 10. you shall finde to be 2. then set downe the saide 2. not onely in the quotient , but also vnder the second pricke towardes the right hande , which together with the former 4. maketh 42. as you sée in this example .

Then multiply that 42. in a place by it selfe by the figure 2. last set downe in the quotient , and it will make 84. which 84. being taken out of 104. there remaineth 20 which you must set ouer the second pricke and cancell the 104. and also the 42. beneath , as you sée in the former example : that done , double the whole quotient , which maketh 44. wherof set the first 4. in the next voide space towards the right hand vnder the figure 6. and the other 4. vnder the second pricke towards the left hand , them aske how many times 4. is comprehended in 20. and you shall find 4. which you must not onely set downe in the quotient , but also vnder the next and last pricke on the right hand , and so the nether Summe shall be 444. which being multiplyed by the Digit 4. last set down in the quotient , will make in all 1776. which being taken out of the vpper number which is 2067. there will remaine 291. so as the square roote of the first number is 224. and the remainder 291. as you may sée in this example .

The square roote whereof being multiplyed in it self amounteth vnto 50176. wherto if you adde the remainder which is 291. the Summe thereof will bee like vnto the first number which is 50467. Moreouer you haue to note that if you finde any number out of the which your quotient being doubled cannot be subtracted , then you must set downe a Cypher in the quotient , and procéede to the next pricke on your right hand , as in this example following .

In which example the first found Digit set in the quotient is 6. which being multiplyed in it selfe maketh 36. and cleane taketh away the first number standing ouer the last pricke on the left hand , that done , I double the quotient , which maketh 12 and I place the 2. in the next voide space towardes the right hand , and the 1. next to that towards the left hand orderly as in this example .

Ouer which 12. there is but 6. remaining , so as I can not take 12. out of 6. and therefore I cancell the 12. and set down a Cypher in the quotient , so as the quotient is now 60. which being doubled maketh 120. the Cypher whereof I set downe in the next voide space towards the right hand , and the other two Digits orderly towards the left hand , as you sée in this example .

Then I aske how many times one is contained in 6 , and I finde 5. which I set downe in the quotient and also vnder the last pricke on the right hand , so as the nether number is 1205. which I multiply aparte by it selfe by the 5. last set downe in the quotient , the product whereof is 6025. which being taken out of the vpper number , which is also 6025. there remaineth nothing , as this example sheweth .

Wherefore I finde the foresaid number to be a iust square number , the roote wherof is 605. which being multiplyed in it selfe maketh 366025. a number like vnto the first number .

The vse of the square roote in setting of battels . Cap. 25.

THe knowledge of finding out the square roote of any number , is very necessary for a Sargent maior in the fielde , that he may the more readily set and range his Squadrans of Battel : And therefore I thinke it not amisse , to giue them here certaine examples of such manner of squares as the Italians were wont to vse in my time , which is foure manner of waies .

Whereof the first teacheth how to set the Battel square of ground , the second a square battel of men , the third a long square battel which we call the Hearse battel , and the fourth teacheth how to set a battel of so much and a third .

Shew by what rules these battels are to be set ?

Before I set downe any rules you haue to vnderstand that all numbers giuen or supposed , are not méete for such purpose , but onely such as may be diuided into 3. equall parts without leauing any remainder : for otherwise all the rules following , take no place , and therefore when any number is giuen you to be brought into any one of the foresaid 4. formes of battell , you shall doe well first to deuide that number by 3. and if there be any remainder , to reiect it , and to take the rest for your number , As for example , A Sargent Maior is commanded by his Generall to set a battaile square of ground , appointing him thereunto 1345. men , here the Sargent by diuiding that number by 3. findeth in the remainder one man too many , which being onely reiected , all the rest of the number which is 1344. is fitte for his purpose .

How to set a battell square of ground , called in Italian Battaglia quadra di terreno .

NOw shew the order of setting such kinds of battell ?

The order is thus , first double the number giuen , then take the square roote thereof , and that roote shall be the Front , that done , diuide the number giuen by the said roote , and the number in the quotient shall be the flanke , now if you multiply the front and the flanke the one into the other , you shall haue your whole number first giuen , vnlesse there be some remainder left in the former Diuision , which remainder must be added to the product to make vp the somme . But if you woulde know how many men are to be put in a rancke , and also how many such ranckes you must haue , then diuide the front into 3. partes , and the quotient will shew you how many men you are to haue in a rancke , that done , multiply the flancke by 3. and the product thereof shall be the number of the ranckes : As for example , take the number before giuen , which was 1345. which being diuided by 3. you finde the remainder to be 1. which you must reiect , and take all the rest which is 1344. and that being doubled maketh 2688. the square roote whereof is 51. by which roote you must diuide the number giuen , Videlicet 1344. and the quotient will be 26. which shall be the flancke , now if you diuide the foresaid front 51. by 3. the quotient will shew how many shall be in a rancke , that is 17. Finally , if you multiply the flancke , which is 26. by 3. the product thereof will be 78. which is the iust number of the rancks which you must haue .

How to set a battell square of men called in Italian Battaglia quadra d'huomini .

TAke the square roote of the number giuen , and that shall bee both the front and the flancke , which if you multiply together , the product thereof will make vp the number first giuen , and if you would know how many men are to bee put in a rancke , and how many such rancks you must haue , then doe as you did before in diuiding the front by 3. and multiplying the flancke by 3.

How to set a long square battell which we call the Hearse battel , and is called in Italian , Battaglia dun tanto emezzo .

TO the number giuen , adde the iust halfe therof , and the square roote of that Somme shall be the front , by which roote , or front you diuide the first number giuen , the quotient shall be the flancke . As for example , let the number giuen be 6144. the halfe thereof is 3072. which being added to the giuen number 6144. maketh in all 9/216. the square roote whereof is 96. which shall bee the front , by which roote or front if you diuide the giuen number 6144. you shall finde in the quotient 64. and that shall bee the flancke , that done , diuide the foresaide front 96. by 3. and you shall finde in the quotient 32. which number sheweth how many men shall be in a rancke , then multiply the foresaid flancke 64. by 3. and the product thereof shall bee 192. which is the number of the ranckes .

How to set a battell of so much and a third , called in Italian , Battaglia dun tanto & dun terzo .

TO the number giuen , adde the third part thereof and the square roote of that Somme shall be the front , then diuide the first number giuen by the foresaid roote or front , and the number in the quotient shall be the flancke : As for example , let the number giuen be 1575. the third part whereof is 525. which being added to 1575. maketh in all 2100. the roote whereof is 45 which must be the front , by which roote or front , if you diuide the number giuen which was 1575. the quotient will be 35. which shall be your flancke , then by diuiding 45. which is the front by 3. the quotient wil shew you how many men you must haue in a rancke that is 15. Againe by multiplying the flancke which is 35. by 3. the product thereof will be 105. which is the number of your ranckes .

How to finde out the Cubique roote of any number . Cap. 26.

AS the square roote is said to bee a number which being multiplyed in it selfe , doth make a square superficiall number , hauing onely length and breadth , so as the Cubique roote is a number which being first multiplyed in it selfe , and the product therof being againe multiplyed by the first number , doth make a Cubique or Corporall number hauing both length breadth , and depth : As for example , 2. times 2. maketh 4. and 2. times 4. maketh 8. Againe 3. times 3. maketh 9. and 3. times 9. maketh 27. and so you may deale with all the rest of the 9. Digits , and make thereby a Table containing both the square numbers and Cubique numbers of euery roote , consisting of any one of the 9. Digits , like vnto this table here following . Rootes Squares Cubiques 1. 1. 1. 2. 4. 8. 3. 9. 27. 4. 16. 64. 5. 25. 125. 6. 36. 216. 7. 49. 343. 8. 64. 512. 9. 81. 729.

Then hauing to finde out the Cubique roote of any number that is greater then 1000. ( for lesser it cannot be to worke vpon ) then set a pricke vnder the first figure on your right hand , and so procéede towards the left hand , omitting alwaies 2. figures betwixt euery two prickes , as in this example 41063625. and looke howe many prickes there be , so many figures shall you haue in the quotient : That done , you haue to finde out thrée seuerall numbers , first such a Cubique number as will cleane take away the number which standeth right ouer the head of the last pricke on the left hand , or so much thereof as may be , which you may easily finde in the foresaid table : The roote of which Cubique number , you must set downe in the quotient , the second number which you haue to find is a number called the Triple , which is easily had by tripling the quotient , and the third number is called the Diuisor which you shall most readily finde by multiplying the quotient into the Triple , both which numbers are to be placed in such order as followeth . First then hauing found the Cubique number as is before said , and taken the same out of the number standing ouer the last pricke on the left hand , write the remainder ( if there be any ) ouer his head , cancelling that which is vnder it , and then place the roote of the said Cubique number in the quotient : that done , Triple the roote set downe in the quotient , and that shall be the Triple which you must place in the second voide space on the left side of the next pricke which is on your right hand , then multiply that Triple by the quotient , and the product thereof shall be the Diuisor which must be placed right vnder the Triple one figure shorter towards the left hand : that done , drawe a line as you sée in the example hereafter following and worke thus : First aske how many times the first figure of the said Diuisor is contained in the number standing right ouer his head : and hauing found an apt Digit for the purpose , put that Digit in the quotient to his fellow , that done , multiply the said Digit into the Diuisor and set the product thereof right vnder the Diuisor : Secondly multiply the said Digit in him selfe , and againe the product thereof into the triple , & set the somme that comes therof right vnder the triple : Thirdly multiply the said Digit in it selfe cubically , and set the product thereof lowest of all , right vnder the next pricke on the right hand : that done , drawe a line , and by Addition bring all the foresaid thrée products into one somme placing euery figure so as you may easily take the said somme out of the vpper number , whereof you séeke the Cubique roote , and write the remainder ( if there be any ) ouer the head , and if such somme will not be subtracted out of the vpper number , then you must séeke out a lesser Digit , reseruing still the former triple and Diuisor , and if there bee more pricks in the number then two , then you must for euery pricke finde out a new Triple , and a new Diuisor by tripling the whole quotient , and by working continually in like order as the rule before teacheth , which rule you shall more plainely perceiue by example thus . First then hauing to finde out the Cubique roote of the number before mentioned , which is 4 1 0 6 3 6 2 5 marked with prickes as before is taught : resorte to the Table wherein you shall finde such a Cubique number as will take away as much of the 41. as may bée , which is 27. the roote whereof is 3. for al the other Cubique numbers in the table are either too small or too great , and therefore you must alwaies haue due consideration therof . Then take 27. which is the Cubique number , out of 41. and there remaineth 14. which remainder you must set ouer the last pricke on your left hand cancelling the 41. as in this example .

Then triple the quotient which maketh 9. which you must set downe vnder the figure 6. next vnto the second pricke which is on your right hand : Then to find out the Diuisor multiply the Triple 9. by 3. which is the quotient , and the product therof shal be the Diuisor : which diuisor you must place right vnder the triple one figure shorter towards the left hand : that done , draw a line as you sée in this example .

Now hauing orderly set downe the Triple and the Diuisor , aske how many times 2. which is the first figure of the Diuisor is contained in 14. standing right ouer his head , and remember to make choise of such a Digit as may not cleane take away the whole number at the first , but rather leaue so much as the saide quotient hauing afterward to be multiplyed diuerse waies as shal be said hereafter , may take away the rest or as much therof as may be , as in this example following , you shall finde the aptest Digit for this purpose to be 4. which you must put in the quotient , whereby the quotient shall be now 34. that done , you must first multiply this last quotient which is 4. into the Diuisor 27. the product whereof is 108. which you must set downe right vnder the Diuisor , beneath the line already drawne , as you sée in this example .

Secondly you must multiply the said 4. in it self quadratly which maketh 16. then multiply that 16. by the triple 9. the product whereof is 144. which is to be set right vnder the triple as in this example .

Thirdly you must multiply the said 4. in it selfe cubically , which maketh 64. which is to be set right vnder the next pricke on the right hand beneath the former summes as you sée in this other example .

So as euery one of the foresaide thrée products doe extend one further then another by one figure towardes the right hand , as you sée in the former example : Now these products being thus placed , drawe another line and bring all the thrée seuerall products contained betwixt the 2. lines , into one summe by Addition , and you shal finde the totall summe to be 12304. which being subtracted out of the number standing right ouer it which is 14063. there will remaine 1759. which you must set downe aboue as you sée in this example .

Now to procéede with the next , you must finde out a new Triple and a new Diuisor .

How is that done ?

Thus , multiply the whole quotient which is 34. by 3. the product whereof is 102. and that is the Triple which is to be placed in the next voide space hard by the next pricke that is on your right hand : Then multiply the whole quotient into the last found Triple 102. and the product thereof which is 3468. shall bee your Diuisor , which is to be placed vnder the Triple one figure shorter towards the left hand as you sée in the example following : which two numbers being thus found and rightly placed , draw a line , then aske how many times the first figure of your Diuisor which is 3. is contained in the number right ouer his head which is 17. and you shall finde it to bee 5. times contained therein , wherefore set downe 5. in the quotient , that doone , multiply the 5. into the Diuisor , and place the product thereof which is 17/340. right vnder the Diuisor beneath the line , as you sée in this example .

Secondly square the same 5 that is to say , multiply it in it selfe , and that maketh 25. which 25. you must multiply againe into the Triple 102. the product whereof is 2550. which number you must set right vnder the saide triple beneath the line . Thirdly multiply the said 5. in it selfe cubically which maketh 125. and place that right vnder the first pricke on the right hande , and drawe a line , that done , adde all the foresayde 3. products together , and you shall finde the totall summe thereof to bée 1759625. which being subtracted out of the vpper number , there remaineth nothing , as this example plainely sheweth .

Whereby you may conclude that the foresaid number whereof you sought the roote , is a perfect Cubique number , for if you multiply the whole quotient in it selfe cubically , it wil produce the selfe same number whereof you sought the roote . But note that if in making the first Subtraction , the first Diuisor is not to be found in the vpper number , then you must set a Cypher in the quotient , that done , triple the whole quotient , and place the saide triple vnder the figure which is next to the next pricke on the right hand , working in such order as before , and to avoide confusion , hauing to deale with a multitud of numbers it shal not be amisse at the finding out of euery new triple , & diuisor to set the remainders in seueral places by themselues , & to worke then in the self same order that is before taught , also note that if you haue to deale with a fewe numbers , and that the Diuisor cannot bee subtracted out of the number standing right ouer his head , then you must set a Cypher in the quotient and so you haue done , as for example , hauing to take the cubicall roote of 8597. here I find 2. to be the quotient , which being cubically multiplyed , doth cleane take away the 8. and now according to my former rule 12. must be my Diuisor , which because I cannot take it out of 5. I set downe therefore a Cypher in the quotient , so as the quotient is now 20. which is the cubicall roote of the foresaide number , for if you multiply 20. cubically and adde therevnto 5 6 7. which is the remainder , the summe thereof will be like vnto the first number , as you may sée in this example .

Of Astronomicall Fractions . Cap. 27

BEcause the vse of these Fractions is very necessarie for those that haue to calculate the motions of the Starres and the difference of time , I thought good to shew here how the same are to be added , subtracted , multiplyed and diuided , for the measure of time falleth not out alwaies to bee a whole yeare , moneth , day , or hower , nor the mouing of the celestiall bodies are to be measured alwaies by whole circles , signes , or whole degrées , & therfore to haue an exact measure of such things it was thought best by the auncient writers to diuide all whole things called Integra into the lessest parts that might bee , for which purpose no number was thought so méete as 60. for there is no number vnder 100. that receiueth so many Diuisions as 60. which may bee diuided many sundrie waies , that is by 2. 3. 4. 5. 6. 10. 12. 15. 20. and by 30. and therefore they diuided euery whole thing that had no vsuall parts into 60. minutes , and euery minute into 60. seconds , and euery second into 60. thirds , and so forth vnto 60. fourthes , fifts , sixts , seuenths , eights , ninthes and tenths , and further if néede were but that seldome chanceth . And you haue to note that minutes are marked with one stréeke ouer the head , seconds with two stréekes , thirds with thrée stréekes , and so forth thus , 23′· 6″· 7‴· 8''''· &c. which do signifie 23. minutes 6. seconds 7. thirds , and 8. fourths .

Of Addition .

WHat is to bee obserued in adding of these kindes of Fractions ?

First that you bring Integrums to Integrums , and Fractions to Fractions , that bee of like denomination , beginning alwaies with the least on the right hand , and if the summe of such Addition doe amount any where to the number of 60. or aboue 60 then you must looke how many 60. are comprehended therein , and for euery 60. adde one to the next greater Fraction that is on the left hand , obseruing still that order vntill you come to the Integrums , of which Integrums , it is also necessary to know their values , that is to say , what parts they containe , and what denomination those parts haue : As for example , if you adde common signes such as the twelue signes of the Zodiaque be , then euery signe containeth 30. degrées , so as euery summe excéeding 30 is to be diuided by 30. but if they be Phisicall signes whereof 6. doe make a whole circle , such as be set downe in the table of Alphonsus , then the summe of those degrées is to be diuided by 6. Moreouer so often as the summe of the common signes doe excéede 12. or the summe of the Phisicall signes doe excéede 6. the ouerplus is alwaies to be reiected , and the remainder to be set in the place of the signes , as you may sée in this example following wherein seconds are reduced to minutes , minutes to degrées , and finally degrées , to signes .

An example of Addition consisting of signes , degrees , minutes and seconds .

Signes Degrees Minutes Seconds 9. 19. 1. 19. 0. 0. 35. 16. 11. 29. 33. 5. 9. 29. 38. 11. 0. 11. 49. 40. 4. 56. The totall summe . 8. 0. 42. 27.

In which example beginning first with the seconds because they are here the least Fractions , you shall finde by Addition that the summe of them amounteth to 147. which being diuided by 60. you shall finde in the quotient 2. and the remainder to bee 27″· which remainder you must set downe vnder the collum of seconds , kéeping the quotient which is 2. in mind to be added to the collum of minutes , the summe whereof is 162. which being diuided by 60. you shall find in the quotient 2. degrées , and the remainder to be 42′· which is to be set downe vnder the collum of minutes , and the quotient 2. kept in minde to bee added to the collum of degrées , the summe whereof is 90. degrées , which being diuided by 30. ( because that 30. degrées doe make one common whole signe ) you shall find in the quotient 3. signes and no remainder : wherefore you must set downe a Cypher vnder the collum of degrées , and adde the 3. signes to the collum of signes , the summe whereof is 32. which it you diuide by twice 12. which maketh 24. you shall finde the remainder to be 8. signes , which is to be set downe vnder the collum of signes , for the quotient here is to be reiected according to the rule before giuen , so as the totall summe of this Addition is 8. signes , 0. degrées 42′· 27″·

Another example of daies , howers , minutes and seconds to be added together .

Daies Howres Minutes Seconds 21. 14. 32. 11. 16. 16. 19. 41. 8. 16. 30. 30. Summa totalis . 46. 23. 22. 22.

In this example one self order as in the other before is to bée obserued as touching the secondes and mynutes , for the excéeding number in both , are diuided by 60. but when you come to the howers you must diuide that number by 24. because that so many howers doe make a whole day , and hauing set downe the remainder vnder the collum of howers , adde that one day which was in the quotient , vnto the collum of daies , and so you shall find the totall summe to be as the example aboue sheweth .

Of Subtraction .

WHat order is to be obserued in Subtraction ?

The selfe same that was before obserued in Addition , so as you alwaies remember that when you haue to take a greater number of Fractions , as of minutes , seconds , thirds and such like , out of a lesser number of Fractions , to borrow 60. and hauing set downe the remainder to adde the one borrowed , vnto the next collum on the left hand , for there the 60. borrowed , is but one , but if you haue to deale with degrées , which are counted Integrums , then you must borrow but 30. for so many degrées doe make one signe , and if you haue to subtract howers , then you must borrow 24. for so many howers doe make one day , as by the example here following you shall more plainely perceiue . Signes Degrees Minutes Secondes 8. 0. 42. 27. 0. 1. 9. 53. 7. 29. 32. 34.

In this example because you can not take 53″· out of 27″· you must borrowe one minute from the next collum on the left hand , which one mynute is 60″· which being added to 27. doe make in all 87″· out of which if you take 53″· there will remaine 34″· that done , you must pay home the minute which you borrowed by adding the same vnto the next 9. on the left hand , which maketh 10. then say , take 10′· out of 42′· and there remaineth 32′· which being set down , procéede to the next . But here to take one out of none that cannot be , and therefore borrow one whole signe from the collum of signes which is 30. degrées , from whence take one , and there remaineth 29. which being set downe , take the one which you borrowed out of 8. and there remaineth 7. so as the whole remainder of this Subtraction is 7 / signes , 29 / degrées , 32′ / and 34″· as the former example sheweth .

Of Multiplycation .

THough there bee more difficultie in multiplying and diuiding Astronomicall Fractions then in adding or subtracting them , yet the greatest difficultie thereof chiefly consisteth in the finding out of the true denomination of the products , for first as touching Multiplycation you must multiply euery number of the multiplyer into all the particular numbers of the sum that is to be multiplyed , and then seuerally to adde together the products that bee of one selfe denomination , and whatsoeuer in that Addition ariseth to the number of 60. or excéedeth 60. it is to be reduced by the sexaginarie Diuision into the greater summe , so shall you collect the whole summe of the Multiplycation . But you haue to note by the way , that if there be any Integrums of diuers denominations in your multiplyer : That then such Integrums must be reduced to one selfe kind of Integrums : As for example , suppose that you would multiply the daily motion of the Moone which according to Alphonsus tables , is 13. degrées , 10′· 35″· and 1‴· by 29. daies , 12. howers , 44′ , and 3″· here because there be in this multiplyer Integrums of diuers denominations , that is to say daies and howers , you must therefore reduce the same into one selfe denomination , before that you can make your Multiplycation .

How is that to be done ?

By multiplying euery number of the said multiplyer by 5. and then by halfing the product thereof , by which halfing you shall reduce mynutes to seconds , and seconds to thirds , and so forth to the smallest Fractions of all , and if any product doe amount to 60. or excéede 60. then you must diuide that product by 60. the ●emainder whereof is to be set vnder his proper denomination , and you must kéepe the quotient in minde to adde the same to the next greater number as this table sheweth : In the front whereof I haue set downe in seuerall spaces not onely the denominations of the two Integrums , as daies and howers , but also the denominations of so many Fractions as I think méete to serue my turne , vnder which from I place the foresaid multiplyer , and then draw a line as you sée in this example following .

Denomination . Daies Howers ′ ″ ‴ ' ' ' ' The multiplyer to be reduced into one selfe denomination . 29. 12. 44 3 The products of the reduction . 29. 31 50 7 30

Now beginning first on the right hand with the least Fraction of the saide multiplyer which is 3″· I multiply 3. by 5. which maketh 15. the one halfe whereof is 7‴· and halfe a third which is 3''''0· wherefore I set downe the said 7‴· and 3''''0· vnder their proper denominations as you sée in the example aboue , then I multiply 44′ by 5. the product whereof is 220. and the halfe thereof is 110. which being diuided by 60. the quotient is 1′· and the remainder 50″· which remainder I set vnder his proper denomination , kéeping the quotient stil in mind , that done , I multiply 12. howers by 5. the product whereof is 60. and the halfe of that is 30′· wherevnto by adding the one which I had in mind , I make it 31′· and so I set it downe vnder his proper denomination , and because there be no more Fractions to be multiplyed , I set downe on the left hand the Integrum 29. and by this meanes I haue brought the foresaid multiplyer to one selfe denomination and to one kind of Integrum , that is to say to 29. daies 31′· 50″· 7‴· 3''''0· which now being the greater number is to be set aboue , and to be Multiplyed by the foresaid daily motion of the Moone , that is 13. degrées , 10′ 35″· and 1‴· but to the intent that in multiplying these numbers together you may set euery product in his true place , that is to say , vnder his proper denomination , it shall not be amisse in the front of your worke to set downe two rowes of numbers , whereof the first must containe many denominations or Fractions as you thinke good , as minutes , seconds , thirds , fourths , and so forth , marked with stréekes and vulgare numbers , and the second rowe shall be the naturall order of numbers written in Arithmeticall figures as this table sheweth . Integra i ii iii iiii v· vi· vii viii The denominations . 0. 1. 2. 3. 4. 5. 6. 7. 8. The naturall numbers .

Vnder which table you must first set the number that is to bée multiplyed , and right vnder that the multiplyer in such sorte as euery particular product may be placed vnder his proper denomination , and then draw a line as you sée in this example following , and when you haue multiplyed 2. numbers the one into the other , and know not where to place the product , then marke vnder which of the naturall numbers in the front , the said two numbers , that is to say the multiplyer and the multiplycand do stand : That done , adde those two naturall numbers of the front together , and the summe thereof will them you vnder what denomination the product is to be placed , as in this example .

Int●gra ′ ″ ‴ ' ' ' ' ᵛ vi vii Denomination 0 1 2 3 4 5 6 7 Naturall numbers . 29 31 50 7 30 The multiplycand . 13 10 35 1 The multiplyer 29 31 50 7 30 The seueral products . 1015 1085 1750 245 1050 290 310 500 70 300 377 403 650 91 390 389 6 24 2 31 12 37 30 The general ●roduct or totall summe . The remainder .

In which example I first multiply 1‴· into 3''''0· the product whereof is 3 vii0· which must be set vnder the denomination vii . because the two naturall numbers that is 3. standing ouer 1. the multiplyer , and 4. standing ouer 30. the multiplycand being added together doe make 7. appointing to the product his proper denomination , then multiply againe the same 1‴· into 7‴· the product whereof is 7 vi· which must be set vnder the denomination vi . because 3. standeth ouer both their heads , and therefore must bée taken twice , that is to say for each number 3. which being added together doe make 6. appointing thereby to the product his proper place of denomination , that done , multiply the said 1‴· into 50″· the product whereof is 50″· which must be set vnder the denomination v. for the 3 which standeth ouer 1. and the 2. ouer 50″· being added together maketh 5. appointing to his proper place of denomination , then multiply the said 1‴· into 31′· the product whereof is 1''''1· here the two naturall numbers that is to say 3. standing ouer 1‴· and one ouer 31′· being added together , doe make 4. appointing to the product his proper place of denomination , then multiply the foresaid 1‴· into 29. Integrums , the product whereof is 2‴9· and must be set vnder the denomination ‴ because 3. standeth ouer 1‴· but 29. being an Integrum , hath no natural number standing ouer him but a Cypher , thus hauing gon through out all the numbers of the multiplycand , with the first number of the multiplyer , procéede in like order with the second number of the multiplyer , which is 35″· which being multiplyed into 3''''0· maketh 10 vi50· to be set vnder the denomination vi . because 2. standeth ouer 35″· the multiplyer , and 4. ouer 3''''0· the multiplycand , which 4. and 2. being added together maketh 6. then multiply 35″· by 7‴· which maketh 245 v· which you must set downe vnder the denomination v. because 2. and 3. maketh 5. that done , multiply 35″· by 50″· which maketh 1750''''· which you must set downe vnder the denomination' ' ' ' because the multiplycand and the multiplyer are both vnder the denomination ″ which being twice repeated , maketh 4. then multiply 35″· by 31′· and that maketh 1085‴· which you must set downe vnder the denomination ‴ because 1. and 2. maketh 3. finally multiply 29. Integrums , by 35″· and that maketh 1015″· thus as you haue gon through with two numbers of your multiplyer , so procéede in like order with the other two numbers of the multiplyer which is 10. and 13. and when you haue ended your Multiplycation , and set euery product in his proper place , and so as euery figure may stand one right vnder another , to auoide confusion when you come to Addition , ( to which end the spaces of collums had néede to be the larger ) then draw a line vnder all the products and beginning on the right hand , adde all the products contained in euery seuer all collum together , and if the summe of any such particular Addition do arise to the summe of 60. or excéed the number of 60. then diuide that summe by 60. and set downe the remainder , kéeping the quotient in mind to be added to the product of the next collum on the lefthand , so shall you find the total summe of your Multiplycation to be 389. degrées , 6′· 24″· 2‴· 31''''· 12 v· 37 vi· and 30 vii· as the former example plainly sheweth . Now if you dsuide 389. degrées , by 30. because euery common signe containeth 30. degrées , you shall find your totall summe to be 12. signes , 29. degrées , 6′· 24″· 2‴· 31''''· 12 v· 37 vi· and 30 vii· and so much the Moone runneth in the space of 2. 9. daies 12. howers , 44′ and 3″· of an hower , which is her full reuolution betwixt euery two changes , but for as much as it chanceth as wel in this example as in many others like , that Integrums of two sundrie denominations are propounded in the question , it may be very well doubted with what denomination the product of such multiplycation is to be named , as in this example hauing multiplyed time by motion , a man may aske whether the product shall be named daies or degrées , the resoluing of which doubt dependeth vpon the nature of the question propounded , for in the foresaid example , because time or daies do comprehend any certaine appointed motion , therefore the product of the Multiplycation is to be referred to the degrées of motion which are comprehended vnder time , and not to time which comprehendeth motion , wherfore this product of Integrums videliz . 389. signifieth here degrées and not daies , so likewise when degrées and minutes are multiplyed by myles and minutes , the product of such Multiplycation taketh his name from myles & not from degrées , because degrées do comprehend myles , for we say in matters of Geography that euery degrée of the great circle comprehendeth 60 myles , thus hauing spoken sufficiently of the Multiplycation of Astronomical Fractions , we wil now procéede to the Diuision of such Fractions .

Of the Diuision of Fractions Astronomicall .

WHat is to be obserued therein ?

First you must consider whether your Diuisor be compound , or simple , I cal that compound which containeth Fractions of diuers denominations , and that simple which consisteth of Integrums , or is one whole number of one selfe denomination , wherein there is no dificultie , for then you haue no more to do but to diuide euery particular number contained in the diuidend by the same Diuisor and to place the product of euery one vnder such denomination , as the little table of denominations sheweth , & therefore it shall not bee amisse to set the foresaid little table ouer your diuidend euen as you did in Multiplycation : Also the Sexagenary progression is alwaies to be vsed , as well in Diuision as in Multiplycation . Moreouer if your Diuisor be not exactly contained in the diuidend , then hauing multiplyed the diuidend by 60. you must adde to the product therof the next Fraction following : As for example , knowing by Alphonsus tables that the daily motion of the Moone is 13. degrées , 10′· 35″· 1‴· 15''''· you would know how much the goeth in the space of an hower , here because that one day containeth 24. howers , the number must be 24. your Diuisor which is simple and not compound , first then set downe in the front of your worke the rowe of denominations onely , and not the natural numbers , because they are not to be vsed in this way of Diuision , that done , right vnder the rowe of denominations place your diuidend , and right vnder th● your Diuisor , as you sée in this example .

Degree● ′ ″ ‴ ' ' ' ' ᵛ The denominations . 13 10 35 1 15 The diuidend . 24 The diuisor . 32 56 27 33 7 The seuerall summ of euery quotient .

In which example because the Diuisor 24. is not contained in 13. therfore I multiply 13. by 60. which maketh 780. wherevnto by adding the next Fraction on the right hand which is 10′· the whole summe is 790′· which being diuided by 24. the quotient is 32′· which because they are minutes , I place them vnder the denomination of minutes , and the remainder is 22′· which being multiplyed by 60. maketh 1320″· wherunto I adde the next figure which is 35. and so the whole summe is 1355″· which being diuided by 24. the quotient is 56″· which I place vnder the denomination of seconds , and the remainder of this Diuision is 11″· which being multiplyed by 60. maketh 660‴· whereto I adde the next Fraction which is 1‴· so that now the whole summe is 661‴· which being diuided by 24. the quotient is 27‴· which I set downe vnder the denomination of thirds , and the remainder is 13‴· which being multiplyed by 60. maketh 780''''· whereunto I adde the next Fraction which is 15''''· which maketh in all 795''''· which being diuided by 24. the quotient is 33''''· which I place vnder the denomination of fourths . and the remainder is 3''''· which being multiplyed by 60. maketh 180 v· whereunto hauing no Fraction to adde , I diuide the same by 24. and so I find in the quotient 7 v· which I set vnder the denomination of fifts , so as I find the howerly motion of the Moone to bée 32′· 56″· 27‴· 33''''· 7 v· and somewhat more , for I leaue to deale any further with the smaller Fractions that would stil grow by multiplying the remainders by 60. thinking this sufficient to shew you in what order you haue to worke , to diuide your diuidend by a simple Diuisor , into as many small parts as you will : but if your Diuisor be compound , then the Diuision is to be done either by reduction into the smallest Fractions , or without reduction : which last way is very hard and tedious , and therefore I will onely shew you how to make your diuision whereof the Diuisor is compound by reduction , and that by this one example here following . Suppose then that the Moone according to her owne course which is from West to East , is distant from some fixed Starre 36. degrées . 30′· 24″· 50‴· and 15''''· and that you would know in what time she will runne that distaunce , according to her daily moouing which as hath béene said before is 13. degrées , 10′· 35″· 1‴· and 15''''· here to make this diuision by reduction , you must doe thus . First reduce all the numbers of your diuidend into the smallest Fractions thereof by the Sexagenarie Multiplycation and Addition of the next Fraction vnto the product of that Multiplycation : that done , reduce all the numbers of your Diuisor by like Multiplycation and Addition , into the smallest Fractions , so as the diuidend & the Diuisor may be both of one selfe denomination , and diuide the one by the other , euen as they were Integrums , as in this example you must first multiply 36. degrées , by 60. and it will make 2160′· whereto by adding 30′· you make the whole summe of minutes to be 2190′· which being multiplyed againe by 60. doe make 131400″· whereto if you adde the 24″· the summe of secondes will bee 131424″· and so procéeding still with the Sexagenarie Multiplycation and Addition of the next Fraction as you did before , you shall find the diuidend to be 473129415''''· Then in like order reduce your Diuisor into the smallest Fraction , and you shall find the totall summe therof to be 170766075''''· this reduction being made , diuide the diuidend by the diuisor , so shal you find in the quotient 2. Integrums , that is to say 2. daies , and the remainder to be 132597365''''· which remainder if you multiply by 60. and diuide the product by the self same Diuisor , you shal haue in the quotient minutes , then multiply againe that remainder by 60. and diuide the product thereof by the same Diuisor , and you shall haue in the quotient seconds , and so by obseruing still that order you shall bring it into as smal Fractions as you will , thus shall you finde that the Moone according to her daily motion , will runne the foresaide space of distance that was betwixt her and the fixed Starre in 2. daies , 46′· and 14″·

How to diuide Astronomicall Fractions when the Diuisor is greater then the diuidennd . Cap. 28.

THough by the last chapter you may learne how to diuide any number in Astronomicall Fractions , whereof the Diuisor is greater then the diuidend , yet I minde once againe , to set downe a generall rule to serue for such purpose , because it commeth often in vse in hauing to deale with Astronomicall tables , and to giue you example thereof : First then hauing to diuide any number , whose Diuisor is greater then the diuidend , doe thus ; multiply the greatest denomination of the diuidend by 60. and if there be any Fractions annexed thereunto which are of the next inferiour kinde , as minutes are to degrées , or to howers , and seconds to minutes , and thirds to seconds &c. Then adde them to the former product , but if such Fractions be not of the next inferior kind , then let them stand as they are vntill you come to deale with them , and hauing diuided according to the common rule of Diuision the first summe of the diuidend by the Diuisor , multiply the quotient into the whole Diuisor , and subtract that product out of the vpper number if it may bee , if not , then make the quotient lesser and lesser , vntill you can finde such a number as will be subtracted out of the said vpper number , and if there be any remainder left , then multiply that remainder by 60. not leauing to follow the former order of working , vntill you haue found the nearest exact quotient that may be . And you haue to note that the denomination of the first quotient must be of the next inferior kind , to that denomination which the Diuisor hath , and to make this rule the plainer , I will set downe and example vsed by Standius ut the 115 page of his Ephemerides who to know the very instant of the sull Moone the second of March 1569. biddeth to diuide the distance of the opposition which was 8. degrées , 46′· by the diurnal excesse of the Moones motion from the sùnne which was then 13. degrées , 48′· which Diuisor because it is greater then the diuidend , you must according to the rule before giuen , worke thus : First multiply the greatest denomination of the diuidend which is 8. degrées by 60′· the product whereof will be 480′· whereunto by adding the Fraction annexed videliz . 46′· it maketh in all 526′· which is to be diuided by 13. degrées , 48′· here in diuiding the last product first by 13. degrées , I finde in the quotient 39. which is one too many , considering that I must take out of the foresaid diuidend 48′· as often as I did take out thereof 13. degrées , wherefore I set downe but 38. in the quotient , and then the remainder will be 96. which because I may easily diuide by the common Diuisor 13. degrées , and 48′· I diuide therefore that 96. first by 13. whereof the quotient is 7. and the remainder is 5. which I reduce into seconds by multiplying that 5. by 60. the product whereof is 300″· that done , I multiply 48′· by 7. the product whereof is 336. which though it bee somewhat too great a number to be taken out of 300. yet I let it stand because it approcheth to a very nigh exactnesse , and by this meanes I finde the whole quotient to bee 38′· and 7″· and you haue to note that if after the first quotient be set downe , there happen any remainder which is lesser then the Diuisor , then you must set downe a Cypher in the quotient , and remooue your Diuisor one place further , euen as you doe in common Diuision , and then to worke as before .

How to take the square roote of Astronomicall Fractions . Cap. 29.

THe greatest difficultie hereof consisteth in finding out the true denomination of the roote , for if the Fraction be seconds , then the roote therof are mynutes , and if the Fraction be fourths , then the roote are seconds , for the Fraction must alwaies haue such denomination as may be halfed , as seconds , fourths , and such like , the one halfe whereof giueth alwaies name to the roote , for if the question bee of thirds , you must first reduce them to fourths before you can take the roote , and you must doe the like with any other Fraction , whose denomination is od and not euen . As for example , if you would take the roote of 43‴· here by multiplying these 43‴· by 60. you shall reduce them into 1580''''· the roote whereof is 50″· Moreouer the Fractions where . with you haue to deale , are either simple or compound , if they bée simple and lesse then minutes , and therewith haue euen denominations and not odde , then you néede to make no further reduction , but to worke as if you had to deale with whole numbers . As in seeking the square roote of 1600″· you find it to be iust 40′· but if the number be compound , that is to say , consisting ot Integrums and Fractions , or of many Fractions hauing diuers denominations , then you must first reduce them all to the smallest Fraction that hath an euen denomination before that you can take the roote : As for example , you would know the roote of 4. degrées , 25′· here you must by the Sexagenarie Multiplycation and Addition of the next Fraction , reduce the degrées to minutes , and the minutes to seconds , as you were taught before in Diuision , and then to worke as you were wont to doe in taking the square roote of whole numbers , and in so doing , you shall finde the summe of seconds to be 15900″· the square roote whereof is 126′· which if you diuide by 60. it will make 2. degrées , 6′· Another example , as to take the roote of 13. degrées , 42′· and 45″· here by reduction as before , you shall bring the degrées and minutes to 49365″· . the square roote whereof is 222′· which being diuided by 60. maketh 3. degrées , 42′· And thus I end with the Astronomicall Fractions , which kinde of Fractions , though they be very learnedly and orderly taught by Reinoldus in the beginning of his Prutenicall tables , yet in mine opinion not in so plaine order , and so fit for euery mans vnderstanding , as I haue here set them downe according to the doctrine of Gemma Frisius , which being once learned , you shall the soner attaine to the other . And without the knowledge of these Fractions , you can neuer truely calculate any thing out of the Astronomicall tables , and therefore such Fractions are most necessarie to be learned .

The description and vse of the Sexagenarie table .

THis table consisteth of two figures , whereof the neather figure hauing foure Angles , is called in Latine Trapezium , marked with the letters A. D. E. B. and the vpper figure is a Triangle , marked with the letters A. B. C. and each figure containeth particular collums of numbers , seruing to finde out the products of Astronomicall Fractions being multiplyed one by another , and also the quotients of the like Fractions being diuided one by another , and also the square rootes of the said Fractions , for which purpose the first collum on the left hand , containeth 59. Fractions , counting from 1. standing aboue , and so procéeding downe ward to 59. and are contained betwixt A. and D. and the foote of the said Trapezium , containeth 30. counting from the lest hand towards the right , which are contained betwixt D. and E. and the rest of the numbers to make vp 59. are to be found in the vpper most front of the Triangle , procéeding from C. towards A. you haue to note also that in the outermost collum of the Trapezium on the right hand , the numbers doe procéede downeward from 30. to 59. that is from B. to E. and the numbers in the outermost collum of the Triangle on the right hand , doe procéede vpward from 31. to 59. contained betwixt B. and C. both which doe serue to fill vp the first multiplyers , and multiplycands , for when you cannot finde them in the Trapezium , then the outermost collum of the Triangle on the right hand , serueth to supply that want , and when you cannot finde the said numbers in the Triangle , then the outermost collum of the Trapezium on the right hand , serueth to supplie that want , and in séeking any multiplyer or multiplycand , either in the Trapezium or in the Triangle , consider alwaies which way they are most readily found out , so as they may directly aunswere one against another .

All the rest of the numbers contained betwixt the two outermost collums and are set downe in square Angles , called common Angles , do signifie either products or diuidends or square numbers , according as occasion shall require . And the outermost collums do signifie sometime multiplyers , sometime multiplycands . sometime quotients , and sometime rootes . All which thinges you shall better vnderstand by the examples hereafter following .

The vse of the Table .

By helpe of this table you may more readily multiply and diuide Astronomicall Fractions , and also find out the square roote of such Fractions , then by those rules which I haue heretofore set downe according to the doctrine of Gemma Frisius . And first I will set downe an example of Multiplycation , then another of Diuision , and thirdly one example of finding out the square roote of the said Fractions , and let the example of Multiplycation be thus : Suppose that you would multiply 29. degrées , 31′· 53″· 7‴· 30''''· by 13. degrées , 10′· 35″· 1‴· which you must set downe in such order as followeth , ( that is to say ) first the denominations , then next vnder them the multiplycand , and next vnder that the multiplyer , and vnder them the seueral products , and lowest of all the totall summe of the said products .

Integra De. ′ ″ ‴ ' ' ' ' ᵛ vi vii Denominations 29 31 50 7 30 The multiplycand . 13 10 35 1 The multiplyer . 29 31 50 7 30 The first product . 17 13 34 14 22 30 The second product . 4 55 18 21 15 0 The third product . 6 23 53 51 37 30 The fourth product , 6 29 6 24 2 31 12 37 30 The total summe

Here beginning with the first number of the multiplyer on the right hand which is 1‴· say thus , one times 30. is 30 which is to bée placed vnder the denomination of 7. because the denominatino ouer 1. is 3. and the denomination ouer 30. is 4. which being added together according to the rule before set downe in the Chapter of Multiplycation of Astronomicall Fractions doe make 7. then say one times 7. is 7. which you must place vnder the sixt denomination . Againe one times 50. is 50. which is to bee set vnder the fift denomination , because 3. and 2. maketh 5 , then say one times 31. is 31. which is to bée set vnder the fourth denomination , for 3. and 1. maketh 4 , then one times 29. is 29. which is to be set vnder the third denomination , and thus you haue the first product : then procéede with the next number of the multiplyer towards your left hand which is 35. and is to be multiplyed into 30. which to doe readily , you must enter the table with these two numbers , and séeking in the first Collum of the Trapezium on the left hand , for 35. in the foote of the said Trapezium , for the number of 30. you shall finde in the common angle the product to be 17. and 30. whereof you must place the 30. vnder the sixt denomination , and kéeping the number 17. still in minde to be added to the next product , multiply 35. into 7. and you shall finde in the Trapezium the product to be 4. and 5. whereunto if you adde the 17. which you had in mind , the product will be 4. and 22. whereof you must set the 22. vnder the fift denomination , and kéeping the 4. in minde , multiply againe the same 35. into 50. the product whereof you shall finde in the Triangle to be 29. and 10. whereto if you adde the 4. in mind , it will make 29. and 14. whereof the 14. is to bee placed vnder the fourth denomination , and kéeping the 29. in minde , multiply againe the said 35. into 31. and you shall finde the product thereof in the Triangle to bee 18. and 5. whereunto if you adde 29. in mind , it will make in all 18. and 34. which 34. is to be set vnder the third denomination , then kéeping 18. in mind , multiply the foresaid 35. into 29. and you shall finde the product thereof in the Trapezium to be 16. and 55. whereunto if you adde the 18. in mind , the product will be 17. and 13. which 13. is to be placed vnder the second denomination . Now because you haue gon through all the numbers of the multiplycand with the number 35. you must place the 17. which you had in mind vnder the denomination of mynutes , and so hauing ended the second product , procéede to the finding out of the third product by multiplying 10. first into 30. the product whereof you shall find in the Trapezium to bée 5. and 0. whereof you must set the Cypher vnder the fift denomination and kéeping the 5 in minde , multiply againe 10. into 7. and you shall finde the product to be 1. and 10. whereunto if you adde the 5. in mind , the product will be 1. and 15. whereof you must set the 15. vnder the fourth denomination , and kéeping the one in mind , multiply 10. into 50. and you shall find the product thereof in the Trapezium to be 8. and 20. whereunto if you adde the 1. in mind , it will make 8. and 21. whereof you must set the 21. vnder the third denomination , and kéeping 8. in mind , multiply 10. into 31. and you shall find the product thereof in the Trapezium to bee 5. and 10. vnto which if you adde the 8. in minde , the product will be 5. and 18. whereof set downe the 18. vnder the second denomination , and kéeping 5. in minde multiply 10. into 29. the product where of you shall finde in the Trapezium to bee 4. and 50. whereunto if you adde the 5. in mind , the product will be 4 and 55. Now because you haue gone through all the numbers of the multiplycand , with 10. you must set downe the 4. which you had in mind vnder the denomination of degrées , and so hauing the third product , procéede to the fourth by multiplying 13. into 30. and you shall find the product in the Trapezium to be 6. and 30. whereof set downe 30. vnder the fourth denomination of Fractions , and kéeping 6. in mind multiply 13. into 7. the product whereof you shall finde in the Trapezium to be 1. and 31. whereunto if you adde the 6. in minde , the product will be 1. and 37. whereof set downe the 37. vnder the third denomination , and kéeping the 1. in mind multiply 13. into 50. and you shall find the product in the Trapezium to be 10. and 50. whereunto if you adde the one in mind , the product will be 10. and 51. whereof set the 51. vnder the second denomination of Fractions , and kéeping 10. in mind multiply 13 into 31. the product whereof you shall find in the Trapezium to bee 6. and 43. whereunto if you adde the 10. in mind , the product will be 6. and 53. whereof set downe 53. vnder the first denomination of Fractions , and kéeping 6. in mind multiply 13. into 29. the product whereof you shall find in the Trapezium to be 6. and 17. whereunto if you adde the 6. in mind , the product will be 6. and 23. whereof set downe 23. vnder the denomination of degrées . Now because you haue gone through all the numbers of the multiplycand with the last number of the multiplyer , you must set downe the 6. which you had in minde vnder the denomination of Integrums , that done , adde all the foure products together , beginning on the right hand , saying thus , 30. and 0 is but 30. which set downe vnder the neathermost line , as you sée in the former figure , then say 30. and 7. maketh 37. which you must set downe vnder the neathermost liue next vnto 30. then say 22. and 50. maketh 72. out of which by subtracting the Sexagenarie number , the remainder is 12 v. which is to bee set vnder the line next vnto the 37. kéeping still the 60. in minde , which in this account maketh but one , and is to be added to the next rancke on the left hand , then say 1. in mind and 5. maketh 6. and 4. maketh 10. and 1. maketh 11. here set downe 1. and kéepe the Article in mind , then say 1. in mind and 3. maketh 4. then 1. and 1. and 3. being added to 4. doe make in all 9. out of which 9. you must subtract 6. which is but one 60. and is to be kept in mind and there remaineth 3. which is to be set downe by the 1. then say 1 in mind and 7. is 8. and 1. is 9. and 4. is 13. and 9. is 22. here set downe 2. and kéepe the 2. Articles in minde , then say 2. and 3. maketh 5. and 2. is 7. and 3. is 10. and 2. maketh 12. tennes , which doe make 2. sixties , and are to be kept in mind , then say 2. and 1. maketh 3. and 8. maketh 11. and 3. maketh 14. wherefore set down 4. kéeping the 1. Article in mind , then say 1. in mind and 5. is 6. and 1. is 7. and 1. is 8. then take 6. from 8. and there remaineth 2. which set downe by the 4. and kéepe one sixtie in mind , then say 1. in mind and 3. is 4. and 5. is 9. and 7. is 16. wherefore set downe 6. and kéeping one in mind , say that 1. and 5. is 6. and 5. is 11. and 1. is 12. which maketh two sixties to be kept in minde , then say 2. in mind and 3. is 5. and 4. maketh 9. which set downe , then say 2. is 2. which set downe by the 9. vnder the denomination of degrées , then say 6. and nothing maketh 6. which set downe vnder the denomination of Integrums , for that 6. in this place signifieth 6. sixties , which is in value 360. degrées , and being diuided by 30. because 30. degrées maketh one whole signe , you shal find in the quotient 12. signes , so as the totall summe of the foure products is 12. signes , 29. degrées , 6′· 24″· 2‴· 31''''· 12 v· 37 vi· 30 vii· as the former figure sheweth , and this is the very same example which I wrought before when I taught you how to multiply Astronomicall Fractions according to Gemma Frisius his rule , and they both doe wholly agrée in euery condition , sauing that to worke by the Sexagenarie table is the readier way of the two .

An example how to diuide Astronomicall Fractions by helpe of the Sexagenarie table .

SVppose then that you would diuide the former totall summe or product found by Multiplycation , which is 6. Integrums , 29. degrées , 6′· 24″· 2‴· 31''''· 12 v· 37 vi· 30 vii· by this Diuisor 29. degrées , 31′· 50″· 7‴· 30''''· which in the former example of Multiplycation was the multiplycand . Now to diuide these two numbers the one by the other , you must doe thus , first you must set downe the rowe of denominations as you did before in Multiplycation , and next vnder that the diuidend , then right vnder that the diuisor , and on the right hand behind a crooked line made like a halfe Moone , all the seuerall quotients are to be set one by another in a right line , as you may sée by the figures hereafter following .

Denominatiō● ▪ Integrum de ′ ″ ‴ ' ' ' ' ᵛ vi vii The diuidend . 6 29 6 24 2 31 12 37 30 The diuisor . 29 31 50 7 30

Heere hauing made your rowe of denominations , and set downe your diuidend , you haue to consider whether the first number of your Diuisor be greater then the first number of your diuidend , for if it be , then you must place your Diuisor one space further towardes your right hand , as in this example , because the first number of your Diuisor 29. cannot be taken out of 6. you set it vnder the second number of the diuidend , and so all the rest of the numbers successiuely towardes the right hand , as the former example sheweth , now the order of working is thus : You must first séeke out the first number of your Diuisor , in the first collum of the Trapezium on the left hand which is 29. then in the rowe right against that 29. on the right hand , you haue to séek out those numbers of the diuidend , which do stand right ouer the first number of your Diuisor which is 6. and 29. and if you cannot finde those numbers iustly , then séeke in the selfe same rowe a number which is somewhat lesse and nearest in value vnto it , as in this example , because you cannot finde 6. and 29. you take 6. and 17. the quotient , wherof you shall find in the foote of the Trapezium right vnder the said number 6. and 17. to bée 13. degrées , which you must set in the quotient line , and that is your first quotient , hauing his proper denomination ouer his head which are degrées , and are to bee found by the rule before taught . Thus you sée here that the number of the Diuisor is to be founde in the outermost collum on the left hand , and the number of the diuidend in that rowe which is right against the said Diuisor , and the quotient in the foote of the Trapezium right vnder the number of the diuidend last found . Then you haue to multiply the whole Diuisor ( that is to say ) euery particular number thereof by the first quotient 13. which you may doe by helpe of the Table , as you did before in the example of Multiplycation , and is to be set downe in this manner . And you shall finde the whole product to be 6. 23. 53. 51. 37. 30.

Which being set right vnder the first diuidend , is to be subtracted out of the same , and the remainder to bee written ouer the head of the diuidend as you doe in common Diuision , first then to multiply all the particular numbers of the Diuisor by the quotient 13. and to find euery product thereof , resort to the Table and séeke for 13. in the foote of the Trapezium , and for 30. which is the multiplycand in the outermost collum of the Trapezium on the left hand , and the common angle will shew the product which is 6. and 30. whereof you must set downe 30. vnder 30. and kéeping 6. in mind , multiply againe 13. into 7. the product whereof you shall finde to bee 1. and 31. vnto 29 31 50 7 30 The Diuisor . 13 The first quotient . 6 23 53 51 37 30 The first product .
which adde the 6. which you kept in mind , and the product shall be 1. and 37. whereof you must set downe 37. and kéeping one in mind , multiply againe 13. into 50. and you shall find the product to be 10. and 50. whereunto if you adde 1. in mind , the product shall be 10. and 51. whereof you must set downe 51. and kéeping 10. in mind , multiply 13. into 31. the product whereof you shall find in the Trapezium to be 6. and 43. whereunto if you adde the 10. which you had in mind , the product shall be 6. and 53. whereof set downe 53. and kéeping 6. in mind , multiply 13. into 29. and you shall find the product to be 6. and 17. whereunto if you adde the 6. in mind , the product shall be 6. & 23. whereof set downe 23 vnder the last number of your multiplycand , and because you haue no more numbers of the Diuisor to be multiplyed , set downe 6. in mind on the left hand , so shall the whole product bee 6. 23. 53. 51. 37. 30. as the former example sheweth , which product is to be subtracted out of the first diuidend , & the remainder is to be set downe ouer the head of that diuidend , as you sée in this example next following , wherein the first diuidend is first set downe , and right vnder that the foresaid product which is the first product , and the remainder aboue the diuidend , and the quotient 13. is set in the quotient line which is your first quotient .

The remainder . 5 12 32 25 1 The first diuidend . 6 29 6 24 2 31 12 37 30 ( 13. The first product , 6 23 53 51 37 30

And remember ( in making your Subtraction ) to begin with the first number of the foresaide product which is on the right hand , and when you cannot take it out of the number standing right ouer his head , to borrow alwaies 60. of the next number on the left hand , and to pay it home againe with 1. for there 60. is but one . This done , remooue your Diuisor one space further towards the right hand ( that is to say ) set the first number of your Diuisor vnder 12 which is the second number of the second diuidend , which together with the first remainder is 5. 12. 32. 25. 1. 12. 37. 30. and al other numbers of the Diuisor orderly towards your right hand , as you sée in this example .

The second diuidend . 5 12 32 25 1 12 37 3● The Diuisor . 29 31 50 7 30

Then aske how many times 29. is in 5. & 12. which number you must séeke in the Trapezium in the rowe that answereth towards the right hand , to the first number of your Diuisor which is 29. standing in the outermost collum of the Trapezium on the left hand , and because you cannot find 5. and 12. in that rowe , you must take in the same rowe the number which is nighest vnto it , but lesse , which you shall find to be 4. and 50. and right vnder that in the foote of the Trapezium you shall find 10. which must be your second quotient , by which quotient you haue to multiply all the particular numbers of the Diuisor in such order as is before set downe and you shall find the product of that Multiplycation to be 4. 55. 18. 21. 15. 0. which product you must place vnder the second diuidend setting 4. vnder 5. and 55. vnder 12. and so forth orderly towards the right hand , that done , subtract the same product out of the numbers of the second diuidend , standing right ouer the said product , and the remainder will be 17. 14. 3. 46. 12. as you sée in this example .

The remainder . 17 14 3 46 12 37 30 The second diuidend . 5 12 32 25 1 12 37 30 ( 13.10 . The second product . 4 55 18 25 15 0

Now remooue your Diuisor one space further towardes the right hand by setting the first number of your Diuisor which is 29 vnder 14. which is the second number of the third diuidend , and so all the rest orderly towardes the right hand , as you sée in this example .

The third diuidend . 17 14 3 46 12 37 30 The Diuisor . 29 31 50 7 30

Then aske how many times 29. is contained in 17. and 14. which diuidend because you can not find it in the rowe that answereth to that 29. which standeth in the outermost collum of the Trapezium on the left hand ( for all those numbers are too little ) you must séeke for it in that collum which standeth right vpon 29. in the foote of the Trapezium , neither shall you find it there , but you shall finde 16. and 55. which is somewhat lesser yet in value nighest vnto 17 and 14. and right against that you shall find in the outermost collum on the right hand 35. which must be your third quotient to bée set in the quotient line , whereby you haue againe to multiply the whole Diuisor in such order as before , the product whereof you shall find to be 17. 13. 34. 14. 22. 30. which product being placed vnder the diuidend by setting 17. vnder 17. and 13. vnder 14 and so forth orderly towards the right hand , subtract the said product ( beginning at the right hand ) out of the numbers which stand right ouer the said product , and the remainder will be 29. 31. 50. 7. 30. which is to be set downe ouer the third diuidend , and the rest to be cancelled as you sée in this example .

The remainder . 29 31 50 7 30 The third diuidend 17 14 3 46 12 37 30 ( 13.10.35 The third product . 17 13 34 14 22 30

And now the remainder of the third diuidend is come to be the fourth diuidend , wherefore remooue your Diuisor one space further towards the right hand by setting 29. vnder 29. and the rest orderly towards the right hand as you sée in the example following . Then aske how many times 29. is contained in 29. which being but once , your fourth quotient is 1. and is to bee set in the quotient line , whereby the whole Diuisor being multiplyed , the product will be 29. 31. 50. 7. 30. which you must place vnder the fourth diuidend , and being subtracted out of the same , nothing will remaine , and so the whole quotient will be 13. degrées , 10′· 35″· 1‴· as you sée in this example .

0 0 0 0 0 The fourth diuidend . 29 31 50 7 30 ( 13. degrees 10′·35″·1‴· The fourth product . 29 31 50 7 30

Another example of Diuision .

IF the daily motion of the Moone be 13. degrées 3′· 53″· 56‴· 23''''· 58 v· in what time shall she make her whole reuolution allowing 360 degrées to that reuolution , which is otherwise called the moneth of Paragration . Heere for so much as this example is to bee wrought by Diuision , & that your diuidend is a simple and whole number ( that is to say ) 360. degrées , without any Fractions of diuers denominations annexed thereunto : you must first set downe the 360. degrées , and next to that towardes your right hand set downe a long rowe of Cyphers with so many denominations ouer their heads as you shall thinke néedful to serue your turne , & right vnder your diuidend set your Diuisor as you sée in this example .

The denominations . ●nte●rum De. ′ ″ ‴ ' ' ' ' ᵛ vi vii viii ix The diuidend . 360 0 0 0 0 0 0 0 0 0 The Diuisor . 13 3 53 56 23 58 ( 27. The first product . 352 45 16 22 47 6

Here you must first aske how many times 13. is in 360. and by the common rule of Diuision you shall finde the quotient to be 27. which you must set downe in the quotient line , and by that quotient you haue to multiply euery particular number of the Diuisor beginning on the right hand , as you doe in Multiplycation , saying that 27. times 58. is 26. and 6. as the Trapezium sheweth , for by séeking 27. in the foote of the Trapezium and for 58. in the outermost collum on the right hand , you shal find in the common Angle 26. and 6. wherefore set downe 6. vnder 58. and kéeping 26. in mind , multiply 23. which is the second number of the Diuisor by 27. and by the Trapezium you shall find the product thereof to be 10. and 21. whereunto if you adde the 26. in mind , it will make 10. and 47. whereof set downe 47. vnder 23. and kéeping 10. in mind , multiply 56. which is the third number of the Diuisor by 27. and in the Trapezium you shall finde the product to bee 25. and 12. whereto if you adde the 10. in minde , it will make 25. and 22. whereof set downe 22. vnder 56. and kéeping 25. in mind , multiply 53. which is the fourth number of the Diuisor by 27. the product whereof you shall find by the Trapezium to be 23. and 51. whereto if you adde 25. in mind , it wil make 76 which is one 60. and 16. whereof set downe 16. vnder 53. and adde the one in mind to 23. and that will make 24. which you must kéepe in mind , then multiply 3. which is the fift number of the Diuisor by 27. the product whereof you shall find in the Trapezium to be 1. and 21. whereto if you adde the 24. in mind , it will make 1. and 45. whereof set downe 45. vnder 3. and kéeping 1. in mind , multiply 13. which is the last number of the Diuisor by 27. the product whereof you shall find in the Trapezium to bee 5. and 51. whereunto if you adde one in mind , it will make 5. and 52. here because the first 5. is 5. sixties , it maketh in all 352. and is to be set vnder 360. so as the first product of this Multiplycation containeth these numbers 352. 45. 16. 22. 47. 6. as you sée them set downe in the former example , and this product is to bee subtracted out of 360 which is the first diuidend . And to auoide confusion , it shall not bee amysse to set downe the first Diuidend and the first product apart by themselues thus .

The first remainder . 7 14 43 37 12 54 0 0 0 The first diuidend . 360 0 0 0 0 0 0 0 0 ( 27. The first product . 352 45 10 22 47 6

Heere beginning on the right hand say thus , take 6. out of nothing which wil not be , wherfore you must borrow 60 then by taking 6. out of 60. there will remaine 54. which you must set aboue that Cypher which standeth right ouer 6. and caucell the 6. kéeping still the one 60. which you borrowed in mind , then say 47. and one in mind maketh 48. which will not be taken out of nothing , and therefore you must borrow againe one 60. as you did before , so shall the remainder be 12. which is to be set aboue the Cypher which standeth right ouer 47. and cancell 47. and so procéede with like order in subtracting all the rest of the numbers of the first product out of the first diuidend , so shall the remainder be 7. 14. 43. 37. 12. 54. as you sée them set downe in the former figure . Now hauing to remoue your Diuisor one space further towards the right hand , you shall doe well to make your first remainder which is 7. 14. 43. 37. 12. 54. to be your second diuidend , and vnder that to set your Diuisor as you sée in this example .

The second diuidend . 7 14 43 37 12 54 0 0 0 ( 27.33 . The Diuisor . 13 3 53 56 23 58

Then aske how many times 13. is contained in 7. and 14. & hauing found 13. in the foote of the Trapezium , séek in that collum for 7. and 14. and not finding it there , take in the self same collum that number which is nighest in value vnto it & lesse , which you shal find to be 7. and 9. right against which in the outermost collum on the right hand is 33. which must be your second quotient , and is to be set in the quotient line , next vnto 27. and by this quotient you haue to multiply euery number of the Diuisor as you did before , the product wherof you shall find to be 7. 11. 8. 40. 1. 10. 54. wherfore you must first set downe your second diuidend , and then the second product right vnder the same as you sée in this example .

The second remainder . 3 34 57 11 43 6 The second diuidend . 7 14 43 37 12 54 0 0 0 0 ( 27.33 . The second product . 7 11 8 40 1 10 54

Which product being subtracted out of the second diuidend , the remainder will be 3. 34. 57. 11. 43. 6. which is to be set aboue the second diuidend , and the product to bee cancelled as you sée in the former example . Here hauing to remoonue agine your common Diuisor one space further towards the right hand , set downe first the last remainder which now must be your third diuidend , and vnder that set the common Diuisor as you sée in this example .

The third diuidend . 3 34 57 11 43 6 0 0 0 ( 27. 33. 16. The Diuisor . 13 3 53 56 23 58

Then aske how many times 13. is contained in 3. and 34. here by séeking in the foote of the Trapezium for 13. though you cannot find in that collum 3. and 34. yet you shall find 3. and 28. which is the nighest , right against which in the outermost collum on the left hand you shall find 16. which must be your third quotient , and is to bee set in the quotient line , by which quotient you must multiply the whole Diuisor as before , the product whereof you shall find to be 3. 29. 2. 23. 2. 23. 28. which being the third product you must set downe vnder the third diuidend which was your last remainder and to be subtracted out of the same as you sée in this example .

5 54 48 40 42 32 0 0 The third diuidend . 3 34 57 11 43 6 0 0 0 The third product . 3 20 2 23 2 23 28

And so the remainder will be 5. 54. 48. 40. 42. 32. which you must set aboue the third diuidend , and al the inferior numbers are to be cancelled as you sée in the former example . Here hauing againe to remoue your Diuisor one space further towardes your right hand , the last remainder must be your fourth diuidend , vnder which the common Diuisor is to be set thus .

The fourth diuidend . 5 54 48 40 42 32 0 0 The Diuisor . 13 3 53 56 23 58 27 daies / 33′/16″ 16″/26‴·

Here asking how many times 13. is contained in 5. and 54. séeke for 13. in the foote of the Trapezium , in whose collum you shal not finde 5. and 54. but 5. and 51. which is nighest vnto it , and right against that in the outermost collum on the left hand you shall find the quotient to be 27. by which if you should multiply the whole Diuisor , the whole product thereof would be 5. 56 21. 24. 1. 47. 6. which is more then the diuidend , and therefore you must make your quotient one lesse , setting downe no more but 26. in the quotient line , by which if you multiply the Diuisor , the product will be 5. 26. 41. 22. 25. 14. 8. which being subtracted out of the fourth diuidend , the remainder will bee 28. 7. 18. 17. 17. 52. which remainder if you will , you may make to be a fift diuidend , and then to worke as before , if you would haue your quotient to extend to smaller denominations , which I leaue to doe because I thinke that thirds be small inough . And as often as the product of any particular quotient shall bee greater then the diuidend , remember to take a lesse quotient euen as you doe in common Diuision . But now you haue to note that though this whole quotient here signifieth time , for the first quotient signifieth daies , the second quotient minutes of daies , the third quotient seconds , and the fourth quotient thirds , yet for so much as the day is to be counted by 24. howers and not by mynutes , you must therefore reduce all the particular quotients sauing the first , into howers and parts of howers , euen to so small denominations as you shall thinke good your selfe , by helpe of this rule which in diuision of Astronomicall Fractions biddeth to multiply the quotient by 2. and to diuide the product thereof by 5. as here if you multiply the 33. which is the second quotient by 2. the product will bee 66. which being diuided by 5. the quotient will be 13. howers and one sixtie remaining to be kept in minde , wherefore set downe in the place of 33′· 13. howers , then multiply the third quotient which is 16. by 2. and that will make 32. whereunto if you adde the one 60. in mind it will make in all 92. which being diuided by 5. the quotient will be 18. and two sixties , which is 120. remaining to be kept in minde , wherefore in stéede of 16″· which was the third quotient , set 18′ of an hower , and then procéede to the fourth quotient which is 26‴· which being multiplyed by 2. maketh 52. whervnto if you adde , 120. it will make 172. which if you diuide by 5. the quotient will bee 34″· of an hower , which is to bee set downe in the place of 26. so shall your whole former quotient containe 27. daies , 13. howers , 18′· of an hower and 34″· of an hower . And note that as in the Diuision of Astronomicall Fractions to bring the quotients to like denomination , you doe multiply by 2. and diuide by 5. so in Multiplycation to reduce the numbers of the multiplyer being of diuers denominations to one selfe denomination , you must by order reuerse , multiply by 5. and diuide by 2. whereof I haue giuen you an example before , whereas I shew you how to multiply Astronomicall Fractions according to Gemma Frisius , without the helpe of the Sexagenarie table .

An example shewing how to extract the square roote out of Astronomicall Fractions .

SVppose the number giuen to be 17′· 12″· 33‴· 4''''· whereof you haue to take the square roote , here hauing set downe the said numbers with their proper denominations ouer their heads , as you sée in the example following , first set a pricke vnder the last number on the right hand , and then pricke euery other number leauing one voide space betwixt euery two prickes , as you doe when you séeke the square roote of whole numbers or Iutegrums , as you sée here in this example .

The number giuen 17′· 12″· 33‴· 4 ' ' ' '

Then resort to the table & séek amongst the products which are placed next to the line A. B. as well in the Trapezium as in the Triangle , and sée whether you can find the number standing ouer the first pricke on the left hand which is 17. and 12. but not finding it there , you must take that which is nighest vnto it and lesse , which you shall find in the Triangle towards B. to be 17. 4. the roote whereof you shall finde both in the head , and also in the outward collum of the Triangle on the right hand to be 32. aunswering to the foresaid square roote , which roote you must place behinde the quotient line , then subtract 17. and 4. out of 17. and 12. and there will remaine 8. which is to be set ouer 12. and the 17. and 12. to be cancelled , that done , double the roote 32. which will be 64. that is to say 1. and 4. and setting 1. vnder 8. and 4. vnder 33. aske how many times 1. is in 8. and there is 8. which is the second quotient by which you must multiply 1. and 4. the product whereof will be 8. and 32. which being subtracted from the vpper number which is 8. and 33. the remainder is 1. which is to be set ouer 33. and the 8. and 33. to be cancelled . Finally multiply the second quotient 8. in it selfe , and the product will be 1. and 4. which being subtracted out of the former remainder of the giuen number nothing remaineth , so as you shall finde 32′· 8″· . to be the roote of the giuen number , as this example sheweth .

Which roote if you multiply into it selfe squarely , the product will be like vnto the number giuen , and by this meanes you shall find the giuen number to be a iust square number . But you haue to note that if the last denomination standing on the right hand be od & not euen , as thirds or fifts , then you must set downe a Cypher beyond the last denomination towards your right hand , and vnder that Cypher set your first pricke , as in this example 3′· 2″· . 9‴· 17''''· . 8 v· here because the last denomination on the right hand is od , that is to say fifts , you must therefore set downe next vnto it towards the right hand a Cypher hauing ouer his head the next denomination which is vi . and is euen , wherefore set your first pricke vnder that Cypher and so procéede towards your left hand pricking euery other number , and then worke as followeth .

First then séeke in the table amongst the square numbers nigh vnto the line A. B. for 3. and 2. which should stand in one selfe square Angle next vnto the line A. B. for those onely are square numbers , and not finding it amongst the square numbers , take that square which is nighest vnto it and lesse , which you shall find to be 2. and 49. the roote whereof is 13. and is to be found as well in the foote of the Trapezium right vnder the foresaid square number , as also in the outermost collum of the said Trapezium on the left hand standing right against the same square number , which roote is to be set in the quotient line , that done , subtract the foresaid square number out of the number standing right ouer the last pricke on the left hand which is 3. and 2. and the remainder will be 13. which is to be set ouer 2. and the 3. and 2. to be cancelled as you sée in the former example , that done , double the foresaid found roote 13. which wil make 26. by which you haue to diuide 13. and 9. wherefore resorte to the Trapezium , in the foote whereof you shall find 26. in whose collum not finding 13. and 9. take that which is nighest vnto it and lesse , which is 13. and 0. right against which you shall find in the outermost collum on the right hand 30. which should be your quotient , but because it is too great and that the square thereof cannot bée taken out of the remainder , you must make the quotient one lesse , and set in the quotient line no more but 29. by which 29. if you multiply 26. the product will bee 12. and 34. as the Trapezium sheweth , which being subtracted out of 13. and 9 , standing ouer 26. the remainder will be 35. which 35. is to bee set ouer 9. then by multiplying the second quotient which is 29. into it selfe , you shall find by the Trapezium the product to be 14. and 1. which being subtracted out of 35. and 17. the remainder will be 21. and 16. which is to be set ouer 35. and 17. as you sée in this example .

That done , double bothe the quotients or whole roote , and the product will be 26. 58. and hauing set 58. vnder 8. and 26. vnder 16. diuide 21. 16. and 8 by 26. 58. wherefore looke for 26. in the foote of the Trapezium , and for 21. 16. in that collum , and not finding it there , take that which is nighest vnto it and lesse , which you shall find to be 21 and 14 , and right against that in the outermost collum on the right hand standeth 49. which being too great to be your quotient by 2. therefore make your quotient no more but 47. which being set downe in the quotient line , multiply thereby 26. 58. and the product will be 21. 7. 26. which being subtracted out of 21. 16. and 8. the remainder will be 8. and 42. then setting 42. ouer 8. and 8. ouer 16. cancell 21. 16. and 8. then multiply 47. into it selfe squarely , and you shall finde the product thereof in the Triangle to be 36. 49. which being subtracted out of 8. 42. and 0. the remainder will be 8. 6. 11. as you sée in this example .

And thus you haue found the roote of your giuen number to be 13. 29 and 47. and whether you haue done rightly or not , you may know by multiplying the said roote into it selfe squarely , and by adding to the product thereof the remainder , and if you would finde a more exact roote , then adde more Cyphers to your giuen number towardes your right hand , together with their denominations ouer their heads as they encrease successiuely , and be sure alwaies that the number of the Cyphers bee euen and not odde as 2. 4. 6. and so forth , & the more Cyphers that you adde to the giuen number ( so as the number of them be euen ) the more exact roote you shal haue .

Now to giue to euery number of the roote his proper denomination , do thus , take halfe of those denominations that are ouer euery pricke , and those shall be the denominations of the roote , as in the former example , whereas seconds doe stand ouer the last pricke on the left hand take halfe of that which is 1′· wherefore the first number of your roote must be minutes , and ouer the rest of the numbers of the roote set seconds , thirds and so forth successiuely vntill you haue brought your roote to as small denominations as you desire .

FINIS .

A BRIEFE DEscription of the tables of the three speciall right lines belonging to a Circle , called Sines , lines Tangent , and lines Secant . Not only shewing why they were first inuented , and defining the proper tearmes thereto belonging : But also shewing diuers necessarie vses thereof , written by Maister Blundeuile 1593.

LONDON . Printed by Iohn Windet , 1594.

THE DESCRIPTION and vse of the Table of Synes.

BEcause there is no proportion , comparison , or likenes betwixt a right line and a crooked , the auntient Philosophers , as Ptolomey and diuers other , were much troubled in séeking to know the measures of a Circle or of any portion thereof by his Diameter , and by knowing the Diameter to find out the length of any chorde in a circle , which is alwaies lesser then the Diameter it selfe , and finding that the more parts whereinto the Diameter was diuided , the nearer they approched to the truth : Some of them therefore , as Ptolomey , diuided the Diameter of a circle into 120. parts , and the Semidiameter into 60. parts , and euery such part into 60′· and euery minute into 60. seconds &c. And in like manner did Arzahel , an auntient Arabian , who diuided the Diameter into 300. parts and the Semidiameter into 150. and euery of those parts into 60′· and so forth as before , according to which computation they made their tables : but because the working by those tables was very tedious and troublesome , by reason that it was néedfull continually to vse the art of numbring by Astronomicall Fractions : therefore Georgius Purbachius , and Regio montanus his Scholer to auoide that trouble of calculating by Astronomicall Fractions , diuided the Diameter of a Circle into a farre greater number of parts , and made such tables as are vsed at this present , the description and vse whereof doth hereafter follow , first of those that are set downe by Monte Regio in Folio , and then of those that were lately Corrected and made perfect by Clauius the Iesuite which are Printed in quarto .

And because that the way to find out the proportion which any chord hath to the whole Diameter , was very hard , therefore the said Purbachius and Monte Regio hauing direction from certaine propositions of Euclyd , as from the 47. proposition of his first booke , and from the third proposition of his third booke , and also from the 15. proposition of his fift booke , they made choise of the halfe chord and Semidiameter of the Circle , calling the halfe chord , Sinum rectum , and the Semidiameter Sinum totum . And because that the proportion of any circumference to his Diameter neuer changeth , how great or how little so euer the Circle be : after that they had calculated for one Circle , they made such tables as might serue for all Circles , and though these tables of sines doe suffice to worke thereby all manner of conclusions , as well of Astronomie , as of Geometrie , yet for more ease , our moderne Geometricians haue of late inuented two other right lines belonging to a Circle called lines Tangent , and lines Secant , and haue made like tables for them that were made for sines , and both tables , that is to say as well of the sines , as of the lines Tangent and Secant , haue one selfe manner of working thereby , as shall plainely appeare hereafter when wee come to describe the same . But first we will beginne with the tables of sines , and plainely define euery terme or vocable of Art , belonging thereunto : The termes are these here following : An arch , a Chord , Sinus rectus Sinus versus , Quadrans , Complementum , and sinus Complementi .

The definitions of the foresaid tearmes .

AN Arch is any part or portion of the circumference of a Circle , which in this practise doth not commonly extend beyond 180. degrées which is one halfe of the circumference of any Circle how great or small so euer it be , for euery Circle containeth 360 degrées .

A Chord is a right line drawne from one end of the Arch to the other end thereof , and note that all chordes are alwaies lesser then the Diameter it selfe , for that is the greatest chord in any Circle .

Sinus rectus is the one halfe of a Chord or string of any Arke which is double to the Arke that is geuen or supposed , and falleth with right Angles vpon that Semidiameter which diuideth the double Arke into two equall parts .

Sinus versus that is to say turned the contrary way , is a right line , and that part of the Semidiamiter , which is intercepted betwixt the beginning of the giuen Arke and the right Sine of the same Arke , and this is also called in Latine Sagitta , in English a Shaft or Arrowe , for the Demonstratiue figure thereof hereafter following , is not vnlike to the string of a bowe ready bent hauing a Shaft in the midst thereof .

Quadrans is the fourth part of a Circle containing 90. degrées .

Complementum arcus , is that portion of the Circle , which sheweth how much the giuen Arke is lesser then the quadrant , if the giuen Arke doe containe fewer degrées then the Quadrant , but if it containe more degrées then the Quadrant , then the difference betwixt such giuen arch , and the halfe Circle is the complement of the said giuen Arke .

Sinus complementi , is the right Sine of that Arch which is the complement of the giuen Arke .

Sinus totus , is the Semidiamiter of the Circle , & is the greatest Sine that may be in the Quadrant of a Circle , which according to the first tables of Monte Regio containeth 6/000/000. and according to the last tables 10/000 000/000 parts , for the more parts that the totall Sine hath , the more true and exact shall your worke be , notwithstanding sometime it shall suffice to attribute vnto the totall Sine but 60/000. parts , which number Appian obserueth in teaching the way to find out the distance of two places differing both in Longitude and Latitude by the tables of Sines , and some doe make the totall Sine to containe 100/000 parts , as Wittikindus in his treatise of Dials , and diuers other doe the like . Also Clauius himselfe saith that in the tables set downe by him in quarto , you may sometime make the totall Sine to be but 100/000. so as you cut off the two last figures on the right hand in euery Sine , but you shall better vnderstand euery thing here aboue mentioned , by the figure Demonstratiue here following .

The figure Demonstratiue .

In this figure you sée first a whol Circle drawne vpon the Centre E. and marked with the letters A. B. C D. which Circle by two crosse Diameters marked with the letters A C. and B. D. & passing both through the Centre E. is diuided into foure Quadrantes or quarters , the vpper Quadrante whereof on the left hand is marked with the letters A. B. E. in which Quadrant , the right perpendicular line marked with the letters F. H. betokeneth the right Sine of the giuen Arke A. F. which right Sine is the one halfe of the chord or string F. G. and the giuen Arke A. F. is the one halfe of the double Arke or bowe G. A. F. and A. H. is the Shaft called in Latine Sinus versus : Againe the letters F. B. doe shew the complement which together with the giuen Arke A. F. doe make the whole Quadrant A. F. B which is diuided into 9. spaces , euery space containing 10. degrées , whereby you may plainely perceiue that in this demonstration , the giuen Arke A. F. is 50. degrées , and the complement F. B. is 40. degrées , both which being added together doe make vp the whole Quadrant of 90. degrées , marked with the letters A. F. B. Now Sinus complementi is the crosse line marked with the letters F. K. the totall Sine which is the whole Semidiameter and greatest right Sine , is marked with the letters B. E. But because it is not enough to know the signification of the things aboue specified to vse the foresaid tables when néede is , vnlesse you know also how to find out those things in the said tables , I thinke it good therefore to shew you the order of the said tables by describing the same as followeth .

You haue then to vnderstand that the tables of Monte Regio printed in Folio , are contained in 18. Pages , and euery Page containeth eleauen partitions , called collums , whereof the first on the left hand containeth 60. minutes , which are to be counted from head to foote , as they stand in order one right vnder another in seuerall places , procéeding from 1. to 60. The second collum containeth Sines . The third containeth onely a portion or part of one second , and from thence forth procéeding towards the right hand all the other collums doe containe in like manner Sines and the portion of one second . And right ouer the head of euery Sine ( the first collum of Sines onely excepted , hauing nothing but a Cypher ouer his head ) are set downe the degrées of the whole Quadrant called arches , in such order as from the first Page to the last , there are in all 89. degrées , or arches , as by perusing the said tables you may plainely sée . Now to find out in these tables the things aboue mentioned , you must doe as followeth .

First to find out the right Sine of any giuen Arke , you must séeke out the number of the said Arke in the front of the tables , and if the giuen Arke hath no minutes ioyned thereunto , then the first number of Sines right vnder the said Arke , is the right Sine thereof . But if it hath any minutes ioyned thereunto , then you must séeke out in that Page , where you found the giuen Arke , the number of the minutes in the first collum of the said Page , on the left hand , and right against those minutes on the right hand , in the square Angle right vnder the said arch , you shall find the right Sine . As for example , you would find out the right Sine of a giuen Arke containing 8. degrées , and 20′· here hauing found out in the front of the second Page the figure of 8. standing right ouer the eight collum , séeke in the first collum on the left hand of the said Page , for 20. minutes , and right against the 20. minutes you shall find on the right hand in the common Angle or square 8695 93. which is the right Sine of the foresaid giuen Arke , so as you make 6/000/000. to be the totall Sine : but if you make 60/000. the totall Sine , then you must alwaies reiect the two last figures standing on the right hand of the said right Sine , and the rest of the figures shall be the right Sine .

Now to find out the complement , there is nothing to be done , but onely to subtract the giuen Arke out of the whole Quadrant which is 90. degrées , and the remainder shall be the complement : as in the former example by subtracting 8. degrées , 20′· out of 90 degrées , you find that there remaineth 81. degrées , 40′· which is the complement of that arch . Againe to find out the Sine of the complement you must doe thus , séeke the complement in the front of the tables of Sines , euen as you doe to find out any giuen arke : as in the former example , the complement being 81. degrées 40′· you must séeke 81. in the front of the 17. Page of the first tables , which being found , séeke out also the 40′· in the first collum of the said Page on the left hand , and right against those 40′· in the common Angle right vnder the Arke 81. you shall find 5/936/649. which number is the right Sine of the foresaid complement , so as you make 6/000/000. to be the totall Sine , for if 60/000. be the total Sine , then you must reiect ( as I said before ) the two last figures on the right hand , and the number remaining shall be the right Sine of the foresaid complement , and therefore in working by these tables , you must alwaies remember what number you make the totall Sine to be .

Sinus versus commeth seldome in vse , notwithstanding if you would know how to find it out , you néede to doe no more but to subtract Sinum complementi of the giuen Arke , out of the totall Sine , and the remainder shall be Sinus versus , as in the former example your Sinus complementi was 5/936/649. which being subtracted out of the totall Sine 6/000/000. there remaineth this 63/351. and that number is Sinus versus : for if you adde this remainder to the number which you subtracted , it will make vp the totall Sine 6/000/000. But there is one thing more necessarie to be knowen then this , because it commeth oftner in vse , and that is vpon some diuision made how to find out the Arke of any quotient , which is to be done thus : Enter with the quotient into the body of the tables , and leaue not séeking amongst the squares of the Sines , vntill you haue found out the iust number of the quotient ( if it be there ) if not , you must take the number of that Sine which is in value most nigh vnto it , whether it be a little more or lesse , it maketh no matter , and hauing found that number , looke in the front of that collum , and you shall find the Arke of your quotient , standing right ouer the head of that collum , and also the mynutes thereof in the first collum of the said Page on the left hand . As for example , hauing diuided one number by another , I find the quotient to be 469/012. whereof I would know the arch , now in séeking this quotient amongst the Sines , I can not find that iust number , but I find in the first Page , and in the tenth collum 469/015. which is the nighest number vnto it that I can sée . In the front of which collum I find the Arke to be 4. degrées , and directly against that Sine on the left hand , I find 29′· belonging to that arch , whereof that quotient is the Sinus , so as I gather hereof that the arch of the foresaid quotient is 4 , degrées , 29′· But you haue to note by the way that the number of your quotient must neuer be much lesse then 1745. for otherwise it is not to be found in these tables , vnlesse you make the totall Sine to be but 60/000. for then by reiecting the last two figures on the right hand , as I haue said before , the first right Sine of these tables shal be no more but 17. and by that account a very small quotient may be found in these tables . And whatsoeuer hath béene said here touching the order that is to be obserued in the first tables of Monte Regio , whose totall Sine is 6/000/000. the like in all points is to be obserued in the last tables , whose totall Sine is 10/000/000. Thus much touching the order of the foresaid tables of Monte Regio Printed in Folio : but for so much as those tables bee not altogether truely Printed , and for that they haue béene lately corrected , and made more perfect by Clauius , who doth set downe the said tables in quarto and not in folio , whereby they are the more portable , and the more commodious , as well for that they are more truely Printed , as also for that the complement of euery Arke is set downe in euery Page at the foote of euery collum , so as you néed to spend no time in subtracting the Arke from 90. I thinke it good therefore to make a briefe description of those tables , and the rather for that I haue requested the Printer to print the like here in quarto , and I doe worke all such conclusions as hereafter follow , by the said tables , the totall Sine whereof is 10/000/000. according to the last tables of Monte Regio . But for so much as some may haue already the tables of Monte Regio Printed in Folio , not knowing perhaps the vse thereof , I will set downe two conclusions to bee wrought by those tables , and all the rest of the conclusions are to be wrought by those tables which I haue here caused to be Printed in quarto like to those of Clauius : and though the two conclusions next following , which are to shew the vse of the foresaid tables , may be wrought by the tables of Sines in what forme so euer they be truely Printed if Folio , or in quarto , yet because I had appointed them to bee done by the tables of Monte Regio , Printed in folio before that euer I saw Clauius his booke , I mind not now to alter them but to let them stand still as they are .

How to find out by the said tables , the distance betwixt two places differing both in Longitude and Latitude , making the totall Sine to be no more but 60/000.

THis is done by finding out two numbers , whereof the one is called in Latine Primum inuentum : that is to say , the first found number , and the other is called Secundum inuentum , that is the second found number in such order as followeth .

First then knowing the Longitude of either place , take the difference of their Longitudes by subtracting the lesser Longitude out of the greater , that done , multiply the right Sine of that difference into the Sine of the complement of the lesser Latitude , and diuide the product of that Multiplycation by the totall Sine , and then séeke out the arch of that quotient according to the rule before taught , so shall you haue the first found number : That done , multiply the right Sine of the lesser Latitude by the totall Sine , and hauing diuided the product thereof by the right Sine of the complement of the first found number , subtract the arch of that quotient out of the greater Latitude , and you shall haue the second found number . Then multiply the right Sine of the complement of the first found number into the right Sine of the complement of the second found number , and hauing diuided the product of that Multiplycation by the totall Sine , séeke the Arke of that quotient in the tables , and take that Arke out of the whole Quadrant , and the degrées that doe remaine , are degrées of the great Circle , which if you multiply by 60. the product of that Multiplycation will shew you how many Italian miles the one place is distant from the other , or if you would haue Germane miles , them multiply the foresaid degrées of the great Circle by 15. or else diuide the product of the Italian miles by 4. and you shall haue your desire . As for example , you would know what distance is betwixt Hierusalem and Noremberg a famous towne in Germanie , Hierusalem according to Appian his tables , hath in Longitude 66. degrées 0′· and in Latitude 31. degrées , 40′· Againe Noremberg hath in Longitude 28. degrées , 20′· and in Latitude 49. degrées , 24′· the difference of their Longitudes is 37. degrées , 40′· the right Sine wherof is 36/664. for in this example Appian maketh 60/000. to be the total Sine , and therefore he reiecteth the two last figures en the right hand found in the first tables of Monte Regio . Now you must multiply 36/664. into the right Sine of the complement of the lesser Latitude which Sine is 51/067. the product of which two Sines being multiplyed the one by the other , amounteth to 1/872 872/320/488. which if you diuide by the totall Sine . 60/000. you shall find in the quotient 31/205. whose arch is 31. degrées , 20′· and this shall be your first found number . This done , multiply the right Sine of the lesser Latitude which is 31/498. by the totall Sine 60/000. and the product thereof wil be 1/889 889/880/000. which summe if you diuide by the Sine of the complement of the first found number which Sine is 51/249. you shall find in the quotient 36876. the Arke whereof is 37. degrées 55′· which arch being subtracted out of the greater Latitude , there will remaine 11. degrées , 29′· and that shall be your second found number , then multiply the foresaid Sine of the complement of the first found number which is 51/249. by the Sine of the complement of the second found number which is 58/798. and the product thereof will amount to 3/013 013/338/702. which if you diuide by the totall Sine , you shall find in the quotient 50/222. the arch whereof is 56. degrées , 50′· which being subtracted out of the whole Quadrant which is 90. degrées , there will remaine 33. degrées , 10′· of the greater Circle , which 33. degrées , if you multiply by 60. it will make 1980. miles , whereunto you must adde for the 10′· 10. miles , so shall you find the distance betwixt the two foresaid places to be 1990. Italian miles , which if you would reduce into Germaine miles , then diuide that number by 4. for 4. Italian miles doe make but one Germaine mile , so shall you haue 497. Germaine miles , and two Italian miles remaining , which is halfe a Germaine mile , which summe agréeth with that which Appian setteth downe in his Geographie , whereas he vseth the selfe same example , and worketh it in like manner Per tabulas sinuum .

The Altitude of the sunne being knowne how to find out the Longitude of the shadow both right and verse of any body yeelding shadow by helpe of the foresaid tables .

FIrst you haue to vnderstand that euery bodily thing yéelding shadow , is diuided into 12. equall partes , and euery such part into 60. minuts , and euery minute into 60. seconds , and so forth : Againe , of shadowes there be 2. kindes , that is , Vmbra recta , and Vmbra versa , Vmbra recta is that which procéedeth from some bodie rightly erected vpon the vpper face of the Horizon , as from some tower or post standing right vp vpon a leauel ground : And that shadowe is called vmbra versa which procéedeth from some right style or pearch being thrust into a wall or post standing right vp and not leaning , in such sort as the sayd style or pearch may be a iust paralell to the vpper face of the Horizon . Now to find out the length of either the foresaid shadowes , you must doe thus .

Multiply the right Sine of the complement of the giuen Solar altitude , by 12. and diuide the product by the right Sine of the said Solar altitude , and you shall haue the Longitude of the right shadow of the said body . Againe if you multiply the right Sine of the foresaid altitude by 12. and diuide the product by the Sine of the complement of the said altitude , you shall haue the Longitude of Vmbra versa , of the said body . As for example , suppose the giuen Solar altitude to be 25. degrées , the complement wherof is 65. and the right Sine of that complement is 54/378. if you make the totall Sine to be 60/000. Then in multiplying the foresaid right Sine by 12. the product will be 652/536. which if you diuide by the Sine of the altitude which is 25/357. you shall find the Longitude of Vmbra recta to be 25. parts 44′ 42″ and 6‴· Now if you multiply the Sine of the altitude which is 25/357. by 12. and diuide the product by the Sine of the complement which is 54/378. you shall find the Longitude of Vmbra versa to bee 5. parts , 35′· and in saying here parts , I meane alwaies such parts as are the 12. parts , whereinto the body yéelding shadow is diuided : but if you worke this example by the first tables of Sines making the totall Sine 6/000/000. though you finde it true in the parts and minutes , yet shall you not find it so in the seconds and thirds , and if you worke the same by the second tables making the totall Sine 10/000/000. you shal find it to agrée only in the parts , but neither in minutes nor secondes , which maketh me to suspect that the Printer hath committed some error therein , for both the tables were made by one selfe rule .

A briefe description of the tables of Sines Printed in quarto like vnto those which Clauius setteth down in his commentaries vpon Theodosius his Spheriques .

HAuing heere before plainely described vnto you the tables of Sines made by Monte Regio , which are Printed in folio , and how to vse the same I will now briefly describe the said tables lately corrected by Clauius , and were Printed in quarto at Rome Anno 1586. the totall Sine of which tables according to the last table of Monte Regio , is 10/000/000. by which tables are to be wrought all the conclusions hereafter following .

First then you haue to vnderstand that these tables are contained in 36. Pages , in the front whereof are set downe the degrées of the Quadrant procéeding from 1. to 89. but because the whole number of minutes belonging to the said degrées , which is 60′· can not be all placed in one selfe Page , but onely 30′· in the left outermost collum of the left hand Page : and other 30′· in the left outermost collum of the right hand Page : therefore the degrées or arches of the Quadrant are faine to be twice repeated in the front of euery two Pages , as you may plainely sée by viewing the said tables , and euery Page containeth seauen collums , whereof the first on the left hand containeth the minutes belonging to the degrées or arches of the Quadrant , which minutes do procéed downward from 1. to 30′· and the seauenth collum on the right hand in euery Page containeth the minutes belonging to the complement of euery arch , which minutes doe procéede backward , that is to say from 1. set downe at the lowest end of the last collum of the second Page , and so procéeding vpward to 30′· which 30′· is also set downe at the lowest end of the last collum of the left hand Page , and so procéedeth vpward to 60′· which 60. minutes doe helpe to make vp the complement that is answerable to euery arch wherevnto no minutes be annexed , for if the arch hath no minutes , then you must adde to the complement thereof 60. minutes , which is one whole degrée , to make vp the complement . As for example , suppose the Arke to be 46. degrées , without any minutes ioyned thereunto , the complement whereof set downe at the foote of the said arch , is but 43. degrées , wherefore you must adde thereunto 60′· which is one degrée , so shall the complement be 44. degrées , which is the true complement indéede , but if you suppose the foresaid Arke to haue 13′· ioyned thereunto you shall find the complement to be 43. degrées , 47′· which is answerable to the foresaid Arke 46. degrées , and 13′· for if you take 46. degrées , 13′· out of 90. degrées , the remainder will be 43. degrées , 47′· which is the complement , so as you néed not to make any Subtraction out of 90. to find the complement of any arch , that hath any minutes annexed thereunto : but when so euer you haue to find out the right Sine of any complement in these tables , you must then make the complement an arch , séeking for the same in the front and not at the foote of the tables , & if the said complement haue any minutes annexed thereunto , you must séeke those minutes in the left outermost collum of euery Page , and not in the outermost right collum belonging to complements , for in this case the complement is an arch and not a complement . The order of working by these tables in all other things differeth not one iotte from that which we haue obserued in working the two former conclusions by the tables of Monte Regio Printed in folio , as you shall easily perceiue by the examples here following .

1 How to find out the declination of the Sunne at any time his place in the Zodiaque being giuen , Per tabulas Sinuum .

KNowing the place of the Sun for the day , consider how much the said place is distant from the first point of Aries if the place of the Sunne be nigher to Aries then to Libra . But if it be nigher to Libra , then take his distance from the first point of Libra , which distance must not excéede 90. degrées , and séeke that distance amongst the Arkes in the front of the tables if they bee degrées , if minutes , you shall find them in the first collum on the left hand , then multiply the Sine of that distance by the Sine of the greatest declination which is 23. degrées , 28′· and diuide the product therof by the totall Sine , and the quotient will shew you the Sine of the declination for that day , the arch whereof , is the very number of the declination it selfe . As for example , you would know the declination of the Sunne the eleauenth of Aprill 1591. when as the Sunne was entred 33′· into Taurus vnto which you must adde according to the rule before giuen 30. degrées , for that is his distance from the first point of Aries , then séeke out the said 30. degrées , in the front of the last tables among the Arkes and the 33′ in the first collum on the left hand right against which you shall find the Sine of the said distance to bee 5082/901. which being multiplyed by 3/982/155. ( which is the Sine of the greatest declination ) the product thereof will be 20/240 240/890 890/631/655. which if you diuide by the totall Sine which is 10/000/000. you shall find the quotient to be 2/024/089. which quotient must be sought out in the said tables , and if you find no such number , then take the nearest number thereunto , which is 2025025. and the Arke thereof together with the minutes that stand right against the said quotient in the first collum on the left hand , is the declination of the Sunne for that day which is 11. degrées , 41′·

2 How to know the right ascention of the Sunne , Per tabulas Sinuum .

FIrst knowing the Sunnes place , you shall learne the right ascention thereof thus . First consider how farre his place is from the Equinoctiall point , as is said before in the last proposition , and multiply the Sine of the complement of that distance by the totall Sine , then knowing the declination of the Sunne for that day by the last rule , deuide the former product by the Sine of the complement of the said declination , and the quotient will shew the Sine of that Arke whereof the complement is the right ascention . As for example , the Sunne being in the 33′· of Taurus and his declination 11. degrées , 41′· as you found by the last proposition and his distance from the first point of Aries to be in all 30. degrées , 33′ the complement whereof is 59. degrées , 27′· the Sine of which complement is 8/611/860. which being multiplyed by the totall Sine which is 10/000/000. the product will be 86/118 118/600 600/000/000. which being diuided by the Sine of the complement of the Sunnes declination which is 78. degrées , 19′· whose Sine is 9792818. by which Sine if you diuide the former product , the quotient will be 8/773/600. the Arke of which quotient is 61. degrées , and 19′· and the complement of that , is 28. degrées , and 41′· which is the right ascention of the place of the Sunne for the foresaid day .

And here note that if the Arke from Aries to the giuen pointe , doe containe iust 90. degrées , the right ascention thereof is also 90. degrées , but if the said Arke be more then 90. degrées , and lesser then 180. degrées , then subtract the same out of 180. and séeke out the right ascention of the remainder as before , whose right ascention if you take out of 180. the remainder shall be the ascention of the propounded Arke . But if the said arch which is comprehended betwixt Aries and the giuen pointe bee greater then 180. degrées , or 6. signes , and lesser then 270. degrées , or 9. signes , then hauing subtracted 180. degrées from the same arch , calculate the right ascention of the arch from the beginning of Libra , as before , and the right ascention thereof being added to 180. degrées , shall be the ascention desired .

Lastly if the arch comprehended betwixt Aries and the pointe giuen , be greater then 270. degrées , or 9. signes , then subtract the same out of 360. degrées , and séeke out the right ascention of that remainder as before , which if you subtract out of 360. degrées , the remainder shall be the right ascention desired .

3 How to finde out the ascentionall difference , Per tabulas Sinuum .

MVltiply the Sine of the Latitude giuen by the total Sine , and diuide the product by the complement of the said Latitude , that done , multiply the quotient by the Sine of the Sunnes declination , and diuide the product by the Sine of the complement of the declination , and the quotient thereof wil shew the signe of the ascentionall difference : and by working according to this rule , you shal find that when the Sunne is entred 33′· into Taurus , at which time his declination is 11. degrées , 41′· ( as hath béene said before ) the ascention all difference will be 15. degrées , 21′· And here note that the ascentional differences for one quarter of the Circle serueth also for all the rest , so that the Latitude be not altered , and that the declination of the pointes in the later quarters be equall to the declination of the points in the first quarter .

4 How to find out the oblique ascention of any pointe of the Ecliptique in any Latitude assigned .

FIrst find the right ascention of the giuen pointe by the second proposition , & also the ascentional difference thereof by the third proposition : then consider whether the declination of the said point be Northward or Southward , for if it be Northward , then subtract the ascentionall difference out of the right ascention , and the remainder shall be the oblique ascention desired : but if the declination be Southward , then adde the ascentionall difference vnto the right ascention , and the summe shall be the oblique ascention . In working thus for the Latitude 52. and supposing the Sunne to be in the 33′· of Taurus , & hauing found by the former proposition his right ascention to be 28. degrées , 41′· and the ascentionall difference to be 15. degrées , 21′· you shall find his oblique ascention in the foresaid Latitude to be 13. degrées , 20′·

5 How to finde out the time of the Sunnes rising and setand thereby the length of the artificial day .

FIrst you must know the ascentionall difference , and conuert the same into howers & minutes , then if the Sunne be in any of the Northern signes , adde those howers to 6. howers which is the one halfe of an Equinoctiall day , and the summe of such Addition shall be the one halfe of the artificial day , which being subtracted out of 12. howers , the remainder shal be the hower of the Sunnes rising : As for example , the Sunne being in the 33′· of Taurus , you found the ascention all difference to be 15. degrées , and 21′· which being turned into howers maketh one hower 1′· 2″· and somewhat more , which being added to 6. maketh 7. howers 1′· 2″· and a little more which is the one halfe of the artificiall day which being doubled , maketh in all 14. howers 2′· and 4″· but if the Sunne be in any of the Southerne signes , you must remember to subtract the howers of the ascentionall difference out of 6. howers , and the remainder shall be the one halfe of the artificiall day , and by subtracting the halfe length of the artificiall day out of 12. you shall know the hower of the Sunnes rising , and hauing the time of his rising , you must néeds know the time of his setting .

6 How to find out the Meridian altitude of the Sunne in any day though he doth not shine at all , the eleuation of the Pole being giuen .

SVbtract the eleuation of the pole from 90. and the remainder shall be the eleuation of the Equinoctiall , then if the Sunne be in any of the Northern signes , adde the declination of the Sunne , vnto the altitude of the Equinoctiall , or else if he be in any of the Southern signes , subtract the declination , and the summe of the Addition , or remainder of the Subtraction , shal shew the Meridian altitude . As for example , the second of May 1591. the Sunne being in the 20. degrées , 42′· of Taurus , and his declination Northward 17. degrées , 56′· 21″· here by subtracting 52. degrées , which is our Latitude from 90. you shall find the remainder to be 38. which is the altitude of the Equinoctiall , whereunto if you adde the Sunnes declination for that day which is 17. degrées , 56′· and 21″· the summe will bee 55. degrées , 56′· and 21″· and that is the Meridian altitude of the Sunne for that day .

Let these few conclusions serue to shew you the vse of the tables of Sines , for it would make a long booke to set downe so many conclusions as are to be wrought by these tables , and therefore I leaue to trouble you any further therewith , minding now briefly to declare vnto you the vse of the tables of lines Tangent & Secant by one example or two as followeth . But first I thinke it necessary to shew you what those lines be , and whereto they serue .

THE DESCRIPTION and vse of the Tables of Tangents and Secants .

EVclid in the second proposition of his third booke defineth the line Tangent in this sort . A right line ( saith hee ) is said to touch a Circle when it toucheth it so , as being drawne out in length , it would neuer cut the said Circle .

The line Secant is not by him any wher defined , but what these two lines are , you shal better vnderstand by this figure Demonstratiue here following , then by any definition that can be made thereof : for a definition ought to bee plaine and briefe , and not long , intricate or doubtfull , which will be hardly performed in shewing the nature of these two lines by way of definition , and therefore marke well this figure following .

In this figure you sée first a Circle drawne vpon the Centre C. from which Centre is extended to the circumference of the Circle a right line , called the Semidiameter , marked with the letters A. C. then there is another right line which toucheth the said Circle , and also the outermost end of the said Semidiameter making therewith a right Angle in the point A. and is called the line Tangent , then there is a third line which procéeding from the Centre C. doth cut the circumference of the Circle in the point B. and also méeteth with the line Tangent in the point D. and therefore is called the line Secant , betwixt which two lines , I meane the Tangent and Secant , is intercepted or included a certaine portion or arch of the foresaide Circle lesse then a Quadrant marked with the letters A. B. of which Arke the line A. D. is the Tangent , and the line C. D. is the Secant thereof , which must néeds méete with the Tangent in the point D. because that the two Angles C. A. D. and D. C. A. are lesser then two right Angles , for the one is right , and the other sharpe , by reason that the Arke is lesse then a Quadrant . And some doe call the line Tangent the line Ascript , because it is ascribed to the Circle , and they call the line Secant the Hipothenuse , because it subtendeth the right Angle A. & they cal the Semidiameter or total Sine , the base of the rectangle Triangle C. A. D which is called a rectangle Triangle because it containeth one right Angle marked with the letter A. and note that whensoeuer any manner of Angle is propounded by thrée letters , that the middle letter doth alwaies signifie the Angle propounded , be it right , sharpe , or blunt . Now if you would know to what end the foresaid two lines were inuented , & wherto they serue , you haue to vnderstand that they chiefly serue in calculating the quantitie of Angles and their sides , as well in right lined Triangles , as in sphericall Triangles , for the sides of Triangles are either right or crooked , and if they haue right sides , such Triangles are either right angled Triangles , or oblique angled Triangles , and you haue to note that the quantity of euery Angle is to be measured by the arch of a Circle subtending that Angle , for the point of euery Angle is imagined to be the Centre of a whole Circle , which you may suppose to be so great or little as you will , for euery Circle ( be it great or little ) is diuided into 360. degrées , and looke how many degrées and minutes the arch subtending that Angle containeth , so much is the quantitie of that Angle , the practise whereof is very well set down by Clauius in his commentaries vpon Theodosius , which I mind ( God willing ) hereafter to translate into our mother tongue : In the meane time my intention here , is onely to shew you by one example or two , the vse of the tables made for the foresaid lines Tangent and Secant .

The vse of the said tables according to Clauius , is thus .

IN séeking out the Tangent or Secant of any Arke giuen , or of the complement of any Arke by either of these tables , you haue to obserue the selfe same order which you did before in finding out the right Sine of any Arke giuen , or of the complement of any Arke , Per tabulas sinuum . As for example , if you would find out the Tangent of an Arke containing 50. degrées , 24′· then resorte to the table of Tangents in the front , whereof looke first for the Arke 50. degrées , and then in the first collum on the left hand of the said table , for 24′· right against which on the right hand vnder the Arke 50. you shall find in the common Angle the Tangent to be 12/087/923. the totall Sine whereof is 10/000/000. but if you would find out the Secant of the foresaid Arke 50. and 24′· then you must resort to the table of Secants , and hauing found out the Arke 50. in the front of the said tables , and the 24′· on the left hand as before , you shall find in the common Angle the Secant to be 15688144. And if you would haue the Tangent of the complement of the said Arke which is 39. degrées , and 36′· you shall find the 39. degrées of the complement in the foote of the table of Tangents right vnder the Arke 50. & the 36′· . in the outermost collum on the right hand of the said table , with which complement you must enter the table of Tangents , séeking for 39. degrées in the front of the table , and 36′· in the first collum on the left hand of the said table , right against which in the common Angle you shall find 8272720. to be the Tangent of 39. degrées , 36′· which is the complement of 50 degrées , 24′·

And you must work in like manner with the table of Secants : As for example , if you would find the Secant of 72. degrées , 36′· first then enter the table of Secants , looking for 72. degrees aboue in the front of the table , and 36′· in the first collum on the left hand of the said table , and in the common Angle you shall find 33/440/240. which is the Secant of 72. degrees , 36′· But if you would haue the Secant of the complement of the said arch 72. degrees , 36′· then looking in the foote of the table right vnder 72. degrees , you shall find 17. degrees , and in the outermost collum on the right hand , iust against 36′· you shall find 24′· so as you see that 17. degrees 24′· is the complement of 72. degrees , 36′· with which complement you must enter the table of Secants , looking for 17. degrees , aboue in the front of the table , and for 24′· in the first collum on the left hand of the said tables , & in the common Angle you shall find 10/479/542. to bee the Secant of the arch 17. degrees , 24′· which is the complement of the arch 72. degrees , 36′·

The vse of which tables in Astronomicall matters , I haue here set downe as followeth .
1 To find out the declination of the Sunne , the place thereof being knowne .

MVltiply the Secant of the complement of the greatest declination by the totall sine , and diuide the product by the sine of the sunnes distance from one of the equinoctiall points , the quotient is the Secant of an arch , whose complement is the declination of the sunne : for example , suppose that the sunne be entering into ♉ to finde the declination therof , first I multiply 25112030 the Secant of 66. degrées 32′· ( which is the complement of 23. degrées 38′ the greatest declination ) by 10000000. the product is 251120300000000. which being diuided by 5000000. ( the sine of 30. degrées the sunnes distance from the equinoctial point ) the quotient is 50224060. for which number I séeke in the table of Secants , the arch answering vnto it , is 78. degrées 32′· the complement whereof is 11. degrées 29′· which is the declination of the sunne .

2 Knowing the declination of the sunne how to finde his distance from the equinoctiall point , and so consequently his place in the Zodiacke .

MVltiply the Secant of the complement of the greatest declination by the totall sine , and deuide the product by the Secant of the complement of the declination giuen , the quotient is the distance of the sun from the equinoctiall point . As for example , the declination of the sunne is supposed to be 11. degrées 29′· then to finde his distance from the equinoctiall , I multiply 25112030. the Secant of 66. degrées 32′· ( which is the complement of the greatest declination ) by the totall sine the product is 251120300000000. which I diuide by 50224350. the Secant of 78. degrées 31′ the complement of 11. degrées 29′· the supposed declination , the quotient is 5000000. the sine whereof is 30. dedegrées 0′ which is the distance of the sunne from the equinoctial : them for his place you must take the same according to the season of the yeare : for if it be in Aprill , then the sunne is entering into Taurus , but if it be in August it is entering into Virgo , and being in October , it is entering into Scorpio , and being in February it is in the beginning of Pisces .

3 To finde out the right ascention of the sunne .

MVltiply the Tangent of the distance of the sunne from the equinoctiall point which is nearest vnto it , by the sine of the complement of the greatest declination , and diuide the product by the totall sine , the quotient is the Tangent of the right ascention of the sunne , for which if you seeke in the table of Tangents , the arch answering vnto it is your desire : For example the sunne being in the first of Taurus , to knowe the right ascention thereof , I multiply 5773502. the Tangent of 30. degrées ( for that 30. degrées is the distance of the sunne from the equinoctiall point ) by 9172920. the sine of 66. degrées 32′· which is the complement of 23 degrées 28′· the greatest declination , the product is 52959871965840. which being deuided by 10000000. the totall sine , the quotient is 5295987. which is the Tangent of 27. degrées , 54′· which is the right ascention of the sunne béeing entered into Taurus .

4 How to finde out the declination of the sunne knowing onely the right ascention thereof .

MVltiply the Tangent of the complement of the greatest declination by the total sine , and deuide the product by the sine of the right ascention giuen , the quotient is the Tangent of the complement of the sunnes declination : for example the right ascention of the sunne being 27. degrées 54′· I would knowe the declination thereof , multiplying 23035062. the Tangent of 66. degrées 32′· the complement of 23. degrées 28′ by 10000000. the product is 230350620000000. which being deuided by 4679298. the sine of the giuen ascention the quotient is 49224856. the arch of which Tangent is 78. degrées 32′· which being subducted out of 90. the remainder is 11. degrées 28′· and so much is the declination of the sunne .

5 How to finde the place of the sunne knowing onely the right ascention thereof .

SVbduct the right ascention giuen out of 90. if it bee lesse then 90. but if the same be more then 90. subtract the ascention giuen out of 180. & being greater then 180. subduct the same out of 270. or being greater then 270. subduct the same from 360. and multiply the Tangent of the remainder by the sine of the complement of the greatest declination , and deuide the product by the totall sine , the quotient is the Tangent of the complement of the suns distance from one of the Equinoctiall points : which distance being knowne , the place of the sunne can not be vnknowne . For example , supposing the right ascention of the sunne to be 27. degrées 54′· the complement thereof is 62. degrées 6′· the Tangent whereof is 18886715. which being multiplyed by 9172920. the product is 173245985757800. which being deuided by 10000000. the quotient is 17324598. for which I looke in the table of Tangents , and I finde the arch thereof to be 60. the complement whereof is 30. which is the distance of the sunne from the equinoctiall point , that is , from Aries , for that the right ascention is lesse then 90. I say then that the sunne is in the beginning of Taurus . But if the right ascention had béene more then 90. and lesse then 180. the place of the sunne had béene betwixt Cancer and Libra 30. degrées from Libra , so should it haue béene in the first point of Virgo , but if the right ascention had béene more then 180. the place of the sunne should be betwixt Libra and Capricorne , that is in the beginning of Scorpio , but being more then 270. the place of the sunne should be betwixt Capricorne and Aries , that in the beginning of Pisces .

6 To find out the ascentionall difference of the sunne or any starre in the firmament , knowing the declination thereof , and also the latitude of your region .

MVltiply the Tangent of the declination of the Sunne or star by the Tangent of the latitude of the place , and deuide the product by the totall sine : the sine of the quotient is the sine of the ascentionall difference , the arch wherof is the desired ascentionall difference . For example let the declination of the sunne , starre , or other point in the firmament be 10. degrées 3′· and suppose the latitude to be 52. degrées , the Tangent of 10. degrées 3′· is 1772268. the tangent of 52. degrées is 12799416. which being multiplyed together , the product will be 22674995395488. which being deuided by 10000000. the totall sine , the quotient is 2267499. for which number I looke amongst the sines , the arch answering thereunto is 13. degrées 6′· which is the ascentionall difference desired .

7 To find out the oblique ascention of the sunne .

KNowing the place of the sunne , finde out the right ascention of the same by the thirde proposition , and finde also the ascentionall difference of the same point , then if the declination of the sunne be North , subduct the ascentionall difference out of the right ascention , the remainder is the oblique ascention . For example , suppose the sunne to be in the beginning of Taurus , now to finde the oblique ascention thereof in the latitude 52. degrées , first by the third proposition I finde out the right ascention of the beginning of Taurus , which I finde to be 27. degrées 54′· then by the sixth proposition I finde the ascentionall difference to bee 15. degrées 4′· which being subducted from 27. degrées 54′· ( for that the declination is North ) the right ascention the remainder is 12. degrées 50′· and so much is the oblique ascention for the latitude of 52. degrées . But if the sunne be in any of the Southerne sines , and that the declination bée South , then the ascentionall difference is to be added vnto the right ascention before giuen .

8 To finde out the oblique descention of the Sunne at any time .

FIrst finde out the right ascention of the sunne by the third proposition , the same shall be the right descention thereof , then finde the ascentionall difference by the sixt proposition , and if the sunne be in any of the Northern signes , adde the ascentional difference vnto the right ascention , the summe shall be the oblique descention of the sunne : But if it be in any of the Southerne signes subduct the ascentionall difference out of the right ascention , the remainder is the desired descention : as for example the sunne being in the beginning of Taurus , the right ascention thereof by the third proposition is 27. degrées 54′· the ascentional difference thereof by the sixth proposition is 15. degrées 4′· for as much then as Taurus is a Northerne signe , I adde the ascentionall difference vnto the right ascention , the summe is 42. degrées 58′· and so much is the oblique descention of the sunne being in the beginning of Taurus .

9 To finde out the length of the day or night .

HAuing found out the ascentionall difference by the sixth proposition , adde the same vnto 90. if the sunne be in any of the Northerne signes , but if it bee in any of the Southerne signes , subduct the ascentionall difference out of 90. then diuide the summe of the addition or the remainder of the subtraction by 15. the quotient wil shew the halfe length of the day in houres and minutes , which being doubled you shall haue the whole length of the artificiall day : For example the sunne being in the beginning of Taurus , the ascentionall difference thereof is 15. degrées 4′· which for that the sunne is in a Northerne signe , I adde vnto 90. degrées , the summe is 105. degrées 4′ which being diuided by 15. the quotient is 7. houres o′· the halfe length of the day which being doubled will be 14. houres the whole length of the artificiall day in the latitude of 52. degrées .

10 To finde the houre of the Sunne his rising or setting in any latitude assigned .

FIrst find out the halfe length of the artificial day by the 9. proposition , and subtract the same from 12. houres , the remainder will shew the houre of the sunnes rising , for example the sunne being in the beginning of Taurus to know the houre of his rising in the latitude of 52. degrées , by the ninth proposition , the half length of the artificiall day I find to be 7. houres , which I subtract from 12. houres the remainder is 5. which sheweth that the sunne riseth at 5. of the clocke in the morning , but the halfe length of the day it selfe is the houre of the sunnes setting .

11 To find out the length of the planetary houres and to find what Planet raigneth at any houre of the day .

FIrst find out the length of the artificiall day by the ninth proposition , and diuide the same by 12. the quotient is the length of one planetary houre : or thus , hauing an houre of the artificiall day giuen , looke what houre the same is from the sunne rising , and multiply the same by 12. diuide the product by the length of the artificiall day , the quotient is the number of the Planetarye houre . For example the sunne being in the beginning of Taurus , and our latitude being 52. degrées , the length of the artificiall day by the ninth proposition is 14. houres : then doe I diuide 14. by 12. the quotient is 1 ⅙ . that is 1. houre 10′· the length of one planetary houre . But if an houre of the artificial day be giuen as that I would know what planetary houre it is at 9. of the clock the sunne being in the first of Taurus in the latitude of 52. degrées , hauing found that the sunne riseth at fiue of the clocke by the tenth proposition , I sée that 4. houres of the artificiall day are gone at nine of the clocke , I therefore multiply 12. by 4. the product is 48. which I diuide by 14. the length of the artificiall day , the quotient is 3 3/7. which is the planetary houre at the time set downe : likewise shall you finde the planetary houre of the night , finding the length thereof , and then worke with it as was shewed before for the day .

Then to know the Planet which raigneth at the appointed time , you must consider the Planet whereof the day taketh his name , for that Planet ruleth the first houre of that day , the next Planet the second houre , and so foorth . But for your ease I haue set down a table whereby to finde the Planet which raigneth at any time .

Houres 1 2 3 4 5 6 7 of the day . 8 9 10 11 12 Houres 10 11 12 1 2 of the night . 3 4 5 6 7 8 9 Sunday . ☉ ♀ ☿ ☽ ♄ ♃ ♂ Munday . ☽ ♄ ♃ ♂ ☉ ♀ ☿ Tuesday . ♂ ☉ ♀ ♂ ☽ ♄ ♃ Wednesday ☿ ☽ ♄ ♃ ♂ ☉ ♀ Thurseday . ♃ ♂ ☉ ♀ ♂ ☽ ♄ Friday . ♀ ♂ ☽ ♄ ♃ ♂ ☉ Saturday . ♄ ♃ ♂ ☉ ♀ ☿ ☽

The vse whereof is this , hauing founde the number of the Planetary houre , looke for the same in the head of the Table , whether it be in the day or night , and right vnder it iust against the name of the day , you shall haue the Planet which rayneth at that time . For example the sunne being in the beginning of Taurus the 11. of April , being Thursday at nine of the clocke in the morning , I find the number of the Planetary houre at that time to be 3 3/7. them looking for thrée amongst the houres of the day , I descend in that collum vntill I come to be iust against thursday , where I sée the Character of Sol to be , I conclude then that the sunne raigneth at that time .

12 To finde the arch of the Equinoctial , comprehended betwixt the Meridian and any Circle of position according vnto Campanus and Gazula .

MVltiply the Sine of the complement of your latitude by the Tangent of the distance of the giuen Circle of position from the Zenith , & diuide the product by the totall Sine , the quotient is the tangent of the arch of the Equinoctiall , which is comprehended betwixt the giuen Circle of position , and the Meridian . For example in the latitude of 52. I would knowe what part of the Equinoctiall is comprehended betwixt the Meridian and that Circle of position , which is 30. degrées from the Zenith : the latitude being 52. degrées , the complement thereof is 38. degrées , the Sine whereof 6156615. and the tangent of 30. degrees , ( for that 30. is the distance betwixt the giuen Circle of position and the Zenith ) is 5773502. which being multiplyed by 6156615. the product is 35545229015730. which being diuided by 10000000. the totall Sine , the quotient is 3554522. for which I séeke amongst the tangents , and I finde the Arke answering thereunto , to be 19. degrées 34′ the Arch of the Equinoctiall , betwixt the Meridian and the giuen Circle of position .

13 Knowing the Latitude of your Region , and also the eleuation of the Pole aboue any Circle of position , howe to finde the inclination of the saide Circle of position vnto the Meridian , and so consequently the Arch of the Equinoctiall , which is betwixt the said Circle of position & the Meridian .

MVltiply the Secant of the complement of the eleuation of the Pole aboue the Circle of position by the Sine of your Latitude , and diuide the product by the totall Sine , the quotient is the Secant of the complement of the inclination of the Circle of position vnto the Meridian , and that is the distance betwixt the Circle of Position and the Zenith , by helpe wherof you shall find the Arch of the Equinoctiall , betwixt the Circle of Position and the Meridian , as in the former proposition . As for example , suppose the eleuation of the Pole aboue a Circle of position , in our Latitude of 52. to be 23. degrées 12′· Now to find out the Inclination of that Circle of Position vnto the Meridian , first I multiply 25384445. the Secant of 66. degrées 48′· ( for that is the complement of 23. degrées 12′· the eleuation of the Pole aboue the Circle of position ) by 7880108. the Sine of 52. the Latitude of our Region , the product is 200032168120060. which being diuided by 10000000. the quotient is 20003216. for which I looke in the Table of Secants , and the Arch therof is 60. degrées 0′· the complement whereof is 30. degrées 0′· which is the Inclination of the Circle of position vnto the Meridian , or the distance of the Zenith from the sayde Circle , then to find the Arch of the Equinoctiall betwixt the sayd Circle of Position and the Miridian , repeate the worke of the former proposition .

14 To finde out the Eleuation of the Pole aboue any assigned Circle of Position in any giuen Latitude .

KNowing the Inclination of the assigned Circle of Position vnto the Meridian , multiply the Secant of the complement thereof by the totall Sine , and diuide the product by the Sine of your Latitude , the quotient is the Secant of the complement of the Eleuation of the Pole aboue the giuen Circle of Position : as for example , suppose the inclination of a Circle of position to be 30. degrées 0′· Now to finde the eleuation of the Pole aboue the same for the Latitude of 52. degrées 0′· First take the complement of 30. degrées 0′· which is 60. degrées 0′· the Secant wherof is 20000000. which being multiplyed by the totall Sine , the product is 200000000000000. whcih being diuided by 7880108. the Sine of 52. the assigned Latitude the quotient is 25380362. for which I séeke in the Table of Secants , and the Arch answering thereunto I finde to be 66. degrées 48′· the complement whereof is , 23. degrées 12′· and so much is the eleuation of the Pole aboue the assigned Circle of Position in your Latitude .

And thus much shall be sufficient to haue béene spoken for the vse of the lines Tangent and Secant at this present , of whose ample and infinite vse , you shall haue further taste in our Directorie tables and Horologies , whereof a beginning is made , and shal be ended so soone as may be possible : in the meane time I shal desire the Reader fauourably to accept of these , vntill better leasure and more fit opportunitie will be offered . And now followe the Tables themselues .

HERE FOLLOWETH THE TABLE OF RIGHT SINES FOR EVERIE MINVTE OF THE QVADRANT , FIRST CALCVLATED BY I. REGIO MONTANVS , BVT NOW EXAMINED AND IN MANY PLACES CORRECTED AND AMENDED BY CLAVIVS . The Table of Sines .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Quadrant . 0 1 2 3 4 The minutes of degrees of the Quadrant for right Sines of the complement of the Arches of the same Quadrant . 0 0000 174524 348995 523360 697565 60 1 2909 177433 351902 526265 700467 59 2 5818 180341 354809 529170 703369 58 3 8727 183250 357716 532075 706270 57 4 11636 186158 360623 534980 709172 56 5 14544 189066 363530 537884 712073 55 6 17453 191975 366437 540789 714975 54 7 20362 194883 369344 543694 717876 53 8 23271 197792 372251 546598 720777 52 9 26180 200700 375158 549503 723678 51 10 29088 203608 378064 552407 726579 50 11 31997 206517 380971 555312 729480 49 12 34906 209425 383878 558216 732381 48 13 37815 212333 386785 561120 735282 47 14 40724 215241 389692 564024 738183 46 15 43632 218149 392598 566928 741084 45 16 46541 221057 395505 569832 743985 44 17 49450 223965 398412 572736 746886 43 18 52359 226873 401318 575640 749787 42 19 55268 229781 404225 578544 752688 41 20 58177 232689 407131 581448 755588 40 21 61086 235597 410038 584352 758489 39 22 63995 238505 412944 587256 761389 38 23 66904 241413 415851 590160 764290 37 24 69813 244321 418757 593064 767190 36 25 72721 247229 421663 595967 770090 35 26 75630 250137 424570 598871 772991 34 27 78539 253045 427476 601775 775891 33 28 81448 255953 430382 604678 778791 32 29 84357 258861 433288 607582 781691 31 30 87265 261769 436194 610485 784591 30 31 90174 264677 439100 613389 787491 29 32 93083 267585 442006 616292 790391 28 33 95992 270493 444912 619196 793291 27 34 98901 273401 447818 622099 796191 26 35 101809 276308 450724 625002 799090 25 36 104718 279216 453630 627905 801990 24 37 107627 282124 456536 630808 804889 23 38 110536 285032 459442 633711 807789 22 39 113445 287940 462348 636614 810688 21 40 116353 290847 465253 639517 813587 20 41 119262 293755 468159 642420 816486 19 42 122171 296663 471065 645323 819385 18 43 125079 299570 473970 648226 822284 17 44 127988 302478 476876 651129 825183 16 45 130896 305385 479781 654031 828082 15 46 133805 308293 482687 656934 830981 14 47 136714 311200 485592 659837 833880 13 48 139622 314108 488498 662739 836778 12 49 142531 317015 491403 665642 839677 11 50 145439 319922 494308 668544 842576 10 51 148348 322830 497214 671447 845474 9 52 151257 325737 500119 674349 848372 8 53 154165 328645 503024 677251 851271 7 54 157074 331552 505929 680153 854169 6 55 159982 334459 508834 683055 857067 5 56 162891 337367 511740 685957 859965 4 57 165799 340274 514645 688859 862863 3 58 168708 343181 517550 691761 865761 2 59 171616 346088 520455 694663 868659 1 60 174524 348995 523360 697565 871557 0 89 88 87 86 85 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Quadrant . 5 6 7 8 9 The minutes of the degrees of the Quadrant for right Sines of the complements of the same Quadrant . 0 871557 1045285 1218693 1391731 1564345 60 1 874455 1048178 1221580 1394612 1567218 59 2 877353 1051071 1224467 1397492 1570091 58 3 880250 1053964 1227354 1400373 1572964 57 4 883148 1056857 1230241 1403253 1575837 56 5 886045 1059749 1233128 1406133 1578709 55 6 888943 1062642 1236015 1409013 1581581 54 7 891840 1065534 1238901 1411893 1584453 53 8 894737 1068426 1241788 1414772 1587325 52 9 897634 1071318 1244674 1417652 1590197 51 10 900531 1074210 1247560 1420531 1593069 50 11 903428 1077102 1250446 1423410 1595941 49 12 906325 1079994 1253332 1426289 1598812 48 13 909222 1082886 1256218 1429168 1601684 47 14 912119 1085778 1259104 1432047 1604555 46 15 915016 1088669 1261990 1434926 1607426 45 16 917913 1091561 1264876 1437805 1610297 44 17 920809 1094452 1267761 1440684 1613168 43 18 923706 1097344 1270647 1443562 1616038 42 19 926602 1100235 1273532 1446441 1618909 41 20 929498 1103126 1276417 1449319 1621779 40 21 932395 1106017 1279302 1452197 1624649 39 22 935291 1108908 1282187 1455075 1627519 38 23 938187 1111799 1285072 1457953 1630389 37 24 941083 1114690 1287957 1460831 1633259 36 25 943979 1117580 1290841 1463708 1636129 35 26 946875 1120471 1293726 1466586 1638999 34 27 949771 1123361 1296610 1469463 1641868 33 28 952667 1126252 1299495 1472340 1644738 32 29 955563 1129142 1302378 1475217 1647607 31 30 958458 1132032 1305262 1478094 1650476 30 31 961354 1134922 1308146 1480971 1653345 29 32 964249 1137812 1311030 1483848 1656214 28 33 967144 1140702 1313914 1486724 1659082 27 34 970039 1143592 1316798 1489601 1661951 26 35 972934 1146482 1319681 1492477 1664819 25 36 975829 1149372 1322564 1495353 1667687 24 37 978724 1152261 1325447 1498229 1670555 23 38 981619 1155151 1328330 1501105 1673423 22 39 984514 1158040 1331213 1503981 1676291 21 40 987408 1160929 1334096 1506857 1679159 20 41 990303 1163818 1336979 1509733 1682027 19 42 993198 1166707 1339862 1512608 1684894 18 43 996092 1169596 1342744 1515484 1687761 17 44 998987 1172485 1345627 1518359 1690628 16 45 1001881 1175374 1348509 1521234 1693495 15 46 1004775 1178263 1351392 1524109 1696362 14 47 1007669 1181151 1354274 1526984 1699229 13 48 1010563 1184040 1357156 1529859 1702095 12 49 1013457 1186928 1360038 1532734 1704962 11 50 1016351 1189816 1362920 1535608 1707828 10 51 1019245 1192704 1365802 1538482 1710694 9 52 1022139 1195592 1368683 1541356 1713560 8 53 1025032 1198480 1371564 1544230 1716426 7 54 1027926 1201368 1374446 1547104 1719292 6 55 1030819 1204255 1377327 1549978 1722157 5 56 1033713 1207143 1380208 1552852 1725022 4 57 1036606 1210031 1383089 1555725 1727887 3 58 1039499 1212918 1385970 1558599 1730725 2 59 1042392 1215806 1388851 1561472 1733617 1 60 1045285 1218693 1391731 1564345 1736482 0 84 83 82 81 80 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Quadrant . 10 11 12 13 14 The minutes of the degrees of the Quadrant for right Sines of the complements of the same Quadrant . 0 1736482 1908090 2079117 2249511 2419219 60 1 1739347 1910945 2081962 2252345 2412041 59 2 1742●11 1913800 2084807 2255179 2424863 58 3 1745075 1916655 2087652 2258013 2427685 57 4 1747939 1919510 2090497 2260847 2430507 56 5 1750803 1922365 2093342 2263680 2433329 55 6 1753667 1925220 2096186 2266513 2436150 54 7 1756531 1928074 2099030 2269346 2438971 53 8 1759394 1950928 2101874 2272179 2441792 52 9 1762258 1933782 2104718 2275012 2444613 51 10 1765121 1936636 2107562 2277844 2447434 50 11 1767984 1939490 2110405 2280676 2450254 49 12 1770847 1942344 2113248 2283508 2453074 48 13 1773710 1945197 2116091 2286340 2455894 47 14 1776573 1948050 2118934 2289172 2458714 46 15 1779435 1950903 2121777 2292004 2461533 45 16 1782298 1953756 2124620 2294835 2464352 44 17 1785160 1956609 2127462 2297666 2467171 43 18 1788022 1959462 2130304 2300497 2469990 42 19 1790884 1962314 2133146 2303328 2472809 41 20 1793746 1965166 2135988 2306159 2475628 40 21 1796608 1968018 2138830 2308989 2478446 39 22 1799469 1970870 2141671 2311819 2481264 38 23 1802331 1973722 2144512 2314649 2484082 37 24 1805192 1976574 2147353 2317479 2486900 36 25 1808053 1979425 2150194 2320309 2489717 35 26 1810914 1982276 2153035 2323138 2492534 34 27 1813774 1985127 2155876 2325967 2495351 33 28 1816634 1987978 2158716 2328796 2498168 32 29 1819495 1990829 2161556 2331625 2500984 31 30 1822355 1993679 2164396 2334454 2503800 30 31 1825215 1996530 2167236 2337282 2506616 29 32 1828075 1999380 2170076 2340110 2509432 28 33 1830935 2002230 2172916 2342938 2512248 7 34 1833795 2005080 2175755 2345766 2515064 26 35 1836654 2007930 2178594 2348594 2517879 25 36 1839513 2010780 2181433 2351421 2520694 24 37 1842372 2013629 2184272 2354248 2523509 23 38 1845231 2016478 2187111 2357075 2526324 22 39 1848090 2019327 2189949 2359902 2529138 21 40 1850949 2022176 2192787 2362729 2531952 20 41 1853808 2025025 2195625 2365555 2534766 19 42 1856666 2027874 2198463 2368381 2537580 18 43 1859524 2030722 2201300 2371207 2540393 17 44 1862382 2033570 2204137 2374033 2543206 16 45 1865240 2036418 2206974 2376859 2546019 15 46 1868098 2039266 2209811 2379684 2548832 14 47 1870956 2042114 2212648 2382509 2551645 13 48 1873813 2044962 2215485 2385334 2554458 12 49 1876670 2047809 2218322 2388159 2557270 11 50 1879527 2050656 2221158 2390983 2560082 10 51 1882384 2053503 2223994 2393808 2562894 9 52 1885241 2056350 2226830 2396632 2565706 8 53 1888098 2059197 2229666 2399456 2568517 7 54 1890954 2062043 2232502 2402280 2571328 6 55 1893810 2064889 2235337 2405104 2574139 5 56 1896666 2067735 2238172 2407927 2576950 4 57 1899522 2070581 2241007 2410750 2579760 3 58 1902378 2073427 2243842 2413573 2582570 2 59 1905234 2076272 2246677 2416396 2585380 1 60 1908090 2079117 2249511 2419219 2588190 0 79 78 77 76 75 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Quadrant . 15 16 17 18 19 The minutes of the degrees of the Quadrant for right Sines of the complements of the same Quadrant . 0 2588190 2756373 2923717 3090170 3255682 60 1 2591000 2759169 2926499 3092936 3258532 59 2 2593809 2761965 2929280 3095702 3261182 58 3 2596618 2764761 2932061 3098468 3263931 57 4 2599427 2767556 2934842 3101234 3266681 56 5 2602236 2770351 2937623 3103999 3269430 55 6 2605045 2773146 2940403 3106764 3272179 54 7 2607853 2775941 2943183 3109529 3274927 53 8 2610661 2778735 2945963 3112294 3277675 52 9 2613469 2781529 2948743 3115058 3280423 51 10 2616277 2784323 2951523 3117822 3283171 50 11 2619084 2787117 2954302 3120586 3285918 49 12 2621891 2789911 2957081 3123349 3288665 48 13 2624698 2792704 2959860 3126112 3291412 47 14 2627505 2795497 2962638 3128875 3294159 46 15 2630312 2798290 2965416 3131638 3296906 45 16 2633118 2801082 2968194 3134400 3299652 44 17 2635924 2803874 2970972 3137162 3302398 43 18 2638730 2806666 2973750 3139924 3305144 42 19 2641536 2809458 2976527 3142686 3307889 41 20 2644342 2812250 2979304 3145448 3310634 40 21 2647147 2815041 2982081 3148209 3313379 39 22 2649952 2817832 2984857 3150970 3316123 38 23 2652757 2820623 2987633 3153731 3318867 37 24 2655562 2823414 2990409 3156491 3321611 36 25 2658366 2826204 2993185 3159251 3354355 35 26 2661170 2828994 2995960 3162011 3327098 34 27 2663974 2831784 2998735 3164770 3329841 33 28 2666777 2834574 3001510 3167529 3332585 32 29 2669580 2837364 3004284 3170288 3335327 31 30 2672383 2840153 3007058 3173047 3338069 30 31 2675186 2842942 3009832 3175805 3340811 29 32 2677989 2845731 3012606 3178563 3343553 28 33 2680792 3848520 3015380 3181321 3346294 27 34 2683595 2851308 3018153 3184079 3349035 26 35 2686397 2854096 3020926 3186837 3351776 25 36 2689199 2856884 3023699 3189594 3354516 24 37 2692001 2859672 3026472 3192351 3357256 23 38 2694802 2862459 3029244 3195108 3359996 22 39 2697603 2865246 3032016 3197864 3362736 21 40 2700404 2868033 3034788 3200620 3365475 20 41 2703205 2870819 3037559 3203375 3368214 19 42 2706005 2873605 3040330 3206130 3370953 18 43 2708805 2876391 3043101 3208885 3373691 17 44 2711605 2879177 3045872 3211640 3376429 16 45 2714405 2881963 3048643 3214395 3379167 15 46 2717204 2884748 3051413 3217150 3381905 14 47 2720003 2887533 3054183 3219904 3384642 13 48 2722802 2890318 3056953 3222658 3387379 12 49 2725601 2893103 3059723 3225412 3390116 11 50 2728400 2895888 3062492 3228165 3392852 10 51 2731198 2898672 3065261 3230918 3395588 9 52 2733996 2901456 3068030 3233671 3398324 8 53 2736794 2904240 3070798 3236423 3401060 7 54 2739592 2907023 3073566 3239175 3403795 6 55 2742389 2909806 3076334 3241927 3406530 5 56 2745186 2912589 3079102 3244679 3409265 4 57 2747983 2915371 3081869 3247430 3411999 3 58 2750780 2918153 3084636 3250181 3414733 2 59 2753577 2920935 3087403 3252932 3417467 1 60 2756373 2923717 3090170 3255682 3420201 0 74 73 72 71 70 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Quadrant . 20 21 22 23 24 The minutes of the degrees of the Quadrant for right Sines of the complements of the same Quadrant . 0 3420201 3583679 3746066 3907311 4067366 60 1 3422934 3586395 3748763 3909989 4070023 59 2 3425667 3589110 3751460 3912666 4072680 58 3 3428400 3591825 3754156 3915343 4075337 57 4 3431133 3594540 3756852 3918020 4077993 56 5 3433865 3597254 3759548 3920696 4080649 55 6 3436597 3599968 3762243 3923372 4083305 54 7 3439329 3602682 3764938 3926048 4085960 53 8 3442060 3605395 3767633 3928723 4088615 52 9 3444791 3608108 3770327 3931398 4091269 51 10 3447522 3610821 3773021 3934072 4093923 50 11 3450253 3613533 3775715 3936746 4096577 49 12 3452983 3616245 3778408 3939420 4099231 48 13 3455713 3618957 3781101 3942093 4101884 47 14 3458442 3621669 3783794 3944766 4104537 40 15 3461171 3624380 3786486 3947439 4107189 45 16 3463900 3627091 3789178 3950112 4109841 44 17 3466629 3629802 3791870 3952784 4112493 43 18 3469357 3632512 3794562 3955456 4115144 42 19 3472085 3635222 3797253 3958128 4117795 41 20 3474813 3637932 3799944 3960799 4120446 40 21 3477540 3640642 3802635 3963470 4123096 39 22 3480267 3643351 3805325 3966140 4125746 38 23 3482994 3646060 3808015 3968810 4128395 37 24 3485721 3648768 3810704 3971480 4131044 36 25 3488447 3651476 3813393 3974149 4133693 35 26 3491173 3654184 3816082 3976818 4136341 34 27 3493899 3656892 3818771 3979487 4138989 33 28 3496624 3659599 3821459 3982155 4141637 32 29 3499349 3662306 3824147 3984823 4144285 31 30 3502075 3665012 3826834 3987491 4146932 30 31 3504799 3667718 3829521 3990159 4149579 29 32 3507523 3670424 3832208 3992826 4152226 28 33 3510247 3673130 3834895 3995493 4154872 27 34 3512971 3675835 3837581 3998159 4157518 26 35 3515694 3678541 3840267 4000825 4160163 25 36 3518417 3681246 3842953 4003491 4162808 24 37 3521140 3683951 3845638 4006156 4165453 23 38 3523862 3686655 3848323 4008821 4168097 22 39 3526584 3689359 3851008 4011486 4170741 21 40 3529306 3692062 3853692 4014150 4173385 20 41 3532027 3694765 3856376 4016814 4176028 19 42 3534748 3697468 3859060 4019478 4178671 18 43 3537469 3700170 3861743 4022141 4181313 17 44 3540190 3702872 3864426 4024804 4183955 16 45 3542910 3705574 3867109 4027467 4186597 15 46 3545630 3708276 3869791 4030130 4189239 14 47 3548350 3710977 3872473 4032792 4191880 13 48 3551070 3713678 3875155 4035454 4194521 12 49 3553789 3716379 3877837 4038115 4197162 11 50 3556508 3719080 3880518 4040776 4199802 10 51 3559227 3721780 3883199 4043437 4202442 9 52 3561945 3724480 3885880 4046097 4205081 8 53 3564663 3727179 3888560 4048757 4207720 7 54 3567380 3729878 3891240 4051416 4210359 6 55 3570097 3732577 3893919 4054075 4212997 5 56 3572814 3735275 3896598 4056734 4215635 4 57 3575531 3737973 3899277 4059392 4218273 3 58 3578247 3740671 3901955 4062050 4220910 2 59 3580963 3743369 3904633 4064708 4223547 1 60 3583679 3746066 3907311 4067366 4226183 0 69 68 67 66 65 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Quadrant . 25 26 27 28 29 The minutes of the degrees of the Quadrant for right Sines of the complements of the same Quadrant . 0 4226183 4383712 4539905 4694710 4848096 60 1 4228819 4386326 4542497 4697284 4850640 59 2 4231455 4388940 4545088 4699852 4853184 58 3 4234090 4391554 4547679 4702419 4855727 57 4 4236725 4394167 4550270 4704986 4858270 56 5 4229360 4396780 4552860 4707553 4860812 55 6 4241994 4399392 4555450 4710119 4863354 54 7 4244628 4402004 4558039 4712685 4865895 53 8 4247262 4404616 4560628 4715250 4868436 52 9 4249895 4407227 4563216 4717815 4870977 51 10 4252528 4409838 4565804 4720380 4873517 50 11 4255161 4412449 4568392 4722944 4876057 49 12 4257793 4415059 4570979 4725508 4878596 48 13 4260425 4417669 4573566 4728071 4881135 47 14 4263056 4420278 4576153 4730634 4883674 46 15 4265687 4422887 4578739 4733197 4886212 45 16 4268318 4425496 4581325 4735759 4888750 44 17 4270949 4428104 4583911 4738321 4891287 43 18 4273579 4430712 4586496 4740882 4893824 42 19 4276209 4433320 4589081 4743443 4896361 41 20 4278838 4435927 4591665 4746004 4898897 40 21 4281467 4438534 4594249 4748564 4901433 39 22 4284096 4441140 4596833 4751124 4903968 38 23 4286724 4443746 4599416 4753683 4906503 37 24 4289352 4446352 4601999 4756242 4909037 36 25 4291979 4448957 4604581 4758801 4911571 35 26 4294606 4451562 4607163 4761359 4914105 34 27 4297233 4454167 4609744 4763917 4916638 33 28 4299859 4456771 4612325 4766474 4919171 32 29 4302485 4459375 4614906 4769031 4921703 31 30 4305111 4461978 4617486 4771588 4924235 30 31 4307736 4464581 4620066 4774144 4926767 29 32 4310361 4467184 4622646 4776700 4929298 28 33 4312986 4469786 4625225 4779255 4931829 27 34 4315610 4472388 4627804 4781810 4934359 26 35 4318234 4474990 4630382 4784365 4936889 25 36 4320858 4477591 4632960 4786919 4939418 24 37 4323481 448019● 4635538 4789473 4941947 23 38 4326104 4482792 4638115 4792026 4944476 22 39 4328726 4485392 4640692 4794579 4947004 21 40 4331348 4487992 4643268 4797132 4949532 20 41 4333970 4490591 4645844 4799684 4952059 19 42 4336591 4493190 4648420 4802236 4954586 18 43 4339212 4495788 4650995 4804787 4957113 17 44 4341833 4498386 4653570 4807338 4959639 16 45 4344453 4500984 4656145 4809888 4962165 15 46 4347073 4503582 4658719 4812438 4964690 14 47 4349693 4506179 4661293 4814988 4967215 13 48 4352312 4508776 4663866 4817537 4969740 12 49 4354931 4511372 4666439 4820086 4972264 11 50 4357549 4513968 4669012 4822635 4974788 10 51 4360167 4516563 4671584 4825183 4977311 9 52 4362785 4519158 4674156 4827731 4979834 8 53 4365402 4521753 4676727 4830278 4982356 7 54 4368019 4524347 4679298 4832825 4984878 6 55 4370635 4526941 4681869 4835371 4987399 5 56 4373251 4529535 4684439 4837917 4989920 4 57 4375867 4532128 4687009 4840462 4992441 3 58 4378482 4534721 4689578 4843007 4994961 2 59 4381097 4537313 4692147 4845552 4997481 1 60 4383712 4539905 4694716 4848096 5000000 0 64 63 62 61 60 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Qudrant . 30 31 32 33 34 The minutes of the degrees of the Quadrant for right Sines of the complement of the same Quadrant . 0 5000000 5150381 5299192 5446390 5591929 60 1 5002519 5152874 5301659 5448829 5594340 59 2 5005038 5155367 5304125 5451268 55967●1 58 3 5007556 5157859 5306591 5453707 55991●1 57 4 5010074 5160351 5309056 5456145 5601571 56 5 5012591 5162843 5311521 5458583 5603981 55 6 5015108 5165334 5313985 5461020 5606390 54 7 5017624 5167825 5316449 5463456 5608798 53 8 5020140 5170315 5318913 5465892 5611206 52 9 5022656 5172805 5321376 5468328 5613614 51 10 5025171 5175294 5323839 5470763 5616021 50 11 5027686 5177783 5326301 5473198 5618427 49 12 5030200 5180271 5328763 5475632 5620833 48 13 5032714 5182759 5331224 5478066 5623239 47 14 5035227 5185246 5333685 5480499 5625644 46 15 5037740 5187733 5336145 5482932 5628049 45 16 5040253 5190220 5338605 5485364 5630453 44 17 5042765 5192706 5341065 5487796 5632857 43 18 5045277 5195192 5343524 5490228 5635260 42 19 5047788 5197677 5345983 5492659 5637663 41 20 5050299 5200162 5348441 5495090 5640066 40 21 5052809 5202646 5350898 5497520 5642468 39 22 5055319 5205130 5353355 5499950 5644869 38 23 5057829 5207614 5355812 5502379 5647270 37 24 5060338 5210097 5358268 5504808 5649670 36 25 5062847 5212580 5360724 5507236 5652070 35 26 5065355 5215062 5363179 5509664 5654469 34 27 5067863 5217544 5365634 5512091 5656868 33 28 5070370 5220025 5368088 5514518 5659266 32 29 5072877 5222506 5370542 5516944 5661664 31 30 5075384 5224986 5372996 5519370 5664062 30 31 5077890 5227466 5375449 5521795 5666459 29 32 5080396 5229949 5377902 5524220 5668856 28 33 5082901 5232425 5380354 5526645 5671252 27 34 5085406 5234904 5382806 5529069 5673648 26 35 5087911 5237382 5385258 5531493 5676043 25 36 5090415 5239860 5387709 5533916 5678438 24 37 5092619 5242337 5390159 5536338 5680832 23 38 5095422 5244814 5392609 5538760 5683226 22 39 5097925 5147290 5395058 5541182 5685619 21 40 5100427 5249766 5397507 5543603 5688012 20 41 5102929 525224● 5399955 5546024 5690404 19 42 5105430 5254716 5402403 5548444 5692796 18 43 5107931 5257191 5404851 5550864 5695187 17 44 5110431 5259665 5407298 5553283 5697578 16 45 5112931 5262139 5409745 5555702 5699968 15 46 5115431 5264612 5412191 5558120 5702358 14 47 5117930 5267085 5414637 5560538 5704747 13 48 5120429 5269557 5417082 5562956 5707136 12 49 5122927 5272029 5419527 5565373 5709524 11 50 5125425 5274501 5421972 5567790 5711912 10 51 5127922 5276972 5424416 5570206 5714299 9 52 5130419 5279443 5426859 5572622 5716686 8 53 5132916 5281913 5429302 5575037 5719072 7 54 5135412 5284383 5431745 5577452 5721458 6 55 5137908 5286852 5434187 5579866 5723844 5 56 5140403 5289321 5436620 5582280 5726229 4 57 5142898 5291789 5439070 5584693 5728613 3 58 5145393 5294257 5441510 5587106 5730997 2 59 5147887 5296725 5443950 5589518 5733381 1 60 5150381 5299192 5446390 5591929 5735764 0 59 58 57 56 55 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Qudrant . 35 36 37 38 39 The minutes of the degrees of the Quadrant for right Sines of the complement of the same Quadrant . 0 5735764 5877852 6018150 6156615 6293204 60 1 5738147 5880205 6020470 6158907 6295464 59 2 5740529 5882558 6022796 6161198 6297724 58 3 5742911 5884910 6025118 6163489 6299983 57 4 5745292 5887262 6027439 6165780 6302242 56 5 5747672 5889613 6029760 6168070 6304501 55 6 5750052 5891964 6032080 6170359 6306759 54 7 5752432 5894314 6034400 6172648 6309016 53 8 5754811 5896664 6036719 6174936 6311273 52 9 5757190 5899013 6039038 6177224 6313529 51 10 5759568 5901361 6041357 6179512 6315784 50 11 5761946 5903709 6043675 6181799 6318039 49 12 5764323 5906056 6045992 6184085 6320293 48 13 5766700 5908403 6048309 6186371 6322547 47 14 5769076 5910750 6050625 6188656 6324800 46 15 5771452 5913096 6052940 6190940 6327053 45 16 5773827 5915442 6055255 6193224 6329305 44 17 5776202 5917787 6057570 6195508 6331557 43 18 5778576 5920132 6059884 6197791 6333808 42 19 5780950 5922476 6062198 6200074 6336059 41 20 5783324 5924820 6064511 6202356 6338310 40 21 5785697 5927163 6066824 6204638 6340560 39 22 5788069 5929505 6069136 6206919 6342809 38 23 5790441 5931847 6071448 6209199 6345058 37 24 5792812 5934189 6073759 6211479 6347306 36 25 5795183 5936530 6076069 6213758 6349553 35 26 5797553 5938871 6078379 6216037 6351800 34 27 5799923 5941211 6080688 6218315 6354046 33 28 5802292 5943551 6082997 6220593 6356292 32 29 5804●61 5945890 6085306 6222870 6358537 31 30 5807030 5948228 6087614 6225146 6360782 30 31 5809398 595056● 6089922 6227422 6363026 29 32 5811766 5952904 6092229 6229698 6365270 28 33 5814133 5955241 6094536 6231973 6367513 27 34 5816499 5957578 6096842 6234248 6309756 26 35 5818865 5959914 6099147 6236522 6371999 25 36 5821230 5962250 6101452 9238796 6374241 24 37 5823595 5964585 6103756 6241069 6376482 23 38 5825959 5966919 6106060 6243342 6378722 22 39 5828323 5969253 6108364 6245614 6380962 21 40 5830687 5971586 6110667 6247885 6383201 20 41 5833050 5973919 6112970 6250156 6385440 19 42 5835412 5976251 6115272 6252426 6387678 18 43 5837774 5978583 6117573 6254696 6389916 17 44 5840136 5980915 6119873 6256966 6392153 16 45 5842497 5983146 6122173 6259235 6394390 15 46 5844858 5985577 6124473 6261503 6396626 14 47 5847218 5987907 6126772 6263771 6398862 13 48 5849578 5990237 6129071 6266038 6401097 12 49 5851937 5992566 6131369 6268305 6403332 11 50 5854295 5994894 6133667 6270572 6405566 10 51 5856653 5997222 6135964 6272838 6407799 9 52 5859010 5999549 6138261 6275103 6410032 8 53 5861367 6001876 6140557 6277368 6412264 7 54 5863724 6004202 6142853 6279632 6414496 6 55 5866080 6006528 6145148 6281895 6416728 5 56 5868436 6008853 6147442 6284158 6418959 4 57 5870791 6011178 6149736 6286420 6421189 3 58 5873145 6013502 6152030 6288682 6423419 2 59 5875499 6015826 6154323 6290943 6425648 1 60 5877852 6018150 6156615 6293204 6427876 0 54 53 52 51 50 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Quadrant . 40 41 42 43 44 The minutes of the degrees of the Quadrant for right Sines of the complements of the same Quadrant . 0 6427876 6560590 6691306 6819984 6946584 60 1 6430104 6562785 6693468 6822111 6948676 59 2 6432331 6564979 6695629 6824237 6950767 58 3 6434558 6567173 9697789 6826363 6952858 57 4 6436785 6569367 6699949 6828489 6954949 56 5 6439011 6571560 6702108 6830614 6957039 55 6 6441236 6573753 6704267 6832738 6959128 54 7 6443461 6575945 6706425 6834861 6961216 53 8 6445685 6578136 6708582 6836984 6963304 52 9 6447909 6580326 6710739 6839107 6965392 51 10 6450132 6582516 6712895 6841229 6967479 50 11 6452355 6584705 6715051 6843350 6969565 49 12 6454577 6586894 6717206 6845471 6971651 48 13 6456799 6589082 6719361 6847591 6973736 47 14 6459020 6591270 6721515 6849711 6975821 46 15 6461240 6593458 6723668 6851830 6977905 45 16 6463460 6595645 6725821 6853949 6979988 44 17 6465679 6597831 6727973 6856067 6982071 43 18 6467898 6600016 6730125 6858184 6984153 42 19 6470116 6602201 6732276 6860301 6986235 41 20 6472333 6604386 6734427 6862417 6988319 40 21 6474550 6606570 6736577 6864533 6990396 39 22 6476766 6608753 6738726 6866648 6992476 38 23 6478982 6610936 6740875 6868762 6994555 37 24 6481198 6613118 6743024 6870876 6996634 36 25 6483413 6615300 6745172 6872989 6998712 35 26 6485628 6617481 6747319 6875102 7000789 34 27 6487842 6619661 6749465 6877214 7002866 33 28 6490055 6621841 6751611 6879325 7004942 32 29 6492268 6624021 6753757 6881436 7007018 31 30 6494480 6626200 6755902 6883546 7009093 30 31 6496692 6628379 6758047 6885656 7011167 29 32 6498903 6630557 6760191 6887765 7013241 28 33 6501114 6632734 6762334 6889874 7015314 27 34 6503324 6634911 6764477 6891982 7017387 26 35 6505533 6637087 6766619 6894089 7019459 25 36 6507742 6639263 6768760 6896196 7021530 24 37 6509950 6641438 6770901 6898302 7023601 23 38 6512158 6643612 6773041 6900408 7025671 22 39 6514365 6645786 6775181 6902513 7027741 21 40 6516572 6647959 6777320 6904617 7029810 20 41 6518778 6650132 6779459 6906721 7031879 19 42 6520984 6652304 6781597 6908824 7033947 18 43 6523189 6654476 6783734 6910927 7036014 17 44 6525394 6656647 6785871 6913029 7038081 16 45 6527598 6658817 6788007 6915131 7040147 15 46 6529801 6660987 6790143 6917232 7042213 14 47 6532004 6663156 6792278 6919332 7044278 13 48 6534206 6665325 6794413 6921432 7046342 12 49 6536408 6667493 6796547 6923531 7048406 11 50 6538609 6669661 6798681 6925630 7050469 10 51 6540809 6671828 6800814 6927728 7052432 9 52 6543009 6673994 6802946 6929725 7054594 8 53 6545208 6676160 6805078 6931922 7056655 7 54 6547407 6678326 6807209 6934018 7058716 6 55 6549606 6680491 6809340 6936114 7060776 5 56 6551804 6682655 6811470 6938209 7062836 4 57 6554001 6684818 6813599 6940303 7064895 3 58 6556198 6686981 6815728 6942397 7066953 2 59 6558394 6689144 6817856 6944491 7069011 1 60 6560590 6691306 6819984 6946584 7071068 0 49 48 47 46 45 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Quadrant . 45 46 47 48 49 The minutes of the degrees of the Quadrant for right Sines of the complements of the same Quadrant . 0 7071068 7193398 7313537 7431448 7547096 60 1 7073125 7195418 7315521 7433394 7549004 59 2 7075181 7197438 7317504 7435339 7550911 58 3 7077236 7199457 7319486 7437284 7552818 57 4 7079291 7201476 7321468 7439229 7554724 56 5 7081345 7203494 7323449 7441173 7556630 55 6 7083399 7205511 7325429 7443116 7558535 54 7 7085452 7207527 7327409 7445058 7560439 53 8 7087504 7209543 7329388 7447000 7562343 52 9 7089556 7211559 7331367 7448941 7564246 51 10 7091607 7213574 7333345 7450882 7566148 50 11 7093658 7215588 7335322 7452822 7568050 49 12 7095708 7217601 7337298 7454761 7569951 48 13 7097757 7219614 7339274 7456690 7571851 47 14 7099806 7221627 7341250 7458637 7573751 46 15 7101854 7223639 7343225 7460574 7575650 45 16 710390● 7225661 7345199 7462511 7577548 44 17 7105949 7227662 7347173 7464447 7579446 43 18 7107995 7229672 7349146 7466382 7581343 42 19 7110041 7231681 7351118 7468317 7583240 41 20 7112086 7233689 7353090 7470251 7585136 40 21 7114131 7235697 7355061 7472184 7587031 39 22 7116175 7237704 7357031 7474117 7588925 38 23 7118218 7239711 7359001 7476049 7590819 37 24 7120261 7241718 7360970 7477981 7592714 36 25 7122303 7243724 7362939 7479912 7594606 35 26 7124344 7245729 7364907 7481842 7596498 34 27 7126385 7247733 7366874 7483771 7598389 33 28 7128425 7249737 7368841 7485700 7600280 32 29 7130465 7251741 7370807 7487629 7602170 31 30 7132504 7253744 7372773 7489557 7604960 30 31 7134543 7255746 7374738 7491484 7605949 29 32 7136581 7257747 7376702 7493410 7607837 28 33 7138618 7259748 7378666 7495336 7609725 ●7 34 7140655 7261749 7380629 7497262 7611612 26 35 7142691 7263749 7382592 7499187 7613498 25 36 7144727 7265748 7384554 7501111 7615384 24 37 7146762 7267746 7386515 7503034 7617269 23 38 7148796 7269744 7388475 7504957 7619153 22 39 7150830 7271741 7390435 7506879 7621037 21 40 7152863 7273737 7392394 7508801 7622920 20 41 7154895 7275733 7394353 7510722 7624802 19 42 7156927 7277728 7396311 7512642 7626683 18 43 7158958 7279722 7398268 7514561 7628564 17 44 7160989 7281716 7400225 7516480 7630445 16 45 7163019 7283710 7402181 7518398 7632325 15 46 7165049 7285703 7404137 7520316 7634204 14 47 7167078 7287695 7406092 7522233 7636082 13 48 7169106 7289687 7408046 7524149 7637960 12 49 7171134 7291678 7410000 7526065 7639838 11 50 7173161 7293668 7411953 7527980 7641715 10 51 7175187 7295658 7413905 7529894 7643591 9 52 7177213 7297647 7415856 7531808 7645466 8 53 7179238 7299635 7417807 7533721 7647341 7 54 7181263 7301623 7419758 7535634 7649215 6 55 7183287 7303610 7421708 7537546 7651088 5 56 7185310 7305597 7423657 7539457 7652961 4 57 7187333 7307583 7425605 7541367 7654833 3 58 7189355 7309568 7427552 7543277 7656704 2 59 7191377 7311553 7429501 7545187 7658575 2 06 7193398 7313537 7431448 7547096 7660445 0 44 43 42 41 40 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Quadrant . 50 51 52 53 54 The minutes of the degrees of the Quadrant for right Sines of the complements of the same Quadrant . 0 7660445 7771460 7880108 7986355 8090170 60 1 7662314 7773290 7881898 7988105 8091879 59 2 7664183 7775120 7883688 7989855 8093588 58 3 7666051 7776949 7885477 7991604 8095296 57 4 7667919 7778777 7887266 7993352 8097004 56 5 7669786 7780605 7889054 7995100 8098711 55 6 7671652 7782432 7890841 7996847 8100417 54 7 7673517 7784258 7892927 7998593 8102122 53 8 7675382 7786084 7894413 8000339 8103827 52 9 7677246 7787909 7896198 8002084 8105531 51 10 7679110 7789833 7897983 8003828 8107234 50 11 7680973 7791557 7899767 8005571 8108936 49 12 7682835 7793380 7901550 8007314 8110638 48 13 7684697 7795202 7903332 8009056 8112339 47 14 7686558 7797024 7905114 8010797 8114040 46 15 7688418 7798845 7906895 8012538 8115740 45 16 7690278 7800665 7908676 8014278 8117439 44 17 7692137 7802485 7910456 8016017 8119137 43 18 7693995 7804304 7912235 8017756 8120835 42 19 7695853 7806123 7914014 8019494 8122532 41 20 7697710 7807941 7915792 8021232 8124229 40 21 7699566 7809758 7917569 8022969 8125925 39 22 7701422 7811574 7919345 8024705 8127620 38 23 7703277 7813390 7921121 8026440 8129314 37 24 7705132 7815205 7922896 8028175 8131008 36 25 7706986 7817020 7924671 8029909 8132701 35 26 7708839 7818834 7926445 8021642 8134393 34 27 7710692 7820647 7928218 8033375 8136084 33 28 7712544 7822459 7929990 8035107 8137775 32 29 7714395 7824271 7931762 8036838 8139465 31 30 7716246 7826082 7933533 3038569 8141155 30 31 7718096 7827892 7935303 3040299 8142844 29 32 7719945 7829702 7937073 8042028 8144532 28 33 7721794 7831511 7938842 8043757 8146220 27 34 7723642 7833320 7940611 8045485 8147907 26 35 7725490 7835128 7942379 8047212 8149593 25 36 7727337 7836935 7944146 8048938 8151278 24 37 7729183 7838741 7945912 8050664 8152963 23 38 7731028 7840547 7947678 8052389 8154647 22 39 7732872 7842352 7949443 8054114 8156330 21 40 7734716 7844157 7951208 8055838 8158013 20 41 7736559 7845961 7952972 8057561 8159695 19 42 7738402 7847764 7954735 8059283 8161376 18 43 7740244 7849566 7956497 8061005 8163057 17 44 7742085 7851368 7958259 8062726 8164737 16 45 7743926 7853169 7960020 8064446 8166416 15 46 7745766 7854970 7961780 8066166 8168094 14 47 7747606 7856770 7963540 8067885 8169772 13 48 7749445 7858569 7965299 8069603 8171449 12 49 7751283 7860368 7967057 8071321 8173126 11 50 7753121 7862166 7968815 8073038 8174802 10 51 7754958 7863963 7970572 8074754 8176477 9 52 7756794 7865759 7972328 8076470 8178151 8 53 7758630 7867555 7974084 8078185 8179825 7 54 7760465 7869350 7975839 8079899 8181498 6 55 7762299 7871145 7977593 8081613 8183170 5 56 7764132 7872939 7979347 8083326 8184841 4 57 7765965 7874732 7981100 8085038 8186512 3 58 7767797 7876525 7982852 8086749 8188182 2 59 7769629 7878317 7984604 8088460 8189851 1 06 7771460 7880108 7986355 8090170 8191520 0 39 38 37 36 35 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Quadrant . 55 56 57 58 59 The minutes of the degrees of the Quadrant for right Sines of the complements of the same Quadrant . 0 8191520 8290376 8386706 8480481 8571673 60 1 8193188 8292002 8388290 8482022 8573171 59 2 8194855 8293628 8389873 8483562 8574668 58 3 8196522 8295253 8391456 8485102 8576164 57 4 8198188 8296877 8393038 8486641 8577660 56 5 8199854 8298501 8394619 8488180 8579155 55 6 8201519 8300124 8396199 8489718 8580649 54 7 8203183 8301746 8397778 8491255 8582142 53 8 8204846 8303367 8399357 8492791 8583635 52 9 8206508 8304987 8400935 8494326 8585127 51 10 8208170 8306607 8402513 8495860 8586619 50 11 8209831 8308226 8404090 8497394 8588110 49 12 8211491 8309844 8405666 8498927 8589600 48 13 8213151 831146● 8407241 8500459 8591089 47 14 8214810 8313079 8408816 8501991 8592577 46 15 8216469 8314696 8410390 8503522 8594064 45 16 8218127 8316312 8411963 8505052 8595551 44 17 8219784 8317927 8413536 8506582 8597037 43 18 8221440 8319541 8415108 8508111 8598523 42 19 8223096 8321155 8416679 8509639 8600008 41 20 8224751 8322768 8418250 8511167 8601492 40 21 8226405 8324380 8419820 8512694 8602975 39 22 8228058 8325991 8421389 8514220 8604457 38 23 8229711 8327602 8422957 8515745 8605939 37 24 8231363 8329212 8424525 8517270 8607420 36 25 8233015 8330822 8426092 8518794 8608901 35 26 8234666 8332431 8427658 8520317 8610381 34 27 8236316 8334039 8429223 8521839 8611860 33 28 8237965 8335646 8430788 8523361 8613338 32 29 8239614 8337252 8432352 8524882 8614815 31 30 8241262 8338858 8433915 8526402 8616292 30 31 8242909 8340463 8435477 8527921 8617768 29 32 8244556 8342067 8437039 8529440 8619243 28 33 8246202 8343671 8438600 8530958 8620718 27 34 8247847 8345247 8440161 8532476 8622192 26 35 8249492 8346877 8441721 8533993 8623665 25 36 8251136 8348479 8443280 8535509 8625137 24 37 8252779 8350080 8444838 8537024 8626608 23 38 8254421 8351680 8446396 8538538 8628079 22 39 8256062 8353279 8447953 8540052 8629549 21 40 8257703 8354878 8449509 8541565 8631019 20 41 8259343 8356476 8451064 8543077 8632488 19 42 8260982 8358073 8452618 8544588 8633956 18 43 8262621 8359670 8454172 8546099 8635423 17 44 8264259 8361266 8455725 8547609 8636889 16 45 8265897 8362862 8457278 8549119 8638355 15 46 8267534 8364457 8458830 8550628 8639820 14 47 8269170 8366051 8460381 8552136 8641284 13 48 8270806 8367644 8461932 8553643 8642748 12 49 8272441 8369236 8463482 8555149 8644211 11 50 8274075 8370828 8465031 8556655 8645673 10 51 8275708 8372419 8466579 8558160 8647134 9 52 8277340 8374009 8468126 8559664 8648595 8 53 8278972 8375599 8469673 9561168 8650055 7 54 8280603 8377188 8471219 8562671 8651514 6 55 8282234 8378776 8472765 8564173 8652973 5 56 8283864 8380363 8474310 8565675 8654431 4 57 8285493 8381950 8475854 8567176 8655888 3 58 8287121 8383536 8477397 8568676 8657344 2 59 8288749 8385121 8478939 8570175 8658799 1 60 8290376 8386706 8480481 8571673 8660254 0 34 33 32 31 30 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Quadrant . 60 61 62 63 64 The minutes of the degrees of the Quadrant for right Sines of the complements of the same Quadrant . 0 8660254 8746197 8829476 8910065 8987940 60 1 8661708 8747607 8830841 8911385 8989215 59 2 8663162 8749016 8832205 8912704 8990489 58 3 8664615 8750425 8833569 8914023 8991762 57 4 8666067 8751833 8834932 8915341 8993035 56 5 8667518 8753240 8836295 8916659 8994307 55 6 8668968 8754646 8837657 8917976 8995578 54 7 8670417 8756051 8839018 8919292 8996848 53 8 8671866 8757456 8840378 8920607 8998117 52 9 8673314 8758860 8841737 8921921 8999386 51 10 8674762 8760263 8843095 8923234 9000654 50 11 8676209 8761665 8844452 8924546 9001921 49 12 8677655 8763067 8845809 8925858 9003187 48 13 8679100 8764468 8847165 8927169 9004453 47 14 8680544 8765868 8848521 8928479 9005718 46 15 8681988 8767268 8849876 8929789 9006982 45 16 8683431 8768667 8851230 8931098 9008245 44 17 8684874 8770065 8852583 8932406 9009508 43 18 8686316 8771462 8853936 8933714 9010770 42 19 8687757 8772859 8855288 8935021 9012031 41 20 8689197 8774255 8856639 8936327 9013292 40 21 8690636 8775650 8857989 8937632 9014552 39 22 8692074 8777044 8859338 8938936 9015811 38 23 8693512 8778437 8860687 8940240 9017069 37 24 8694949 8779830 8862035 8941543 9018326 36 25 8696386 8781222 8863383 8942845 9019582 35 26 8697822 8782613 8864730 8944146 9020838 34 27 8699257 8784003 8866076 8945446 9022093 33 28 8700691 8785393 8867421 8946746 9023347 32 29 8702124 8786782 8868765 8948045 9024600 31 30 8703557 8788171 8870108 8949344 9025853 30 31 8704989 8789559 8871451 8950642 9027105 29 32 8706420 8790946 8872793 895193● 9028356 28 33 8707851 8792332 8874134 8953235 9029606 27 34 8709281 8793717 8875475 8954530 9030856 26 35 8710710 8795102 8876815 8955824 9032105 25 36 8712138 8796486 8878154 8957117 9033353 24 37 8713565 8797869 8879492 8958410 9034600 23 38 8714992 8799251 8880830 8959702 9035847 22 39 8716418 8800633 8882167 8960994 9037093 21 40 8717844 8802014 8883503 8962285 9038338 20 41 8719269 8803394 8884838 8963575 9039582 19 42 8720693 8804773 8886172 8964864 9040825 18 43 8722116 8806152 8887506 8966152 9042068 17 44 8723538 8807530 8888839 8967440 9043310 16 45 8724960 8808907 8890171 8968727 9044551 15 46 8726●81 8810283 8891502 8970013 9045791 14 47 8727801 8811659 8892833 8971299 9047031 13 48 8729221 8813034 8894163 8972584 9048270 12 49 8730640 8814408 8895492 8973868 9049508 11 50 8732058 8815782 8896821 8975151 9050746 10 51 8733475 8817155 8898149 8976433 9051983 9 52 8734891 8818527 8899476 8977715 9053219 8 53 8736307 8819898 8900802 8978996 9054454 7 54 8737722 8821268 8902127 8980276 9055688 6 55 8739137 8822638 8903452 8981555 9056922 5 56 8740551 8824007 8904776 8982833 9058155 4 57 8741964 8825375 8906099 8984111 9059387 3 58 8743376 8826743 8907422 8985388 9060618 2 59 8744787 8828110 8908744 8986664 9061848 1 60 8746197 8829476 8910065 8987940 9063078 0 25 26 27 28 29 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Quadrant . 65 66 67 68 69 The minutes of the degrees of the Quadrant for right Sines of the complements of the same Quadrant . 0 9063078 9135455 9205049 9271839 9335804 60 1 9064307 9136638 9206185 9272928 9336846 59 2 9065535 9137820 9207321 9274017 9337887 58 3 9066763 9139001 9208456 9275105 9338928 57 4 9067990 9140181 9209590 9276192 9339968 56 5 9069216 9141361 9210723 9277278 9341007 55 6 9070441 9142540 9211855 9278363 9342045 54 7 9071665 9143718 9212986 9279448 9343082 53 8 9072889 9144895 9214117 9280532 9344119 52 9 9074112 9146072 9215247 9281615 9345155 51 10 9075334 9147248 9216376 9282697 9346190 50 11 9076555 9148423 9217504 9283778 9347224 49 12 9077775 9149597 9218631 9284859 9348257 48 13 9078995 9150770 9219758 9285939 9349289 47 14 9080214 9151943 9220884 9287018 9350321 46 15 9081432 9153115 9222010 9288096 9351352 45 16 9082649 9154286 9223135 9289173 9352382 44 17 9083866 9155457 9224259 9290250 9353411 43 18 9085082 9156627 9225382 9291326 9354440 42 19 9086297 9157796 9226504 9292401 9355468 41 20 9087512 9158964 9227625 9293476 9356495 40 21 9088726 9160131 9228746 9294550 9357521 39 22 9089939 9161297 9229866 9295623 9358546 38 23 9091151 9162463 9230985 9296695 9359571 37 24 9092362 9163628 9232103 9297766 9360595 36 25 9093572 9165792 9233220 9298836 9361618 35 26 9094781 9165955 9234337 9299905 9362640 34 27 9095990 9167117 9235453 9300974 9363662 33 28 9097198 9168279 9236568 9302042 9364683 32 29 9098406 9169440 9237682 9303109 9365703 31 30 9099613 9170601 9238795 9304176 9366722 30 31 9100819 9171761 9239908 9305242 9367740 29 32 9102024 9172920 9241020 9306307 9368758 28 33 9103228 9174078 9242131 9307371 9369775 27 34 9104432 9175235 9243242 9308434 9370791 26 35 9105636 9176391 9244352 9309497 9371806 25 36 9106837 9177547 9245461 9310559 9372820 24 37 9108038 9178702 9246569 9311620 9373834 23 38 9109238 9179856 9247676 9312680 9374847 22 39 9110438 9181009 9248782 9313739 9375859 21 40 9111637 9182161 9249888 9314798 9376870 20 41 9112835 9183313 9250993 9315856 9377880 19 42 9114032 9184464 9252097 9316913 9378889 18 43 9115229 9185614 9253200 6317969 9379898 17 44 9116425 9186763 9254303 9319024 9380906 16 45 9117620 9187912 9255405 9320079 9381913 15 46 9118814 9189060 9256506 9321133 9382919 14 47 9120007 9190207 9257606 9322186 9383925 13 48 9121200 9191353 9258706 9323238 9384930 12 49 9122392 9192499 9259805 9324290 9385934 11 50 9123584 9193644 9260963 9325241 9386937 10 51 9124775 9194788 9262000 9326391 9387939 9 52 9125965 9195931 9263096 9327440 9388941 8 53 9127154 9197073 9264192 9328488 9389942 7 54 9128342 9198215 9265287 9329535 9390942 6 55 9129529 9199356 9266381 9330582 9391941 5 56 9130716 9200496 9267474 9331628 9392940 4 57 9131902 9201635 9268566 9332673 9393938 3 58 9133087 9202774 9269658 9333717 9394935 2 59 9134271 9203912 9270749 9334761 9395931 1 60 9135455 9205049 9271839 9335804 9396926 0 24 23 22 21 20 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Quadrant . 70 71 72 73 74 The minutes of the degrees of the Quadrant for right Sines of the complements of the same Quadrant . 0 9396926 9455186 9510565 9563048 9612617 60 1 9397921 9456133 9511464 9563898 9613418 59 2 9398915 9457079 9512362 956474● 9614219 58 3 9399908 9458024 9513259 9565596 9615019 57 4 9400900 9458968 9514155 9566444 9615818 56 5 9401891 9459911 9515050 9567291 9616616 55 6 9402882 9460854 9515944 9568137 9617413 54 7 9403872 9461796 9516838 9568982 9618209 53 8 9404861 9462737 9517731 9569826 9619005 52 9 9405849 9463677 9518623 9570670 9619800 51 10 9406836 9464616 9519514 9571513 9620594 50 11 9407822 9465555 9520404 9572355 9621387 49 12 9408808 9466493 9521294 9573196 9622179 48 13 9409793 9467430 9522183 9574036 9622971 47 14 9410777 9468366 9523071 9574875 9623762 46 15 9411760 9469301 9523958 9575714 9624552 45 16 9412742 9470236 9524844 9576552 9625341 44 17 9413724 9471170 9525730 9577389 9626129 43 18 9414705 9472103 9526615 9578225 9626917 42 19 9415685 9473035 9527499 9579061 9627704 41 20 9416665 9473967 9528382 9579896 9628490 40 21 9417644 9474898 9529264 9580730 9629275 39 22 9418622 9475828 9530146 9581563 9630059 38 23 9419599 9476757 9531027 9582395 9630843 37 24 9420575 9477685 9531907 9583226 9631626 36 25 9421550 9478612 9532786 9584057 9632408 35 26 9422525 9479539 9533664 9584887 9633189 34 27 9423499 9480465 9534541 9585716 9633969 33 28 9425472 9481390 9535418 9586544 9634748 32 29 9425444 9482314 9536294 9587371 9635527 31 30 9426415 9483237 9537169 9588197 9636305 30 31 9427380 9484160 9538043 9589023 9637082 29 32 9428356 9485082 9538917 9589848 9637858 28 33 9429325 9486003 9539790 9590672 9638633 27 34 9430293 9486923 9540662 9591495 9639408 26 35 9431260 9487842 9541533 9592318 9640182 25 36 9432227 9488761 9542403 9593140 9640955 24 37 9433193 9489679 9543272 9593961 9641727 23 38 9434158 9490596 9544141 9594781 9642498 22 39 9435122 9491512 9545009 9595600 9643268 21 40 9436085 9492427 9545876 9596419 9644038 20 41 9437048 9493341 9546742 9597237 9644807 19 42 9438010 9494255 9547607 9598054 9645575 18 43 9438971 9495168 9548472 9598870 9646342 17 44 9439931 9496080 9549336 9599685 9647108 16 45 9440890 9496991 9550199 9600499 9647873 15 46 9441849 9497902 9551061 9601313 9648638 14 47 9442807 9498812 9551922 9602126 9649402 13 48 9443764 9499721 9552783 9602938 9650165 12 49 9444720 9500629 9553643 9603749 9650927 11 50 9445676 9501536 9554502 9604559 9651689 10 51 9446631 9502443 9555360 9605368 9652450 9 52 9447585 9503349 9556217 9606177 9653210 8 53 9448538 9504254 9557074 9606985 9653969 7 54 9449490 9505158 9557930 9607792 9654727 6 55 9450441 9506061 9558785 9608598 9655484 5 56 9451392 9506963 9559639 9609403 9656240 4 57 9452342 9507865 9560492 9610208 9656996 3 58 9453291 9508766 9561345 9611012 9657751 2 59 9454239 9509666 9562197 9611815 9658505 1 60 9455186 9510565 9563048 9612617 9659258 0 19 18 17 16 15 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Quadrant . 75 76 77 78 79 The minutes of the degrees of the Quadrant for right Sines of the complements of the same Quadrant . 0 9659258 9702957 9743600 9781476 9816272 60 1 9660011 9703660 9744355 9782080 9816827 59 2 9660163 9704363 9745008 9782684 9817381 58 3 9661514 9705065 9745660 9783281 9817934 57 4 9662264 9705766 9740312 9783889 9818486 56 5 9663013 9706466 9746963 9784490 9819037 55 6 9663761 9707165 9747013 9785090 9819587 54 7 9664508 9707863 9748262 9785689 9820137 53 8 9665255 9708561 9748910 9786288 9820686 52 9 9666001 9709258 9749557 9786886 9821234 51 10 9666746 9709954 9750203 9787483 9821781 50 11 9667490 9710649 9750849 9788079 9822327 49 12 9668233 9711343 9751494 9788674 9822872 48 13 9668976 9712036 9752138 9789268 9823417 47 14 9669718 9712729 9752781 9789862 9823961 46 15 9670459 9713421 9753424 9790455 9824504 45 16 9671199 9714112 9754065 9791047 9825046 44 17 9671938 9714802 9754706 9791638 9825587 43 18 9672677 9715491 9755346 9792228 9826128 42 19 9673415 9716180 9755985 9792818 9826668 41 20 9674152 9716868 9756623 9793407 9827207 40 21 9674888 9717555 9757260 9793995 9827745 39 22 9675623 9718241 9757897 9794582 9828282 38 23 9676357 9718926 9758533 9795168 9828818 37 24 9677091 9719610 9759168 9795753 9829354 36 25 9677824 9720294 9759802 6796337 9829889 35 26 9678556 9720977 9760435 9796921 9830423 34 27 9679287 9721659 9761067 9797504 9830956 33 28 9680017 9722340 9761699 9798086 9831488 32 29 9680747 9723020 9762330 9798667 9832019 31 30 9681476 9723699 9762960 9799247 9832549 30 31 9682804 9724378 9763589 9799827 9833079 29 32 9682931 9725056 9764217 9800406 9833608 28 33 9683657 9725733 9764845 9800984 9834136 27 34 9684383 9726409 9765472 9801561 9834663 26 35 9685108 9727085 9766098 9802137 9835189 25 36 9685832 9727760 9766723 9802712 9835714 24 37 9686555 9728434 9767347 9803287 9836239 23 38 9687277 9729107 9767970 9803861 9836763 22 39 9687998 9729779 9768593 9804434 9837286 21 40 9688719 9730450 9769215 9805006 9837808 20 41 9689439 9731120 9769836 9805577 9838329 19 42 9690158 9731789 9770456 9806147 9838850 18 43 9690879 9732458 9771075 9806716 9839370 17 44 9691593 9733126 9771693 9807285 9839889 16 45 9692309 9733793 9772311 9807853 9840407 15 46 9693025 9734459 9772928 9808420 9840924 14 47 9693740 9735124 9773544 9808986 9841440 13 48 9694454 9735789 9774159 9809551 9841956 12 49 9695167 9736453 9774773 9810116 9842471 11 50 9695879 9737116 9775387 9810680 9842985 10 51 9696590 9737778 9776000 9811243 9843498 9 52 9697301 9738439 9776612 9811805 9844010 8 53 9698011 9739099 9777223 9812366 9844521 7 54 9698720 9739759 9777833 9812926 9845032 6 55 9699428 9740418 9778442 9813486 9845542 5 56 9700135 9741076 9779050 9814045 9846051 4 57 9700842 9741733 9779658 9814603 9846559 3 58 9701548 9742389 9780265 9815160 9847066 2 59 9702253 9743045 9780871 9815716 9847572 1 60 9702957 9743700 9781476 9816272 9848078 0 14 13 12 11 10 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Quadrant . 80 81 82 83 84 The minutes of the degrees of the Quadrant for right Sines of the complements of the same Quadrant . 0 9848078 9876883 9902681 9925461 9945219 60 1 9848583 9877338 9903085 9925816 9945523 59 2 9849087 9877792 9903489 9926109 9945826 58 3 9849590 9878245 990389● 9926521 9946128 57 4 9850092 9878697 9904294 9926873 9946429 56 5 9850593 9879148 9904695 9927224 9946729 55 6 9851093 9879598 9905095 9927574 9927028 54 7 9851593 9880048 9905494 9927923 9947337 53 8 9852092 9880497 9905893 9928271 9947625 52 9 9852590 9880945 9906291 99●8618 9947622 51 10 9853087 9881392 9906688 9928965 9948218 50 11 9853583 9881838 9907084 9929311 9948513 49 12 9854079 9882283 9907479 9929656 9948807 48 13 9854574 9882728 9907873 9930000 9949100 47 14 9855068 9883172 9908266 9930343 9949393 46 15 9855561 9883615 9908659 9930685 9949685 45 16 9856053 9884057 9909051 9931026 9949976 44 17 9856544 9884498 9909442 9931367 9950266 43 18 9857035 9884938 9909832 9931707 9950555 42 19 9857525 9885378 9910221 9932046 9950844 41 20 9858014 9885817 9910610 9932384 9951132 40 21 9858502 9886255 9910998 9932721 9951419 39 22 9858989 9886692 9911385 9933057 9951705 38 23 9859475 9887128 9911771 9933393 9951990 37 24 9859961 9887564 9912156 9933728 9952274 36 25 9860446 9887999 9912540 9934062 9952557 35 26 9860930 9888433 9912923 9934395 9952840 34 27 9861413 9888866 9913306 9934727 9953122 33 28 9861895 9889298 9913688 9935058 9953403 32 29 9862376 9889729 9914069 9935389 9953683 31 30 9862856 9890159 9914449 9935719 9953962 30 31 9863336 9890588 9914828 9936048 9954240 29 32 9863815 9891017 9915206 9936376 9954518 28 33 9864293 9891445 9915584 9936703 9954795 27 34 9864770 9891872 9915961 9937029 9955071 26 35 9865246 9892298 9916337 9937355 9955346 25 36 9865722 9892723 9916712 9937680 9955620 24 37 9866197 9893147 9917086 9938004 9955893 23 38 9866671 9893571 9917459 9938327 9956165 22 39 9867144 9893994 9917832 9938649 9956437 21 40 9867616 9894416 9918204 9938970 9956708 20 41 9868087 9894837 9918575 9939290 9956978 19 42 9868557 9895257 9918945 9939609 9957247 18 43 9869027 9895677 9919314 9939928 9957515 17 44 9869496 9896096 9919682 9940246 9957782 16 45 9869964 9896514 9920049 9940563 9958049 15 46 9870431 9896931 9920416 9940879 9958315 14 47 9880897 9897347 9920782 9941194 9958580 13 48 9871362 9897762 9921147 9941509 9958844 12 49 9871827 9898177 9921511 9941823 9959307 11 50 9872291 9898591 9921874 9942136 9959370 10 51 9872754 9899004 9922236 9942448 9959632 9 52 9873216 9899416 9922598 9942759 9959893 8 53 9873677 9899827 9922959 9943069 9960153 7 54 9874137 9900237 9923319 9943379 9960412 6 55 9874597 9900646 9923678 9943688 9960670 5 56 9875056 9901055 9924036 9943996 9960927 4 57 9875514 9901463 9924393 9944303 9961183 3 58 9875514 9901463 9924393 9944303 9961183 3 58 9875971 4901870 9924750 9944609 9961438 2 59 9876427 9902276 9925106 9944914 9961693 1 60 9876883 9902681 9925461 9945219 9961947 0 9 8 7 6 5 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for right Sines of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for right Sines of the Arches of the same Quadrant . 85 86 87 88 89 The minutes of the degrees of the Quadrant for right Sines of the complements of the same Quadrant . 0 9961947 9975640 9986295 9993908 9998477 60 1 9962200 9975843 9986447 9994009 9998527 59 2 9962452 9976045 9986598 9994109 9998576 58 3 9962703 9976246 9986748 9994208 9998625 57 4 9962954 9976446 9986897 9994307 9998673 56 5 9963204 9976645 9987045 9994405 9998720 55 6 9963453 9976843 9987193 9994502 9998766 54 7 9963701 9977040 9987340 9994598 9998811 53 8 9963948 9977237 9987486 9994693 9998855 52 9 9964194 9977433 9987631 9994787 9998899 51 10 9964440 9977628 9987775 9994881 9998942 50 11 9964685 9977822 9987918 9994974 9998984 49 12 9964929 9978015 9988061 9995066 9999025 48 13 9965172 9978207 9988203 9995157 9999065 47 14 9965414 9978398 9988344 9995247 9999104 46 15 9965655 9978589 9988484 9995336 9999143 45 16 9965895 9978779 9988623 9995424 9999181 44 17 9966235 9978968 9988761 9995512 9999218 43 18 9966374 9979156 9988899 9995599 9999254 42 19 9966612 9979343 9989036 9995685 9999289 41 20 9966849 9979530 9989172 9995770 9999323 40 21 9967085 9979716 9989307 9995854 9999356 39 22 9967320 9979901 9989441 9995937 9999389 38 23 9967555 9980085 9989574 9996019 9999421 37 24 9967789 9980268 9989706 9996101 9999452 36 25 9968022 9980450 9989837 9996182 9999482 35 26 9968254 9980631 9989968 9996262 9999511 34 27 9968485 9980811 9990098 9996341 9999539 33 28 9968715 9980992 9990227 9996419 9999560 32 29 9968944 9981170 9990355 9996496 9999593 31 30 9969173 9981348 9990482 9996573 9999616 30 31 9969401 9981525 9990608 9996649 9999644 29 32 9969628 9981701 9990734 9996724 9999668 28 33 9969854 9981877 9990859 9996798 9999691 27 34 9970079 9982052 9990983 9996871 9999713 26 35 9970304 9982226 9991106 9996943 9999735 25 36 9970528 9982399 9991228 9997014 9999756 24 37 9970751 9982571 9991349 9997085 9999776 23 38 9970973 9982742 9991470 9997155 9999795 22 39 9971194 9982912 9991590 9997224 9999813 21 40 9971414 9983082 9991709 9997292 9999830 20 41 9971633 9983251 9991827 9997359 9999846 19 42 9971851 9983419 9991944 9997425 9999862 18 43 9972069 9983586 9992060 9997491 9999877 17 44 9972286 9983752 9992175 9997556 9999891 16 45 9972502 9983917 9992290 9997620 9999904 15 46 9972717 9984081 9992404 9997683 9999916 14 47 9972931 9984245 9992517 9997745 99999●7 13 48 9973145 9984408 9992629 9997806 9999938 12 49 9973358 9984570 9992740 9997867 9999948 11 50 9973570 9984731 9992850 9997927 9999957 10 51 9973781 9984891 9992960 9997986 9999965 9 52 9973991 9985050 9993069 9998044 9999972 8 53 9974200 9985209 9993177 9998101 9999978 7 54 9974408 9985367 9993284 9998157 9999984 6 55 9974615 9985524 9993390 9998212 9999989 5 56 9974822 9985680 9993495 9998267 9999993 4 57 9975028 9985835 9993599 9998321 9999996 3 58 9975233 9985989 9993703 9998374 9999998 2 59 9975437 9986143 9993806 9998426 9999999 1 60 9975640 9986295 9993908 9998477 1000000 0 4 3 2 1 0 The degrees of the Quadrant for right Sines of the complements of the Arches of the same Quadrant .

THE TABLE OF TANGENTS , OTHERWISE CALLED THE FRVITFVLL TABLE . The Table of Tangents .

The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadarnt . 0 1 2 3 4 The minutes of the degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant . 0 0000 174550 349207 524078 699269 60 1 2909 177459 352120 526995 702193 59 2 5818 180369 355033 529911 705116 58 3 8727 183279 357945 532828 708039 57 4 11636 186189 360858 535745 710962 56 5 14544 189100 363770 538663 713886 55 6 17452 192010 366683 541580 716809 54 7 20361 194920 369596 544498 719733 53 8 22270 197820 372508 547415 722657 52 9 2●17● ●00●●● 37●421 550●33 725580 51 10 2●08● ●03●50 37●334 553251 728504 50 11 31906 206561 381247 556169 731428 49 12 34905 302471 384100 559087 734353 48 13 37814 212381 387073 562005 737277 47 14 40723 215291 389987 764923 74020● 46 15 43632 218201 392900 567841 743127 45 16 46541 221111 39●814 570759 746052 44 17 49450 224022 398727 573678 748978 43 18 52359 226932 401641 576596 751903 42 19 55268 229842 404554 579514 754829 41 20 58177 232752 407468 582433 757754 40 21 61086 235663 410382 585352 760680 39 22 63995 238574 413295 588270 763606 38 23 66904 241485 416209 591189 766532 37 24 69813 244395 419123 594108 769459 36 25 72722 247306 422037 597028 772385 35 26 75631 250217 424951 599947 775311 34 27 78540 253128 427866 602866 778238 33 28 81450 256038 430780 605786 781164 32 29 84359 258949 433694 608705 784091 31 30 87268 261859 436609 611625 787017 30 31 90177 264770 439523 614544 789944 29 32 93086 267681 442438 617464 792871 28 33 95995 270592 445353 620384 795799 27 34 98904 273503 448267 623304 798726 26 35 101814 276414 451182 626225 801653 25 36 104723 279325 454097 629145 804581 24 37 107632 282237 457012 632066 807509 23 38 110541 285148 459927 634986 810437 22 39 113450 288059 462842 637907 813365 21 40 116360 290970 465757 640828 816293 20 41 119269 293882 468672 643749 819221 19 42 122178 296794 471588 646671 822150 18 43 125088 299705 474503 649592 825079 17 44 127997 302617 477419 652514 828008 16 45 130906 305528 480335 655435 830937 15 46 133816 308439 483251 658357 833866 14 47 136725 311351 486166 661278 836795 13 48 139635 314262 489082 664200 839724 12 49 142544 317174 491997 667121 842653 11 50 145454 320085 494913 670043 845583 10 51 148363 322997 497829 672965 848513 9 52 151273 325909 500745 675888 851443 8 53 154182 328821 503662 678810 854374 7 54 159092 331733 506578 681733 857304 6 55 160001 334645 509495 684656 860234 5 56 162911 337558 512411 687578 863164 4 57 165820 340470 515328 690501 866095 3 58 168730 343382 518244 693423 869025 2 59 171640 346295 521161 696346 871956 1 60 174550 349207 524078 699269 874886 0 89 88 87 86 85 The degrees of the Quadrant for the Tangent of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . 5 6 7 8 9 The minutes of the degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant . 0 874886 1051042 1227846 1405408 1583844 60 1 877817 1053983 1230798 1408374 1586826 59 2 880748 1056924 1233751 1411341 1589808 58 3 883680 1059866 1236704 1414308 1592791 57 4 886611 1062808 1239658 1417275 1595774 56 5 889543 1065750 1242612 1420242 1598757 55 6 892475 1068692 1245566 1423210 1601740 54 7 895407 1071634 1248520 1426178 1604723 53 8 898339 1074576 1251474 1429146 1607707 52 9 901271 1077518 1254428 1432115 1610691 51 10 904204 1080461 1257383 1435084 1613675 50 11 907137 1083404 1260338 1438053 1616660 49 12 910070 1086347 1263293 1441022 1619645 48 13 913003 1089291 1266249 1443992 1622630 47 14 915936 1092234 1269205 1446961 1625615 46 15 918870 1095178 1272161 1449931 1627601 45 16 921804 1098122 1275117 1452901 1631587 44 17 924738 1101066 1278073 1455871 1634573 43 18 927771 1104010 1281029 1458842 1637560 42 19 930605 1106954 1283986 1461813 1640547 41 20 933539 1109899 1286943 1464784 1643534 40 21 936473 1112844 1289900 1467755 1646522 39 22 939407 1115789 1292857 1470727 1649510 38 23 942342 1118734 1295815 1473699 1652499 37 24 945277 1121680 1298773 1476671 1655488 36 25 948212 1124625 1301731 1479644 1658477 35 26 951147 1127571 1304689 1482617 1661466 34 27 954083 1130517 1307648 1485590 1664456 33 28 957019 1133463 1310607 1488563 1667446 32 29 959954 1136409 1313566 1491536 1670436 31 30 962890 1139355 1316525 1494510 1673426 30 31 965826 1142302 1319485 1497484 1676417 29 32 968763 1145249 1322445 1500458 1679408 28 33 971699 1148196 1325405 1503433 1682399 27 34 974636 1151144 1328365 1506408 1685390 26 35 977573 1154092 1331325 1509383 1688382 25 36 980509 1157040 1334285 1512358 1691374 24 37 983446 1159988 1337246 1515334 1694366 23 38 986383 1162936 1340207 1518310 1697358 22 39 989320 1165884 1343168 1521286 1700351 21 40 992257 1168832 1346129 1524262 1703344 20 41 995195 1171781 1349091 1527239 1706337 19 42 998133 1174730 1352053 1530216 1709331 18 43 1001072 1177679 1355015 1533193 1712325 17 44 1004010 1180628 1357977 1536170 1715319 16 45 1006949 1183577 1360940 1539148 1718313 15 46 1009887 1186527 1363903 1542126 1721308 14 47 1012825 1189477 1366866 1545104 1724304 13 48 1015763 1192427 1369830 1548082 1727300 12 49 1018702 1195377 1372793 1551061 1730296 11 50 1021641 1198328 1375757 1554040 1733292 10 51 1024580 1201279 1378721 1557019 1736287 9 52 1027519 1204230 1381686 1559999 1739284 8 53 1030459 1207181 1384650 1562979 1742281 7 54 1033399 1210132 1387615 1565959 1745278 6 55 1036339 1213084 1390580 1568939 1748275 5 56 1039279 1216036 1393545 1571920 1751273 4 57 1042219 1218988 1396510 1574901 1754271 3 58 1045160 1221940 1399476 1577882 1757270 2 59 1048101 1224892 1402442 1580863 1760269 1 60 1051042 1227845 1405408 1583844 1763268 0 84 83 82 81 80 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . 10 11 12 13 14 The minutes of the degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant . 0 1763268 1943803 2125565 2308682 2493280 60 1 1766268 1946822 2128605 2311746 2496370 59 2 1769268 1949841 2131646 2314810 2499411 58 3 1772268 1952861 2134687 2317875 2502552 57 4 1775269 1955881 2137729 2320940 2505643 56 5 1778270 1958901 2140771 2324006 2508735 55 6 1781271 1961922 2143814 2327072 2511827 54 7 1784272 1964943 2146857 2330139 2514920 53 8 1787274 1967964 2149900 2333206 2518013 52 9 1790276 1970985 2152944 2336273 2521106 51 10 1793278 1974007 2155988 2339341 2524200 50 11 1796281 1977029 2159032 2342419 2527294 49 12 1799284 1980052 2162077 2345478 2530389 48 13 1802287 1983075 2165122 2348547 2533484 47 14 1805291 1986098 2168167 2351616 2536580 46 15 1808295 1989122 2171213 2354686 2539676 45 16 1811299 1992146 2174259 2357757 2542773 44 17 1814303 1995171 2177306 2360828 2545870 43 18 1817308 1998196 2180352 2363899 2548968 42 19 1820313 2001221 2183400 2366971 2552066 41 20 1823318 2004247 2186448 2370043 2555165 40 21 1826324 2007273 2189496 2373116 2558264 39 22 1829329 2010299 2192544 2376189 2561364 38 23 1832335 2013326 2192544 2379263 2564464 37 24 1835342 2016353 2195593 2382337 2567564 36 25 1838349 2019380 2201692 2385411 2570665 35 26 1841357 2022408 2204742 2388486 2573766 34 27 1844365 2025436 2207792 2391561 2576868 33 28 1847373 2028464 2210843 2394636 2579970 32 29 1850382 2031493 2213894 2397712 2583073 31 30 1853301 2034522 2216946 2400788 2586176 30 31 1856400 2037552 2219998 2403865 2589280 29 32 1859409 2040582 2223051 2406942 2592384 28 33 1862419 2043612 2226104 2410020 2595489 27 34 1865429 2046643 2229157 2413098 2598594 26 35 1868439 2049674 2232211 2416176 2601700 25 36 1871449 2052705 2235265 2419255 2604806 24 37 1874460 2055737 2238319 2422334 2607912 23 38 1877471 2058769 2241374 2425414 2611019 22 39 1880482 2061801 2244429 2428494 2614126 21 40 1883494 2064834 2247485 2431574 2617234 20 41 1886506 2067867 2250541 2434655 2620342 19 42 1889516 2070900 2253597 2437736 2623451 18 43 1892531 2073934 2256654 2440818 2626560 17 44 1895544 2076968 2259711 2443900 2629670 16 45 1898558 2080002 2262769 2446983 2632780 15 46 1901572 2083037 2265827 2450066 2635891 14 47 1904586 2086073 2268885 2453150 2639002 13 48 1907601 2089109 2271944 2456234 2642114 12 49 1910616 2092145 2275003 2459319 2645226 11 50 1913632 2095182 2278063 2462404 2648339 10 51 1916648 2098219 2281123 2465490 2651452 9 52 1999664 2101256 2284183 2468576 2654566 8 53 1922680 2104293 2287244 2471662 2657680 7 54 1925697 2107331 2290305 2474749 2660795 6 55 1928714 2110369 2293367 2477836 2663910 5 56 1931731 2113407 2296429 2480924 2667026 4 57 1934749 2116446 2299492 2484012 2670142 3 58 1937767 2119485 2302555 2487101 2673258 2 59 1940785 2122525 2305618 2490191 2676375 1 60 1943803 2125565 2308682 2493280 2679492 0 79 78 77 76 75 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . 15 16 17 18 19 The minutes of the degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant . 0 2679492 2867453 3057307 3249197 3443276 60 1 2662610 2870601 3060487 3252413 3446530 59 2 2685728 2873749 3063669 3255630 3449785 58 3 2688847 2876898 3066851 3258848 3453040 57 4 2691966 2880048 3070034 3262066 3456296 56 5 2695086 2883198 3073218 3265285 3459553 55 6 2698206 2886349 3076402 3268504 3462810 54 7 2701327 2889501 3079587 3271724 3466068 53 8 ●704448 2892653 3082772 3274944 3469326 52 9 2707570 2895806 3085958 3278165 3472585 51 10 2710693 2898960 3085144 3281387 3475845 50 11 2713816 2902114 3092331 3284609 3479105 49 12 2716940 2905268 3095518 3287832 3482366 48 13 2720064 2908423 3198706 3291055 3485628 47 14 2723189 2911578 3101895 3294280 3488891 46 15 2726314 2914734 3105084 3297505 3492154 45 16 2729439 2917890 3108274 3300731 3495418 44 17 2732565 2921047 3111464 3303957 3498683 43 18 2735691 2924204 3114655 3307184 3501949 42 19 2738818 2927362 3117846 3310411 3505215 41 20 2741945 2930520 3121038 3313639 3508482 40 21 2745073 2933679 3124230 3316868 3511749 39 22 2748201 2936839 3127423 3320097 3515017 38 23 2751330 2939999 3130617 3323327 3518286 37 24 2754459 2943160 3133811 3326558 3521555 36 25 2757589 2946321 3137006 3329789 3524825 35 26 2760729 2946483 3140201 3333020 3528096 34 27 2763850 2952645 3143397 3336252 3531368 33 28 2766981 2955808 3146594 3339485 3534640 32 29 2770113 2958971 3149791 3342719 3537913 31 30 2773245 2962135 3152989 3345953 3541186 30 31 2776378 2965299 3156187 3349188 3544460 29 32 2779511 2968464 3159386 3352423 3547735 28 33 2782645 2971629 3162585 3355659 3515010 27 34 2785779 2974795 3165785 3358896 3554286 26 35 2788914 2977962 3168986 3362133 3557563 25 36 2792050 2981129 3172187 3365371 3560840 24 37 2795186 2984297 3175389 3368610 3564118 23 38 2798323 2987465 3178591 3371850 3567397 22 39 2801460 2990634 3181794 3375090 3570676 21 40 2804597 2993804 3184998 3378331 3573956 20 41 2807735 2996973 3188202 3381572 3577237 19 42 2810873 3000143 3191407 3384814 3580519 18 43 2814012 3003314 3194613 3388057 3583801 17 44 2817151 3006486 3197819 3391300 3587084 16 45 2820291 3009658 3201026 3394544 3590367 15 46 2823432 3012831 3204233 3397798 3593651 14 47 2826573 3016004 3207441 3401033 3596936 13 48 2829714 3019178 3210649 3404279 3600221 12 49 2832856 3022353 3213858 3407525 3603507 11 50 2835999 3025528 3217067 3410772 3606794 10 51 2839142 3028703 3220277 3414020 3610082 9 52 2842286 3031879 3223488 3417268 3613370 8 53 2845430 3035055 3226699 3420517 3616659 7 54 2848575 3038232 3229911 3423766 3619949 6 55 2851720 3041410 3233124 3427016 3623239 5 56 2854866 3044588 3236337 3430267 3626530 4 57 2858012 3047767 3239551 3433518 3629822 3 58 2861159 3050946 3242766 3436770 3633115 2 59 2864306 3054126 3245981 3440023 3636408 1 60 2867453 3057307 3249197 3443276 3639702 0 74 73 72 71 70 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . 20 21 22 23 24 The minutes of the degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant . 0 3639702 3838640 4040262 4244748 4452286 60 1 3642997 3841978 4043647 4248182 4455772 59 2 3646293 3845316 4047031 4251617 4459259 58 3 3649589 3848655 4050416 4255052 4462747 57 4 3652886 3851995 4053802 4258488 4466236 56 5 3656183 3855336 4057189 4261925 4469726 55 6 3659481 3858678 4060577 4265363 4473216 54 7 3662780 3862020 4063966 4268801 4476707 53 8 3666079 3865363 4067356 4265363 4480199 52 9 3669379 3868707 4070747 4268801 4483692 51 10 3672680 3872052 4074139 4272240 4487186 50 11 3675982 3875397 4077531 4275680 4490681 49 12 3679284 3878743 4080924 4279121 4494177 48 13 3682587 3882090 4084318 4282563 4497674 47 14 3685891 3885438 4087713 4286006 4501172 46 15 3689195 3888787 4091109 4289450 4504671 45 16 3692500 3892136 4094506 4292895 4508171 44 17 3695806 3895486 4097903 4296340 4511672 43 18 3699113 3898837 4101301 4299786 4515173 42 19 3702420 3902188 4104699 4303233 4518675 41 20 3705728 3905540 4108097 4306681 4522178 40 21 3709037 3908893 4111497 4310130 4525682 39 22 3712347 3912247 4114898 4313580 4529187 38 23 3715657 3915601 4118300 4317031 4532693 37 24 3718968 3918956 4121703 4327387 4536200 36 25 3722279 3922312 4125107 4330841 4539708 35 26 3725591 3925669 4128511 4334296 4543217 34 27 3728904 3929027 4131916 4337752 4546727 33 28 3732218 3932385 4135322 4341209 4550238 32 29 3735533 3935744 4138728 4344666 4553750 31 30 3738848 3939104 4142135 4348124 4557264 30 31 3742164 3942465 4145544 4351583 4560778 29 32 3745480 3945826 4148953 4355043 4564293 28 33 3748797 3949188 4152363 4358504 4567809 27 34 3752115 3952551 4155773 4361966 4571326 26 35 3755434 3955915 4159184 4365429 4574843 25 36 3758753 3959280 4162596 4368893 4578361 24 37 3762073 3962646 4166009 4372357 4581880 23 38 3765394 3966012 4169423 4375822 4585400 22 39 3768716 3969379 4172838 4379288 4588921 21 40 3772038 3972746 4176255 4382755 4592443 20 41 3775361 3976114 4179672 4386223 4595966 19 42 3778685 3979483 4183090 4389692 4599490 18 43 3782010 3982853 4186509 4393162 4603015 17 44 3785335 3986224 4189928 4396633 4606541 16 45 3788661 3989596 4193348 4400105 4610068 15 46 3791988 3992969 4196769 4403578 4613596 14 47 3795315 3996342 4200191 4407051 4617125 13 48 3798643 3999716 4203613 4410525 4620654 12 49 3801972 4003090 4207036 4414000 4624184 11 50 3805302 4006465 4210460 4417476 4627715 10 51 3808632 4009841 4213885 4420953 4631247 9 52 3811962 4013217 4217311 4424432 4634780 8 53 3815295 4016594 4220738 4427910 4638314 7 54 3818628 4019972 4224165 4431390 4641849 6 55 3821961 4023351 422759● 4434871 4645385 5 56 3825295 4026731 4231022 4438352 4648922 4 57 3828630 4030112 4234452 4441834 4652460 3 58 3831966 4033494 4237883 4445317 4655999 2 59 3835303 4036877 4241315 4448801 4659540 1 60 3838640 4040262 4244748 4452286 4663081 0 69 68 67 66 65 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . 25 26 27 28 29 The minutes of the degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant . 0 4663081 4877328 5095254 5317094 5543090 60 1 4666623 4880930 5098919 5320826 5546893 59 2 4670166 4884533 5102585 5324559 5550697 58 3 4673710 4888137 5106252 5328293 5554503 57 4 4677255 4891742 5109920 5332028 5558310 56 5 4680801 4895347 5113589 5335765 5562118 55 6 4684348 4898953 5117259 5339503 5565927 54 7 4687896 4902560 5120930 5343242 5569738 53 8 4691444 4906168 5124602 5346982 5573550 52 9 4694993 4909777 5128275 5350723 5577363 51 10 4698543 4913387 5131949 5354465 5581177 50 11 4702094 4916998 5135625 5358209 5584993 49 12 4705646 4920610 5139302 5361954 5588810 48 13 4709199 4924223 5142980 5365700 5592628 47 14 4712753 4927838 5146659 5369447 5596447 46 15 4716308 4931454 5150339 5373195 5600268 45 16 4719864 4935071 5154020 5376944 5604090 44 17 4723422 4938689 5157702 5380694 5607913 43 18 4726981 4942308 5161385 5384445 5611737 42 19 4730541 4945928 5165069 5388198 5615562 41 20 4734102 4949549 5168755 5391952 5619388 40 21 4737664 4953171 5172442 5395707 5623216 39 22 4741227 4956794 5176130 5399463 5627045 38 23 4744790 4960418 5179819 5403221 5630875 37 24 4748354 4964043 5183509 5406980 5634707 36 25 4751919 4967669 5187200 5410740 5638540 35 26 4755485 4971296 5190892 5414501 5642374 34 27 4759052 4974924 5194585 5418263 5646210 33 28 4762620 4978553 5198279 5422026 5650047 32 29 4766189 4982184 5201974 5425791 5653885 31 30 4769759 4985816 5205670 5429557 5657725 30 31 4773330 4989448 5209368 5433324 5661566 29 32 4776902 4993081 5213067 5437092 5665408 28 33 4780475 4996716 5216767 5440861 5669251 27 34 4784049 5000352 5220468 5444632 5673096 26 35 4787624 5003989 5224170 5448404 5676942 25 36 4791200 5007627 5227873 5452177 5680789 24 37 4794777 5011266 5231577 5455951 5684637 23 38 4798355 5014906 5235283 5459726 5688486 22 39 4801934 5018547 5238990 5463503 5692337 21 40 4805515 5022189 5242698 5467281 5696189 20 41 4809096 5025832 5246407 5471060 5700043 19 42 4812678 5029476 5250117 5474840 5703898 18 43 4816261 5033121 5253828 5478621 5707754 17 44 4819845 5036767 5257540 5482404 5711611 16 45 4823430 5040414 5261254 5486188 5715469 15 46 4827016 5044062 5264969 5489973 5719329 14 47 4830603 5047712 5268685 5493759 5723190 13 48 4834191 5051363 5272402 5497546 5727052 12 49 4837780 5055015 5276120 5501335 5730916 11 50 4841371 5058668 5279839 5505125 5734781 10 51 4844962 5062322 5283959 5508916 5738647 9 52 4848554 5065977 5287280 5512708 5742515 8 53 4852147 5069633 5291003 5516501 5746384 7 54 4855741 5073290 5294727 5520296 5750254 6 55 4859336 5076948 5298452 5524092 5754125 5 56 4862932 5080607 5302178 5527889 5757998 4 57 4866529 5084267 5305905 5531687 5761872 3 58 4870127 5087928 5309633 5535487 5765747 2 59 4873727 5091590 5313363 5539288 5769624 1 60 4877328 5095254 5317094 5543090 5773502 0 64 63 62 61 60 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . 30 31 32 33 34 The minutes of the degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant . 0 5773502 6008606 6248693 6494076 6745085 60 1 5777381 6012566 6252738 6498212 6749318 59 2 5781262 6016528 6256785 6502350 6753553 58 3 5785144 6020491 6260834 6506489 6757789 57 4 5789027 6024455 6264884 6510630 6762027 56 5 5792911 6028420 6268935 6514773 9766267 55 6 5796797 6032387 6272988 6518917 6770508 54 7 5800684 6036355 6277042 6523063 6774751 53 8 5804572 6040324 6281098 6527200 6778996 52 9 5808462 6044295 6285155 6531359 6783243 51 10 5812353 6048267 6289214 6535510 6787491 50 11 5816245 6052241 6293274 6539662 6791741 49 12 5820139 6056216 6297336 6543816 6795993 48 13 5824034 6060193 6301399 6547971 6800246 47 14 5827930 6064171 6305464 6552128 6804501 46 15 5831828 6068150 6309530 6556287 6808758 45 16 5835727 6072131 6313598 6560447 6813016 44 17 5839627 6076113 6317667 6564609 6817276 43 18 5843528 6080096 6321738 6568772 6821538 42 19 5847431 6084081 6325810 6572937 6825801 41 20 5851335 6088067 6329883 6577103 6830066 40 21 5855241 6092055 6333958 6581271 6834333 39 22 5859148 6096044 6338034 6585440 6838602 38 23 5863056 6100035 6342112 6589611 6842872 37 24 5866966 6104027 6346191 6593784 6847144 36 25 5870877 6108020 6350272 6597958 6851417 35 26 5874489 6112015 6354355 6602134 6855692 34 27 5878702 6116011 6358439 6606312 6859969 33 28 5882617 6120009 6362525 6610491 6864247 32 29 5886533 6124008 6366613 6614672 6868527 31 30 5890450 6128008 6370702 6618855 6872809 30 31 5894369 6132010 6374792 6623039 6877093 29 32 5898289 6136013 6378884 6627225 6881379 28 33 5902211 6140018 6382977 6631413 6885666 27 34 5906134 6144024 6387072 6635603 6889955 26 35 5910058 6148032 6391169 6639792 6894246 25 36 5913984 6152041 6395267 6643984 6898539 24 37 5917911 6156052 6399366 6648178 6902833 23 38 5921839 6160064 6403467 6652373 6907129 22 39 5925769 6164077 6407569 6656570 6911426 21 40 5929700 6168092 6411673 6660768 6915725 20 41 5933633 6172108 6415779 6664968 6920026 19 42 5937567 6176126 6419886 6669170 6924329 18 43 5941502 6180147 6423995 6673373 6928634 17 44 5945438 6184168 6428105 6677578 6932940 16 45 5949376 6188190 6432216 6681785 6937248 15 46 5955315 6192213 6436329 6685994 6941558 14 47 5957255 6196237 6440444 6690204 6945869 13 48 5961197 6200263 6444560 6694416 6950182 12 49 5965140 6204290 6458678 6698630 6954497 11 50 5969084 6208319 6452798 6702845 6958813 10 51 5973030 6212350 6456919 6707062 6963131 9 52 5976776 6216382 6461042 6711281 6967451 8 53 5980926 6220416 6465166 6715501 6971773 7 54 5984876 6224451 6469292 6719723 6976097 6 55 5988827 6228488 6473419 6723946 6980423 5 56 5992780 6232526 6477548 6728171 6984750 4 57 5996734 6246566 6481678 6732397 6989079 3 58 6000690 6240607 6485809 6736625 6993409 2 59 6004647 6244649 6489942 6740854 6997741 1 60 6008606 6248693 6494076 6745085 7002075 0 95 58 57 56 55 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadarnt . 35 36 37 38 39 The minutes of the degrees of the Quadrant for the Tangents of the complement of the Arches of the same Quadrant . 0 7002075 7265424 7535541 7812856 8097840 60 1 7006411 7269869 7540103 7817542 8102658 59 2 7010749 7274316 7544667 7822230 8107478 58 3 7015088 7278765 7549233 7826920 8112300 57 4 7019429 7283216 7553801 7831612 8117124 56 5 7023772 7287669 7558371 7836306 8121951 55 6 7028117 7292124 7562943 7841002 8126780 54 7 7032463 7296581 7567517 7845700 8131611 53 8 7036811 7301040 7572093 7850400 8136444 52 9 7041161 7305501 7576670 7855102 8141280 51 10 7045513 7309563 7581249 7859807 8146118 50 11 7049867 7314427 7585830 7864514 8150958 49 12 7054223 7318893 7590413 7869223 8155801 48 13 7058581 7323361 7594999 7873934 8160646 47 14 7062940 7327831 7599587 7872647 8165493 46 15 7067301 7332303 7604177 7883363 8170343 45 16 7071664 7336777 7608769 7888081 8175195 44 17 7076029 7341253 7613363 7892801 8180049 43 18 7070395 7345731 7617959 7897523 8184905 42 19 7084763 7350210 7622557 7902247 8189764 41 20 7089133 7354691 7627157 7906973 8194625 40 21 7093505 7359174 7631759 7911702 8199488 39 22 7097879 7363659 7636363 7916433 8204354 38 23 7102254 7368146 7640969 7921166 8209222 37 24 7106631 7372635 7645577 7925901 8214092 36 25 7111010 7377126 7650187 7930638 8218965 35 26 7115391 7381619 7654799 7935378 8223840 34 27 7119773 7386114 7659413 7940120 8228717 33 28 7124167 7390611 7664030 7944864 8233597 32 29 7128543 7395110 7668649 7949610 8238479 31 30 7132931 7399610 7673270 7954358 8243363 30 31 7137321 7404112 7677893 7959109 8248250 29 32 7141713 7408616 7682518 7963862 8253139 28 33 7146106 7413122 7687145 7968617 8258031 27 34 7150501 7417630 7691774 7973374 8262925 26 35 7154898 7422140 7696405 7978133 8267821 25 36 7159297 7426652 7701038 7982895 8272720 24 37 7163698 7431167 7705673 7987659 8277621 23 38 7168100 7435684 7710310 7992425 8282524 22 39 7172504 7440203 7714949 7997193 8287429 21 40 7176910 7444724 7719590 8001963 8292337 20 41 7181318 7449246 7724233 8006736 8297247 19 42 7185728 7453770 7728878 8011511 8302160 18 43 7190140 7458296 7733525 8016288 8307075 17 44 7194554 7462824 7738175 8021067 8311992 16 45 7198970 7476354 7742827 8025849 8316912 15 46 7203387 74718●6 7747481 8030633 8321834 14 47 7207806 7476420 7752137 8035419 8326759 13 48 7212227 7480956 7756795 8040207 8331686 12 49 7216650 7485494 7761455 8044997 8336615 11 50 7221075 7490033 7766117 8049790 8341547 10 51 7225502 7494574 7770781 8054585 8346481 9 52 7229931 7499117 7775447 8059382 8351418 8 53 7234362 7503663 7780116 8064181 8356357 7 54 7238794 7508211 7784787 8068983 8361298 6 55 7243228 7512761 7789460 8073787 8366242 5 56 7247664 7517313 7794135 8078593 8371188 4 57 7252102 7521867 7798812 8083401 8376136 3 58 7256541 7526423 7803491 8088212 8381087 2 59 7260982 7530981 7808172 8093025 8386040 1 60 7265424 7535541 7812856 8097840 8390996 0 54 53 52 51 50 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for Tangents of the Arches of the same Quadrant . 40 41 42 43 44 The minutes of the degrees of the Quadrant for the Tangents of the complement of the Arches of the same Quadrant . 0 8390996 8692867 9004040 9325151 9656888 60 1 8395954 8697975 9009308 9330591 9662511 59 2 8400915 8703085 9014579 9336034 9668137 58 3 8405878 8708198 901985● 9341480 9673766 57 4 8410844 8713344 9025130 9346929 9679398 56 5 8415812 8718433 9030410 9352381 9685034 55 6 8420782 8723555 9035693 9357835 9690674 54 7 8425754 8728679 9040978 9363292 9696315 53 8 8430729 8733806 9046266 9368752 9701960 52 9 8435706 8738935 9051557 9374215 9707609 51 10 8440686 8744067 9056850 9379682 9713261 50 11 8445668 8749201 9062146 9385152 9718916 49 12 8450653 8754338 9067445 9390625 9724574 48 13 8455640 8759478 9072747 9396101 9730235 47 14 8460630 8764620 9078052 9401580 9735900 46 15 8465622 8769764 9083360 9407062 9741568 45 16 8470617 8774911 9088670 9412547 9747239 44 17 8475614 8780061 9093983 9418034 9752913 43 18 8480614 8785214 9099299 9423524 9758591 42 19 8485617 8790369 9104618 9429017 9764272 41 20 8490622 8795527 9109940 9434513 9769956 40 21 8495629 8800688 9115265 9440012 9775643 39 22 8500639 8805851 9120593 9445514 9781334 38 23 8505651 8811017 9125923 9451019 9787028 37 24 8510666 8816186 9131256 9456528 9792725 36 25 8515683 8821357 9136592 9462040 9798425 35 26 8520703 8826531 9141930 9467555 9804128 34 27 8525725 8831708 9147271 9473072 9809835 33 28 8530750 8836887 9152615 9478594 9815545 32 29 8535777 8842069 9157962 9484118 9821258 31 30 8540806 8847253 9163312 9489645 9826974 30 31 8545838 8852440 9168665 9495175 9832694 29 32 8550872 8857630 9174021 9400708 9838417 28 33 8555909 8862822 9179380 9506244 9844143 27 34 8560949 8868017 9184741 9511783 9849872 26 35 8565991 8873015 9190105 9517325 9855605 25 36 8571036 8878415 9195472 9522870 9861341 24 37 8576083 8883628 9200842 9528419 9867180 23 38 8581133 8888824 9206215 9533971 9872922 22 39 8586185 8899033 9211590 9539526 9878668 21 40 8591239 8899244 9216968 9545084 9884317 20 41 8596296 8904458 9222349 9550645 9890070 19 42 8601355 8909675 9227733 9556209 9895826 18 43 8606417 8914894 9233120 9561776 9901585 17 44 8611482 8920116 9238510 9567346 9907347 16 45 8616549 8925341 9243903 9572919 9913113 15 46 8621619 8930568 9249299 9578495 9918882 14 47 8626692 8935798 9254698 9584074 9924654 13 48 8631767 8941031 9260100 9589656 9930430 12 49 8636845 8946267 9265505 9595241 9936209 11 50 8641926 8951506 9270913 9600830 9941991 10 51 8647009 8956747 9276324 9606422 9947777 9 52 8652095 8961991 9281738 9612017 9953566 8 53 8657683 8967238 9287155 9617615 9959359 7 54 8662273 8972487 9292574 9623216 9965155 6 55 8667366 8977739 9297996 9628820 9970954 5 56 8672461 8982994 9303421 9634427 9976756 4 57 8677559 8988252 9308849 9640037 9982562 3 58 8682659 8993512 9314280 9645651 9988371 2 59 8687762 8998775 9319714 9651268 9994184 1 60 8692867 9004040 9325151 9656888 10000000 0 49 48 47 46 45 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for Tangents of the Arches of the same Quadrant . 45 46 47 48 The minutes of the degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant . 0 10000000 10355302 10723686 11106124 60 1 10005820 10367332 10729942 11112623 59 2 10011643 10367365 10736202 11119126 58 3 10017469 10373402 10742466 11125634 57 4 10023299 10379443 10748734 11132146 56 5 10029132 10385487 10755006 11138662 55 6 10034968 10391535 10761282 11145182 54 7 10040808 10397587 10767562 11151706 53 8 10046651 10403643 10773845 11158235 52 9 10052497 10409702 10780132 11164768 51 10 10058347 10415765 10786423 11171305 50 11 10064201 10421832 10792718 11177846 49 12 10070058 10427902 10799017 11184392 48 13 10075918 10433976 10805320 11190942 47 14 10081782 10340054 10811627 11197496 46 15 10087649 10446135 10817938 11204054 45 16 10093520 10452220 10824253 11210617 44 17 10099394 10458309 10830572 11217184 43 18 10105272 10464401 10836895 11223755 42 19 10111153 10470497 10843222 11230330 41 20 10117038 10476597 10849554 11236910 40 21 10122926 10482701 10855889 11243494 39 22 10128818 10488808 10862228 11250082 38 23 10134713 10494919 10868571 11256675 37 24 10140611 10501034 10874918 11263272 36 25 10146513 10507153 10881269 11269873 35 26 10152418 10513275 10887624 11276478 34 27 10158327 10519401 10893983 11283088 33 28 10164239 10525531 10900346 11289702 32 29 10170154 10531664 10906713 11296321 31 30 10176073 10537801 10913084 11302944 30 31 10181996 10543942 10919459 11309571 29 32 10187922 10550087 10925838 11316203 28 33 10193852 10556235 10932221 11322899 27 34 10199785 10562387 10938608 11329480 26 35 10205722 10568543 10945000 11336125 25 36 10211663 10574703 10951396 11342774 24 37 10217607 10580867 10957796 11349428 23 38 10223555 10587034 10964200 11356086 22 39 10229506 10593205 10970608 11362748 21 40 10235460 10599280 10977020 11369415 20 41 10241418 10605559 10983436 11376086 19 42 10247380 10611742 10989856 11382762 18 43 10253345 10617929 10996280 11389442 17 44 10259314 10624119 11002708 11396126 16 45 10265286 10630313 11009140 11402815 15 46 10271262 10636511 11015577 11409508 14 47 10277242 10642713 11022028 11416206 13 48 10283225 10648919 11028463 11422908 12 49 10289212 10655128 11034912 11429615 11 50 10295202 10661341 11041365 11436326 10 51 10301196 10667558 11047832 11443042 9 52 10307193 10673779 11054283 11449762 8 53 10313194 10680004 11060748 11456487 7 54 10319199 10686233 11067218 11463216 6 55 10325207 10692466 11073692 11469950 5 56 10331219 10698702 11080170 11476688 4 57 10337234 10704942 11086652 11483431 3 58 10343253 10711186 11093138 11490178 2 59 10349276 10717434 11099629 11496929 1 60 10355302 10723686 11106124 11503684 0 44 43 42 41 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadarnt . 49 50 51 52 The minutes of the degrees of the Quadrant for the Tangents of the complement of the Arches of the same Quadrant . 0 11503684 11917537 12348972 12799416 60 1 11510444 11924580 12356320 12807093 59 2 11517208 11931628 12363673 12814776 58 3 11523977 11938680 12371032 12822465 57 4 11530751 11945737 12378394 12830159 56 5 11537529 11952799 12385762 12837859 55 6 11544312 11959866 12393136 12845565 54 7 11551100 11966938 12400515 12853277 53 8 11557893 11974015 12407999 12860994 52 9 11564691 11981097 12415288 12868717 51 10 11571494 11988183 12422683 12876445 50 11 11578301 11995274 12430083 12884179 49 12 11585112 12002370 12437489 12891919 48 13 11591928 12009471 12444900 12899665 47 14 11598748 12016578 12452317 12907417 46 15 11605572 12023690 12459739 12915175 45 16 11612401 12030807 12467167 12922939 44 17 11619234 12037929 12474600 12930709 43 18 11626072 12045056 12482039 12938485 42 19 11632915 12052188 12489484 12946267 41 20 11639763 12059325 12496934 12954055 40 21 11646615 12066467 12504389 12961848 39 22 11653472 12073614 12511850 12969647 38 23 11660334 12080766 12519316 12977457 37 24 11667200 12087923 12526787 12985263 36 25 11674071 12095085 12534264 12993080 35 26 11680947 12102252 12541746 13000903 34 27 11687827 12109424 12549233 13008732 33 28 11694712 12116601 12556725 13016567 32 29 11701602 12123783 12564222 13024407 31 30 11708497 12130970 12571724 13032253 30 31 11715396 12138162 12579232 13040105 29 32 11722300 12145359 12586746 13047963 28 33 11729208 12152561 12594265 13055827 27 34 11736121 12159768 12601790 13063697 26 35 11743039 12166981 12609321 13071573 25 36 11749962 12174199 12616858 13079455 24 37 11756989 12181422 12624400 13087343 23 38 11763821 12188650 12631948 13095237 22 39 11770758 12195883 12639501 13103138 21 40 11777700 12203121 12647060 13111045 20 41 11784646 12210364 12654624 13118958 19 42 11791597 12217613 12662194 13126877 18 43 11798553 12224867 12669769 13134802 17 44 11805514 12232126 12677350 13142732 16 45 11812479 12239390 12684937 13150668 15 46 11819449 12246659 12692530 13158610 14 47 11826424 12253933 12700128 13166558 13 48 11833404 12261212 12707732 13174512 12 49 11840388 12268496 12715341 13182472 11 50 11847377 12275786 12722956 13190438 10 51 11854371 12283081 12730577 13198411 9 52 11861370 12290381 12738203 13206390 8 53 11868374 12297687 12745835 13214375 7 54 11875383 12304998 12753473 13222367 6 55 11882397 12312314 12761116 13230365 5 56 11889417 12319635 12768765 13238369 4 57 11896438 12326961 12776420 13246379 3 58 11903466 12334293 12784080 13254396 2 59 11910499 12341630 12791745 13262419 1 60 11917537 12348972 12799416 13270448 0 40 39 38 38 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadarnt . 53 54 55 56 The minutes of the degrees of the Quadrant for the Tangents of the complement of the Arches of the same Quadrant . 0 13270448 13763820 14281480 14825610 60 1 13278483 13772243 14290325 14834916 59 2 13286524 13780673 14299177 14844230 58 3 13294571 13789109 14308037 14853553 57 4 13302624 13797552 14316905 14862884 56 5 13310683 13806002 14325780 14872223 55 6 13318749 13814459 14334662 14881570 54 7 13326821 13822922 14343552 14890925 53 8 13334899 13831392 14352451 14909288 52 9 13342984 13839869 14361354 14909659 51 10 13351075 13848352 14370266 14919038 50 11 13359172 13856842 14379186 14928426 49 12 13367276 13865339 14388113 14937822 48 13 13375386 13873843 14397048 14947226 47 14 13383502 13882354 14405990 14956638 46 15 13391624 13890872 14414939 14966058 45 16 13399753 13899397 14423896 14975486 44 17 13407888 13907930 14432861 14984923 43 18 13416029 13916470 14441833 14994368 42 19 13424177 13925017 14450812 15003821 41 20 13432331 13933571 14459799 15013283 40 21 13440492 13942131 14468794 15022753 39 22 13448659 13950698 14477797 15032231 38 23 13456832 13959272 14486807 15041717 37 24 13465011 13967853 14495825 15051211 36 25 13473197 13976441 14504850 15060714 35 26 13481390 13985035 14513883 15070225 34 27 13489589 13993636 14522924 15079744 33 28 13497794 14002244 14531972 15089271 32 29 13506006 14010859 14541028 15078807 31 30 13514224 14019481 14550091 15108351 30 31 13522449 14028110 14559162 15117903 29 32 13530680 14036746 14568241 15127464 28 33 13538918 14045389 14577327 15137034 27 34 13547162 14054040 14586421 15146612 26 35 13555413 14062698 14595523 15156199 25 36 13563670 14071363 14604633 15165794 24 37 13571834 14080035 14613750 15175398 23 38 13580104 14088715 14622875 15185011 22 39 13588381 14097402 14632007 15194632 21 40 13596764 14106097 14641146 15204261 20 41 13605054 14114798 14650293 15213899 19 42 13613350 14123506 14659449 15223545 18 43 13621653 14132221 14668613 15233200 17 44 13629963 14140923 14677785 15242863 16 45 13638279 14149672 14686965 15252535 15 46 13646602 14158409 14696153 15262216 14 47 13654932 14167153 14705349 15271905 13 48 13663268 14175904 14714553 15281603 12 49 13671610 14184663 14723765 15291309 11 50 13679959 14193429 14732985 15301024 10 51 13688315 14202202 14742212 15310748 9 52 13696677 14210982 14751447 15320481 8 53 13705046 14219769 14760690 15330222 7 54 13713422 14228563 14769941 15339972 6 55 13721805 14237365 14779200 15349730 5 56 13730194 14246174 14788466 15349497 4 57 13738590 14254990 14797740 15369273 3 58 13746993 14263813 14807022 15379057 2 59 13755403 14272643 14816312 15388850 1 60 13763820 14281480 14825610 15398651 0 36 35 34 33 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . 57 58 59 60 The minutes of the degrees of the Quadrant for the Tangents of the complement of the Arches of the same Quadrant . 0 15398651 16003347 16642794 17320508 60 1 15408461 16013710 16653766 17332150 59 2 15418280 16024083 16664749 17343804 58 3 15428108 16034466 16675742 17355469 57 4 15437945 16044859 16686746 17367146 56 5 15447791 16055261 16697760 17378834 55 6 15457646 16065673 16708785 17390534 54 7 15467510 16076095 16719820 17402246 53 8 15477382 16086527 16730866 17413969 52 9 15487263 16096968 16741922 17425704 51 10 15497153 16107419 16752989 17437451 50 11 15507052 16117880 16764067 17449210 49 12 15516960 16128351 16775156 17460981 48 13 15526877 16138832 16786256 17472764 47 14 15536803 16149322 16797367 17484559 46 15 15546738 16159822 16808489 17496366 45 16 15556682 16170332 16819621 17508185 44 17 15566636 16180852 16830764 17520026 43 18 15576599 16191381 16841918 17531869 42 19 15586571 16201920 16853083 17543724 41 20 15596552 16212469 16864259 17555591 40 21 15606542 16224028 16875446 17567470 39 22 15616541 16233597 16886644 17579362 38 23 15626549 16244176 16897853 17591266 37 24 15636566 16254766 16909074 17603182 36 25 15646592 16265366 16920306 17615111 35 26 15656627 16275976 16931549 17627052 34 27 15666671 16286596 16942803 17639006 33 28 15676724 16297226 16954068 17650972 32 29 15686786 16307866 16965344 17662951 31 30 15696857 16318516 16976631 17674941 30 31 ●5706938 16329176 16987929 17686945 29 32 15717028 16339847 16999239 17698960 28 33 1572127 16350528 17010560 17710987 27 34 15737235 16361219 17021892 17723027 26 35 15747353 16371920 17033236 17735079 25 36 15757480 16382631 17044591 17747143 24 37 15767616 16393352 17055957 17759220 23 38 15777761 16404083 17067325 17771309 22 39 15787915 16414824 17078714 17783410 21 40 15798078 16425575 17090115 17795524 20 41 15808251 16436337 17101527 17808651 19 42 15818433 16447109 17112950 17819790 18 43 15828625 16457892 17124384 17831942 17 44 15838827 16468685 17135829 17844107 16 45 15849038 16479488 17147285 17856285 15 46 15859259 16490302 17158752 17868475 14 47 15869489 16501126 17170231 17880678 13 48 15879729 16511960 17181721 17892894 12 49 15889979 16522805 17193222 17905123 11 50 15900238 16533660 17204734 17917364 10 51 15910507 16544526 17216258 17929618 9 52 15920785 16555402 17227794 17941885 8 53 15931073 16566289 17239342 17954164 7 54 15941370 16577186 17250902 17966456 6 55 15951676 16588094 17262473 17978761 5 56 15961992 16599013 17274056 17991079 4 57 15972317 16609942 17285651 18003410 3 58 15982651 1662088● 17297258 18015753 2 59 15992994 16631833 17308877 18028109 1 60 16003347 16642794 17320508 18040478 0 32 31 30 29 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . 61 62 63 64 The minutes of the degrees of the Quadrant for the Tangents of the complement of the Arches of the same Quadrant . 0 18040478 18807265 19626104 20503034 60 1 18052860 18820471 19640225 20518180 59 2 18065255 18833691 19654362 20533344 58 3 18077663 18846925 19668516 20548526 57 4 18090084 18860174 19682686 20563726 56 5 18102518 18873437 19696872 20578945 55 6 18114966 18886715 19711074 20594182 54 7 18127427 18900007 19725293 20609437 53 8 18139901 18913314 19739528 20624711 52 9 18152388 18926636 19753780 20640003 51 10 18164889 18939972 19768048 20655313 50 11 18177403 18953323 19782333 20670642 49 12 18189930 18966689 19796634 20685989 48 13 18202470 18980070 19810951 20701355 47 14 18215024 18993466 19825285 20716739 46 15 18227591 19006876 19839635 20732142 45 16 18240171 19020301 19854002 20747564 44 17 18252765 19033741 19868386 20763004 43 18 18265372 19047196 19882786 20778463 42 19 18277992 19060665 19897203 20793941 41 20 18290626 19074149 19911637 20809438 40 21 18303273 19087648 19926088 20824953 39 22 18315934 19101162 19940555 20840487 38 23 18328608 19114691 19955039 20856040 37 24 18341296 19128235 19669540 26871612 36 25 18353997 19141795 19984057 20887202 35 26 18366712 19155370 19998591 20902811 34 27 18379440 19168960 20013142 20918439 33 28 18392182 19182565 20027709 20934086 32 29 18404938 19196185 20042297 20949752 31 30 18417707 19209821 20056898 20965436 30 31 18430490 19223472 20071516 20981140 29 32 18443287 19237138 20086152 20996863 28 33 18456098 19250819 20100805 21012605 27 34 18468922 19264516 20115475 21028367 26 35 18481760 19278228 20130163 21044148 25 36 18494612 19291955 20144868 21059949 24 37 18507478 19305698 20159590 21075769 23 38 18520357 19319456 20174329 21091609 22 39 18533250 19333230 20189086 21107468 21 40 18546157 19347019 20203860 21123347 20 41 18559078 19360824 20218651 21139246 19 42 18572013 19374644 20233460 21155164 18 43 18584962 19388480 20248286 21171102 17 44 18597925 19402331 2026●130 21187059 16 45 18610902 19416198 20277991 21203036 15 46 18623894 19430081 20292870 21219032 14 47 18636900 19443980 20307767 21235048 13 48 18649920 19457894 20322681 21251083 12 49 18662954 19471824 20337613 21267138 11 50 18676002 19485770 20352563 21283213 10 51 18689064 19499732 20367531 21299308 9 52 18702140 19513710 20382516 21315423 8 53 18715231 19527704 20397519 21331558 7 54 18728335 19541714 20412539 21347713 6 55 18741454 19555739 20427577 21363888 5 56 18754587 19569780 20442633 21380083 4 57 18767735 19583837 20457706 21396298 3 58 18780897 19597910 20472797 21412534 2 59 18794074 19611999 20487906 21428790 1 60 18807265 19626104 20503034 21445067 0 28 27 26 25 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . 65 66 67 68 The minutes of the degrees of the Quadrant for the Tangents of the complement of the Arches of the same Quadrant . 0 21445067 22460371 23558529 24750869 60 1 21461364 22477965 23577595 24771613 59 2 21477681 22495582 23596687 24792387 58 3 21494019 22513222 23615805 24813191 57 4 21510377 22530885 23634950 24834024 56 5 21526756 22548571 23654121 24854887 55 6 21543155 22566281 23673318 24875780 54 7 21559575 22584014 23692542 24896704 53 8 21576015 22601771 23711793 24917659 52 9 21592475 22619951 23731071 24938644 51 10 21608956 22637355 23750375 24959659 50 11 21625458 22655183 23769706 24980705 49 12 21641981 22673034 23789064 25001782 48 13 21658525 22690909 23808448 25022890 47 14 21675090 22708808 23827859 25044029 46 15 21691676 22726730 23847297 25065198 45 16 21708283 22744676 23866762 25086398 44 17 21724911 22762646 23886254 25107629 43 18 21741559 22780639 23905773 25128991 42 19 21758228 22798656 23925320 25150183 41 20 21774918 22816696 23944895 25171506 40 21 21791629 22834760 23964496 25192861 39 22 21808362 22852848 23984124 25214248 38 23 21825116 22870960 24003779 25235666 37 24 21841892 22889096 24023462 25257116 36 25 21858689 22907256 24043172 25278597 35 26 21875508 22925441 24062910 25300115 34 27 21892348 22943650 24082675 25321655 33 28 21909210 22961883 24102468 25343232 32 29 21926094 22980141 24122289 25364841 31 30 21944000 22998424 24142137 25386482 30 31 21959926 23016731 24162013 25408154 29 32 21976874 23035062 24181917 25429858 28 33 21993843 23053418 24201849 25451594 27 34 22010834 23071798 24221809 25473362 26 35 22027846 23090203 24241798 25495162 25 36 22044879 23108632 24261815 25516995 24 37 22061934 23127086 24281860 25538860 23 38 22079011 23145565 4301934 25560758 22 39 22096109 23164068 14322037 25582688 21 40 22113229 23182597 24342169 25604651 20 41 22130372 23201151 24362329 25626647 19 42 22147537 23219730 24382518 25648675 18 43 22164725 23238335 24402735 25670736 17 44 22181935 23256965 24422981 25692830 16 45 22199168 23275621 24443256 25714957 15 46 22216424 23294302 24463559 25737218 14 47 22233703 23313008 24483891 25759312 13 48 22251004 23331740 24504252 25781540 12 49 22268328 23350498 24524641 25803801 11 50 22285675 2336928● 24545061 25826096 10 51 22303044 23388092 24565509 25848424 9 52 22320435 23406927 24585986 25870786 8 53 22337848 23425788 24606492 25893181 7 54 22355284 23444674 24627028 25915610 6 55 22372742 23463586 24647594 25938073 5 56 22390223 23482523 24668189 25960569 4 57 22407726 23501486 24688814 25983099 3 58 22425252 23520475 24709469 26005663 2 59 22442800 23539489 24730154 26028261 1 60 22460371 23558529 24750869 26050893 0 24 23 22 21 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . 69 70 71 72 The minutes of the degrees of the Quadrant for the Tangents of the complement of the Arches of the same Quadrant . 0 26050893 27474777 29042105 30776834 60 1 26073559 27499665 29069569 30807323 59 2 26096260 2752459● 29097080 30837866 58 3 26118996 27549559 29124638 30868465 57 4 26141766 27574565 29152243 30890119 56 5 2616457● 27599612 29179895 30929828 55 6 26187411 27624699 29207595 30960593 54 7 26210286 27649827 ●9235343 30991413 53 8 26233196 27674995 29263139 31022289 52 9 26256141 27700204 29290382 31053221 51 10 26279120 27725453 29318873 31084208 50 11 26302135 27750742 29346811 31115252 49 12 26325185 27776072 29374797 31146352 48 13 26348270 27802443 29402831 31177508 47 14 26371390 27826855 29430913 31208720 46 15 26394546 27852308 29459043 31239989 45 16 26417738 27877803 29487221 31271315 44 17 26440966 27903339 29515446 31302698 43 18 26464229 27928917 29543719 31334138 42 19 26487528 27954536 29572041 31365636 41 20 26510863 27980196 29600411 31397191 40 21 26534234 28005898 29628831 31428805 39 22 26557641 28031642 29657301 31460470 38 23 26581084 28057429 29685820 31492205 37 24 26604563 28083258 29714388 31523992 36 25 26628079 28109129 29743006 31555838 35 26 26651631 28135043 29771674 31587742 34 27 26675220 28160999 29800392 31619705 33 28 26698845 28186998 29829160 31651727 32 29 26722507 28213040 29857978 31683807 31 30 26746206 28239125 29886847 31715946 30 31 26769942 28265253 29915765 31748144 29 32 26793716 28291424 29944734 31780401 28 33 26816527 28317638 29973753 31812717 27 34 26841375 28343895 30002823 31845093 26 35 26865260 28379195 30031943 31877528 25 36 26889183 28396539 30061113 31910 24 24 37 26913143 28422926 30090334 31942580 23 38 26937141 28449357 30119605 31975197 22 39 26961177 28475832 30148927 32007875 21 40 26985251 28502350 30178299 32040613 20 41 27009362 28528913 30207723 32073413 19 42 27033511 28555520 30237200 32106275 18 43 27057698 28582172 30266730 32139200 17 44 27081922 28608868 30296312 32172187 16 45 27106184 28635608 30325947 32205237 15 46 27130484 28662393 30355635 32238349 14 47 27154823 28689222 30385375 32271524 13 48 27179200 28716096 30415169 32304762 12 49 27203616 28743015 30445015 32338064 11 50 27228070 28769979 30474915 32371430 10 51 27252563 28796987 30504867 32404858 9 52 27272095 28824040 30534872 32438348 8 53 27301667 28851139 30564930 32471901 7 54 27326278 28878283 30595041 32505517 6 55 27350929 28905472 30625205 32539196 5 56 27375620 28932707 30655423 32572937 4 57 27400350 28959988 30685695 32606741 3 58 27425120 28987315 30716020 32640907 2 59 27449929 29014687 30746400 32674536 1 60 27474777 29042105 30776834 32708528 0 20 19 18 17 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . 73 74 75 76 The minutes of the degrees of the Quadrant for the Tangents of the complement of the Arches of the same Quadrant . 0 32708528 34874151 37320517 40107808 60 1 32745286 34912477 37363987 40157569 59 2 32776709 34950881 37407551 40207446 58 3 32810898 34989364 37451210 40257440 57 4 32845153 35027925 37494964 40307552 56 5 32879747 35066565 37538814 40357781 55 6 32913862 35105283 37582760 40408129 54 7 32948317 35144080 37626803 40458596 53 8 32982839 35182956 37670943 40509183 52 9 33017427 35221911 37715180 40559890 51 10 33052082 35260945 37759515 40610718 50 11 33086802 35300059 37803948 40661665 49 12 33121588 35339253 37848479 40712731 48 13 33156441 35378528 37893109 40763917 47 14 33191362 35417883 37937838 40815224 46 15 33226351 35457320 37982666 40866652 45 16 33261408 35496838 38027592 40918201 44 17 33296534 35536438 38072616 40969871 43 18 33331728 35576121 38117740 41021663 42 19 33366990 35615888 38162963 41073577 41 20 33402321 35655739 38208285 41125614 40 21 33437720 35695672 38253708 41177775 39 22 33473188 35735689 38299232 41230062 38 23 33508725 35775789 38344857 41282475 37 24 33544330 35815973 38390584 41335015 36 25 33580005 35856241 38436414 41387683 35 26 33615750 35896593 38482347 41440480 34 27 33651566 35937029 38528384 41493407 33 28 33687453 35977550 38574525 41546464 32 29 33723410 36018156 38620772 41599653 31 30 33759438 36058848 38667125 41652974 30 31 33795535 36099623 38713580 41706424 29 32 33831703 36140483 38760139 41760003 28 33 33867942 36181427 38806801 41813712 27 34 33904252 36222456 38853567 41867550 26 35 33940634 36263570 38900438 41921518 25 36 33977088 36304771 38947416 41975617 24 37 34013615 36346060 38994501 42029848 23 38 34050215 36387437 39041695 42084211 22 39 34086888 36428903 39088998 42138706 21 40 34123634 36470459 39136409 42193334 20 41 34160453 36512103 39183929 42248096 19 42 34197345 36553836 39231557 42302993 18 43 34234310 36595659 39279294 42358025 17 44 34271348 36637572 39327139 42413193 16 45 34308459 36679574 39375094 42468497 15 46 34345644 36721666 39423158 42523937 14 47 34382903 36763849 39471331 42579514 13 48 34420237 36806121 39519614 42635228 12 49 34457647 36848483 39568006 42691080 11 50 34495132 36890936 39616509 42747070 10 51 34532692 36933479 39665124 42803199 9 52 34570327 36976114 39713852 42859468 8 53 34608038 37018840 39762695 42915878 7 54 34645824 37061659 39811654 42972429 6 55 34683686 37104570 39860729 43029122 5 56 34721625 37147574 39909917 43085958 4 57 34759640 37190670 39959218 43142937 3 58 34797733 37233859 40008633 43200060 2 59 34835903 37277141 40058103 43257328 1 60 34874151 37320517 40107808 43314742 0 16 15 14 13 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . 77 78 79 80 The minutes of degrees of the Quadrant for the Tangents of the complement of the Arches of the same Quadrant 0 43314742 47046295 51445543 56712854 60 1 43372301 47113680 51525561 56809480 59 2 43430006 47181249 51605820 56906425 58 3 43487857 47249003 51686321 57003690 57 4 43545855 47316942 51767065 57101277 56 5 43604000 47385067 51848053 57199188 55 6 43662293 47453380 51929285 57297425 54 7 43720733 47521882 52010762 57395990 53 8 43779321 47590575 52092485 57494885 52 9 43838057 47659460 52174455 57594111 51 10 43896942 47728538 52256673 57693670 50 11 43955977 47797809 52339140 57793564 49 12 44015163 47867274 52421857 57893795 48 13 44074501 47936934 52504826 57994366 47 14 44133992 48006790 52588048 58095279 46 15 44193637 48076841 52671525 58196536 45 16 44253435 48147088 52755259 58298138 44 17 44313387 48217531 52839251 58400087 43 18 44373494 48288171 52923503 58502385 42 19 44433756 48359008 53008016 58605934 41 20 44494174 48430043 53092792 58708035 40 21 44554749 48501278 53177831 58811388 39 22 44615481 48572714 53263134 58915095 38 23 44676371 48644352 53348702 59019157 37 24 44737419 48716193 53434536 59123576 36 25 44798626 48788238 53520637 59228353 35 26 44859993 48860488 53607006 59333490 34 27 44921521 48932945 53693644 59438989 33 28 44983211 49005610 53780552 59544852 32 29 45045065 49078483 53867731 59651081 31 30 45107083 49151565 53955183 59757678 30 31 45169263 49224856 54042909 59864646 29 32 45231607 49298357 54130911 59971987 28 33 45294114 49372069 54219190 60079703 27 34 45356785 49445993 54307748 60187796 26 35 45419621 49520130 54396586 60296268 25 36 45482623 49594481 54485705 60405112 24 37 45545790 49669047 54575107 60514358 23 38 45609123 49743829 54664793 60623981 22 39 45672623 49818827 54754764 60733992 21 40 45736291 49894042 54845022 60844392 20 41 45800128 49969475 54935569 60955184 19 42 45864135 50045127 55029406 61066370 18 43 45928314 50120999 55117535 61177952 17 44 45992666 50197092 55208958 61289930 16 45 46057192 50273407 55300676 61402307 15 46 46121892 50349935 55392692 61515085 14 47 46186767 50246707 55485007 61628267 13 48 46251817 50503605 55577622 61741856 12 49 46318043 50580910 55670539 61855854 11 50 46382445 50658353 55763759 61970263 10 51 46448023 50736025 55857283 62085085 9 52 46513778 50813927 55951112 62200323 8 53 46579711 50892060 56045247 62315979 7 54 46645823 50970425 56139689 62432056 6 55 46712115 51049023 56234439 62548556 5 56 46778587 51127855 56329498 62665481 4 57 46845240 51206922 56424868 62782833 3 58 46912075 51286225 56520550 62900615 2 59 46979093 51365765 56616545 63018829 1 60 47046295 51445543 56712854 63137478 0 12 11 10 9 The degrees of the Quadrant for the Tangent of the complement of the Arches of the same Quadrant .

The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadarnt . 81 82 83 84 The minutes of the degrees of the Quadrant for the Tangents of the complement of the Arches of the same Quadrant . 0 63137478 71153707 81443502 95143611 60 1 63256564 71304198 81639821 95410585 59 2 63376089 71455313 81837074 95679034 58 3 63496056 71607058 82035268 95948971 57 4 63616468 71759440 82234410 96220411 56 5 63737327 71912459 82434508 96493467 55 6 63858635 72066117 82635570 96767939 54 7 63980394 72220422 82837603 97044063 53 8 64102607 72375376 83040614 97321646 52 9 64225276 72530983 83244610 97600890 51 10 64348404 72687247 83449598 97881716 50 11 64471994 72844173 83655585 98164135 49 12 64596049 73001766 83862572 98448162 48 13 64720571 73160031 84070565 98733810 47 14 64845563 73318972 84279571 99021104 46 15 64971028 73478593 84489598 99310047 45 16 65096969 73638898 84700687 99600655 44 17 65223388 73799892 84912817 99893042 43 18 65350287 73961579 85125995 100187022 42 19 65477669 74123964 85340229 100482822 41 20 65605537 74287052 85555525 100780346 40 21 65733894 74450847 85771891 101079507 39 22 65862743 74615354 85989335 101380525 38 23 65992087 74780577 86207866 101683314 37 24 66121928 74946521 86427493 101987889 36 25 66252268 75113189 86648225 102294266 35 26 66383110 75280586 86870072 102602473 34 27 66514457 75448716 87093043 102912514 33 28 66646313 75617584 87317150 103224405 32 29 66778681 75787195 87542404 103538166 31 30 66911564 75957554 87768816 103853919 30 31 67044965 76128666 87996394 104171468 29 32 67178887 76300536 88225146 104491055 28 33 67313334 76473170 88455079 104812581 27 34 67448309 76646573 88686196 105136063 26 35 67583815 76820751 88918508 105461519 25 36 67719855 76995710 89152021 105788969 24 37 67856423 77171455 89386745 106118428 23 38 67993549 77347991 89622688 106449917 22 39 68131209 77525324 89859858 106783466 21 40 68269416 77703459 90098268 107119198 20 41 68408173 77882402 90337927 107456902 19 42 68547438 78062159 90578848 107796712 18 43 68687350 78242737 90821043 108138767 17 44 68827777 78424142 91064526 108482852 16 45 68968769 78606379 91309309 108829233 15 46 69110326 78789454 91555401 109177805 14 47 69252455 78973371 91802810 109528589 13 48 69395158 79158136 92051546 109881598 12 49 69538439 76343754 92301618 110236864 11 50 69682302 79530231 92553036 110594415 10 51 69826751 79717572 92805759 110954264 9 52 69971789 79905783 93059875 111316432 8 53 70117419 80094869 93315361 111680940 7 54 70263645 80284835 93572238 112047814 6 55 70410470 80475688 93830595 112417202 5 56 70557898 80667435 94090270 112788878 4 57 70705932 80860083 94351448 113163656 3 58 70854576 81053639 94614055 113539681 2 59 71003833 81248110 94878103 113918875 1 60 71153706 81443502 95143611 114300579 0 8 7 6 5 The degrees of the Quadrant for the Tangent of the complement of the Arches of the same Quadrant .

The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadarnt . 85 86 87 The minutes of the degrees of the Quadrant for the Tangents of the complement of the Arches of the same Quadrant . 0 114300579 143006601 190811200 60 1 114684819 143606943 191879163 59 2 115071619 144212307 192959095 58 3 115461005 144822757 194051200 57 4 115853017 145438358 195155685 56 5 116247668 146059175 196273146 55 6 116644985 146685275 197403054 54 7 117044995 147316726 198545993 53 8 117447864 147953611 199702191 52 9 117853346 148595987 200871878 51 10 118261757 149244148 202055705 50 11 118672834 149897753 203253093 49 12 119086890 150557233 204464726 48 13 119503669 151222301 205691260 47 14 119923488 151893462 206932111 46 15 120346233 152570581 208188402 45 16 120771937 153253487 209459545 44 17 121200643 153942729 210746693 43 18 121632370 154638158 212049271 42 19 122067151 155339855 213368214 41 20 122505017 156047923 214704085 40 21 122946003 156762433 216056022 39 22 123390142 157483474 217425507 38 23 123837634 158211136 218812405 37 24 124288195 158945509 220217049 36 25 124742169 159686753 221639784 35 26 125199280 160434770 223080983 34 27 125659878 161189849 224540987 33 28 126123842 161952305 226020167 32 29 126591211 162721698 227518902 31 30 127062036 163498660 229037584 30 31 127536341 164282764 230576614 29 32 128014165 165074651 232136427 28 33 128495548 165873906 233717425 27 34 128980531 166681172 235320041 26 35 129469305 167496287 236945285 25 36 129961652 168319085 238592501 24 37 130457692 169150247 240262714 23 38 130957670 169989613 241957021 22 39 131461286 170837304 243674732 21 40 131968930 171693461 245417543 20 41 132480297 172558198 247184785 19 42 132995769 173431641 248978216 18 43 133515636 174313925 250797165 17 44 134038804 175205183 252643455 16 45 134566419 176105555 254517088 15 46 135098153 177015180 256417991 14 47 135634096 177934219 258348100 13 48 136174272 178862806 260307416 12 49 136718731 179801085 262296605 11 50 137267523 180749537 264316358 10 51 137820702 181707670 266366704 9 52 138378319 182676299 268449755 8 53 138940429 183654941 270565570 7 54 139507087 184644417 272714927 6 55 140078545 185644562 274898633 5 56 140654481 186655202 277117516 4 57 141235334 187677207 279372435 3 58 141820765 188710414 281664304 2 59 142411234 189755028 283994009 1 60 143006601 190811200 286362498 0 4 3 2 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadarnt . 88 89 The minutes of the degrees of the Quadrant for the Tangents of the complement of the Arches of the same Quadrant . 0 286362498 572899830 60 1 288770746 582610421 59 2 291219764 592655713 58 3 293710598 603057015 57 4 296244357 613825994 56 5 298823024 624990311 55 6 301445987 636564040 54 7 304115322 648591509 53 8 306833212 661050728 52 9 309599077 674016435 51 10 312416191 687500725 50 11 315283945 701531474 49 12 318204757 716149676 48 13 321181137 731385593 47 14 324212583 747289264 46 15 327302782 763899813 45 16 330451272 781259259 44 17 333661982 799432199 43 18 336934467 818463792 42 19 340272744 838430438 41 20 343677949 859395374 40 21 347150587 881427652 39 22 350695255 904627361 38 23 354312962 929081086 37 24 358006024 954893332 36 25 361776788 982180553 35 26 365626388 1011062679 34 27 369560062 1041705454 33 28 373579199 1074263399 32 29 377686614 1108922084 31 30 381885288 1145891136 30 31 386178258 1185395877 29 32 390568737 1227736470 28 33 395060088 1273213435 27 34 399655828 1322188681 26 35 404359642 1375082163 25 36 409175388 1432363027 24 37 414111295 1494645462 23 38 419159137 1562590046 22 39 424335793 1637005697 21 40 429641796 1718863124 20 41 435082056 1809337410 19 42 440661780 1909864971 18 43 446386310 2022219818 17 44 452261453 2148619711 16 45 458293185 2291873854 15 46 464487853 2455533838 14 47 470852152 2644433955 13 48 477393195 2864819229 12 49 484118353 3125276745 11 50 491038024 3437829002 10 51 498155754 3819696333 9 52 505482730 4297181900 8 53 513030946 4911098124 7 54 520805157 5729633839 6 55 528821258 6875680006 5 56 537085003 8594003953 4 57 545610968 11457529506 3 58 554414914 17188033688 2 59 563504309 34376070815 1 60 572899830 Infinita . 0 1 0 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

THE TABLE OF SECANTS OTHERWISE CALLED THE BENEFICIALL TABLE . The Table of Secants .

The degrees of the Quadrant for Secants . of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 0 1 2 3 The minutes of the Quadrant for the Scants of the Complements of the Arches of the same Quadrant . 0 10000000 10001524 10006095 10013723 60 1 10000001 10001574 10006198 10013875 59 2 10000002 10001626 10006301 10014029 58 3 10000004 10001679 10006405 10014184 57 4 10000008 10001733 10006509 10014339 56 5 10000010 10001788 10006615 10014495 55 6 10000014 10001844 10006721 10014653 54 7 10000020 10001900 10006828 10014811 53 8 10000027 10001957 10006936 10014970 52 9 10000034 10002015 10007045 10015130 51 10 10000042 10002074 10007155 10015291 50 11 10000051 10002134 10007265 10015453 49 12 10000060 10002195 10007376 10015615 48 13 10000071 10002256 10007488 10015778 47 14 10000083 10002318 10007601 10015942 46 15 10000095 10002381 10007716 10016107 45 16 10000108 10002445 10007831 10016273 44 17 10000122 10002510 10007946 10016440 43 18 10000137 10002576 10008062 10016608 42 19 10000152 10002642 10008179 10016777 41 20 10000168 10002709 10008298 10016946 40 21 10000186 10002777 10008417 10017116 39 22 10000204 10002846 10008537 10017287 38 23 10000223 10002916 10008658 10017459 37 24 10000243 10002987 10008779 10017632 36 25 10000264 10003058 10008902 10017806 35 26 10000285 10003130 10009025 10017981 34 27 10000308 10003203 10009149 10018157 33 28 10000332 10003277 10009274 10018333 32 29 10000357 10003352 10009400 10018510 31 30 10000381 10003428 10009527 10018687 30 31 10000407 10003505 10009655 10018865 29 32 10000433 10003582 10009783 10019044 28 33 10000461 10003660 10009912 10019224 27 34 10000489 10003739 10010043 10019405 26 35 10000518 10003819 10010174 10019587 25 36 10000548 10003900 10010306 10019770 24 37 10000579 10003982 10010439 10019954 23 38 10000611 10004060 10010572 10020138 22 39 10000643 10004148 10010706 10020324 21 40 10000677 10004232 10010841 10020510 20 41 10000711 10004317 10010977 10020698 19 42 10000746 10004403 10011114 10020886 18 43 10000782 10004490 10011252 10021086 17 44 10000819 10004578 10011390 10021266 16 45 10000857 10004666 10011529 10021456 15 46 10000895 10004755 10011670 10021649 14 47 10000934 10004845 10011811 10021842 13 48 10000975 10004936 10011952 10022035 12 49 10001016 10005028 10012098 10022239 11 50 10001058 10005122 10012238 10022424 10 51 10001100 10005216 10012383 10022620 9 52 10001144 10005310 10012528 10022817 8 53 10001188 10005405 10012674 10023015 7 54 10001233 10005501 10012822 10023213 6 55 10001280 10005598 10012970 10023412 5 56 10001327 10005696 10013119 10023612 4 57 10001375 10005795 10013269 10023813 3 58 10001423 10005894 10013419 10024014 2 59 10001473 10005994 10013570 10024217 1 60 10001524 10006095 10013723 10024420 0 89 88 87 86 The degrees of the Quadrant for the Tangents of the complements of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The degrees of the Quadrant for Secants . of the Arches of the same Quadrant . 4 5 6 7 The minutes of degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant . 0 10024420 10038198 10055082 10075098 60 1 10024625 10038454 10055390 10075459 59 2 10024830 10038710 10055699 10075820 58 3 10025036 10038968 10056009 10076182 57 4 10025242 10039226 10056320 10076545 56 5 10025450 10039486 10056632 10076909 55 6 10025658 10039747 10056944 10077274 54 7 10025868 10040008 10057256 10077639 53 8 10026078 10040269 10057570 10078005 52 9 10026289 10040532 10057884 10078372 51 10 10026500 10040796 10058200 10078740 50 11 10026713 10041061 10058517 10079009 49 12 10026927 10041326 10058834 10079479 48 13 10027141 10041592 10059153 10079850 47 14 10027357 10041859 10059472 10080222 46 15 10027573 10042128 10059792 10080595 45 16 10027790 10042397 10060113 10080968 44 17 10028009 10042667 10060435 10081332 43 18 10028227 10642936 10060757 10081717 42 19 10028447 10043207 10061080 10082093 41 20 10028667 10043479 10061405 10082470 40 21 10028889 10043752 10061730 10082848 39 22 10029111 10044025 10062056 10083226 38 23 10029334 10044300 10062383 10083606 37 24 10029559 10044576 10062711 10083987 36 25 10029784 10044853 10063039 10084368 35 26 10030009 10045130 10063369 10084750 34 27 10030236 10045409 10063700 10085134 33 28 10030463 10045689 10064031 10085518 32 29 10030692 10045969 10064364 10085903 31 30 10030920 10046250 10064690 10086289 30 31 10031150 10046532 10065035 10086677 29 32 10031381 10046815 10065365 10087065 28 33 10031614 10047098 10065701 10087454 27 34 10031846 10047383 10066038 10087843 26 35 10032079 10047669 10066376 10088243 25 36 10032314 10047954 10066715 10088623 24 37 10032550 10048241 10067054 10089015 23 38 10032786 10048529 10067394 10089408 22 39 10033023 10048818 10067735 10089802 21 40 10033261 10049107 10068076 10090196 20 41 10033500 10049398 10068419 10090592 19 42 10033740 10049690 10068763 10090988 18 43 10033981 10049983 10069107 10091385 17 44 10034223 10050276 10069452 10091783 16 45 10034465 10050571 10069808 10092182 15 46 10034708 10050865 10070155 10092582 14 47 10034952 10051160 10070493 10092983 13 48 10035196 10051456 10070842 10093385 12 49 10035441 10051753 10071192 10093787 11 50 10035688 10052051 10071543 10094190 10 51 10035936 10052350 10071895 10094624 9 52 10036184 10052649 10072247 10095030 8 53 10036434 10052951 10072600 10095406 7 54 10036684 10053252 10072954 10095813 6 55 10036934 10053555 10073310 10096221 5 56 100371●5 10053858 10073666 10096630 4 57 10037438 10054162 10074023 10097040 3 58 10037690 10054468 10074380 10097451 2 59 10037944 10054775 10074737 10097863 1 60 10038198 10055082 10075098 10098275 0 85 84 83 82 The minutes of degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The degrees of the degrees of the Quadrant for the Secants . of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 8 9 10 11 The minutes of the Quadrant for the Secants of the Complements of the Arches of the same Quadrant . 0 10098275 10124650 10154264 10187166 60 1 10098698 10125117 10154786 10187743 59 2 10099103 10125585 10155308 10188320 58 3 10099518 10126054 10155831 10188899 57 4 10099934 10126524 10156356 10189478 56 5 10100351 10126994 10156881 10190058 55 6 10100769 10127465 10157407 10190639 54 7 10101188 10127947 10157934 10191221 53 8 10101607 10128410 10158462 10191804 52 9 10102028 10128684 10158991 10192387 51 10 10102450 10129358 10159520 10192972 50 11 10102872 10129634 10160051 10193557 49 12 10103295 10130311 10160582 10194144 48 13 10103720 10130788 10161114 10194732 47 14 10104144 10131266 10161648 10195320 46 15 10104570 10131746 10162182 10195910 45 16 10104996 10132226 10162707 10196500 44 17 10105423 10132707 10163252 10197092 43 18 10105851 19133189 10163789 10197684 42 19 10106286 10133672 10164327 10198277 41 20 10106710 10134156 10165865 10198872 40 21 10107140 10134641 10165495 10199467 39 22 10107572 10135127 10165944 10200063 38 23 10108005 10135614 10166485 10200060 37 24 10108438 10136102 10167028 10201258 36 25 10108873 10136591 10167571 10201857 35 26 10109309 10137080 10168116 10202457 34 27 10109755 10137571 10168661 10203058 33 28 10110182 10138163 10169207 10203659 32 29 10110620 10138555 10169765 10204262 31 30 10111059 10139048 10170303 10204867 30 31 10111509 10139543 10170852 10205470 29 32 10111940 10140038 10171401 10206075 28 33 10112482 10140534 10171952 10206681 27 34 10112825 10141036 10172504 10207289 26 35 10113279 10141528 10173056 10207897 25 36 10113713 10142027 10173609 10208506 24 37 10114159 10142526 10174163 10209116 23 38 10114606 10143026 10174718 10209727 22 39 10115053 10143528 10175274 10210339 21 40 10115501 10144030 10175831 10210952 20 41 10115951 10144533 10176389 10211566 19 42 10116401 10145037 10176947 10211180 18 43 10116852 10145542 10177507 10212796 17 44 10117303 10146048 10178068 10213412 16 45 10117754 10146554 10178630 10214030 15 46 10118209 10147062 10179193 10214668 14 47 10118663 10147572 10179756 10215268 13 48 10119118 10148082 10180321 10215889 12 49 10119574 10148593 10180886 10216510 11 50 10120031 10149104 10181453 19217113 10 51 10120489 10149615 10182021 10217756 9 52 10120948 10150128 10182589 10218380 8 53 10121408 10150642 10183158 10219015 7 54 10121868 10151156 10183728 10219631 6 55 10122330 10151672 10184299 10220258 5 56 10122792 10152188 10184870 10220885 4 57 10123256 10152705 10185443 10221514 3 58 10123720 10153224 10186017 10222143 2 59 10124275 10153744 10186591 10222774 1 60 10124650 10154264 10187166 10223405 0 81 80 79 78 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Secants . of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 12 13 14 15 The minutes of degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant . 0 10223405 10263040 10306136 10352762 60 1 10224037 10263730 10306884 10353569 59 2 10224671 10264420 10307633 10354377 58 3 10225305 10265112 10308383 10355186 57 4 10225941 10265804 10309134 10355996 56 5 10226577 10266498 10309886 10356807 55 6 10227215 10267192 10310639 10357619 54 7 10227854 10267888 10311393 10358433 53 8 10228493 10268584 10312148 10359247 52 9 10229134 10269281 10312903 10360063 51 10 10229775 10269979 10313660 10360880 50 11 10230417 10270688 10314417 10361698 49 12 10231060 10271379 12315176 10362517 48 13 10231644 10272080 10315935 10363337 47 14 10232288 10272782 10316696 10364158 46 15 10232994 10273485 10317457 10364980 45 16 10233641 10274190 10318220 10365802 44 17 10234289 10274895 10318984 10366626 43 18 10234938 10275601 10319749 10367450 42 19 10235587 10276318 10320525 10368276 41 20 10236238 10277016 10321282 10369102 40 21 10236889 10277726 10322050 10369930 39 22 10237541 10278436 10322819 10370758 38 23 10238195 10279148 10323589 10371588 37 24 10238849 10279860 10324359 10372418 36 25 10239505 10280573 10325131 10373250 35 26 10240161 10281287 15325903 10374092 34 27 10240818 10282002 10326677 10374916 33 28 10241476 10282717 10327451 10375750 32 29 10242135 10283434 10328127 10376586 31 30 10242795 10284151 10329003 10377422 30 31 10243456 10284870 10329781 10378260 29 32 10244118 10285589 10330559 10379098 28 33 10245782 10286310 10331339 10379938 27 34 10245445 10287032 10332119 10380778 26 35 10246110 10287754 10332902 10381620 25 36 10246776 10288478 10333684 10382463 24 37 10247442 10289202 10334467 10383307 23 38 10248110 10289928 10335252 10384153 22 39 10248778 10290654 10336037 10384999 21 40 10249448 10291381 10336824 10385846 20 41 10250119 10292119 10337612 10386694 19 42 10250790 10292838 10338400 10387543 18 43 10251461 10293569 10339189 10388393 17 44 10252136 10294300 10339980 10389244 16 45 10252811 10295043 10340771 10390096 15 46 10253482 10295766 10341564 10390949 14 47 10254162 10296501 10342347 10391803 13 48 10254839 10297237 10343152 10392657 12 49 10255517 10297973 10343947 10393513 11 50 10256196 10298710 10344743 10394370 10 51 10256876 10299449 10345541 10395228 9 52 10257557 10300188 10346340 10396087 8 53 10258239 10300928 10347139 10396947 7 54 10258922 10301669 10347940 10397808 6 55 10259606 10302411 10348741 10398670 5 56 10260291 10303154 10349544 10899533 4 57 10260977 10303898 10350347 10400397 3 58 10261661 10304643 10351151 10401262 2 59 10262351 10305390 10351956 10402128 1 60 10263040 10306136 10352762 10402994 0 77 76 75 74 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Secants . of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 16 17 18 19 The minutes of degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant . 0 10402994 10456917 10514621 10576207 60 1 10403862 10457847 10515616 10577267 59 2 10404730 10458779 10516612 10578328 58 3 10405590 10459711 10517609 10579400 57 4 10406471 10460645 10518607 10580463 56 5 10407343 10461580 10519606 10581518 55 6 10408216 10462516 10520606 10582583 54 7 10409091 10463453 10521607 10583650 53 8 10409966 10464391 10522608 10584717 52 9 10410843 10465330 10523611 10585795 51 10 10411721 10466270 10524615 10586855 50 11 10412600 10467211 10525620 10587925 49 12 10413479 10468153 10526626 10588997 48 13 10414360 10469096 10527633 10590070 47 14 10415241 10470041 10528642 10591145 46 15 10416124 10470986 10529651 10592220 45 16 10417007 10471933 10530662 10593297 44 17 10417892 10472880 10531673 10594375 43 18 10418778 10473829 10532686 10595455 42 19 10419665 10474778 10533699 10596534 41 20 10420553 10475729 10534714 10597615 40 21 10421442 10476680 10535730 10598697 39 22 10422333 10477633 10536747 10599780 38 23 10423224 10478587 10537765 10600865 37 24 10424116 10479542 10538785 10601950 36 25 10425009 10480498 10539805 10603037 35 26 10425903 10481454 10540826 10604125 34 27 10426798 10482412 10541848 10605214 33 28 10427694 10483371 10542872 10606304 32 29 10428591 10484331 10543897 10607395 31 30 10429489 10485292 10544923 10608487 30 31 10430388 10486254 10545950 10609580 29 32 10431288 10487217 10546977 10610675 28 33 10432189 10488181 10548006 10611770 27 34 10433091 10489146 10549036 10612867 26 35 10433995 10490113 10550067 10613964 25 36 10434899 10491080 10551099 10615063 24 37 10435805 10492049 10552133 10616163 23 38 10436711 10493018 10553168 10617264 22 39 10437619 10493989 10554204 10618366 21 40 10438528 10494961 10555241 10619469 20 41 10439436 10494934 10556279 10620574 19 42 10440346 10496908 10557318 10621680 18 43 10441257 10497883 10558359 10622787 17 44 10442170 10498059 10559400 10623895 16 45 10443083 10499836 10560443 10625004 15 46 10443998 10500814 10561496 10626114 14 47 10444913 10501793 10562531 10627226 13 48 10445830 10502773 10563577 10628338 12 49 10446749 10503754 10564623 10629451 11 50 10447668 10504736 10565670 10630566 10 51 10448588 10505719 10566719 10631682 9 52 10449509 10506704 10567769 10632799 8 53 10450431 10507689 10568820 10633917 7 54 10451354 10508676 10569872 10635037 6 55 10452279 10509664 10570925 10636157 5 56 10453204 10510653 10571980 10637279 4 57 10454131 10511643 10573034 10638402 3 58 10455058 10512635 10574091 10639526 2 59 10455987 10513627 10575149 10640651 1 60 10456917 10514621 10576207 10641777 0 73 72 71 70 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 20 21 22 23 The minutes of the Quadrant for the Secants of the Complements of the Arches of the same Quadrant . 0 10641777 10711449 10785347 10863603 60 1 10642905 10712646 10786616 10864945 59 2 10644034 10713888 10787885 10866289 58 3 10645164 10715042 10789155 10867633 57 4 10646295 10716242 10790427 10868979 56 5 10647427 10717444 10791700 10870326 55 6 10648560 10718647 10792974 10871675 54 7 10649694 10719850 10794250 10873024 53 8 10650829 10721056 10795527 10874374 52 9 10651965 10722261 10796805 10875626 51 10 10653103 10723469 10798085 10877079 50 11 10654242 10724677 10799365 10878434 49 12 10655381 10725887 10800647 10879790 48 13 10656522 10727098 10803214 10881147 47 14 10657664 10728310 10803214 10882506 46 15 10658807 10729524 10804500 10883865 45 16 10659951 10730738 10805787 10885226 44 17 10661097 10731953 10807074 10886588 43 18 10662244 10733170 10808363 10887952 42 19 10663392 10734387 10809652 10889317 41 20 10664541 10735606 10810942 10890683 40 21 10665692 10736826 10812234 15892051 39 22 10666844 10738048 10813528 10893417 38 23 10667996 10739270 10814823 10894788 37 24 10669150 10740494 10816119 10896159 36 25 10670304 10741719 10817417 10897531 35 26 10671460 10742945 10818715 10898905 34 27 10672617 10744173 10820015 10900280 33 28 10673776 10745401 10821316 10901656 32 29 10674936 10746631 10822617 10903033 31 30 10676096 10747864 10823920 10904413 30 31 10677258 10749094 10825225 10905790 29 32 10678420 10750327 10826531 10907171 28 33 10679584 10751561 10827838 10908553 27 34 10680749 10752797 10829146 10909936 26 35 10681915 10754034 10830455 10911322 25 36 10683082 10755273 10831766 10912709 24 37 10684250 10756513 10833078 10914096 23 38 10685420 10757753 10834391 10915484 22 39 10686591 10758995 10835706 10916874 21 40 10687763 10760237 10837023 10918265 20 41 10688936 10761481 10838341 10919657 19 42 10690111 10762726 10839660 10921051 18 43 10691287 10763972 10840980 10922436 17 44 10692464 10765220 10842301 10923833 16 45 10693642 10766469 10843623 10925241 15 46 10694821 10767720 10844947 10926641 14 47 10696001 10768971 10846272 10928041 13 48 10697182 10770224 10847597 10929442 12 49 10698364 10771477 10848924 10930846 11 50 10699548 10772732 10850252 10932249 10 51 10700732 10773988 10851583 10933654 9 52 10701918 10775244 10852914 10935061 8 53 10703105 10776502 10854246 10936469 7 54 10704294 10777761 10855578 10937879 6 55 10705483 10779022 10856912 10939290 5 56 10706674 10780284 10858247 10940702 4 57 10707866 10781547 10859584 10942115 3 58 10709059 10782802 10860922 10943527 2 59 10710254 10784078 10862262 10944945 1 60 10711449 10785347 10863603 10946362 0 69 68 67 66 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 24 25 26 27 The minutes of degrees of the Quadrant for the Secants of the Complements of the Arches of the same Quadrant . 0 10946362 11033783 11126021 11223262 60 1 10947781 11035280 11127601 11224927 59 2 10949201 11036779 11129182 11226593 58 3 10950622 11038279 11130765 11228260 57 4 10952045 11039780 11132349 11229929 56 5 10953469 11041283 11133933 11231599 55 6 10954898 11042787 11135519 11233270 54 7 10956320 11044293 11137106 11234943 53 8 10957747 11045799 11138694 11236617 52 9 10959175 11047306 11140284 11238292 51 10 10960605 11048815 11141875 11239969 50 11 10962036 11050325 11143467 11241648 49 12 10963469 11051837 11145061 11243329 48 13 10964903 11053350 11146656 11245011 47 14 10966338 11054865 11148254 11246694 46 15 10967775 11056381 11149853 11248378 45 16 10969213 11057898 11151453 11250064 44 17 10970652 11059420 11153055 11251751 43 18 10972092 11060939 11154658 11253440 42 19 10973533 11062461 11156262 11255130 41 20 10974976 11063985 11157868 11256822 40 21 10976420 11065510 11159475 11258516 39 22 10977865 11067037 11161084 11260211 38 23 10979312 11068564 11162694 11261907 37 24 10980760 11070092 1116430● 11263605 36 25 10982210 11071621 11165919 11265304 35 26 10983661 11073152 11167533 11267005 34 27 10985113 11074684 11169149 11268707 33 28 10986567 11076218 11170766 11270410 32 29 10988022 11077753 11172385 11272114 31 30 10989480 11079289 11174006 11273820 30 31 10990938 11080827 11175627 11275528 29 32 10992398 11082366 11177249 11277238 28 33 10993859 11083906 11178873 11278949 27 34 10995321 11085448 11180499 11280661 26 35 10996783 11086990 11182125 11282374 25 36 10998247 11088536 11183753 11284089 24 37 10999712 11090082 11185383 11285805 23 38 11001179 11091629 11187014 11287524 22 39 11002647 11093178 11188647 11289244 21 40 11004116 11094729 11190281 11290965 20 41 11005587 11096280 11191916 11292688 19 42 11007059 11097833 11193553 11294412 18 43 11008533 11099387 11195191 11296132 17 44 11010008 11100943 11196831 11297864 16 45 11011484 1110250● 11198472 11299593 15 46 11019262 11104058 11200114 11301324 14 47 11014441 11105618 11201758 11303056 13 48 11015921 11107179 11203404 11304789 12 49 11017402 11108741 11205051 11306523 11 50 11018884 11110306 11206700 11308259 10 51 11020367 11111871 11208550 11309996 9 52 11021852 11113438 11210001 11311735 8 53 11023338 11115006 11211654 11313476 7 54 11024826 11116575 11213308 11315218 6 55 11026315 11118145 11214963 11316961 5 56 11027806 11119717 11216620 11318706 4 57 11029298 11121290 11218278 11319452 3 58 11030791 11122865 11219938 11322199 2 59 11032287 11124442 11221599 11323949 1 60 11033783 11126021 11223262 11325700 0 65 64 63 62 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 28 29 30 31 The minutes of the Quadrant for the Scants of the Complements of the Arches of the same Quadrant . 0 11325700 11433540 11547004 11666331 60 1 11327452 11435384 11548944 11668371 59 2 11329206 11437230 11550886 11670413 58 3 11330961 11439078 11552829 11672457 57 4 11332718 11440927 11554774 11674502 56 5 11334479 11442777 11556720 11676548 55 6 11336237 11444629 11558669 11678597 54 7 11337999 11446483 11560619 11680647 53 8 11339762 11448339 11562570 11682698 52 9 11341526 11450196 11564523 11684752 51 10 11343292 11452054 11566480 11686807 50 11 11345060 11453915 11568434 11688864 49 12 11346830 11455776 11570393 11690923 48 13 11348601 11457639 11572353 11692984 47 14 11350373 11459503 11574314 11695046 46 15 11352149 11461370 11576277 11697110 45 16 11353923 11463238 11578242 11699176 44 17 11355698 11465107 11580208 11701243 43 18 11357475 11466978 11582175 11703312 42 19 11359255 11468850 11584145 11705383 41 20 11361036 11470723 11586116 11707455 40 21 11362819 11472599 11588089 11709530 39 22 11364603 11474483 11590064 11711606 38 23 11366389 11476354 11592040 11713684 37 24 11368177 11478235 11594018 11715764 36 25 11369966 11480117 11595998 11717845 35 26 11371756 11482001 11597979 11719928 34 27 11373548 11483887 11599961 11722012 33 28 11375341 11485774 11601946 11724099 32 29 11377136 11487662 11603932 11726187 31 30 11378933 11489353 11605919 11728276 30 31 11380731 11491445 11607909 11730367 29 32 11382530 11493338 11609900 11732460 28 33 11384331 11495233 11611893 11734555 27 34 11386134 11497140 11613888 11736652 26 35 11387938 11499028 11615876 11738751 25 36 11389744 11500928 11617882 11740851 24 37 11391551 11502829 11619881 11742953 23 38 11393359 15504731 11621882 11745057 22 39 11395169 11506626 11623885 11747162 21 40 11396981 11508532 11625889 11749269 20 41 11398794 11510450 11627996 11751378 19 42 11400609 11512360 11629904 11753489 18 43 11402425 11514271 11631913 11755603 17 44 11404243 11516183 11633924 11757718 16 45 11406063 11518097 11635937 11759834 15 46 11407884 11520013 11637952 11761951 14 47 11409706 11521930 11639968 11764069 13 48 11411530 11523849 11641986 11766190 12 49 11413356 11525770 11644005 11768312 11 50 11415183 11527692 11646026 11770437 10 51 11417012 11529616 11648049 11772564 9 52 11418842 11531542 11650075 11774696 8 53 11420673 11533469 11652099 11776822 7 54 11422507 11535398 11654127 11778954 6 55 11424342 11537328 11656156 11781088 5 56 11426178 11539260 11658188 11783223 4 57 11428016 11541193 11660221 11785361 3 58 11429856 11543128 11662256 11787500 2 59 11431689 11545065 11664292 11789640 1 60 11433540 11547004 11666331 11791783 0 61 60 59 58 The degrees of the Quadrant for the Secants of the complement of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 32 33 34 35 The minutes of the Quadrant for the Scants of the Complements of the Arches of the same Quadrant . 0 11791783 11923633 12062179 12207745 60 1 11793927 11925886 12064546 12210233 59 2 11796073 11928141 12066916 12212723 58 3 11798221 11930397 12069286 12215214 57 4 11800371 11932656 12071660 12217708 56 5 11802522 11934917 12074036 12220204 55 6 11804675 11937180 12076413 12222702 54 7 11806830 11939445 12078792 12225201 53 8 11808987 11941701 12081174 12227703 52 9 11811145 11943979 12083558 12230207 51 10 11813306 11946250 12085943 12232713 50 11 11815468 11948522 12088330 12235221 49 12 11817632 11950796 12090720 12237732 48 13 11819797 11953071 12093111 12240245 47 14 11821965 11955349 12095504 12242759 46 15 11824134 11957629 12097899 12245275 45 16 11826306 11959910 12100296 12247794 44 17 11828479 11962194 12102696 12250315 43 18 11830654 11964479 12105097 12252837 42 19 11832830 11966766 12107500 12255361 41 20 11835008 11969055 12109905 12257888 40 21 11837188 11971346 12112312 12260417 39 22 11839369 11973638 12114722 12262948 38 23 11841552 11975932 12117133 12265481 37 24 11843737 11978229 12119546 12268016 36 25 11845924 11980527 12121960 12270553 35 26 11848114 11982828 12124377 12273093 34 27 11850305 11985131 12126796 12275634 33 28 11852498 11987435 12129216 12278187 32 29 11854693 11989741 12131638 12280722 31 30 11856890 11992050 12134063 12283270 30 31 11859088 11994360 12136490 12285820 29 32 11861288 11996672 12138919 12288372 28 33 11863489 11998986 12141350 12290925 27 34 11865693 12001303 12143783 12293481 26 35 11867899 12003619 12146218 12296039 25 36 11870107 12005938 12148656 12298599 24 37 11872316 12008259 12150095 12301161 23 38 11874527 12010582 12153536 12303725 22 39 11876739 12012907 12155978 12306291 21 40 11878954 12015233 12158423 12308859 20 41 11881171 12017562 12160870 12311430 19 42 11883389 12019893 12163319 12314003 18 43 11885609 12022226 12165770 12316578 17 44 11887831 12024560 12168223 12319156 16 45 11890054 12026897 12170677 12321736 15 46 11892280 12029236 12173135 12324317 14 47 11894508 12031576 12175594 12326900 13 48 11896737 12033919 12178055 12329486 12 49 11898968 12036264 12180518 12332074 11 50 11901202 12038610 12182983 12334664 10 51 11903437 12040958 12185450 12337256 9 52 11905674 12043309 12187919 12339851 8 53 11907912 12045661 12190390 12342448 7 54 11910153 12048016 12192864 12345046 6 55 11912395 12050372 12195340 12347646 5 56 11914640 12052730 12197817 12350●49 4 57 11916886 12055089 12200296 12352854 3 58 11919133 12057451 12202777 12355460 2 59 11921382 12059814 12205260 12358068 1 60 11923633 12063179 12207745 12360678 0 57 56 55 54 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 36 37 38 39 The minutes of the Quadrant for the Scants of the Complements of the Arches of the same Quadrant . 0 12360678 12521357 12690184 12867599 60 1 12363290 12524103 12693070 12870632 59 2 12365906 12526851 12695957 12873667 58 3 12368524 12529601 12698847 12876704 57 4 12371144 12532354 12701739 12879744 56 5 12373766 12535110 12704634 12882787 55 6 12376391 12537867 12707531 12885832 54 7 12379018 12540627 12710430 12888879 53 8 12381647 12543389 12713332 12891929 52 9 12384278 12546152 12716236 12894982 51 10 12386911 12548918 12719143 12898035 50 11 12389546 12551686 12722052 12901094 49 12 12●92183 12554456 12724964 12904155 48 13 12394822 12557229 12727878 12907218 47 14 12397464 12560005 12730794 12910283 46 15 12400108 12562783 12733713 12913351 45 16 12402754 12565563 12736635 12916422 44 17 12405402 12568345 12739559 12919494 43 18 12408053 12571130 12742485 12922569 42 19 12410705 12573917 12745413 12925647 41 20 12413359 12576706 12748344 12928727 40 21 12416015 12579597 12751277 12931809 39 22 12418674 12582912 12754213 12934895 38 23 12421335 12585087 12757151 12937983 37 24 12423998 12587885 12760092 12941073 36 25 12426663 12590685 12763035 12944166 35 26 12429331 12593488 12765981 12947262 34 27 12432001 12596293 12768929 12950360 33 28 12434673 12599101 12771880 12953461 32 29 12437348 12601911 12774833 12956565 31 30 12440024 12604724 12777788 12959671 30 31 12442702 12607539 12780746 12962780 29 32 12445383 12610356 12783707 12965892 28 33 12448066 12613175 12786670 12969007 27 34 12450751 12615997 12789635 12972124 26 35 12453438 12618821 12792602 12975243 25 36 12456128 12621648 12795573 12978366 24 37 12458821 12624477 12798546 12981491 23 38 12461516 12627308 12801521 12984618 22 39 12464213 12630141 12804498 12987747 21 40 12466913 12632977 12807478 12990880 20 41 12469614 12635815 12810460 12994015 19 42 12472317 12638655 12813445 12997153 18 43 12475022 12641597 12816432 13000293 17 44 12477730 12644343 12819422 13003436 16 45 12480440 12646191 12822415 13006582 15 46 12483152 12650041 12825410 13009730 14 47 12485866 12652893 12828407 13012881 13 48 12488583 12655748 12831407 13016034 12 49 12491302 12658605 12834409 13019189 11 50 12494022 12661464 12837414 13022348 10 51 12496743 12664325 12840421 13025509 9 52 12499469 12667189 12843431 13028673 8 53 12502197 12670055 12846443 13031839 7 54 12504927 12672924 12849458 13035008 6 55 12507659 12675795 12852475 13038180 5 56 12510394 12678668 12855495 13041354 4 57 12513132 12681543 12858517 13044530 3 58 12515871 12684421 12861542 13047710 2 59 12518613 12687301 12864569 13050892 1 60 12521357 12690184 12867599 13054077 0 53 52 51 50 The degrees of the Quadrant for the Secants of the complement of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 40 41 42 43 The minutes of degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant . 0 13054077 13250131 13456326 13673275 60 1 13057264 13253482 13459851 13676986 59 2 13060455 13256835 13463380 13680700 58 3 13063646 13260192 13466912 13684417 57 4 13066843 13263582 13470447 13688138 56 5 13070041 13266915 13473985 13691861 55 6 13073242 13270282 13477527 13695587 54 7 13076445 13273651 13481071 13699316 53 8 13079651 13277023 13484618 13703048 52 9 13082859 13280397 13488168 13706783 51 10 13086071 13283775 13491721 13710523 50 11 13089285 13287155 13495276 13714266 49 12 13092502 13290538 13498835 13718012 48 13 13095721 13293924 13502397 13721761 47 14 13098944 13297313 13505962 13725514 46 15 13102169 13300704 13509530 13729270 45 16 13105397 13304098 13513101 13733029 44 17 13108627 13307495 13516675 13736790 43 18 13111861 13310896 13520252 13740555 42 19 13114098 13314299 13523832 13744322 41 20 13118337 13317705 13527416 13748092 40 21 13121578 13321114 13531003 13751867 39 22 13124823 13324526 13534593 13755644 38 23 13128070 13327941 13538185 13759424 37 24 13131320 13331359 13541781 13763209 36 25 13134572 13334779 13545380 13766997 35 26 13134828 13338203 13548981 13770788 34 27 13141085 13341629 13552585 13774582 33 28 13144346 13345058 13556193 13778380 32 29 13147509 13348490 13559803 13782181 31 30 13150874 13351924 13563417 13785985 30 31 13154142 13355361 13567034 13789792 29 32 13157413 13358802 13570654 13793603 28 33 13160687 13362245 13574277 13797416 27 34 13163964 13365691 13577903 13801233 26 35 13167243 13369140 13581532 13805053 25 36 13170526 13372592 13585164 13808876 24 37 13173811 13376057 13588799 13812703 23 38 13177099 13379505 13592438 13816534 22 39 13180389 13382966 13596079 13820368 21 40 13183682 13386430 13599723 13824205 20 41 13186978 13389897 13603370 13828045 19 42 13190276 13393367 13607021 13831889 18 43 13193577 13396839 13610975 13835736 17 44 13196882 13400315 13614332 13839586 16 45 13200189 13403794 13617992 13843439 15 46 13203499 13407275 13621656 13847296 14 47 13206812 13410759 13625323 13851156 13 48 13210128 13414247 13628993 13855019 12 49 13213447 13417738 13632666 13858885 11 50 13216769 13421232 13636342 13862755 10 51 13220093 13424728 13640021 13866628 9 52 13223421 13428227 13643704 13870505 8 53 13226750 13431729 13647390 13874385 7 54 13230082 13435234 13651078 13878268 6 55 13233417 13438742 13654769 13882154 5 56 13236754 13442253 13658464 13886044 4 57 13240094 13445767 13662162 13889636 3 58 13243437 13449284 13665863 13893833 2 59 13246783 13452804 13669567 13897733 1 60 13250131 13456326 13673275 13901636 0 49 48 47 46 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 44 45 46 47 The minutes of degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant . 0 13901636 14142135 14395564 14662790 60 1 13905542 14146251 14399901 14667366 59 2 13909451 14150371 14404242 14671946 58 3 13913365 14154494 14408587 14676530 57 4 13917281 14158621 14412937 14681119 56 5 13921201 14162751 14417290 14685712 55 6 13925126 14166884 14421647 14690309 54 7 13929052 14171021 14426008 14694910 53 8 13932982 14175162 14430374 14699514 52 9 13936916 14179306 14434743 14704122 51 10 13940854 14183454 14439116 14708735 50 11 13944795 14187606 14443493 14713352 49 12 13948739 14191761 14447874 14717973 48 13 13952686 14195919 14452259 14722598 47 14 13956638 14200082 14456648 14727228 46 15 13960592 14204248 14461040 14731862 45 16 13964550 14208418 14465437 14736500 44 17 13968511 14212591 14469838 14741142 43 18 13972476 14216769 14474242 14745788 42 19 13976444 14220950 14478650 14750438 41 20 13980416 14225135 14483062 14755094 40 21 13984391 14229324 14487478 14759753 39 22 13988370 14233517 14491898 14764416 38 23 13992352 14237713 14496322 14769083 37 24 13996338 14241913 14500750 14773755 36 25 14000327 14246115 14505182 14778430 35 26 14004319 14250321 14509617 14783110 34 27 14008315 14254531 14514056 14787794 33 28 14012314 14258745 14518500 14792482 32 29 14016316 14262961 14522946 14797174 31 30 14020322 14267182 14527397 14801871 30 31 14024332 14271407 14531852 14806571 29 32 14028345 14275635 14536311 14811276 28 33 14032361 14279867 14540773 14815985 27 34 14036381 14284103 14545250 14820698 26 35 14040404 14288343 14549711 14825416 25 36 14044431 14292587 14554186 14830139 24 37 14048461 14296834 14558665 14834866 23 38 14052494 14301086 14563148 14839597 22 39 14056531 14305331 14567635 14844332 21 40 14060572 14309599 14572126 14849072 20 41 14064616 14313861 14576621 14853815 19 42 14068664 14318127 14581120 14858563 18 43 14072715 14322396 14585624 14863315 17 44 14076770 14326670 14590131 14868071 16 45 14080829 14330947 14594642 14872831 15 46 14084891 14335228 14599157 14877597 14 47 14088956 14339513 14603676 14882377 13 48 14093026 14343802 14608199 14887141 12 49 14097099 14348095 14612725 14891919 11 50 14101175 14352391 14617256 14896701 10 51 14105255 14356691 14621791 14901487 9 52 14109339 14360995 14626330 14906278 8 53 14113427 14365303 14630873 14911073 7 54 14117518 14369615 14635421 14915873 6 55 14121612 14373930 14639973 14920677 5 56 14125709 14378350 14644528 14925486 4 57 14129810 14382573 14649087 14930299 3 58 14133915 14386900 14653651 14935116 2 59 14138023 14391230 14658218 14939938 1 60 14142135 14395564 14662790 14944764 0 45 44 43 42 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 48 49 50 51 The minutes of degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant . 0 14944764 15242532 15557239 15890158 60 1 14949594 15247634 15562635 15895869 59 2 14954429 15252741 15568036 15091586 58 3 14959268 15257852 15573441 15907307 57 4 14964112 15262969 15578852 15913034 56 5 14968960 15268090 15584267 15918766 55 6 14973812 15273216 15589688 15924504 54 7 14978668 15278347 15595114 15930247 53 8 14983530 15283484 15600545 15936095 52 9 14988396 15288626 15605981 15941748 51 10 14993266 15293773 15611422 15947508 50 11 14998104 15298924 15616868 15953273 49 12 15003020 15304080 15622319 15959044 48 13 15007903 15309240 15627775 15964820 47 14 15012791 15314405 15633237 15970603 46 15 15017683 15319574 15639704 15976390 45 16 15022580 15324748 15644177 15982184 44 17 15027481 15329926 15649655 15987983 43 18 15032387 15335109 15655138 15993788 42 19 15037297 15340297 15660626 15999599 41 20 15042212 15345491 15666119 16005416 40 21 15047131 15350689 15671617 16011237 39 22 15052054 15355892 15677121 16017065 38 23 15056982 15361100 15682630 16022898 37 24 15061915 15366313 15688144 16028736 36 25 15066852 15371530 15693663 16034579 35 26 15071791 15376753 15699188 16040429 34 27 15076739 15381980 15704717 16046283 33 28 15081690 15387212 15710252 16052143 32 29 15086645 15392449 15715791 16058008 31 30 15091605 15397692 15721337 16063878 30 31 15096569 15402939 15726887 16069754 29 32 15101538 15408191 15732443 16075637 28 33 15106571 15413447 15738003 16081524 27 34 15111490 15418708 15743569 16087418 26 35 15116472 15423974 15749141 16093318 25 36 15121459 15429246 15754718 16099224 24 37 15126451 15434522 15760300 16105135 23 38 15131447 15439803 15765887 16111053 22 39 15136447 15445089 15771479 16116976 21 40 15141453 15450380 15777077 16122905 20 41 15146463 15455675 15782680 16128839 19 42 15151478 15460976 15788289 16134779 18 43 15156497 15466182 15793903 16140424 17 44 15161520 15471593 15799523 16146676 16 45 15166548 15476908 15805147 16152634 15 46 15171581 15482229 15810777 16158598 14 47 15176619 15487554 15816412 16164567 13 48 15181661 15492885 15822052 16170542 12 49 15186708 15498220 15827697 16176522 11 50 15191760 15503560 15833349 16182509 10 51 15196816 15508905 15839005 16188501 9 52 15201877 15514256 15844667 16194499 8 53 15206943 15519611 15850335 16200503 7 54 15212013 15524972 15856008 16206513 6 55 15217088 15530338 15861676 16212528 5 56 15222168 15535710 15867370 16218550 4 57 15227253 15541083 15873058 16224577 3 58 15232342 15546463 15878753 16230610 2 59 15237435 15551848 15884453 16236648 1 60 15242532 15557239 15890158 16242692 0 41 40 39 38 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 52 53 54 55 The minutes of degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant . 0 16242692 16616401 17013017 17434469 30 1 16248742 16622819 17019832 17441715 29 2 16254799 16629243 17026654 17448968 28 3 16260861 16635673 17033482 17456229 27 4 16266929 16642109 17040318 17463499 26 5 16273003 16648551 17047160 17470775 25 6 16279083 16655001 17054010 17478059 24 7 16285169 16661457 17060866 17485351 23 8 16291261 16667919 17067729 17492650 22 9 16297358 16674408 17074599 17499957 21 10 16303461 16680864 17081476 17507272 20 11 16309570 16687345 17088359 17514594 19 12 16315685 16693834 17095250 17521924 18 13 16321806 16700328 17102148 17529262 17 14 16327934 16706829 17109053 17536607 16 15 16334067 16713336 17115965 17543959 15 16 16340197 16719850 171●2885 17551319 14 17 16346353 16726362 17129812 17558687 13 18 16352505 16732877 17136747 17566063 12 19 16358663 16739430 17143689 17573446 11 20 16364827 16745970 17150638 17580837 10 21 16370996 16752517 17157593 17588236 9 22 16377172 16759070 17164556 17595643 8 23 16383359 16765629 17171525 17603057 7 24 16389542 16772195 17178502 17610480 6 25 16395736 16778767 17185485 17617909 5 26 16401936 16785347 17192476 17625347 4 27 16408152 16791933 17199472 17632793 3 28 16414365 16798525 17206477 17640246 2 29 16420573 16805124 17213488 17647707 1 30 16426798 16811729 17220507 17655175 60 31 16433027 16818341 17227532 17662651 59 32 16439263 16824960 17234565 17670136 58 33 16445505 16831585 17241605 17677627 57 34 16451754 16838217 17248653 17685127 56 35 16458008 16844856 17255708 17692635 55 36 16464269 16851502 17262770 17700151 54 37 16470536 16858154 17269839 17707674 53 38 16476809 16864813 17276917 17715206 52 39 16483089 16871479 17284002 17722744 51 40 16489385 16878151 17291095 17730290 50 41 16495668 16884830 17298194 17737844 49 42 16501967 16891515 17305300 17745407 48 43 16508272 16898207 17312413 17752978 47 44 16514582 16904907 17319514 17760555 46 45 16520898 16911613 17326662 17768142 45 46 15527220 16918326 17333798 17775740 44 47 16533548 16925046 17340941 17783343 43 48 16539883 16931772 17348091 17790955 42 49 16546224 16948504 17355249 17798575 41 50 16552571 16945244 17362415 17806203 40 51 16558920 16951990 17369587 17813838 39 52 16565286 16958743 17376767 17821481 38 53 16571642 16965498 17383954 17829132 37 54 16578026 16972270 17391148 17836792 36 55 16584406 16979044 17398350 17844460 35 56 16590792 16985824 17405560 17852135 34 57 16597184 16992611 17412776 17859818 33 58 16603584 16999406 17420000 17867509 32 59 16609989 17006208 17427231 17875209 31 60 16616401 17013017 17434469 17882917 30 37 36 35 34 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 56 57 58 59 The minutes of degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant . 0 17882917 18360816 18870800 19416039 60 1 17890632 18369014 18879589 19425445 59 2 17898356 18377251 18888389 19434862 58 3 17906089 18385497 18897196 19444290 57 4 17913830 18393753 18906018 19453727 56 5 17921579 18402017 18914846 19463175 55 6 17929337 18410291 18923685 19472635 54 7 17937102 18418574 18932534 19482114 53 8 17944876 18426865 18941393 19491595 52 9 17952658 18435165 18950261 19501076 51 10 17960448 18443454 18959139 19510578 50 11 17968247 18451792 18968027 19520091 49 12 17976054 18460120 18976926 19529615 48 13 17983869 18468456 18985834 19539150 47 14 17991693 18476802 18994752 19548697 46 15 17999525 18485157 19003680 19558254 45 16 18007365 18493521 19012618 19567822 44 17 18015214 18501895 19021516 19577401 43 18 18023071 18510278 19030523 19586991 42 19 18030936 18518670 19039491 19596592 41 20 18038811 18527072 19048468 19606204 40 21 18046693 18535483 19057455 19615827 39 22 18054584 18543903 19066453 19625462 38 23 18062482 18552332 19075461 19635107 37 24 18070389 18560770 19084480 19644765 36 25 18078305 18569217 19093509 19654434 35 26 18086229 18577674 19102549 19664114 34 27 18094161 18586139 19111598 19673805 33 28 18102102 18594614 19120658 19683507 32 29 18110051 18603098 19129727 19693220 31 30 18118009 18611591 19138807 19702945 30 31 18125975 18620094 19147897 19712680 29 32 18133950 18629606 19156998 19722428 28 33 18141934 18637127 19166109 19732186 27 34 18149926 18645658 19175231 19741956 26 35 18157927 18654198 19184362 19751738 25 36 18165937 18662748 19193504 19761531 24 37 18173956 18671507 19202656 19771335 23 38 18181984 18679875 19211818 19781141 22 39 18190021 18688452 19220990 19790968 21 40 18198065 18697038 19230172 19800808 20 41 18206118 18705634 19239365 19810658 19 42 18214179 18714239 19248569 19820320 18 43 18222249 18722854 19257783 19830393 17 44 18230328 18731480 19267008 19840277 16 45 18238416 18740115 19276242 19850172 15 46 18246513 18748760 19285488 19860079 14 47 18254618 18757414 19294744 19869997 13 48 18262732 18766078 19304010 19879927 12 49 18270854 18774752 19313287 19889068 11 50 18278986 18783436 19322574 19899820 10 51 18287126 18792130 19331872 19909784 9 52 18295276 18800833 19341181 19919760 8 53 18303434 18809546 19350501 19929748 7 54 18211601 ●8818268 19359831 19939749 6 55 18319776 18826999 19369172 19949760 5 56 18327961 18835741 19378524 19959784 4 57 18337154 18844492 19387886 19966820 3 58 18344356 18853252 19397260 19979868 2 59 18352567 18862021 19406644 19989928 1 60 18360816 18870800 19416939 20000000 0 33 32 31 30 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 60 61 62 63 The minutes of the Quadrant for the Scants of the Complements of the Arches of the same Quadrant . 0 20000000 20626654 21300545 22026892 60 1 20010083 20637484 21312206 22039475 59 2 20020179 20648338 21323882 22052074 58 3 20030285 20659184 21335570 22064690 57 4 20040404 20670054 21347275 22077322 56 5 20050534 20680937 21358993 22089970 55 6 20060676 20691834 21370727 22102635 54 7 20070832 20702744 21382475 22115316 53 8 20080995 20713667 21394238 22128014 52 9 20091172 20724603 21407016 22140728 51 10 20101361 20735554 21417808 22153459 50 11 20111562 20746517 21429615 22166204 49 12 20121776 20757494 21441438 22178971 48 13 20132001 20768484 21453275 22191751 47 14 20142239 20779488 21465128 22204548 46 15 20152489 20790505 21476995 22217361 45 16 20162751 20801535 21488877 22230191 44 17 20173035 20812579 21500774 22243038 43 18 20183321 20823636 21512686 22255902 42 19 20193619 20834706 21524612 22268782 41 20 20203930 20845791 21536553 22281680 40 21 20214252 20856888 21548509 22294595 39 22 20224588 20868000 21560481 22307526 38 23 20234936 20879125 21572467 22320474 37 24 20245296 20890264 21584469 22333439 36 25 20255669 20901416 21596487 22346420 35 26 20266054 20912582 21608520 22359419 34 27 20276452 20923761 21620568 22372434 33 28 20286863 20934955 21632631 22385466 32 29 20297286 20946162 21644710 22398418 31 30 20307721 20957383 21656804 22411584 30 31 20318170 20968618 21668913 22424667 29 32 20328630 20979867 21681038 22437768 28 33 20339102 20991130 21693178 22450886 27 34 20349587 21002406 21705334 22464022 26 35 20360084 21013696 21717505 22477175 25 36 20370594 21025001 21729691 22490346 24 37 20381116 21036319 21741893 22503543 23 38 20391751 21047651 21754111 22516748 22 39 20402198 21058997 21766344 22529965 21 40 20412758 21070357 21778593 22543201 20 41 20423331 21081731 21790858 22556358 19 42 20433916 21093119 21803138 22569723 18 43 20444514 21104522 21815434 22583025 17 44 20455126 21115938 21827745 22596336 16 45 20465750 21127368 21840072 22609663 15 46 20476387 21138814 21852415 22623009 14 47 20487037 21150273 21864774 22636372 13 48 20497700 21161747 21877149 22649754 12 49 20508376 21173235 21889539 22663152 11 50 20519064 21184737 21901946 22676569 10 51 20529765 21196253 21914369 22690004 9 52 20540479 21207783 21926808 22703456 8 53 20551205 21219328 21939263 22716924 7 54 20561945 21230887 21951734 22730414 6 55 20572697 21242460 21964220 22743919 5 56 20583463 21254048 21976722 22757443 4 57 20594242 21265650 21989240 22770984 3 58 20605033 21277267 22001775 22784543 2 59 20615837 21288899 22014325 22798120 1 60 20626654 21300545 22026892 22811726 0 29 28 27 26 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 64 65 66 67 The minutes of the Quadrant for the Scants of the Complements of the Arches of the same Quadrant . 0 22811726 23662013 24585936 25593051 60 1 22825329 23676784 24602010 25610602 59 2 22838962 23691575 24618107 25628180 58 3 22852612 23706387 24634227 25645783 57 4 22866281 23721220 24650370 25663414 56 5 22879968 23736073 24666536 25681071 55 6 22893674 23750947 24682727 25698754 54 7 22907387 23765842 24698940 25716464 53 8 22921140 23780757 24715178 25734201 52 9 22934901 23795692 24731439 25751965 51 10 22948680 23810648 24747724 25769755 50 11 22962478 23825625 24764033 25787582 49 12 22976294 23840623 24780365 25805417 48 13 22990129 23805642 24796721 25823287 47 14 23003983 23870683 24813101 25841185 46 15 23017855 23885844 24829504 25859104 45 16 23031747 23900827 24845932 25877061 44 17 23045657 23915931 24862383 25895040 43 18 23059586 23931055 24879958 25913046 42 19 23073534 23946200 24895356 25931080 41 20 23087501 23961366 24911878 25949142 40 21 23101486 23976553 24928423 25967230 39 22 23115490 23991762 24944993 25985345 38 23 23129513 24006992 24961587 26003487 37 24 23143556 24022245 24978●05 26021658 36 25 23157616 24037518 24994847 26039855 35 26 23171696 24052814 25011514 26058081 34 27 23185795 24068130 25028205 26076333 33 28 23199913 24083469 25044920 26094614 32 29 23214050 24098850 25061660 26112923 31 30 23228205 24114213 25078426 26131259 30 31 23242380 24129616 25095216 26149623 29 32 23256574 24145041 25112030 26168015 28 33 23270797 24160487 25128869 26186436 27 34 23285021 24175956 25145732 26204884 26 35 23299273 24191445 25162620 26223361 25 36 23313546 24206956 25179532 26241867 24 37 23327838 24222488 25196469 26260400 23 38 23342150 24238043 25213432 26278963 22 39 23356481 24253619 25230418 26297555 21 40 23370832 24269217 25247431 26316176 20 41 23385203 24284838 25264468 26334825 19 42 23399593 24300481 25281531 26353503 18 43 23414003 24316147 25298620 26372209 17 44 23428433 24331835 25315734 26390945 16 45 23442882 24347546 25332874 26409709 15 46 23457351 24363281 25350039 26428502 14 47 23471840 24379038 25367229 26447323 13 48 23486348 24394818 25384445 26366174 12 49 23500876 24410620 25401687 26485053 11 50 23515424 24426446 25418956 26503962 10 51 23529992 24442294 25436250 26522890 9 52 23544580 24458164 25453570 26541867 8 53 23559188 24474056 25470915 26560863 7 54 23573817 24489973 25488286 26579889 6 55 23588565 24505908 25505683 26598945 5 56 23603134 24521869 25523005 26618030 4 57 23617822 24537851 25540553 26637145 3 58 23632532 24553857 255580●7 26656251 2 59 23647262 24569885 25575526 26675466 1 60 23662013 24585936 25593051 26694672 0 25 24 23 22 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 68 69 70 71 The minutes of the Quadrant for the Scants of the Complements of the Arches of the same Quadrant . 0 26694672 27904284 29238045 30715531 60 1 26713907 27925445 29261433 30741500 59 2 26733172 27946642 29284861 30767516 58 3 26752467 27967873 29308328 30793579 57 4 26771791 27989139 29331835 30819689 56 5 26791145 28010440 29355382 30845846 55 6 26810529 28031776 29378970 30872051 54 7 26829942 28053147 29402599 30898304 53 8 26849390 28074553 29426268 30924605 52 9 26868867 28095994 29449978 30950953 51 10 26888373 28117469 29473728 30977350 50 11 26907910 28138980 29497519 31003793 49 12 26927479 28160527 29521350 31030285 48 13 26947078 28182108 29545222 31056824 47 14 26966709 28203725 29569136 31083412 46 15 29986370 28225378 29593090 31110047 45 16 27006062 28247067 29617087 31136731 44 17 27025785 28268793 29641124 31163462 43 18 27045539 28290553 29665204 31190241 42 19 27065323 28312349 29689326 31217019 41 20 27085138 28334181 29713488 31243945 40 21 27104985 28356049 29737692 31270871 39 22 27124864 28377954 29761938 31297848 38 23 27144774 28399894 29786227 31324873 37 24 27164717 28421871 29810558 31351948 36 25 27184690 28443884 29834931 31379072 35 26 27204686 28465934 29859347 31406247 34 27 27224734 28488021 29883705 31433472 33 28 27244804 28510144 29908306 31460747 32 29 27264906 28532304 29932850 31488072 31 30 27285040 28554501 29957438 31515448 30 31 27305205 28576735 29982069 31542873 29 32 27325402 28599007 30006743 31570349 28 33 27345631 28621316 30031460 31597875 27 34 27365893 28643662 30056220 31625453 26 35 27386186 28666045 30081023 31653080 25 36 17406513 28688467 30105870 31680758 24 37 27426872 28710925 30130760 31708486 23 38 27447264 28733422 30155714 31736265 22 39 27467688 28755956 30180672 31764094 21 40 27488145 28778549 30205694 31791974 20 41 27508635 28801139 30230760 31819906 19 42 27529157 28823787 30255871 31847891 18 43 27549722 28846473 30281026 31875929 17 44 27570301 28869196 30306226 31904019 16 45 27590922 28891957 30331460 31932164 15 46 27611578 28914756 30356759 31960358 14 47 27632266 28937594 30382092 31988606 13 48 27952989 28960471 30407470 32016909 12 49 27673745 28983386 30432893 32045263 11 50 27694535 29006340 30458361 32073672 10 51 27715358 29029332 30483873 32102132 9 52 27736215 29052363 30509430 32130649 8 53 27757105 29075435 30535033 32159212 7 54 27778029 29098546 30560682 32187832 6 55 27798987 29121697 30586375 32216504 5 56 27819978 29144888 30612115 32245231 4 57 27841003 29168●18 30637890 32274012 3 58 27862060 29191388 30663732 32302846 2 59 27883156 29214697 30689608 32331735 1 60 27904284 29238045 30715531 32360678 0 21 20 19 18 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 72 73 74 75 The minutes of the Quadrant for the Scants of the Complements of the Arches of the same Quadrant . 0 32360678 34203038 36279559 38637042 60 1 32389676 34235609 36316402 38679033 59 2 32418726 34268245 36353333 38721117 58 3 32447837 34300947 36390323 38763296 57 4 32477001 34333716 36427401 38805571 56 5 32506219 34366553 36464558 38847941 55 6 32535494 34399452 36501793 ●8890408 54 7 32564823 34432420 36539107 38932971 53 8 32594209 34465456 36570511 38975632 52 9 32623651 34498557 36613973 39018390 51 10 32653148 34531726 36651525 39061246 50 11 32682701 34564959 36689156 39104200 49 12 32712311 34598259 36726868 39147252 48 13 32741977 34631626 36764660 39190423 47 14 32771699 34665061 36802533 39233653 46 15 32801478 34698564 36840488 39277002 45 16 32831314 34732135 36878524 39320449 44 17 32861207 34765775 36916641 39363994 43 18 32891157 34799483 36954842 39407640 42 19 32921165 34833259 36993127 39451384 41 20 32951231 34867105 37031496 39495228 40 21 32981355 34901024 37069947 39539172 39 22 33011537 34935005 37108482 39583218 38 23 33041776 34966052 37147101 39627364 37 24 33072074 35003172 37185803 39671613 36 25 33102431 35037361 37224589 39715965 35 26 33131846 35071621 37263459 39760420 34 27 33163320 35105952 37302413 39804979 33 28 33193853 35140354 37341453 39849642 32 29 33224444 35174826 37380577 39894411 31 30 33255094 35209369 37419788 39939226 30 31 33285803 35243981 37459081 39984263 29 32 33316571 35278664 37498460 40029344 28 33 33347398 35313418 37537923 40074528 27 34 33378286 35348244 37577471 40119816 26 35 33409132 35383140 37617104 40165289 25 36 33440240 35418110 37656824 40210709 24 37 33471307 35453152 37696632 40256316 23 38 33502436 35488268 37736518 40302033 22 39 33533625 35523456 37776513 40347858 21 40 33564875 35558718 37816588 40393792 20 41 33596187 35594052 37856751 40439834 19 42 33627561 35629460 37897004 40485985 18 43 33658998 35664940 37937146 40532245 17 44 33690497 35700494 37977779 40578613 16 45 33722059 35736121 38018300 40625091 15 46 33753683 35771822 38058912 40671678 14 47 33785370 35807597 38099614 40718374 13 48 33817120 35843447 38140406 40765180 12 49 33848934 35879373 38181288 40812093 11 50 33880813 35915374 38222261 40859121 10 51 33912753 35951451 38263324 40906259 9 52 33944756 35987602 38304479 40953510 8 53 33976821 36023829 38345725 41004876 7 54 34008950 36060132 38387064 41048358 6 55 34041141 36096510 38428495 41095957 5 56 34073395 36132966 38470019 41143668 4 57 34105712 36169497 38511635 41191492 3 58 34138091 36206107 38553344 41239431 2 59 34170523 36242794 38595146 41287425 1 60 34203038 36279559 38637042 41335654 0 17 16 15 14 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 76 77 78 79 The minutes of the Quadrant for the Scants of the Complements of the Arches of the same Quadrant . 0 41335654 44454097 48097335 52408433 60 1 41383937 44510183 48163151 52486983 59 2 41432338 44566415 48229350 52565774 58 3 41480856 44622793 48295633 52644807 57 4 41529492 44679318 48362102 52724084 56 5 41578245 44735990 48428756 52803604 55 6 41627117 44792810 48495599 52883368 54 7 41676108 44849777 48562631 52963377 53 8 41725219 44906892 48629854 53043632 52 9 41774450 44964155 48697269 53124134 51 10 41823802 45021567 48764877 53204885 50 11 41873273 45079129 48832678 53285884 49 12 41922863 45136843 48900673 53367134 48 13 41972573 45194707 48968853 53448635 47 14 42022405 45252726 49037249 53530390 49 15 42072357 45310898 49105830 53612399 45 16 42122431 45369223 49174607 53694666 44 17 42172625 45427703 49243590 53777191 43 18 42222942 45486338 49312751 53859976 42 19 42273380 45545127 49382118 53943022 41 20 42323942 45604073 49451684 54026331 40 21 42374627 45663175 49521449 54109903 39 22 42425439 45722435 49591416 54193739 38 23 42476377 45781853 49661584 54277840 37 24 42527442 45841429 49731956 54362207 36 25 42578635 45901164 49802532 54446842 35 26 42629957 45961059 49873313 54531744 34 27 42681409 46021115 49944301 54616915 33 28 42732991 46081333 50015497 54702356 32 29 42784705 46141715 50086901 54788068 31 30 42836551 46202261 50158514 54874053 30 31 42888527 46262969 50230335 54960312 29 32 42940631 46323841 50302367 55046847 28 33 42992865 46384877 50374610 55133659 27 34 43045229 46446076 50447065 55220751 26 35 43097722 46507440 50519732 55308122 25 36 43150347 46568970 50592614 55395775 24 37 43203103 46630665 50665711 55483710 23 38 43255992 46692527 50739024 55571930 22 39 43309012 46754555 50812553 55660434 21 40 43362166 46816752 50886299 55749226 20 41 43415454 46879117 50960263 55838300 19 42 43468877 46941653 51034447 55927677 18 43 43522435 47004361 51108850 56017340 17 44 43576129 47067242 51183475 56107297 16 45 43629●5● 47130297 51258321 56197549 15 46 43683925 47193526 51333391 56288099 14 47 43738728 47256930 51408684 56378948 13 48 43792268 47320509 51484204 56470097 12 49 43846646 47384264 51559951 56561548 11 50 43901162 47448195 51635936 56653302 10 51 43955817 47512302 51712129 56745360 9 52 44000612 47576586 51788563 56837723 8 53 44065548 47641048 51865227 56930392 7 54 44120625 47705689 51942124 57023369 6 55 44175844 47770510 52019254 57116653 5 56 44231207 47835511 52096618 57210246 4 57 44286712 47900693 52174216 57304150 3 58 44342362 47966058 52252051 57398367 2 59 44398156 48031605 52330123 57492896 1 60 44454097 48097385 52408433 57587740 0 13 12 11 10 The degrees of the Quadrant for the Secants of the complement of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 80 81 82 83 The minutes of the Quadrant for the Scants of the Complements of the Arches of the same Quadrant . 0 57587740 63924495 71852975 82055127 60 1 57682901 64042118 72002006 82249986 59 2 57778381 64160180 72151659 82445779 58 3 57874180 64278683 72301942 82642513 57 4 57970302 64397632 72452863 82840196 56 5 58066748 64517028 72604421 83038833 55 6 58163520 64636873 72756618 83238436 54 7 58260619 64757168 72909461 83439009 53 8 58358049 64877918 73062954 83640561 52 9 58455●10 64999124 73217100 83843097 51 10 58553904 65120789 73371903 84046626 50 11 58652333 65242916 73527367 84251153 49 12 58751099 65365508 73683499 84456680 48 13 58850205 65488566 73840302 84663213 47 14 58949653 65612095 73997782 84870760 46 15 59049444 65736097 74155942 85079327 45 16 59149581 65859675 74314786 85288957 44 17 59250065 65985531 74474318 85499628 43 18 59350898 66110967 74634544 85711347 42 19 59452082 66246886 74795468 85924121 41 20 59553618 66363291 74957095 86137958 40 21 59655506 66490185 75119429 86352864 39 22 59757728 66617572 75282475 86568849 38 23 59860346 66745453 75446238 86785921 37 24 59963291 66873831 75610721 87004089 36 25 60066612 67002708 75775928 87223362 35 26 60170285 67132088 75941864 87443750 34 27 60274319 67261972 76108533 87665261 33 28 60378718 67392365 76275941 87887909 32 29 60483482 67523270 76444091 88111704 31 30 60588615 67654691 76612989 88336657 30 31 60694118 67786629 76782641 88562776 29 32 60799995 67919089 76953050 88790069 28 33 60906246 68052073 77124223 89018543 27 34 61012875 68185585 77296165 89248201 26 35 61119882 68319630 77468882 89479054 25 36 61227271 68454208 77642381 89711108 24 37 61335043 68589313 77816665 89944373 23 38 61443202 68724977 77991740 90178856 22 39 61551749 68861175 78167612 90414568 21 40 61660686 68997920 78344287 90651519 20 41 61770013 69135315 78521769 90889717 19 42 61879735 69273018 78700066 91129181 18 43 61989853 69411469 78879183 91369917 17 44 62100367 69550434 79059128 91611941 16 45 62211280 69689963 79239905 91855265 15 46 62322594 69830059 79421520 92099899 14 47 62434312 69970726 79603976 92345849 13 48 62546437 70111967 79787381 92593126 12 49 62658971 70253786 79971439 92841739 11 50 62771918 70396188 80156456 93091699 10 51 62885274 70539174 80342336 93342963 9 52 62999049 70682751 80529087 93595620 8 53 63113241 70826919 80716713 93849647 7 54 63227855 70971684 80905219 94105066 6 55 63342890 71117047 81094612 94361964 5 56 63458352 71263014 81284899 94620181 4 57 63574240 71409586 81476087 94879901 3 58 63690559 71556760 81668183 95141050 2 59 63807309 71704564 81861195 95403639 1 60 63924495 71852975 82055127 95667689 0 9 8 7 6 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 84 85 86 The minutes of the Quadrant for the Scants of the Complements of the Arches of the same Quadrant . 0 95867689 114737188 143355808 60 1 95933204 115●19970 143954694 59 2 96200195 115505313 144558602 58 3 96468673 115893242 145167595 57 4 96738655 116283797 145781740 57 5 97010253 116676991 146401101 55 6 97283267 117072851 147025745 54 7 97557932 117471403 147655740 53 8 97834057 117872815 148291169 52 9 98111843 118276840 148932108 51 10 98391211 118683794 149578791 50 11 98672171 119093414 150230942 49 12 98954738 119506013 150888966 48 13 99236930 119921335 15155●578 47 14 99524766 120339695 152222283 46 15 99812250 120760985 152897946 45 16 100101400 121185232 153579394 44 17 100392329 121612482 154267179 43 18 100684851 122042752 154961155 42 19 100979193 122476076 155661396 41 20 101275259 122912485 156368008 40 21 101572962 123352014 157081063 39 22 101872522 123794696 157800648 38 23 102173854 124240732 158526854 37 24 102476971 124689836 159259771 36 25 102781890 125142353 159999560 35 26 103088639 125598007 160746121 34 27 103397202 126057149 161499724 33 28 103707656 126519656 162260744 32 29 104019959 126985568 163028671 31 30 104334254 127454936 163804188 30 31 104650345 127927785 164586836 29 32 104968474 128404152 165377268 28 33 105288542 128884078 166175067 27 34 105610566 129367604 166980877 26 35 105934564 129854921 167794536 25 36 106260557 130345812 168615879 24 37 106588558 130840395 169445585 23 38 106918589 131338917 170283495 22 39 107250680 131841076 171129820 21 40 107584955 132347264 171984431 20 41 107921201 132857174 172847712 19 42 108259554 133371390 173719700 18 43 108600151 133889600 174600528 17 44 108942779 134411312 175490331 16 45 109287702 134937471 176389247 15 46 109634817 135467749 177297417 14 47 109984143 136002235 178215000 13 48 110335695 136540955 179142131 12 49 110689503 137083887 180078954 11 50 111045597 137631223 181025951 10 51 111403988 138183016 181982628 9 52 111764699 138739177 182949802 8 53 112127750 139299830 183926988 7 54 112493167 139865032 184915009 6 55 112861097 140435034 185913698 5 56 113231316 141009514 186922883 4 57 113604036 141588910 187943432 3 58 113979204 142172885 188975184 2 59 114356941 142761897 190018342 1 60 114737188 143355808 191073059 0 5 4 3 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant . 87 88 89 The minutes of the Quadrant for the Scants of the Complements of the Arches of the same Quadrant . 0 191073059 286537048 572987098 60 1 192139567 288943841 582696234 59 2 193218044 291391404 592740072 58 3 194308693 293880683 603139919 57 4 195411723 296413087 613907444 56 5 196527729 298990299 625070305 55 6 197656182 301611807 636642580 54 7 198797665 304279687 648655621 53 8 199952408 306996123 661126359 52 9 201120639 309760533 674090521 51 10 202303011 312576192 687573461 50 11 203498943 315442491 701612741 49 12 204709121 318361849 716229489 48 13 205934200 321336774 731453951 47 14 207173596 324366765 747356168 46 15 208428431 327455509 763965262 45 16 209698119 330602545 781323254 44 17 210983811 333811800 799494739 43 18 212284914 337082830 818524878 42 19 213602421 340419652 838490069 41 20 214936837 343823403 859453551 40 21 216287319 347294586 881484374 39 22 217655350 350837799 904682629 38 23 219040792 354454051 929134899 37 24 220443981 358145679 954945691 36 25 221865261 361914968 982231437 35 26 223305005 365763113 1011112129 34 27 224763453 369095332 1041753449 33 28 226241278 373713015 1074309940 32 29 227738558 377818975 1108967170 31 30 229255785 382016194 1145934768 30 31 230793360 386307709 1185438054 29 32 232351718 390696734 1227777193 28 33 233931261 395186630 1273252703 27 34 235532422 399780916 1322226495 26 35 237156211 404483275 1375118522 25 36 238801972 409397566 1432297932 24 37 240470730 414227875 1494678912 23 38 242163582 419278406 1562622042 22 39 243879838 424453607 1637036239 21 40 245621193 429758156 1718892212 20 41 247386980 435196961 1809365043 19 42 249178956 440775230 1909891150 18 43 250996450 446498305 2022234532 17 44 252841285 452371994 2148642981 16 45 254713463 458402271 2291895669 15 46 256612911 464595485 2455554199 14 47 258541565 470958329 2644450861 13 48 260499426 477497828 2864894681 12 49 262487160 484221619 3125282743 11 50 264505458 491139838 3437843546 10 51 266554348 498256113 3819709423 9 52 268635944 505581634 4297193536 8 53 270750304 513128395 4911255640 7 54 272898206 520901152 5729642566 6 55 275080457 528915798 6875687278 5 56 277297985 537178089 8594018365 4 57 279551349 545702599 11458691197 3 58 281841763 554505091 17188036598 2 59 284170013 563593031 34376072269 1 60 286537048 572987098 INFINITA 0 2 1 0 The degrees of the Quadrant for the Secants of the complements of the Arches of the same Quadrant .

A plaine Treatise of the first principles of Cosmographie , and specially of the Spheare , representing the shape of the whole world : Together with all the chiefest and most necessarie vses thereof , written by M. Blundevill of Newton Flotman , Anno Dom. 1594

The heauens declare the glory of God , and the firmament sheweth his handy worke . Psal. 19. depiction of astrolabe

LONDON Printed by John Windet . 1594.

The exposition of certaine termes or principles of Geometrie .

MInding to treate of the principles of Cosmographie , and especially of the Sphere , I thinke it good first to expound vnto you certaine termes of Geometry , without the which the vnlearned shall hardly vnderstand the Contents of this Treatise , which tearmes are certaine principles of Geometrie , called definitions : For there are but thrée kinds of principles , whereon the demonstration of all Geometricall conclusions dependeth : that is , definitions , petitions , and maxims , but I mind here to deale onely with certaine definitions , Whereof I will set downe as many as I thinke néedful for this purpose , in such order as followeth .

1 A point called in Latine punctus , is a thing supposed to be indiuisible , hauing neither length , breadth , nor deapth , as the point or pricke a.

2 A line called in Lataine linea , is a supposed length , hauing neither breadth nor thicknesse , as the line a. b.

3. The ends or bounds of a line , are two points , and of lines some be right , and some be crooked .

4 A right line is that which goeth right from one point to another , and not bowingly as doth the line c. d. but so straight as is possible , as the line a. b. Againe the crooked line is either a whole Circle , or portion of a Circle , or else goeth winding in and out , as a serpent called in Latine Linea tortuosa , or else winding about like the shell of a Snaile , called of the Latines , linea Spiralis , as these figures here doe shew .

A whole Circle

A Portion of a circle

Linea tortuosa .

Linea spiralis .

5 Superficies or vpperface , is that which onely hath length and breadth , without deapth , which is twofould , that is to say , plaine and crooked .

6 The bounds of superficies are lines .

7 A plaine superficies is that which lyeth straight betwixt his lines , as the figure a. & a crooked superficies is that which goeth bowing and lyeth not straight betwixt his lines as the figure b. Againe superficies being considered in an hollow body , as a barrell , tunne , or vault , may be deuided into two other kinds , that is Conuexe , and Concaue , for the vpper part of such vault is said to be conuex , and the inward part cancaue as the figure b. sheweth .

Conuex . Concaue .

Right . Crooked . Mixt.

9 Of plaine Angles , some are called right line angles , because that both the lines whereof it consisteth are right , and some are called crooked line angles , because that both lines are crooked , & some are said to be mixt , because the one line is crooked and the other right , as you may perceiue by the thrée sundry shapes thereof made in the Margent .

10 If one right line standing vpon another right line do make two equall angles , that is to say , of ech side one , then eyther of those Angles is a right Angle , and the line so standing vpon his fellow , is called the perpendicular or plumbe line , as the line a. b. standing vpon the line c. d.

11 A blunt Angle , is that which is greater then a right Angle , as the Angle e.

12 A sharpe Angle , is that which is lesser then a right Angle , as f. so as there be in all thrée Angles , that is right , blunt , and sharpe , in Latine rectus , obtusus , & acutus : And besides these Angles , there be also certaine Sphericall , that is to say , round Angles , which consist of two circular lines , drawne vppon a Sphericall superficies , which doe crosse one another in some point , eyther rightly , or obliquely : if rightly , then they make right Angles , on ech side of the point where they crosse , as the figure E. sheweth , if obliquely then they make sharpe Angles of the inside , and blunt Angles on the out side of the point , where they crosse as the figure F. doth partly shew , for such Angles cannot be so well described in Plano , as vpon the superficies of some Sphericall body .

13 A terme called in Latine terminus , is the bound or linute of any thing , as points are the bounds of lines , & lines the bounds of superficieses , and superficieses the bounds of a body , which is that which hath imaginatiuely , but not materially , both length , bredth , and depth , and if such body haue many faces or sides , then it is bounded with many superficieses , as the figure g. made like a sixe square dye . But if such body be round as a bowle , Sphere , or Globe , then it is bounded or couered with one superficies onely , as the figure H. doth shewe .

14 A figure is that which is comprehended within one bound , or many bounds .

Circumference .

15 A Circle is a plaine figure bounded with one circular line , which is called the circumference , vnto the which as many right lines as are drawne from the point standing in the midst , are all equall one to the other , that is to say , one is as long as another , as the figure I. here doth shew .

diam●ter .

16 And the middle point of a Circle is called the Center , as the point A. in figure I.

17 The Diameter of a Circle , is a right line passing through the Center , from one side of the Circumference to the other , which deuideth the Circle into two equall parts , as the line B. C.

18 A Semicircle is a figure , contayned within the Diameter and halfe the Circumference of a Circle as the figure D. C. E.

The cord .

19 The portion of a Circle , is a part of a Circle , greater or lesser than the Semicircle , as the figures F. and G. doe shewe , the right line in eyther of which figures is called in Latine chorda , in English the string , and the Circular line in Latine is called arcus in English the bowe , but the Diameter is alwayes the greatest cord in any Circle .

The cord .

20 Right line figures , are those which are bounded with right lines .

21 Triangles or thrée cornerd figures , are those which are bounded with thrée right lines as the figure H.

22 Foure square figures , are those which are bounded with foure right lines , as the figure I.

23 Many square figures , are those which are bounded with more right lines then foure as K.

24 Of Triangles or thrée cornerd figures , there be sixe kindes , whereof the first ●s called Isoplurus , hauing thrée equall sides and thrée equal Angles as A.

25 The second is called Isosceles , ha●ing but two equall sides and angles as B.

26 The thirde called scalenos hauing ●o side equall , one with the other , but one ●horter or longer then another as C.

27 The fourth is called Ort hogonius , ●auing one right Angle , as D.

28 The fift is called ampligonius , because it hath one blunt Angle , as E.

29 The sixt is called oxigonius , because 〈◊〉 hath thrée sharpe Angles , but not equall ●●des as F.

30 Of foure square figures there is one iust foure square , hauing foure equall sides , and is right angled as G.

31 Another is called a long Square , which is right angled , but not hauing foure equall sides , as H.

32 There is another called rombus , th●● is to say a Turbut , in shape like to Diamant , which hath foure equall sides , but is not right angled , as I.

33 There is another called romboides , which though it hath sides and Angles one right against another , yet hath it neither foure equall sides , nor yet is right angled , as K.

34 All other kinds of foure squares are called trapezia , as L. or such like .

Right line paralels .

Circuler paralels .

35 Paralels are two right lines equally distant one from another , which béeing drawne foorth infinitely , would neuer tou●● or méete one another in any part , as th● two lines M. N. And though the lines be● circular , yet if they be equally distant in a●● places one from another , they are also call●● paralels , as the figure P. sheweth , y●● though the lines be winding in and out 〈◊〉 Serpents , yet if they be equally distant in 〈◊〉 places one from another , they are paralel as well as if they were right lines as the ●●gure Q. sheweth .

Serpentine paralels .

The order and contents of this Treatise , touching the first principles of Cosmography , and specially of the Spheare .

HAuing set downe the exposition of certaine termes of Geometry , for the better vnderstanding of this Treatise , I doe first define what Cosmography is , and therewith doe briefely showe what Sciences it comprehendeth , and who were first inuentors therof : That done , because the Spheare doth represent the shape of the whole world , I do define what a Spheare is , and then I doe deuide the world into two essentiall parts , that is , the Celestiall and Elementall part , according to which two parts I do also deuide this Treatise into two Bookes , the first whereof treateth of the Celestiall parts , and the second of the Elementall part of the world , with what order the Chapters of ech Booke hereafter following , do plainely shew .

The Chapters and Contents of the first Booke .

OF Cosmography , what it is , and what kinde of Sciences it comprehendeth , and who were first inuentors thereof , Chap. 1.

The definition of a Spheare , and of the vnitie , roundnesse , and capacitie of the worlde , also of the Poles and Axletree thereof . Chap. 2.

Of the diuision of the world , and of the two Essentiall parts thereof , and what things ech part conteyneth . Chap. 3.

A figure of the whole world , wherein are set forth the two Essentiall parts before mentioned , that is to say , the eleauen heauens and the foure Elements .

A demonstration to prooue the pluralitie of the heauens , Chap. 4.

Of the highest Spheare or heauen , called the Imperial heauen . Chap. 5.

Of the tenth Spheare or heauen called in Latine Primum m●bile , that is , the first moueable , & what motion it hath . Chap. 6.

Of the ninth heauen , what motions & names it hath , & whether there be any waters aboue the firmament or not . Chap. 7.

Of the eight heauen what motions it hath , and what circles are imagined by the Astronomers to be in that heauen , and to what vse and purpose they serue : Also in what time euery one of the seuen Planets maketh his reuolution , & of what thicknes their Spheares be . Chap. 8.

Of the circles whereof a materiall Spheare consisteth , and of their diuers diuisions . Chap. 9.

Of the Equinoctiall line , why it is so called , and of the diuerse vses thereof . Chap. 10.

A figure shewing the Equinoctiall line , the two poles and the Axeltree of the world .

Of the Zodiaque , why it is so called , and of the 12. signes therein conteined . Also of the Latitude , Longitude and declination thereof . Chap. 11.

How much the Zodiaque declineth from the Equinoctial● towards either of the Poles , and of the greatest declinatio● of the sunne , what it is at this present , and what it hath been● in times past . Chap. 12.

How to know the quantitie of the sunnes declination , be it Northward or Southward , euery day throughout the yeare aswell by a Table as by helpe of a materiall Spheare or globe ▪ Chap. 13.

An instrument to knowe thereby in what signe or degree the sunne is euery day throughout the yeare .

A Table shewing the declination of the sunne euery day throughout the yeare lately calculated .

Vpon what poles the Zodiaque turneth about , also of the Ecliptique line and of the diuerse vses thereof . Chap. 14.

Of the Eclipses both Solar and Lunar , & of the head & tail of the Dragon with certaine figures shewing the same . cap. 15.

Of the two Colures , why they are so named , and wheret● they serue : also of the foure Cardinall points , that is , the tw● Equinoctial , and the two Solsticiall points , and of the entrance of the sunne into any of those points , or into any other signe . Chap. 16.

Of the Horizon both right and oblique , making thereby three kinds of Spheares , that is , the right , the paralell , and the oblique Spheare . Chap. 17.

A figure shewing the Latitude of any place to be equall vnto the eleuation of the Pole .

Of the Meridian , and of the vses thereof . Chap. 18.

Of the verticall circle and vses thereof , whereof no mention is commonly made by those that write of the Spheare . Chap. 19.

Of the foure lesser circles , that is to say , the circle Artique , the circle Antartique , the Tropique of Cancer , and the Tropique of Capricorne , and also of the fiue Zones , that is to say , two cold , two temperate , and one extremely hoat . Chap. 20.

A figure shewing the fiue foresaid Zones .

A Table shewing how many minutes are requisite to make one degree , in euery lesser circle answerable to one degree of the Equinoctiall .

Of the starres and celestiall bodies contained in the firmament , and first of their substance . Chap. 21.

Of the moouing and shape of the starres . Chap. 22.

Of the number of the starres , and of their magnitude and greatnes , and into how many Images they are deuided , & how many starres euery Image contayneth . Chap. 23.

Of the xij . Images or signes of the Zodiaque . Chap. 24.

Of the xxj . Northerne Images . Chap. 25.

Of the 15. Southerne Images . Chap. 26.

Of the longitude of the fixed starres , and of the procession of the vernall Equinoctiall point , and what it is . Chap. 27.

Of the Latitude of the fixed starres . Chap. 28.

Of the Declination of the fixed starres . Chap. 29.

Of the ascention and discention , that is the rising and setting of the starres , aswell according to the Astronomers , as according to the Poets . Chap. 30.

Of the Astronomicall ascention and discention in generall both right , meane , and oblique , & what a giuen ark is . Cap. 31.

Of the right , oblique , and meane ascention in particular , and of the chiefe causes of such diuersitie of ascentions . cap. 32.

How to know the diuersities of the ascentions and discentions , as well in the right as oblique Spheare . Chap. 33.

Of the ascentionall difference and vses thereof . Chap. 34.

Of the threefold Poeticall rising and setting of the starres , Chap. 35.

Of time , what it is , and into what parts it is deuided . chap. 36.

Of the yeare , and of his diuerse kindes , and of the diuerse computations had thereof in diuerse ages , and amongst diuers Nations . Chap. 37.

Of the sunnes yeare called in Latine annus solaris , and of the diuerse kinds thereof , and first of the Tropicall yeare , both equall and vnequall . Chap. 38.

Of the Syderall yeare , & how much it containeth . chap. 39.

Of the Politicall yeare , and diuerse kinds thereof . chap. 40.

Of the Iulian yeare , why it is so called , and of the 2. kinds thereof , that is , the common yeare , and the bisextile yeare , otherwise called the leape yeare . Chap. 41.

Of the Egyptian yeare , and how many daies it containeth . Chap. 42.

How many Moones the Iewes yeare , and the Athenian yeare doe containe . Chap. 43.

Of the yeare Lunar , and of the kinds thereof . Chap. 44.

Of the diuers kinds of monthes , and into what parts euery Solar moneth is deuided according to the Romanes , that is , into Kallends , Nones , and Ides . Chap. 45.

Of the diuerse kindes of moneths Lunar . Chap. 46.

Of a weeke . Chap. 47.

Of dayes and nights both naturall and artificiall , as well in the right Spheare , as in the oblique Spheare . Chap. 48.

Two figures shewing the right and oblique Spheare , together with the arckes of the dayes and nightes in ech Spheare . Chap. 49.

How to find out by the materiall Spheare or Globe , and by helpe of the ascentionall difference before defined , the increase & decrease of euery day throughout the yeare in euery seueral latitude , & at what houre the sun riseth & setteth . c. 50

How to know by the materiall Spheare or Globe , in what part of the Horizon the sunne riseth and setteth euery day , and thereby the length of the day . Also how to know the Meridian altitude of the sunne euery day throughout the yeare , & being at his Meridian altitude , to know how farre distant he is from the Zenith euery day , Chap. 51.

Of houres as well equall as vnequall , and into what partes they are deuided . Chap. 52.

How and in what manner the Iewes doe deuide the artificiall day and night , ech of them into foure quarters . chap. 53.

How to knowe what Planet raigneth in euery houre of the day or night artificiall , as well by helpe of a table , as by a rule contained in one verse . chap. 54.

Here ende the Contents of the first Booke .
The Chapters and Contents of the second Booke , containing the Elementall part of the world .

OF the Elementall part of the world . Chap. 1 :

Of the fire , his nature and motion , Chap: 2 :

Of the Aire , and into howe many Regions it is deuided , Chap. 3.

A figure shewing the three seuerall Regions of the aire .

Of the water , and whether it be round or not . Chap. 4.

A figure shewing that the water is round .

Of the Earth , and whether it be in all parts round or not . Chap. 5.

Of the compasse of the earth , and of the diuersitie of measures , according to diuerse countries . chap. 6.

Of the Longitude and Latitude of the earth . chap. 7.

A figure shewing the Longitude and Latitude of the world .

How to knowe the Latitude of any place , as well in the day as in the night . chap. 8.

How to knowe the true Longitude of any place . chap. 9.

A readie way to finde out the Longitude of anie place inuented by Gemma Phrizius . chap. 10.

Another way taught by Appian to find out the Longitude of any place with the Crosse-staffe , by knowing the distance betwixt the Moone and some knowne starre that is situated nigh vnto the Ecliptique line . chap. 11.

How to know the distance of places , that is to say , how many miles one place is distant from another , and howe many wayes one place is sayd to differ in distance one from another . Chap. 12.

A Table shewing how many miles are answereable to one degree of euery seuerall Latitude .

How to knowe by the helpe of the foresaide Table , the distance of two places differing only in Longitude . chap. 13.

How to finde out the true distance of two places differing both in Longitude and Latitude , by the Arithmeticall way . chap. 14.

Also how to find out the same by helpe of a Demicircle diuided into 180. parts , without any Arithmetique .

How to finde out the distance of places by the Geometricall way . chap. 15.

Of the fiue Zones . chap. 16.

A figure of the foresaid fiue Zones , that is to say , two cold , two temperate , and one extremely hoat .

Of Paralels . chap : 17.

A figure shewing the 21. Paralels .

The Table of Paralels , shewing how many degrees and minutes euery one is distant from the Equinoctiall , made according to the rule of Ptolomie .

Of Climes both old and new . chap. 18 :

A figure shewing the order of the seuen auncient common clymes whereunto Appian addeth two more , making in all nine clymes .

A Table shewing the degrees of Latitude in the beginning , midst , and end , of euery one of the foresaid nine clymes .

A Table shewing the longest day in euery degree of Latitude , proceeding orderly from the Equinoctiall to the North pole , by whole degrees without minutes from one degree to 90.

A Table shewing how many miles euery one of the seuen clymes doe containe , aswell in breadth , as in length .

Of the diuerse seasons and shadowes incident to diuerse climes and paralels , and first what seasons and shadowes they haue that dwell right vnder the Equinoctiall . chap. 19.

Of the seasons & shadowes which they haue that dwell betwixt the Equinoctiall and the Tropique of Cancer . chap. 20.

Of the seasons and shadowes which they haue that dwell right vnder the Tropique of Cancer . chap. 21.

Of the seasons and shadowes which they haue that dwell betwixt the Tropique of Cancer , and the circle Artique . chap. 22.

Of the seasons and shadowes which they haue that dwell right vnder the circle Artique , and howe long their day is . chap. 23 :

What seasons , shadowes , and length of day they haue that dwell betwixt the circle Artique , and the pole Artique . chap. 24.

Of those that dwell right vnder the Pole . chap. 25.

By what names certaine inhabitants of the earth are called , according to the diuersitie or likenes , as well of their shadow , as of their situation . chap. 26.

By what names certaine partes of the earth are called by reason of their diuerse shapes . chap. 27 :

Of the wind , what it is , what motion it hath , and of the diuerse names and diuisions thereof . chap. 28.

Of the nature and qualitie of the auncient twelue windes . chap. 29.

Of the moderne diuision of the windes . chap. 30.

A figure of the 32. windes representing the Marriners compasse .

The first part of the Spheare .
Of Cosmography , what it is , and what kinds of Sciences it comprehendeth , and who were first inuentors thereof . Chap. 1.

WHat is Cosmography ?

Cosmography is the description of the whole world , that is to say , of heauen and earth , and all that is contained therein .

What speciall kindes of knowledge are comprehended vnder this Science .

These foure , Astronomie , Astrologie , Geographie , and Chorographie .

What is Astronomy ?

Astronomy is a Science , which considereth and describeth the magnitudes and motions of the celestiall or superlour bodies .

Which call you superiour bodies ?

The Spheares or Heauens , and the Starres as well fixed as moueable , which we shall define hereafter in their proper places .

What is Astrologie ?

It is a Science which by considering the motions , aspects , and influences of the starres , doth foresée and prognosticate things to come .

What is Geographie ?

It is a knowledge teaching to describe the whole earth , and all the places contained therein , whereby vniuersall Maps and Cardes of the earth and sea are made .

What is Chorographie ?

It is the description of some particular place , as Region , I le , Citie , or such like portion of the earth seuered by it selfe from the rest .

Who were first inuentors of these Sciences .

Some say that Atlas was the first inuentor , whom the Poets saine to beare vp the heauens with his shoulders , hauing his head placed in the North Pole , and his féete in the South Pole , and his right hand bearing the East part , and his left hande the West part of the world . Albeit that some apply this fiction of the Poets to an high mountaine in Affrica called Atlas , which for his great height surmounting the cloudes , is sayde to beare vp the skyes . But some say that Adam our first Parent , was the first inuentor thereof : And some others affirme that Abraham was the first inuentor thereof . But whosoeuer was the first inuentor , it well appeareth by Ptolomy his booke , called Almagesti , that he hath béene no small furtherer thereof , and since his time in these latter dayes Georgius Purbacchius , Iohannes de Monte Regio , Copernicus , and diuerse others haue learnedly treated thereof . But leauing to speake of the first inuentors , or of the furtherers of these sciences , I will speake of the shape , capacitie , and vnitie of the world , and because the shape thereof is likned to a round bodie called a Spheare , I will first define what a Spheare is .

The definition of a Spheare , and of the vnitie , roundnesse , and capacitie of the worlde , also of the poles and axeltre thereof . Cap. 2.

WHat is a Spheare ?

A Spheare is a round body , contained or couered with one only superficies or vpperface , in the midst of which bodie is a point or pricke , called the Centre , from which all right lines being drawne to any part of the superficies or vpper face therof , are of equal length , as you may perceiue by this figure here following , the Centre whereof is marked with the letter A.

And this Spheare before defined representeth the frame of the whole world , containing all thinges , and is conteyned of none .

How prooue you that there is but one world ?

By the authoritie of Aristotle , who saieth , that if there were any other world out of this , then the earth of that worlde would mooue towardes the Centre of this world , which cannot be , vnlesse it should first ascend from his own Centre to the superficies or vpper face of the same , which is against nature , for euery heauy thing naturally tendeth downeward , and euery light thing vpward , which common experience sheweth : for a stone being cast vpwards , will fall downewardes , and the flame of a candle being turned downeward , will fly vpward .

How proue you the frame of the world to be round ?

By thrée reasons , First by comparison , for the likenes which it hath with the chiefe Idea or shape of Gods minde , which hath neither beginning nor ending , and therefore is compared to a Circle . Secondly by aptnesse aswell of mouing as of containing , for if it were not round of shape , it should not be so apt to turne about as it continually doth , nor to containe so much as it doth , for the round figure is of greatest capacitie and containeth most . Thirdly , necessitie prooueth it to be round , for if it were with angles or corners , it should not be so apt to turne about , and in turning about , it should leaue voide and emptie places , which nature abhorreth , for no place by nature can be without a bodie , nor a body without a place .

Whereupon is this Spheare or great round frame turned ?

Vpon two most firme and immoueable hookes called in Latin Cardines mundi , and in Gréeke Poli , deriued of the verbe Polo , which is asmuch to say , as to turne , for as a doore turneth vppon his hookes , so the world turneth vpon his two Poles , whereof the one is fixed in the North , and the other in the South , & the North pole is called in Latine Polus arcticus , and the South pole is called Antarcticus , through which Poles from the one to the other passeth a right immaginatiue line called of the Astronomers the Axeltrée of the world , about the which the worlde continually turneth like a Cart whéele , which you may sée liuely expressed in a materiall Spheare made for the purpose , for in Plano it cannot be so well described .

Of the diuision of the world , and of the two essentiall parts thereof , and what things ech part contayneth . Cap. 3.

HOw is the world deuided ?

Most writers do make a twofold diuision therof , that is according to substance , and according to accidentes , which accidents because they are none other thing , but points , lines , and circles , and so incident and necessarily belonging ( by position though not by nature ) to the substantiall parts , as the one can not be described without the other , neither the subiect without the accidents , nor the accidents without their subiect , I will therefore make but one diuision of the world , diuiding the same into two essentiall parts , that is , the celestiall part , and the Elementall part , in the description of which parts , it shal also plainely appeare what those lines and Circles be , and to what end they serue .

What doth the celestiall part containe ?

The eleuen Heauens and Spheares .

Make you no difference betwixt these Spheares , and the great Spheare which you last defined to be couered with one superficies or vpperface .

Yes verily , for these haue two superficies or vpper faces , that is , the Conuexe and Concaue .

Which call you Conuexe and which Concaue ?

The Conuexe is the outermost vpper face of any thing that is round , and therewith hollow , as of an ouen or vault , and the Concaue is the innermost or hollow superficies of the saide ouen or vault , or of any such like rounde bodie hauing concauitie or hollownesse .

Which be those eleuen Spheares or Heauens whereof you did last speake ?

In ascending orderly vpwardes from the Elements they bee these , The first is the Spheare of the Moone , The seconde the Spheare of Mercury , The third the Spheare of Venus , The fourth the Spheare of the Sunne , The fift the Spheare of Mars , The sixt the Spheare of Iupiter , The seuenth the Spheare of Saturne , The eight the Spheare of the fixed Starres , commonly called the firmament . The ninth is called the second moueable or Christall heauen , The tenth is called the first moueable , and the eleueuth is called the Emperiall heauen , where God and his Angels are said to dwell .

What doth the Elementall part containe ?

It containeth the foure Elements .

Which be those ?

The Element of the fire which is next to the Spheare of the Moone , and next to that more downeward is the Element of the aire , and next to that is the Element of the water , and next to that is the earth , which is the lowest of all . All which things you may sée here plainly set foorth in the figure following on the other side .

A figure of the whole world , wherein are set forth the two essentiall partes before mentioned , that is to say , the eleuen heauens and the foure Elements .

EARTHE AER FYER 1 ☽ TE MONE COLDE AN MOYST BENEVOLENT SILVER 2 ☿ MERCVRI SVCHE AS HE IS IOYNED WYTHE QVCKSYLVER . 3 ♀ VENVS COLD AN MOYST BENEVOLENT COPPER 4 ☉ THE SONNE HOT AN DRYE BENEVOLENT GOLDE 5 ♂ MARS HOT AN DRYE . MALEVOLENT IREN 6 ♃ IVPITER HOT AND MOYST BENEVOLENT TYN 7 ♄ SATVRNE · COVLDE AN DRYE MALEVOLENT LEDE 8 LEO * 4 / ♋ * 3 ♊ * 2 ♉ * 1 ♈ * 12 ♓ * 11 ♒ * 10 ♑ * 9 ♐ * 8 ♏ * 7 ♎ * 6 ♍ * 9 THE CRISTALINE HEAVEN ● 10 · THE FIRST · MOVEABLE . 11 · THE EMPYREAL HEAVEN . THE ABITATION · OF · THE BLESED·

A demonstration to prooue the pluralitie of the heauens . Chap. 4.

HOw is it to be prooued that there is such multiplicitie of heauens , sith to the eye it seemeth one whole bodie ?

If there were not diuerse heauens and diuerse motions , there should neither bee generation nor corruption of any thing , for all things then should be as one , and euer in one estate , but besides that reason men haue found out by experience in obseruing the diuerse rising and setting of diuerse wandring starres called the Planets , that euery such Planet hath a seuerall Spheare or heauen by it selfe , for otherwise they should continually kéepe their places as the fixed starres doe , which do mooue altogether according to the moouing of the viii . heauen , wherein they are placed .

I graunt that there may be diuerse heauens or Spheares . But how prooue you that there be tenne moueable heauens , besides the imperiall heauen before mentioned , and specially sith of the olde writers , some affirme that there bee but nine , and some haue saide that there be no more then eight .

The latter writers haue found by good obseruation ( as they say ) that the firmament or eight heauen hath thrée seuerall motions wherewith it is mooued and turned about , which could not be , vnlesse there were two mooueable heauens higher then that , as shall be declared when we come to shewe the thréefold motion of the saide eight heauen , for here I mind briefely to describe vnto you all the heauens beginning at the highest , and so procéede to the lowest .

Of the highest Spheare or Heauen , called the imperiall heauen . Chap. 5.

THe imperiall heauen ( as our auncient Deuines affirme ) is immooueable and being created by God the first day that he began the first creation of the world , was by him immediatly replenished with his ministers the holy Angels , and this heauen being the foundation of the world , is most fine and pure in substance , most round of shape , most great in quantitie , most cleare in qualitie , and most high in place , where God and his Angels are said to dwell .

Of the tenth Spheare or heauen , called in Latine primum mobile , that is , the first moueable , and what motion it hath . Chap. 6.

THis heauen is also of a most pure and cleare substance and without starres , and it continually mooueth with an equall gate from East to West , making his reuolution in 24. houres , which kind of moouing is otherwise called the diurnall or daily moouing , & by reason of the swiftnesse therof , it violently caryeth & turneth about all the other heauens that are beneath it from East to West , in the selfe same space of 24. houres , whether they will or not , so as they are forced to make their owne proper reuolutions , which is contrary from West to East , euery one in longer or shorter time , according as they be far or neare placed to the same : yet some writers doe not admit any violence in the heauens , but that euer the heauen hath his proper and naturall moouing without any violence .

I doe not well vnderstande by what reason the first moueable should haue such force ouer the rest .

For your better vnderstanding how this may be , suppose that you turne with your hande from East to West a Grindstone , or some other turning whéele , whereon is placed a Fly or some other créeping worme which commeth towardes you from West to East , & you shal easily conceiue that by your swift turning of the whéele , you shal turne about the said Fly or Worme many times contrary to her owne course , whether she will or not , before she can get about the said stone or whéele .

That is true , proceede therefore to the rest , and tell the nature and moouing of the ninth heauen .

Of the ninth heauen , what motions and names it hath , and whether there be any waters aboue the firmament or not . Chap. 7.

THe ninth heauen is also cleare of substance , and without starres , hauing two moouinges , the one from East to West vpon the poles of the world , according to the daily moouing of the first mooueable , and the other from West to East vpon his owne poles , according to the succession of the signes of the Zodiaque , which is in the first moueable , turning so slowly about as it maketh but one degrée in 100. yeares , and accomplisheth his full reuolution in 36000. yeares , or as Alfonsus saith in 49000. yeares .

Is there then in the first moueable any Zodiaque , that is to say , a broad Circle carying the 12. signes and yet no starres ?

Yea though there be no stars , yet many superstitious fooles doe imagine that there be diuerse Characters and liniaments not to be séene , but with a most sharpe and subtill sight ( by vertue of which Characters & fained constellations ) they imagine that they can worke wonders and strange effects , inough to deceiue themselues and others too .

What is the cause that the ninth heauen is so long in making his proper reuolution .

Because as I sayd euen now , this heauen is placed next to the first moueable , which carieth him about contrary to his owne course with such violence as he cannot make his owne proper reuolution so soone as the other heauens which are placed further off .

If the ninth heauen be so long in making his course , it will neuer be compleat whilst the world lasteth , for the whole age of the world according to some , is but 6000. yeares ?

Ye Plato was of an other opinion , and therefore this reuolution was called , magnus annus Platonis , that is to say , the great yeare of Plato , because hee affirmed that when this reuolution was once complet , all things should be in the same estate wherein they were at the first , and that he should then stand reading to his Schollers in the selfe same chaire wherein he stood at that present , which fond opinion S. Augustine confuteth in his 12. Booke de Ciuitate Dei , speaking in this maner , God forbid ( saith he ) that we should credit these things , for Christ dyed once for our sinnes , and being risen againe from the dead , he is no more to die , neithet can death haue any more power ouer him , by vertue of whose resurrection , we also that beléeue shall rise againe , and dwell with God for euer .

With what names is this ninth heauen called ?

Some do call it the Christalline heauen , because of the cleareues thereof , and some the watrie heauen , because the Scripture affirmeth that there be waters aboue the firmament as we reade in Gen. cap. 1. let the firmament be made , and deuide the waters from waters . Againe we read in the Psalmes , all ye waters that be aboue the firmament , blesse ye the Lord .

The naturall Philosophers allow no waters to dwell aboue the firmament .

That is true , yet notwithstanding if the holy Scriptures doe manifestly affirme that there be waters aboue the firmament , it behooueth a Christian to beléeue it , but question perhaps may bée mooued , what manner of waters they are that are aboue the firmament , whether such as bréed raine , or whether they are onely to be referred to the Christalline heauen , because it is of a watrie substance , and therefore of some is called the watry heauen , affirming it to be placed next to the primum mobile , or first moueable , to the intent that with the coldnesse thereof , it might asswage and represse the extreame heat of the same primum mobile , which otherwise ( as some affirm ) with his swift & violent moouing , would set all the heauens on fire , and yet no raine is bred therein , for the great raine that drowned the world in Noahes time , did not fall from aboue the firmament , but from the Aire , which in the holy Scripture is many times signified by this Latine world Coelum , that is to say , Heauen , as when the Scripture saith , the flood gates of heauen were opened , is as much to say as the flood-gates of the Aire were opened , &c. but leauing this question , let vs procéede to the eight heauen .

Of the eight heauen what motions it hath , and what circles are imagined by the Astronomers to be in that heauen , and to what vse and purpose they serue : Also in what time euery one of the seuen Planets maketh his reuolution , & of what thicknes their Spheares be . Chap. 8.

THe eight heauen otherwise called the firmament , is a most glorious heauen adorned with all the fixed starres .

Why are they called fixed ?

Because they are fastned in this heauen , like knots in a knotty board , hauing no moouing of themselues , but are mooued according to the moouing of this eight Spheare or heauen wherein they are fixed .

How many fold is the moouing of this heauen ?

The moouing of this heauen ( as hath béene said before ) is thréefold , for first it turneth about euery day from East to West in 24. houres , according to the moouing of the first mooueable , otherwise called the diurnall moouing . Secondly it mooueth from West to East , according to the moouing of the ninth heauen which maketh but one degrée in 100. yeares , and this moouing is called motus angium Stellarum fixarum .

What are Anges ?

They be certaine imagined points in the heauen , notifying the furthest distance of any Orbe or Spheare from the Centre of the world : Thirdly it mooueth sometime towardes the South , and sometime towards the North , by vertue of his owne proper moouing , called in Latine motus trepidationis , that is to say , the trembling moouing , whereby it is mooued vpon two little Circles , the poles whereof are in the beginning of that Aries and that Libra which are imagined to be in the ninth heauen , the Semidiameter of which little Circles is 4. degrées 18′· and 43″· & maketh his whole reuolution in 7000. yeares , and this moouing is otherwise called motus accessus & recessus , that is to say , the moouing of approching and retiring , and is onely proper to the 8. Spheare : for euery naturall body by order of nature , can haue but one proper moouing , and by reason of this moouing the fixed stars be not alwayes equally distant from the immooueable poles of the tenth Spheare , nor alwayes vnder the Ecliptique of the said 10. Sphere , but oftentimes cleane without it . Nor the fixed stars are in equall times equally distant from the beginning of that Aries & Libra , which are imagined to be in the tenth heauen , but they séeme to mooue sometimes towards the East , and sometimes towards the West , now more swiftly , and now more slowly . For Ptolomy in his time found them to be mooued in 100. yeares one degrée . And the latter obseruers haue found them when they be in their swift motion to be mooued one degrée in 63. yeares , all which thrée moouings are easie ( as some affirme ) to be demonstrated by an instrument made of purpose , shewing euery seueral moouing of this heauen by it selfe , but such demonstrations as some others thinke be not altogether necessary conclusions : for it may be that the daily moouing is common to the whole frame , and that the moouing of the Angles of the fixed starres , is proper to the eight Spheare , hauing some other moouer , and the like obiection may be made touching the trembling moouing , called motus trepidationis . But now you haue to vnderstande that for the better describing , deuiding , and measuring of this heauen , and all the apparances thereof , the Astronomers with the Pensill of imagination , or rather of most necessarie inuention , doe ( as it were ) trace the same with certaine Circles both greater and lesser , whereof wee shall speake so soone as we haue briefely shewed the mouing of the rest of the heauens , contained in the celestiall part of the world , in such order as they follow one another . For next to the firmament is the heauen of Saturne , which maketh his reuolution from West to East in 30. yeares , next to him is the heauen of Iupiter , who maketh his reuolution from West to East in 12. yeares , next to him is the heauen of Mars , which maketh his reuolution also from West to East in 2. yéeres , next to him is the heauen of the Sunne , which maketh his reuolution from West to East in 365. dayes , and 6. houres lacking certaine minutes , next to him is the heauen of Venus , which maketh her reuolution from West to East in like time as doth the Sunne , next to Venus is the heauen of Mercury , which maketh his reuolution from West to East in the like space that Venus doth : And next to Mercury is the heauen of the Moone , which maketh her reuolution from West to East in 28. dayes and certaine minutes . Thus hauing briefely described the eleuen heauens , I will now treat of the Circles that are imagined ( as I sayd before ) by the Astronomers to be in the eight heauen or firmament , to the intent that the measures and distances of the starres , images , signes , and other apparances therein contayned , might be the better demonstrated .

Before you deale with these Circles , I would gladly vnderstand some reason why all these seuerall heauens seeme to the eye but as one entire bodie .

That is because they are all cleare and transparent like fine birall glasse , or Christall , through the which the sight doth easily pearce , though there were neuer so many coates of such cleare substance , couering one another like the skales of an Onion , for so the heauens doe couer and enclose one another , and euery one is of an excéeding great thicknesse .

Why how thicke is euery such heauen ?

105/222 2/33 miles . 253372 ⅔ . 3274494 6/11. 343996 4/11 26/308/800. 1899654 6/11. 19604454 6/11.

What neede haue the heauens to be so thicke ?

Because otherwise they could not containe ech one his starre or starres , for there is no fixed starre so little , but that it is farre greater in compasse then the earth , neither is there any wandring starre , but that it is bigger then the earth , the Moone , Venus , and Mercury excepted .

166. times . 95. times . 91. times . 2. times . 39. times . 3143. times .

You haue well satisfied mee touching these matters , and therefore nowe proceede with the Circles if you thinke so good .

Before that I describe vnto you the Circles whereof a Sphere is made , or declare the vses thereof : I thinke it not amisse here to set downe the shape of a Spheare , together with the names of euery Circle written vpon the same , to the intent that you may be acquainted with all the partes of a Spheare before you come to vse the same : which shape or figure was first drawne by Master Blagraue , and is set downe in his booke called the Mathematicall Iewell .

depiction of astrolabe
Of the Circles whereof a materiall Spheare consisteth , and of their diuerse diuisions . Chap. 9.

OF Circles which are imagined to be in the firmament , and whereof a Spheare representing the shape of the world is commonly made , be in all 10. that is to say , the Equinoctiall , the Zodiaque , the two Colures , the Horizon , the Meridian , the two Tropiques and the two polar Circles , of which Circles some be greater and some lesser .

Which call you greater and which lesser ?

Those are called the greater Circles , which passing through the Centre or midst of the firmament , do deuide the whole circuit thereof into two equall parts , and of such Circles there be sixe , that is to say , the Equinoctiall , the Zodiaque , the Colure of the Solstices , the Colure of the Equinoxes , the Horizon and the Meridian . The lesser Circles are the foure last mentioned , that is the two Tropiques , and the two polar Circles , which are called lesser , because they do not deuide the world into two equall parts , but into vnequall portions .

What other diuisions doe the Astronomers make of the said Circles ?

Diuerse , for some are said to be paralels , some right , some oblique , some moueable , and some immoueable .

Which are said to be paralels ?

The two polar Circles , and the two Tropiques , and the Equinoctiall which is in the very midst of them all .

Which are said to be right and which oblique ?

The right be the two Colures , the right Horizon , & the Meridian , because they cut the Spheare or Globe with right Angles : And the oblique are these two , that is , the Zodiaque , and the oblique Horizon , which are said to be oblique because they cut the Spheare or globe with oblique Angles .

Which are said to be moueable , and which immoueable ?

The moueable are these , the Equinoctiall , the Zodiaque , the two Tropiques , & the 2. polar Circles which are said to be moueable , because they continually mooue with the firmament & are like in al places . And these also hauing respect to a materiall Sphere , are said to be intrinsicall or inward . The immoueable are these 2. that is , the Horizon , & the Meridian , which are said to be immoueable because in the turning of the Spheare they remaine vnmooued , for though we change both Meridian and Horizon by going from one habitation to another vpon the earth , yet euery place hath still his owne Meridian and Horizon , which doe remaine immoueable , and these two Circles hauing respect to a materiall Spheare , are said to be extrinsicall or outward , because they do inclose on the outside all the other Circles of a materiall Spheare . Thus hauing sette downe the diuisions of the sayde Circles , I will now describe them all in order , and shew to what vses they serue , beginning first with the great Circles , and so procéede to the lesser , and first I will speake of the Equmoctiall .

Of the Equinoctiall line , why it is so called , and of the diuerse vses thereof . Chap. 10.

WHat is the Equinoctiall ?

The Equinoctiall is a great Circle , which being in euery part equally distant from the two poles of the world , deuideth the Spheare in the very midst thereof , into two equall parts , and therefore it is called of some , the girdle of the world .

Wherefore is it called by the name of the Equinoctiall ?

Because that when the Sunne toucheth this Circle , which is twice in the yeare , the day and night is of equall length throughout the whole world .

At what time of the yeare is it Equinox ?

In the spring of the yeare , about the 11. day of March when as the Sunne entereth into the first degrée of Aries , and againe in Autumne about the 13. of September when as the Sun entereth into the first degrée of Libra , And by reason that this Circle deuideth the world in the very midst , those that dwell right vnder it , are said to haue no Latitude either Northward or Southward , to whom the dayes and nights are alwayes equall . But if they dwell any thing distant from the Equinoctiall , towardes any of the poles , then they are said to haue Latitude more or lesse , eyther Southward or Northward , as shall be declared hereafter more at large , when we come to treat of the Longitude and Latitude of the earth , both which are to be knowne by helpe of the Equinoctiall .

To what other vses serueth this Circle ?

This Circle hath many most necessary vses , for first it sheweth the daily moouing of the first moueable , which maketh his reuolution in 24. houres , which houres are equall , and are to be measured by the degrées of the Equinoctiall , by allowing 15. degrées thereof to an houre . And therefore the degrées of the Equinoctiall are commonly called in Latine of the Astronomers tempora , of which degrées it containeth 360. which being deuided by 15. do make 24. houres , which is a naturall day , contayning both day and night : Moreouer it sheweth the declination of the fixed starres and their right ascentions , whereof we shall speake when we come to treat of the starres , and by his equall motion , all the inequalities of the Zodiaque , & of all the signes conteyned therein , are measured : And to be short , this Circle hath many other good vses not to be declared vntill you haue some knowledge of the rest of the other Circles hereafter described .

A figure shewing the Equinoctiall line , the two Poles and Axletree of the world .
Of the Zodiaque why it is so called , and of the 12. signes therein contained : Also of the Latitude , Longitude , and declination thereof . Chap. 11.

WHat is the Zodiaque ?

It is a broad , oblique , or stope Circle , hauing a circular line in the midst thereof , called the Ecliptique line , and deuideth the Spheare into two equall parts , by crossing the Equinoctiall with oblique Angles in two points , that is , in the beginning of Aries , and in the beginning of Libra , so as the one halfe of this broad Circle declineth towards the North , and the other halfe towards the South : in which Circle many thinges are to be considered , first the name , then what breadth it hath , and why it hath such breadth , and into what parts it is deuided aswelt touching the breadth or Latitude , as circuit or Longitude thereof , also into how many parts the firmament is deuided by the spaces described in the Zodiaque , and appointed to the 12. signes . Also how much it declineth from the Equinoctiall towards the South or North , and vpon what poles it turneth about , and why the line in the midst is called the Ecliptique line , and many other necessary points , which I mind here briefely to touch .

Why is this Circle named the Zodiaque ?

It is named the Zodiaque either of this Gréeke worde zoe , which is asmuch to say as life , because the sunne being mooued vnder this Circle , giueth life to the inferiour bodies , or else of this Gréeke word Zodion , which is as much to say as a beast , because that 12. Images of stars , otherwise called the 12. signes , named by the name of certaine beasts , are formed in this Circle : and therefore the Latines doe call this Circle Signifer because it beareth the 12. signes .

How are these signes called , and with what Characters are they marked .

Their names and Characters this table here following doth shew , and also which be opposite one to another , as Aries to Libra , Taurus to Scorpio , and so foorth .

The sixe Northerne signes : The sixe Southerne signes . Aries ♈ Libra ♎ Taurus ♉ Scorpio ♏ Gemini ♊ Sagittarius ♐ Cancer ♋ Capricornus ♑ Leo ♌ Aquarius ♒ Virgo ♍ Pisces ♓

Of which signes the first sixe on the left hande are called the Northerne signes , because they are conteyned in that halfe of the Zodiaque , which declineth towardes the North. And the other sixe on the right hand being right opposite to the first 6. are called the Southerne signes , because they are conteyned in the other halfe of the Zodiaque , declyning towardes the South : And against every signe is set his proper Character .

What diuisions doe Astronomers make of the 12. signes ?

Diuerse , as followeth : for some are said to be ascendent , and some descendent .

Ascendent are those that rise from the South towards our Zenith , tending Northward , which be these , Capricornus , Aquarius , Pisces , Aries , Taurus , and Gemini .

The descendent are these , Cancer , Leo , Virgo , Libra , Scorpius , and Sagittarius .

Againe some are saide to be vernall , as Aries , Taurus , and Gemini .

Some Estiuall as Cancer , Leo , Virgo .

Some Autumnall as , Libra , Scorpio and Sagittarius .

And some Hiemall or Brumall , as Capricornus , Aquarius , and Pisces , signifying the foure seasons of the yeare , that is to say , the Spring , Sommer , Autumne , and Winter .

And some doe make diuerse other diuisions , which because they appertame to Astrologie rather then to the Treatise of a Spheare , I willingly omit them .

Now tell what breadth the Zodiaque hath , and why it is imagined to haue such breadth ?

It hath ( according to the ancient writers ) 12. degrées in bredth , that is to say , 6. degrées on the one side of the Ecliptique line , & 6. degrées on the other side of the Ecliptique line , but according to the moderne writers , it hath 16. degrées in breadth , that is , eight degrées on ech side of the Ecliptique line , which breadth was necessarily imagined , first to the intent that by the measures and degrées thereof it might be knowne , how much the Planets ( otherwise called the wandring starres , whose course is to passe continually vnder this broad Circle ) doe wander at any time on either side of the Ecliptique line , for they all wander , but some more , some lesse , the Sunne onely excepted , which neuer swarueth from that line , but alwayes goeth right vnder the same , and therefore the saide line is called of some writers the way of the Sunne : And secondly it hath such breadth to the intent it may containe within the same , the 12. signes aforesaid , by meanes of which signes the whole circuit or longitude of the saide Circle is deuided into 12. equall parts , and euery such part is deuided into 30. degrées , and euery degrée into 60. minutes , and euery minute into 60. seconds , &c. so as the whole Longitude thereof contayneth 360. degrées , according vnto which diuision , all the rest of the Circles both greater and lesser described in the Spheare , are made to containe the like number of degrées , and euery halfe Circle to containe 180. degrées , and euery quarter of a Circle to containe 90. degrées , and by this diuision aswell of the breadth as of the length of the Zodiaque , it appeareth that euery one of the 12. Signes hath 30. degrées in length , and 12. degrées in bredth , and thereof the Planets , Starres , and all other Celestiall bodies are said to haue both Longitude and Latitude , the sunne onely excepted .

How is such Longitude and Latitude to be vnderstood ?

The Longitude of any Planet or Starre is to be counted in the Ecliptique line containing in circuit 360. degrées , reckoning from the first point of Aries , and so to Taurus , Gemini , and Carcer , & so foorth according to the succession of the signes , vntill you come againe vnto the first point of Aries , at which point such Longitude both endeth & begimeth . The Latitude is counted from the said Ecliptique line towards any of the Poles of the Zodiaque . And hereof looke how many degrées any fixed starre or Planet is distant from the Ecliptiqueline towards any of the said poles , so much Latitude it is said to haue either Northerne or Southerne : moreouer , by this diuision of the signes the whole firmament is deuided into 12. parts by reason of 6. Circles imagined to passe through the poles of the Zodiaque , and also through the beginning of euery signe , wherby we knowe vnder what signe euery star is situated though it be cleane out of the Zodiaque as this figure here plainely sheweth , marked with these letters A. B. C. D. A signifieth the North pole of the worlde , B. the North pole of the Zodiaque , C. the South pole of the world , D. the South pole of the Zodiaque .

How much the Zodiaque declineth from the Equinoctiall towards either of the Poles , and of the greatest declination of the Sunne , what it is at this present , and what it hath beene in times past . Chap. 12.

YOu haue to vnderstande that the Zodiaque or rather the Ecliptique line , declineth from the Equinoctiall towardes eyther of the poles , 23. degrées 28′· and that is said in these dayes to be the greatest declination of the sunne , which declination is twofold , that is Northerne and Southerne , for like as the Sunne entring into the first point of Aries , beginneth then to decline from the Equinoctiall Northward , to the quantitie of 23. degrées and 28′· so entring into the first point of Libra , he declineth againe from the Equinoctiall as much Southward . And note by the way that by reason of his slowe motion , when he is in the Northerne signes , he spendeth 7. dayes , and ⅗ . of a day more in making his North declination then in making his South declination , because he is then in his swift motion , and the time hath béene that he hath spent aboue ten dayes more in making his North declination , then in making his South declination : neither is the greatest declination of the Sunne in all ages of like quantitie . For in Ptolomies time it was 23. degrées , 51′· and 20″· euer since whose time it hath alwaies continually decreased vntill this present , so as now the greatest declination is no more but 23. degrées and 28′· And Copernicus maketh the declination of the sunne in respect of quantitie to be twofold , that is , greatest and least , affirming the greatest to be 23. degrées and 52′· and the least to be 23. degrées and 28′· as it is now accounted , the difference whereof is 24′· and whilst the Ecliptique departeth from the Equinoctiall , and turneth againe towardes the Equinoctiall , there do runne ( as he sayth ) 34●4 . yeares .

How to know the quantitie of the Sunnes declination be it Northward or Southward , euery day throughout the yeare , as well by a Table as by helpe of the Spheare . Chap. 13.

THis is chiefely to be knowne by Tables calculated of purpose , which Tables most commonly are eyther made to answere euery day of the moneth , or else to the degrée of the signe wherein the sunne is euery day , which kind of Table is contained in lesser roome then the other , but to worke by such a Table , you must first knowe in what signe and degrée the sunne is euery day .

How is that to be done ?

It is to be knowne most truely by the Ephemerides or such like Table caluclated of purpose , shewing not onely the degrée of the signe , but also the very minute wherein the sunne is euery day , and for want of such a Table , you may without consideration of the minutes , know it by such an instrument or figure as this following , which consisteth of diuerse Circles , whereof the outermost conteyneth the degrées of the 12. signes , together with the names of the said signes , & the next the dayes of ech moneth , together with the names 〈◊〉 ●he said moneths , much like the backside of an Astrolabe , in the center or midst of which ●●●●rument or figure is a thred , which if you lay vpon the day of the month which you seek , 〈◊〉 straight direct you to the degrée of the signe wherein the Sun is that day , as for ex●●ple , if you would know in what signe and degrée the Sun is the 4. of May , then by draw●●● the thred right vpon the said day ouer and beyond the outermost circle , you shal finde 〈◊〉 it will fall right vpon the 23. degrée of Taurus .

An instrument to know thereby in what signe and degree the Sunne is euery day throughout the yeare .

Then hauing found the degrée of the Sun you must resort therewith to this Table following , made for the declination of the sun .

ATableshewingthedeclinationofthesuneuerydaythroughouttheyear Degrees of the Signes . ♈ ♎ ♉ ♏ ♊ ♐ Degrees of the signes . D M S D M S D M S 1 0 23 53 11 50 6 20 22 57 29 2 0 47 46 12 10 56 20 35 7 28 3 1 11 39 12 31 34 20 46 55 27 4 1 35 30 12 51 59 20 58 20 26 5 1 59 20 13 12 12 21 9 21 25 6 2 23 8 13 32 12 21 19 59 24 7 2 46 54 13 51 58 21 30 13 23 8 3 10 37 14 11 30 21 40 3 22 9 3 34 18 14 30 48 21 49 29 21 10 3 57 54 14 49 51 21 58 29 20 11 4 21 28 15 8 40 22 7 6 19 12 4 44 57 15 27 13 22 15 17 18 13 5 8 22 15 45 30 22 23 3 17 14 5 31 42 16 3 32 22 30 24 16 15 5 54 57 16 21 17 22 37 19 15 16 6 18 6 16 38 44 22 43 48 14 17 6 41 9 16 55 55 22 49 50 13 18 7 4 6 1● 12 48 22 55 27 12 19 7 26 57 ●● 29 23 22 0 38 11 20 7 49 40 17 45 40 23 5 22 10 21 8 12 16 18 1 39 23 9 39 9 22 8 34 45 18 17 18 23 13 29 8 23 8 57 5 18 32 37 23 16 53 7 24 9 16 16 18 47 38 23 19 50 6 25 9 41 19 19 2 18 23 22 19 5 26 10 3 12 19 16 37 23 24 22 4 27 10 24 56 19 30 36 23 25 57 3 28 10 46 30 19 44 14 23 27 5 2 29 11 7 53 19 57 30 23 27 46 1 30 11 29 5 20 10 25 23 28 9 0 ● D M S D M S D M S D ♍ ♓ ♌ ♒ ♋ ♑

The description and vse of the Table .

This Table consisteth of 5. collums , whereof the first on the left hand , and the last on the right hand do containe the degrées of the 12. signes of the Zodiaque , counting from one to 30. And the thrée middle collums do contain the degrées minutes , and seconds of declination , ouer the head of which thrée middle collums are set downe the characters of these 6. opposit signes , Aries and Libra , Taurus and Scorpio , Gemini and Sagittarius , and at the foot of the said midlde colums are set down the characters of the other 6. opposit signes , that is , Virgo and Pisces , Leo & Aquarius , Cancer and Capricornus , for euery 2. opposit signes , aswell aboue as beneath , haue like declination , the vse of which Table is thus : first hauing found out the degrée of the Sun by the instrument before described , or rather by some true Ephemerides , you must séeke out the number of the said degrée , either in the first or last collum , according as the character of the signe is placed . For if the signe or character be aboue , then you must séeke for the said number in the first collum on the left hand , which numbers do discend from 1. to 30. but if the signe be beneath , then you must find it out in the the outermost colum on the right hand , the numbers wherof do ascend from 1. to 30. and the common angle or square , standing right against the said number will shew you the degrée , minutes and seconds of the declination , as for example , hauing found by the former instrument that the 4. day of May the Sun is in the 23. degrée of Taurus , I séeing the character of Taurus to stand aboue , doe séeke my foresaid number of 23. degrées in the first collum on the left hand , and in the common angle or square right against that number , and vnder the signe Taurus , I find the declination of the Sun to be 18. degrées 32′· and 37″· But this Table cannot serue alwayes : yea rather such tables are to bee renewed as our Astronomers say euery 30. yeares . Also you may know the daylie declination of the Sun , by helpe of a materiall Spheare or globe : thus hauing set the Spheare at your latitude , bring the degrée of the signe wherein the Sanne is that present day vnto the moouable Meridian , and staying it there , marke whether it falleth on the South side or on the North side of the Equinoctiall : for if it be in any of the Northern signes , it will fall on the North side of the Equinoctial , and if it be in any of the Southern signes , it will fall on the south side of the Equinoctiall , and by counting the degrées vpon the Meridian , contained betwixt the degrée of the Sun and the Equinoctial , you shal know what declination the Sun hath that day , as for example in the latitude 52. in the yeare 1590. the fift day of May , I find the Sunne by the Ephemerides to be in the 23. degrée and 48′· of Taurus , which point I bring to the moouable Meridian , and there staying it , I find that point to be distant from the Equinoctiall northward 18. degrées & certaine minutes , and so much of North declination I conclude the Sun to haue that day .

Vpon what Poles the Zodiaque turneth about , also of the Ecliptique line and of the diuers vses thereof . Chap. 14.

THe Zodiaque turneth about vpon his own proper Poles from West to East , whereof the one being placed in the Colure of the Solstices towards the North , is distant from the pole Arctique 23. degrées and 28′· and the other is placed in the said Colure towards the South , being of like distance from the pole Antarctique , whereof the Astronomers haue a generall rule , affirming that the distance of the two Poles of the world from the Poles of the Zodiaque , is alwaies equal to the greatest declination of the sun , which as hath bene said before , is 23. degrées & 28′· as you may plainly sée by the Sphear . And note that these 2. Poles are otherwise called the Poles of the Ecliptique , for in considering the declination of the Sun or of the Zodiaque from the Equinoctiall , you must haue respect onely to the Ecliptique line , which is in the midst of the Zodiaque , and not to any other part of the Zodiaque . And as the Equinoctiall line sheweth the moouing of the first moouable , which is from East to West , so the Ecliptique line sheweth the moouing of the second mooueable , which is from West to East , clean contrary to the first moouable , the causes whereof haue bene before declared .

What other vses hath this line more than you haue already declared ?

It hath diuerse , for in this line or circle are noted the degrées , wherewith any starre riseth or goeth downe , either rightly or obliquely , for all the apparances of the heauens are chiefly referred to this circle . Againe , by this circle the chiefest distinctions and parts of times , as yeares and moneths are knowne , and also the foure seasons of the yeare , as Spring , Summer , Autumne , and Winter : Moreouer , the obliquitie of this circle vnder which the Sun continually walketh , is cause that the dayes both naturall and artificiall are vnquaell . Finally , this circle doth shew the places and times of the Eclipses both Solar and Lunar , from whence this line taketh his name , of which Eclipses we minde here briefly to treate .

Of the Eclipses both Solar and Lunar , and of the head and taile of the Dragon , with certaine figures shewing the same . Chap. 15.

WHat signifieth this word Eclipsis ?

It is as much to say , as to want light , & to be darkened or hidden from our sight , as when the Sun & Moon are both at one self time right vnder the Ecliptique line , the one of these 2. lights most commonly is eclipsed and darkned : for there be two Eclipses , the one of the Sunne , and the other of the Moone , but sith that neither the eclipse of the Sun or of the Moone doth chance , but when they méet either in the head or tail of the Dragon , I think it good to shew first what is meant by the head and tayle of the Dragon . The Dragon then signifieth none other thing but the intersection of 2. circles , that is to say , of the Ecliptique , & of the circle that carieth the Moon , called her Defferent ▪ cutting one another in 2. pointes , whereof that intersection which is westward when as the Moon goeth towards the North , is called the head , and that which is Eastwardes when the Moone goeth towards the South is called the taile , marked with such characters as you sée in the figure following , and that part towards the South is called of some the belly of the Dragon . And note that the Defferent of the Moone is at no time distant from the Ecliptique aboue 5. degrées at the most .

The figure of the Dragon .

This being presupposed , I will speake first of the Eclipse of the Moone , and then of the Sunne , both which may bee eclipsed either totally or in part .

When is the Moone said to be totally eclipsed ?

When the Sunne and Moone are opposite one to the other diame trallie , and the earth in the verie midst betwéen them both , for the bodie of the earth being thick and not transparent , casting his shadow to that point which is opposite to the place of the Sun , will not suffer the Moone to receiue any light from the Sun , from whome she alwaies borroweth her light .

At what time is the Moone said to be diamtrallie opposit to the Sunne ?

When a right line drawne from the center of the Sun to the center of the Moone , passeth through the center of the earth : & note that euery time that she is at the ful , she is opposit to the Sun , and yet the earth is not at euery such full diametrally betwixt her and the Sun , for then she should be eclypsed at euery ful , which indéed cannot be but when she is either in the head or tail of the Dragon .

When is the Moone said to be eclipsed in part ?

When is the Sunne saide to be eclipsed ?

When the Moone is betwixt the Sunne and the Earth , which chanceth in a Coniunction , and yet not in euery coniunction , but when it falleth either in the head or taile of the Dragon , which may chance , as I saide before , either totally or in part : totallie I say , in respect of those parts of the earth whereon the shadowe directly falleth . For sith the Moone is farre lesser than the Earth , she cannot shadow all the Earth , and therefore the Eclipse of the Sunne cannot be vniuersall , but yet to some partes of the Earth totally , and to some partlie , and to other some nothing at all , as you may plainlie sée by this figure following .

Yet all the histories doe affirme that the Eclipse of the Sun was vniuersall at the death of Christ .

Yea , that was miraculous , and also it was then at the full of the Moone , which was also as miraculous : and therefore Dionisius a Senator of Athens , beholding that Eclipse cryed out , saying these words , Either God this day suffereth , or els the world must needs perish for euer : which Dionisius was the first that conuerted the Frenchmen to the faith of Christ , doing there great miracles : in honour of whome was erected the rich Abbey of S. Denise , not far from Paris , whereas the Kinges and Princes of France were woont to be buried .

How is it to be prooued that the Eclipse at Christ his death was at the full of the Moone ?

Aswell by ancient historie , as by S. Augustine , who saith that the Iewes were woont to kéepe their feast of Passouer ( at which time God suffered ) alway at the full of the Moone .

If the Sun and Moone be eclipsed but in part , how are such partes to be accounted ?

By the parts of the Diameter of the bodies of those two Planets , for the Astronomers doe diuide the Diameter aswell of the Sunne , as of the Moone into 12. and some into 24. parts , which they call points , and therefore are woont to say , that the Sunne or Moone are darkned or eclipsed 7. points , 8. points , 9. points , &c.

Of the two Colures , and why they are so named , and wherto they serue : also of the foure Cardinall points , that is , of the two Equinoctiall , and the two Solstitiall points , and of the entrance of the Sunne into any of those points or into any other signe . Chap. 16.

WHat be Colures ?

They be great moouable circles passing through both the Poles of the world , which the Astronomers do otherwise call Circles of declination , wherof they make 180. which are halfe so manie as there be degrées in the Equinoctiall applying them to diuers vses not néedfull here to be rehearsed , for sith that there are but two Colures accustomablie set downe in the Spheare , without the which a materiall Spheare cannot bée made , I minde héere therefore onelie to treate of them .

Shew first what ●his name Colure signifieth .

This word Colure béeing compounded of Colos and Oura , is as much to say as vnperfect or maymed , the taile being cut off , because none of those Circles are euere séene whole aboue our Horizon , but parte thereof , for some part is alwayes séene , and some part is alwaies hidden , as that part which is aboue the Horizon , and nigh vnto the Pole , is alwayes séene , because it neuer goeth downe vnder the Horizon : likewise that which is nigh vnto the South pole is alwayes hidden from vs , because it neuer riseth aboue our Horizon , as by turning the Spheare about , you may easilie perceiue the same .

Which be those Colures that are commonly set downe in the Spheare , and how are they named ?

They are two great moouable circles , passing through the Poles of the world , crossing one another in the said Poles with right Sphearicall angles , by meanes whereof they deuide the whole Spheare into foure equall partes , of which two Colures the one is called the Colure of the Equinoxes , and the other the Colure of Solstices .

Describe these two Circles , & shew why they are so named .

The Colure of the Equinoxes is so called because it cutteth the Zodiaque in the beginning of Aries , which is called the vernal Equinoxe : and also in the beginning of Libra , which is called the Autumnall Equinoxe , at which two times the dayes and nightes be equall , as hath bene said before when we did speake of the Equinoctiall circle , and this circle deuideth the Ecliptique into two halfes , the one Septentrionall , and the other Meridionall and thereby sheweth the signes ▪ wherein the Sunne maketh the dayes longer and shorter than the nightes , for whilest hee is an the 6. Northern signes , he maketh the dayes with vs longer than the nights , and when he is in the 6. Southern signes he maketh the nights longer than the dayes : now you haue to vnderstand that the Colure of the Solstice , is so called because it cutteth the Zodiaque in the two Solistitiall points , that is to say , in the beginning of Cancer , and the beginning of Capricorne , as you may sée in beholding and turning the Spheare about with your hand .

Why are these two points called Solstitiall ?

They take their name of these two Latine words Sol and stano , that is to say , the Sun and standing , for when the Sunne is in any of the two points , hee séemeth to stand still , or at the least mooueth so little , as his proper moouing from West to East cannot be easilie perceiued , during the space of twelue dayes . And you haue to note , that when the Sunne entreth into the first degrée of Cancer , which is about the 12. of Iune , then hée is at the highest , and the dayes be at the longest , and therefore , it is called the Summer Solstice . Againe , when hee entreth into the first degrée of Capricorne , which is about the 12. of December , then the Sunne is at the lowest , and the nightes are at the longest , and therefore it is called the Winter Solstice . And in this Colure there are set downe the two Poles of the Ecl●ptique line being distant from the Poles of the worlde 23. degrées and 28′· Moreouer on this colure is measured the greatest declination of the Sunne , which is alwayes equall to the distance of the Pole of the Ecliptique , from the Pole of the world , as hath bene said before . And you haue to note that the 4. former points , that is to say , the 2. Equinoxes , and the 2. Solistices , are commonlie called the foure cardinall or principall pointes , and of some they are called , the foure points or Change , signifying the 4. beginnings of the four diuers seasons of the yeare : for betwixt the beginning of Aries and beginning of Cancer , is contained the Spring time , and betwixt the beginning of Cancer and the beginning of Libra , is the Summer time : and from the beginning of Libra to the beginning of Capricorne is the time called Autumne , or fal of the leaf : and from the beginning of Capricorne to the beginning of Aries is contained the winter season , albeit the Sun entreth not into any of these signes alwaies at one self day or time of the yeare , for at Christ his incarnation , the Sun entred into Aries the 25. of March , and into Cancer the 24. of Iune , and into Libra the 27. of September , and into Capricorn the 25. of December , which was then the shortest day in the yéer , and the beginning of Winter , and therefore is called of the Latines , dies brumalis , on which day Christ our Sauiour was born , so as from the time of his birth vnto this present yeare , there are run almost 13. daies , wherefore , vnlesse the kalenders bee reformed as well here in England as els where ( for the Romane reformation is not so exactlie true as it might be ) wee shall haue in processe of time , the Spring in Winter , and the Winter in Autumne .

How shall I know this present yeare , or any yeare to come hereafter , at what day and houre the Sunne entreth into any of the 12. Signes ?

First you must learne by some good Ephemerides , or other Table , the true entrance of the Sunne into euerie signe in any yeare passed before , then from the time of the entrance of the Sun into the signe which you desire to know , consider how manie years are betwixt , & how many leap years are in the same contained , and subtract for so many times as there be leape yeares , 44′· of an houre , and ad to the hours remaining , so many times fiue houres , and 49′· as there be yeares remaining ouer and besides the leape yeares , and that summe shall shewe you the day , houre , and minutes of the true entrance of the Sun into that signe in the same year that you desire to knowe .

Of the Horizon both right and oblique , making thereby three kindes of Spheares , that is , the right , the Paralell , and the oblique Spheare . Chap. 17.

WHat is the Horizon ?

It is a great immoouable circle which deuideth the vpper Hemisphear , which is as much to say , as the vpper half of the world which we sée , from the nether Hemispheare which we see not , for standing in a plaine field , or rather vpon some high mountaine voyd of bushes & trées , and looking round about , you shall see your selfe inuironed as it were with a circle , and to be in the verie midst or centre thereof , beneath or beyond which circle , your sight cannot passe , and therefore this circle in Gréeke is called Horizon , and in Latine Finitor , that is to say , that which determineth , limitteth or boundeth the sight , the Poles of which circle are imagined to be two points in the firmament , whereof the one standeth right ouer your head , called in Arabick Zenith : and the other directlie vnder your féete , called in the same tongue Nadir , that is to say the point opposite , and from point to point you must imagine that there goeth a right line passing through the centre of the world , and also through your bodie both head and féet , which is called the Axletrée of the Horizon , and you haue to vnderstand that of Horizons ther be 2. kinds , that is , right , & oblique , making 3. kinds of Sphears , that is to say , the right Spheare , the paralel Spheare , and the oblique Spheare ,

When is the Horizon said to be right , and thereby to make a right Spheare ?

It may be said to be right two manner of waies , first , when the Horizon passeth through both the Poles of the world , cutting the Equinoctiall with right angles , in which Spheare they that dwel haue their Zenith in the Equinoctiall , which passeth right ouer their heads , to whom the daies and nights are alwaies equal . Secondly , they are said to haue a right Horizon , & to dwell in a right Spheare , to whom one of the Poles of the world is their Zenith , and their Horizon is all one with the Equinoctial , cutting the Axletrée of the world in the very midst with right angles , and because the Horizon & the Equinoctial are Paralels , this kind of Spheare is called a paralel Spheare , in which Sphear they that dwel haue 6. moneths day , and 6. moneths night , as you may easily perceiue by placing the Sphear , so as one of the Poles may stand right vp in the midst of the Horizon , by means wherof you shall sée 6. signes of the Zodiaque to be alwayes aboue the Horizon , and 6. signes to be alwayes vnder the Horizon : Againe by placing the Sphear so as both the Poles may lie vpon the Horizon , you shall sée the shape of the first right Spheare , wherein the Horizon passeth throught both the Poles of the worlde , and the Equinoctiall passeth through the Poles of the Horizon , which are the two points called before the Zenith and Nadir .

When is it said to be an oblique Horizon , and thereby to make an oblique Spheare ?

When the Pole of the world is eleuated aboue the Horizon , be it neuer so little , so as the Horizon do cut the Equinoctiall with oblique angles , and look how much the Pole of the world is eleuated aboue your Horizon , so much is your Zenith distant from the Equinoctial , and the nigher that your Horizon approcheth to the Pole , the nigher your Zenith approcheth to the Equinoctial . Againe , look how much the Equinoctiall is eleuated aboue your Horizon , so much is your Zenith distant from the Pole , all which things this figure here following doth plainly shew , whereby you may easily perceiue that the latitude , which is the distance of your Zenith from the Equinoctial , is alwaies equal to the altitude of the Pole , which is the distance betwixt your Horizon & the Pole , as for example , knowing the latitude of Norwich to be 52. degr. lay the zenith of this figure vpon the 52. degrées , reckoning from the Equinoctial towards the pole Arctique on your left hand , and look what distance is betwixt the said zenith and the Equinoctial , the self same distance you shall find to be betwixt the Horizon and the foresaid Pole on your right hand , and you may doe the like vpon the Spheare it self by raising the moouable Meridian aboue the Horizon at that altitude , so as the 52. degr. may be euen with the Horizon .

A Figure shewing the latitude of any place to bee equall to the eleuation of the Pole .

What other vses hath this circle ?

In this circle are set down the foure quarters of the world , as East , West , North and South , and the rest of the winds : Againe , this circle deuideth the a●●ificial day from the artificial night for al the while that the Sun is aboue the Horizon it is day , and whilest it is vnder the same it is night . And by this circle we knowe what starres doe continually appeare , and which are continually hidden , also what starres doe rise and goe downe . Againe , in taking the eleuation of the Pole , this circle is chieflie to be considedered , for when we know how many degrées the Pole is raised aboue the Horizon , then we haue the eleuation therof for that place . For to euery seueral place , yea to euery little moment of the earth in an oblique Spheare , belongeth his proper Horizon and seueral altitude of the Pole , whereby it appeareth that the Horizons are infinite and without number .

How shall I knowe in any place , hauing an oblique Horizon , how much the Pole is eleuated aboue the Horizon ?

That is declared in the second booke of this Treatise , whereas I speake of the latitude and longitude of the earth , in the eight chapter .

Of the Meridian , and of the vses thereof . Chap. 18.

WHat is the Meridian ?

It is a great immoouable circle passing through the Poles of the world , and through the Poles of the Horizon .

Why is it called the Meridian ?

Because that when the sun rising aboue the Horizon in the East , commeth to touch this line with the center of his body , them it is mid-day or noontide to those , through whose Zenith that circle passeth And when the sun after his going down in the west commeth to touch the self line again in the point opposit , it is to them midnight , & note that diuers cities , hauing diuers latitudes , that is to say , being distant one from another North and South bee it neuer so farre , may haue one selfe Meridian : but if they be distant one from another East and West , bee it neuer so little , then they must needes haue diuerse Meridians , and such distance betwixt the two seuerall Meridians , is called the difference of longitude , whereof we shall speake hereafter more at large when we come to treat of the longitude and latitude of the earth , which something differeth from the longitude and latitude of the Starres or Planets , whereof we haue alreadie spoken in the 11. chapter .

How many Meridians be there ?

The Astronomers doe appoint for euery two degrées of the Equinoctiall a Meridian , so as they make in al 180. Albeit most commonly in the Spheare they set downe but one , which serueth for all by turning the bodie of the Spheare to it , which for that cause is called the moouable Meridian . And in such Spheares as haue not a foote and a standing Horizon , there is no Meridian at all , but the two Colures are faine to supply their want , but all terrestriall Globes are commonly described with twelue Meridians , cutting the Equinoctiall in 24. pointes , and deuiding the lame into 24. spaces , euerie space containing 15. degrées , which is an houre , by meanes whereof we know how much sooner or latter it is noonetide in any place , for it is noontide sooner to those whose Meridian is more Eastward then to them whose Meridian is more Westward . And contrariwise the Eclpise of the Sun or Moone appeareth sooner to those whose Meridian is more Westward .

What other vses hath this circle ?

This circle deuideth the East part of the world from the West and also it sheweth both the North and South , for by turning your face towardes the East , you shall find the Sunne being in that line at noone tide to be on your right hand right South , the opposit part of which circle sheweth on your left hand the North. Also this circle by reason that it passeth through both the Poles of the world , deuideth both the Equinoctiall and all his Paralels into two equall parts aswell aboue the Horizon as vnder the Horizon , and by that means it deuideth the artificial day and artificiall night each of them into two parts , that is to say , into two semidiurnal & into two seminocturnal partes . For betwixt that part of the Horizon where the sun riseth , mounting stil vntil he come to this circle , which is at noontide , is contayned the first halfe of the day , & the other half is from the same circle to the going down of the Sun vnder the Horizon . And the first part of the night is the space betwixt the Suns going down and his comming again to the Meridian , which is at midnight , and from thence to the time of his rising is the other half of the night , and also the Astronomers take the beginning of their naturall day from this circle , counting either from noontide to noontide , or els from midnight to midnight . Againe , this circle sheweth the right ascentions and declinations of the starres , and the highest altitude , otherwise called the Meridian altitude of the Sun or of any starre , or degrée of the Ecliptique , or of any other point in the firmament , all which vses and many others more you shal better vnderstand hereafter , when we come to shew the vses of the globe aswell terrestriall as celestiall .

Of the vertical circle , and vses thereof . Chap. 19.

BVt here you haue to note that though the most part of Geographers do set down in their Spheares but 6. great circles , yet there is another great circle called the circle Verticall , which passeth right ouer our heads through our zenith , wheresoeuer we be vpon the land or sea , crossing our Horizon in 2. points opposite , and deuiding the same into two equal parts , and such kind of circles are called in Arabicke Azimuthes , whereof you may imagine that there be so many as there be rombes or winds in the Marriners Compasse , which are in number 32. yea , and if you will , you may make halfe so many as there be degrees in the Horizon , which are in number 360. the halfe whereof is 180. If you be right vnder the Equinoctiall , and doe goe or saile right East or West , then the Equinoctiall is your Verticall circle , and if you goe or saile right North or South , then the Meridian is your verticall circle , which two circles notwithstanding do alwayes kéepe their names . But in sayling by any other rombe , that circle which is imagined to passe from the true East point right ouer your head vnto the true West point , or which crosseth your Meridian in the zenith point with right Sphericall angles , is most properly called the verticall circle , and the learned seamen haue great respect to two speciall kinds of Verticall circles , that is , the Magnetical Meridian , and the Azimuth of the Sun .

What manner of verticall Circles be those , and whereto serue they ?

M. Borrough in his discourse of the variation of the Compasse , defineth the Magneticall Meridian to be a great Circle , which passeth through the Zenith and the Pole of the load stone called in Latine Magnes , and deuideth the Horizon into two equall parts , by crossing the same in two points opposite . Againe the Azimuth of the sunne is a great Circle , passing through the Zenith & the Centre of the sunne in what part of the heauen so euer he be , so as he be aboue the Horizon , which Circle deuideth the Horizon into two equall parts , by crossing the same in two points opposite , And by helpe of these two Circles and a certaine instrument made of purpose to giue a true shadowe , he teacheth to finde out the true Meridian of any place : And also to know how much any Mariners Compasse doth varie from the true North and South , in Northeasting or Northwesting , whereof I shall speake more at large hereafter in my treatise of Nauigation .

What vse is there of the verticall Circle or Azimuthes ?

The verticall Circle sheweth what time the Sunne or any other starre rising beyond the true East point , is passed before the Sunne or saide starre , commeth to the true East or any other rombe . Also in what Coast or part of heauen , the Sunne , Moone , or any other starre is at any time being mounted aboue the Horizon , as whether it be Southeast or Northeast , or in any other rombe : Also by helpe of the verticall Circle most properly so called , are the twelue houses of heauen set , according to Campanus and Gazula . And by helpe of these Circles you may also knowe how any place vpon the earth beareth one from another eyther Eastward or Westward , and so foorth , for euery place hath his seuerall Azimuth answerable to the Horizon and Zenith of the saide place .

Of certaine Circles called Almicanterathes .

Since I haue spoken here somewhat of the verticall Circles called Azimuthes , it shal not be amisse to shew you also that there be other Circles to be considered of in the Spheare aswell as in the Astrolabe called Almicanterathes , that is to say , Circles of Altitude , which though they be not all great Circles , for euery one is lesser then other procéeding from the oblique Horizon of anie place to the Zenith of the saide place , yet the first Almicanterath which is the verie oblique Horizon it selfe , is a great Circle deuiding the Spheare into two equall partes , and all the rest are lesser and lesser , vntill you come to the very Zenith , and are paralels to the Horizon , euen as the Tropiques and the other lesser Circles are paralels vnto the Equinoctiall and the Zenith in Sphericall bodies is the Centre of them all , though it be not so in Astrolabes , for there euery Almicanterath is fame to haue his seuerall Centre , of which Circles there be in all 90. according to the number of 90. degrées contayned betwixt the oblique Horizon and the Zenith , and these Circles doe serue to shewe the Altitude of the Sunne or Moone , or of any other starre fixed or wandring , being mounted at any time aboue the oblique Horizon , which is easie to be founde by any Quadrant , Crosse-staffe , or Astrolabe . But leauing to speake any further of these Circles , because they are not vsed to bee described in Spheares but onely in Astrolabes , I will nowe treate of the foure lesser Circles before mentioned , which are commonly set downe in euery Spheare or Globe .

Of the foure lesser circles , that is to say , the circle Arctique , the circle Antarctique , the Tropique of Cancer , and the Tropique of Capricorne , and also of the fiue Zones , that is to say , two cold , two temperate , and one extremely hoat . Chap. 20.

WHich call you the lesser Circles ?

They are those that doe not deuide the Spheare into two equall parts , as the great Circles doe , and of such there be foure , that is the two Polar circles , and the two Tropiques , that is to say , the Tropique of Cancer , and the Tropique of Capricorne , of which Polar circles the one is called Arctique , and the other Antarctique , and are made by the turning about of the two Poles of the Zodiaque , which Poles being situated in the Colure of the Solstices are so farre distant from the Poles of the world , as is the greatest declination of the Sunne from the Equinoctial , which is 23. degrées , 28′· as hath béene said before .

Which is the Arctique Circle , and why is it so called ?

The Arctique Circle is that which is next to the North pole , and hath his name of this worde Arctos which is the great beare or Charles wayne , which are seuen starres placed next to this Circle on the outside thereof , and it is otherwise called the Septentrionall Circle of this worde Septentrio , which is as much to say as seuen Oxen , signified by the seuen starres of the little Beare , which doe mooue slowely like Oxen , and are placed all within the sayde Circle , and the bright starre that is in the tippe of the tayle of the sayde little Beare , is called of the Mariners the loade starre or North starre , whereby they sayle on the Sea , and the Centre of this Circle is the North Pole of the world which is not to be séene with mans eye .

What is the Antarctique Circle ?

It is that which is next vnto the South Pole , and it is so called , because it is opposite or contrarie to the Circle Arctique .

Now describe the two Tropiques .

The Tropique of Cancer is a Circle imagined to bée betwixt the Equinoctiall and the Circle Arctique , which Circle the Sunne maketh when he entreth into the first degrée of Cancer , which is about the twelfth or thirtéenth day of Iune being then in his greatest declination from the Equinoctiall Northwarde , and nighest to our Zenith , béeing ascended to the highest point that hee can goe , at which time the dayes with vs be at the longest , and the nightes at the shortest . And so from thence hée declineth to the other Tropique called the Tropique of Capricorne , which is a Circle imagined to bée betwixt the Equinoctiall and the Circle Antarctique , which the Sunne maketh when he entreth into the first degrée of Capricorne , which is about the twelfth or thirtéenth day of December , at which time he is againe in his greatest declination from the Equinoctiall Southwarde , and furthest from our Zenith : whereby the dayes with vs bée then at the shortest , and the nights at the longest : And note that these two Circles are called Tropiques of this Gréeke worde Tropos , which is as much to say as a conuersion or turning , for when the Sunne arriueth to any of these two Circles , hée turneth backe againe eyther ascending or descending , by reason of which foure Circles as well the firmament as the earth is deuided into fiue Zones , that is to say , two colde , two temperate , and one extremely hoat , otherwise called the burnt Zone , of which fiue Zones , the foresaide foure Circles are the true boundes . For of the two colde Zones , the one lyeth betwixt the North pole and the Circle Arctique , and the other lyeth betwixt the South Pole and the Circle Antarctique , and of the two temperate Zones , the one lyeth betwixt the Circle Arctique , and the Tropique of Cancer , and the other lyeth betwixt the Circle Antarctique , and the Tropique of Capricorne , and the extreme hoat Zone lyeth betwixt the two Tropiques , in the middest of which two Tropiques , is the Equinoctiall line , as you may see in this figure , and also in the Spheare or Globe it selfe .

A figure shewing the fiue foresaid Zones .

Of which Zones the auncient men were wont to say that thrée were vnhabitable , that is , the two colde , and the extreame hoat , which experience sheweth in these latter dayes to bee vntrue , as we shall declare more at large when we come to treate of the diuision of the earth : Againe you haue to vnderstande that euerie one of these lesser Circles doth containe in length , 360. degrées as well as euery one of the greater Circles , but the degrées are not of like bignesse , no more then the Circles themselues are like in compasse or circuit , for the lesser the Circles are in circuit , the lesser their degrées must néedes be .

Sith euery of the lesser Circles differ one from another in circuit , and thereby the degrees of euery Circle be lesser then other , how shall I know the true quantitie of euery degree in ech Circle , and how many minutes are required in euery lesser degree proportionally to answere one degree of the Equinoctiall .

For the better knowledge hereof , you must first imagine that there may be as many Circles made from the Equinoctiall towards any of the Poles , as there be degrées of Latitude , which are in number 90. as hath béene saide before : And the nigher that any Circle is to the Equinoctiall , the greater it is in circuit , and the further from the Equinoctiall towards any of the Poles , the lesser in circuit , and therefore more or lesse minutes are requisite to answere to one degrée of the Equinoctiall , as you may easily perceiue by this Table following , consisting of 6. collums , euery front or head whereof is noted with three great letters , D. M. S. signifying degrées , minutes and seconds , sixe times repeated , and in the beginning of the first colum on the left hand is set downe one degrée , which is the first degrée of 90. & nighest vnto the Equinoctiall , right against which one degrée is placed towards the right hand , 59. minutes , and 59. seconds : and so procéeding from degrée to degrée successiuely , vntill you come to 90. you shall finde how many minutes and seconds doe answere to one degrée of the Equinoctiall , and this Table will also serue to shew the difference of miles in euery sundry clyme or paralell , whereof we shall speake hereafter when we come to treat of the earth .

ATableshewinghowmanyminutesarerequisitetomakeonedegree,ineuerylessercircleanswerabletoonedegreeoftheEquinoctiall. D M S 1 59 59 2 59 58 3 59 53 4 59 51 5 59 46 6 59 40 7 59 33 8 59 25 9 59 16 10 59 5 11 58 54 12 58 41 13 58 28 14 58 13 15 57 57 16 57 41 17 57 23 18 57 4 19 56 44 20 56 23 21 56 1 22 55 38 23 55 14 24 54 49 25 54 23 26 53 56 27 53 28 28 52 59 29 52 29 30 51 58 31 51 26 33 50 19 34 49 45 35 49 9 36 48 32 37 47 55 38 47 17 39 46 38 40 45 58 41 45 17 42 44 35 43 43 53 44 43 10 45 42 26 46 41 41 47 40 55 48 40 9 49 39 22 50 38 34 51 37 46 52 36 56 53 36 7 54 35 16 55 34 25 56 33 33 57 32 41 58 31 48 19 30 54 50 30 0 61 29 5 62 28 10 63 27 14 64 26 18 65 25 21 66 24 24 67 23 27 68 22 29 69 21 30 70 20 31 71 19 32 72 18 32 73 17 33 74 16 32 75 15 32 76 14 31 77 13 30 78 12 28 79 11 27 80 10 25 81 9 23 82 8 21 83 7 19 84 6 16 85 5 14 86 4 11 87 3 8 88 2 5 89 1 3 90 0 0

Of the starres and celestiall bodies contained in the firmament , and first of their substance . Chap. 21.

HAuing briefely described all the Circles as well greater as lesser that are imagined to be in the 8. heauen , I thinke it good now to speake somewhat of the starres and celestiall bodies placed in the saide heauen , And first of their substance , & then of their moouing , figure , shape , number , magnitude or greatnes , also of their Longitude , Latitude , declination , ascention , descention both right and oblique , and of the ascentionall difference , and finally of the thréefold Poeticall rising and going downe of the starres , but first of their substance .

Of what substance are the starres ?

The starres bee of the same substance that the heauens are wherein they are placed , differing onely from the same in thicknes , and therefore some defining a starre doe say , that it is a bright and shining bodie , and the thickest part of his heauen , apt both to receiue and to reteine the light of the Sunne , and thereby is visible and obiect to the sight : for the heauen it selfe being most pure , thinne , transparent , and without colour is not visible , and for this cause the milke-white impression in heauen like vnto a white way called of the Astronomers Galaxia , and of the common people our Ladies way is visible to the eye , by reason that it is thicker then any other part of the heauen .

Why are not the starres seene as well in the day , as in the night .

Because they are darkened by the excellent brightnesse of the Sunne from whome they borrowe their chiefest light .

Of the moouing and shape of the starres . Chap. 22.

WHat moouing haue the starres ?

The selfe same moouing that the heauen hath wherein they are placed .

Whereby are the heauens mooued ?

Some saye that the first mooueable is turned by God himselfe , and all the rest of the heauens euery one by his proper intelligence , which though it turneth his heauen about , yet it giueth neither life , sense , nor vnderstanding thereunto , as some haue vntruely holden , affirming the heauens to be liuing and intelligible bodies .

If the starres haue no moouing of themselues , wherof commeth it then , that some seeme to our sight sometime nigher and sometime further off .

All the fixed starres of the firmament are alwayes of like distance , notwithstanding by reason of the manifold moouing of the firmament , wherein they are placed , they séeme to change their places , and sometime to bee more towardes the East or West , North or South . And whereas the vii . Planets called the wandring starres , do change their places now here now there , that chanceth not by their owne moouing , but by the moouing of the heauens wherein they are placed : for a starre being round of shape hath no members méete to walke from one place to another , but onely changeth his place through the motion of his Spheare or heauen wherein such Planet is fixed .

Of the number of the starres , and of their magnitude or greatnesse , and into how many Images they are deuided , and how many starres euery image containeth . Chap. 23.

MAy the starres be numbred by man ?

No , for as Dauid saith , that belongeth onely to God , who as he created them , so he can number them and call them all by their names , notwithstanding the Astronomers by their industry and diligent obseruation , haue attained to the knowledge of many : as first they know the seuenplanetes , otherwise called the wandringe starres , and haue made manifest demonstrations of their motions , and by continuall obseruation haue found out the manifold vertues , powers and influences of the same , but of the fixed starres they could neuer finde more then 1022. and because the starres are not equall in greatnes or bignes , they make fixe differences of greatnesse , appointing to the first difference 15. starres , which are bigger then all the rest , whereof euery one containeth the earth 207. times , to the second difference 45. starres , whereof euery one containeth the earth 90. times . To the third they appoint 208. starres , whereof euerie one containeth the earth 72. times . To the fourth difference they appoint 474. starres , wherof euery one containeth the earth 54. times . To the fift they assigne 217. starres , whereof euery one containeth the earth 57. times . To the sixt or last greatnesse they appoint 49. small stars , whereof euery one containeth the earth 18. times , and some say 20. times . Besides these there be 14. others , whereof 5. be called clowdy and the other darke , because they are not to be séene but of a very quicke and sharpe sight . And you haue to note that the antient Astronomers do deuide all the fixed starres to them knowne into 48. images , whereof they liken some to liuing things as to men , women , beastes , monsters , foules , fishes , and créeping wormes , and some to things without life , hauing some artificiall shape , of which 48. images , they appoint 12. to the Zodiaque , commonly called the 12. signes , as Aries , Taurus , Gemini , Cancer , Leo , Virgo , Libra , Scorpio , Sagittarius , Capricornus , Aquarius and Pisces . Againe they place in the North part of the firmament 21. images , and in the South part thereof 15. images , which make in all 48. The description of all which images , together with their names hereafter followeth : and first I will describe vnto you the twelue Images contained in the Zodiaque .

Of the xij . Images or signes of the Zodiaque . Chap. 24.

THe twelue signes ( as some affirme ) doe containe of the foresaide number of fixed stars 273. For the first signe called Aries , that is to say , the Ramme , contayneth 13. starres , which Image or signe being placed in the coniunction of the Zodiaque with the Equinoctiall , hath his backe turned towards the North , and his head towards the East , and riseth with his head , and goeth downe with his féete . The second signe called Taurus , that is to say , the Bull , containeth 33. starres , whereof there is one bright starre of the first bignesse called Oculus Tauri , that is to say , the Bulles eye , who hath his head enclyning towardes the West as though hée looked towardes the earth , and riseth and goeth downe with his héeles vpwarde . The thirde Signe called Gemini , that is to say the twinnes , doe contayne 18. starres , their heades looking towards the North , and their backes being ioyned together doe embrace one another , and doe rise lying , and doe goe downe with their féete . The fourth Signe called Cancer , that is to say , the Crabbe contayneth nine starres , extending his féete towardes both the Poles , and looking towards Leo , hath his bellie turned towardes the earth , and hée riseth and falleth with his hinder part or backe part of his bodie . The fifth Signe called Leo , that is to say the Lyon , contayneth ten starres , whereof there be two bright starres of the first bignesse , the one in his breast called Cor Leonis , and Regulus , that is to say the Lyons heart , and the other in his tayle called Cauda Leonis , that is to say , the Lyons tayle , who looketh towardes Cancer , and hauing his backe turned towardes the North , he riseth and goeth downe with his head . The sixth Signe called Virgo , that is to say , the Uirgine , whose head is behinde the Lyon and toucheth the Equinoctiall line with her left hande , holding in the same hande an eare of corne , shée both ryseth and goeth downe with her head : this image contayneth sixe and twentie starres , whereof there is one bright starre of the first bignesse called Spica Virginis , that is to say , an eare of corne . The seuenth Signe called Libra , that is to saye the Ballance , contayning eight starres hath two skales , whereof the one hangeth towardes the North , and the other towardes the South . The eight signe called Scorpius , that is to say the Scorpion , contayneth one and twentie starres , who looketh towardes , Virgo , and extendeth his féete towardes both the Poles , he boweth his tayle towardes the North , hauing his belly turned towardes the earth , and he riseth and goeth downe bowing . The ninth Signe called Sagittarius , that is to say the Archer , contayning one and thirtie starres , hath his head towardes the North , and looketh towards the Scorpion , hauing a bowe and a shaft , whereof the bowe toucheth his left hande and left foote , he riseth right vppe , and goeth downe headlong . The tenth Signe called Capricornus , that is to say , the Goate contayning eight and twentie starres , hath his backe turned towardes the North , and his heade towardes the Archer , and turning himselfe towardes Aquarius , he riseth right vp , and goeth downe headlong . The eleuenth Signe called Aquarius , that is to saye , the water-bearer contayning two and fourtie starres , hath his heade towardes the North , extending his left hande vppon the backe of Capricorne , and with his right hande powreth out water out of his potte , which bendeth towardes the East , runneth euen to Pisces , hée riseth and goeth downe with his heade before anie other of his members . The twelueth Signe called Pisces , that is to say , two Fishes , doe containe foure and thirtie starres , whereof the backe of the first is towardes the North , and the backe of the seconde towardes the West arme of Andromeda , and one of the Fishes looketh towardes Aquarius , and the other towardes the North , and betwixt these two Fishes is a certaine little line wherewith their tayles are bounde together as it were with a bond , the lower part of which Fishes , doth alwayes both ryse and goe downe , and not the vpper part : And though the 12. Signes of the Zodiaque are saide to bée equall both in length and breadth , that is to say , hauing thirtie degrées in length , and twelue degrées in breadth , as hath béene sayde before , yet these Images are not equall , for some doe extende further then the Zodiaque in breadth , and some are more then thirtie degrées in length ; As the Tables of Alphonsus doe manifestlie shewe , who sayeth there that the twelue Signes doe contayne thrée hundred and fiftie starres , for he appointeth to Aries eightéene , to Taurus fourtie foure , to Gemini twentie fiue , to Cancer thirtéene , to Leo thirtie fiue , to Virgo thirtie two , to Libra seuentéene , to Scorpio twentie foure , to Sagittarius thirtie and one , to Capricornus twentie eight , to Aquarius fourtie fiue , to Pisces thirtie eight , which make in all thrée hundred and fiftie , in which Tables are also set downe the Longitude , Latitude , and Magnitude of the saide starres , but the Longitude of the sayde starres , is farre altered from that Longitude which they had in his tune , whereof wée shall speake héereafter more at large .

Of the xxj . Northerne Images . Chap. 25.

WHich be they ?

These here following , first Vrsa minor , that is to say , the lesser Beare containing 7. starres , the taile starre wherof being a bright starre of the third bignesse is called the load-starre : The second is called Vrsa maior , that is to say , the great Beare containing 27. starres , whereof there be 7. principall , making a shape like vnto a Cart with foure whéeles , and therefore it is commonly called Charles waine . The third is called Draco , that is to say the Dragon that kept Iunos Orchard from robbing , containing 31. stars . The fourth is called Cepheus , the proper name of a King of Ethiope , containing 11. starres . The fifth is called Bootes , that is to say the roaring kéeper of the Beare , containing 22. starres , whereof there is one bright starre betwixt his legges , of the first bignesse called Arcturus . The sixth is called Corona Ariadnae , that is to say the Crowne of Ariadna the daughter of king Minos , containing 8. starres . The seuenth is Hercules , who lyeth groueling with his héeles towardes the North pole , holding a clubbe in his right hand , and a Lions skinne hanging on his left arme containing 28. starres . The eight is called Lira , that is to say an instrument of musicke like a Harpe placed in heauen in the memorie of Orpheus , containing 10. starres . The ninth is called Cignus , that is to say the Swanne , into which Iupiter transformed himselfe to deceaue the Nymph Laeda , containing 17. starres . The tenth is called Cassiopea sitting in her chaire , wife to Cepheus , and mother to Andromeda , containing 13. starres . The eleuenth is called Perseus , holding his sworde in the one hand , and the head of Gorgon or Medusa in the other hand , whose haires were all Serpents , containing 26. starres . The twelfth is Auriga & Ericthonius , who was the inuentor of the first Chariot that euer was made , containing 13. starres . The 13. is called Serpentarius and Anguitenens , that is to say , he that holdeth the Serpent , who as some thinke was Aesculapius the famous Phisitian , containing 24. starres . The 14. is called Serpens or Anguis , that is to say , the Serpent of Esculapius containing 18 starres . The 15. is called Sagitta or Telum , that is to say the shaft or dart wherewith Hercules slew the Eagle tormenting Prometheus in the mount Caucasus , containing 5. stars . The 16. is called Aquila , that is to say , the Eagle which caried Ganimedes into heauen containing 9. starres . The 17. is called Delphinus that is to say the Dolphyn , which saued Arion that excellent Musitian , being cast by Pyrats into the sea , containing 10. stars . The 18. is called minor Equus , that is to say , the lesser Horse , containing onely 4. little darke starres in his head . The 19. is the great Horse called Paegasus , that is to say the winged Horse , wherewith Bellerophon conquered the monstrous beast called Chimera , which was halfe a Lyon and halfe a Dragon , and is adorned with 20. starres . The 20. is called Andromeda the daughter of Cepheus by Cassiopeia , and wife to Perseus , who for her constant loue towards her husband , was placed in heauen nigh vnto him , and was adorned with 23. starres . The 21. is called Triangulum , that is to say a Triangle or thrée cornerd figure , which being like in shape to the Ile Cicilia the Goddesse Ceres obtained to be placed in heauen , and was adorned with 4. starres , that is to say , euery corne one , and the fourth in the midst of the shortest side : To these ( in mine opinion ) ought to be added Bernices haire , called crinis Bernices , which in al celestial globes is placed not farre from the right hinder foote of the great Beare , and this Image containeth 4. little starres .

Of the 15. Southerne Images . Chap. 26.

WHich be they ?

These hereafter following , whereof the first is called Caetus , that is to say the Whale , that monstrous fish which by the appointment of Neptune , would haue destroyed Andromeda , whom Perseus deliuered by killing the Fishe , and afterwardes tooke Andromeda to wife , which Fish Neptune placed in heauen , adorning the same with two and twentie starres . The second is called Orion with his sword by his side , who afterward was slaine by Diana by mis-hap against her will , for the which she placed him in heauen , adorning him with 38. starres , whereof there be two bright starres of the first bignesse , the one in his right shoulder called Bed Alguze , and the other in his left foote called Rigell Alguze . The third is the flood called Eridanus , into the which Phaeton the sonne of Apollo was stroken with a thunder bolt by Iupiter , for burning the earth by rashly driuing his fathers Chariot , which he was not able to guide , in memorie whereof the flood was placed in heauen , and adorned with 34. starres , whereof one is a bright starre of the first bignesse called Acarnar . The fourth is called Lepus , that is to say the Hare , placed nigh vnto Orion , because he was a hunter , adorned with 12. starres . The fift is called Canis maior , that is to say the great Dogge , passing all others in swiftnesse , which was giuen by Aurora to Cepheus the sonne of Eolus , and is placed next to the Hare , being adorned with 18. starres , whereof there is in his mouth a very faire starre of the first bignesse called Syrius . The sixt is called Canis minor , that is to say , the lesser Dogge , without the which Orion his master would not be placed in heauen , which hath but two starres , whereof the one is in his flanke , and is a bright starre of the first bignesse called Canicula and Procion . The seuenth is the ship called Argos , in the which Iason and his companions sayled to Cholcos to winne the Golden Fléese , which is adorned with 45. starres , whereof there is one bright starre of the first bignesse in the left oare called Canopus . The eight is called Hydra , that is to say , the Water-serpent which Hercules slewe , or as some say which kept the water-bowle , and would not suffer the thirstie Crowe to drinke , which Crowe Apollo sent for water to doe sacrifice , and is adorned with 25. starres . The ninth is called Crater , that is to say the Cuppe or Bowle which the Crowe brought too late vnto Apollo , and therefore his feathers were made all blacke whereas before they were white , contayning 7. starres . The tenth is called Coruus that is to say the Crowe before mentioned , adorned with 7. starres . The 11. is called Chiron siue Centaurus , the son of Saturne passing all others in iustice and religion , and therefore is figured in heauen as though he were offring sacrifice vppon an altar adorned with 37. starres whereof there is one bright starre of the first bignesse in his right foote called Chiron and Centaurus . The twelfth is called Lupus , that is to say the Wolfe , into which Licaon the cruell Tirant was turned by Iupiter , or as some say that Wolfe which Centaurus killed to doe sacrifice vpon the altar containing 19. starres . The thirtéenth is called Ara , that is to say the Altar made by the Smithes of Vulcane , wherevpon all the Goddes sware to reuenge the insolencie and pride of the Giants , which altar is placed next to Centaurus being adorned with 7. starres . The fourtéenth is called Corona Australis that is to say , the Southerne Crowne which Bacchus did weare when hee fetched his mother Semele from hel , which is placed in heauen , & is adorned with thirtéene starres . The fiftéenth and last of the Southerne signes is called Piscis Australis , that is to say , the Southerne Fish which was placed in heauen in memoriall of the fishes which the people of Syria did worship as their Goddes , and is adorned with 12. starres , whereof there is one bright starre of the first bignesse in his mouth called Fomahant : All which Images and Signes before mentioned , aswell Northerne as Southerne , you may sée plainely described in euery celestiall Globe , and also set foorth in Plano in the nether ende of Vopellius his vniuersall Map , that is to say the Northerne signes on the left side , and the Southerne signes on the right side of the Mappe : Or in the front of Planctius his great vniuersall Mappe , who in the rondle representing the Southerne halfe of the celestiall Globe , setteth down also certaine Southerne starres lately found out by the trauellers into the Indies , as the Crosse , the Southerne Triangle , Noe his Doue or Pigeon , and another in shape of a man called Polophilax , so as there be now in all 19. Southerne Images .

Of the longitude of the fixed starres , and of the procession of the vernall Equinoctiall point , and what it is . Chap. 27.

WHat is the Longitude of a starre ?

The Longitude of any starre , is that Arke or portion of the Ecliptique line which is contayned betwixt the first point of Aries , and that Circle which passeth through the Poles of the Zodiaque , and also through the bodie of the starre , as for example the starre called Cor Leonis is distant in these dayes from the vernall Equinoctiall point of the Ecliptique line 143. degrées and 32′· and thereby is found to be in the 23. degrée and 32′· of Leo , againe the starre called Spica Virginis in these dayes is distant from the first point of Aries 198. degrées , and so is found to be in the eightéenth of Libra .

Why do you say in these dayes ?

Because the fixed starres in processe of time , doe change their Longitude by reason of their proper moouing vpon the Poles of the Zodiaque , which is from West to East , for whereas Spica Virginis in Ptolomies time was in the 26. degrée of Virgo , it is found now to be in the 18. of Libra , the cause whereof is the precession of the Equinoctiall point or section .

Define what that Precession is ?

It is an Arke or portion of the Ecliptique line , contayned betwixt two great Circles , both passing through the Poles of the Zodiaque , in such sort as the one passeth through the first minute of the vernall Equinoctiall point of the saide Ecliptique , and the other Circle passeth through the first or former starre of the Rams horne , from which starre the Astronomers doe make all the celestiall motions and reuolutions to take their first beginning , and this starre in olde time past was knowne to be before the vernall Equinoctiall point , which is the first moment of Aries , but now it is found to haue passed that point so farre towardes the Solsticiall point , as in these dayes it is knowne to be in the 27. degrée and 42′· of Aries , and in processe of time it will be cleane out of Aries , and enter into Taurus .

Of the Latitude of the fixed starres . Chap. 28.

WHat is the Latitude of a starre ?

The Latitude is none other thing , but the distance of any starre from the Ecliptique line either towards the North or South pole of the Zodiaque , and such Latitude neuer changeth or altereth , for as the starre Spica Virginis is at this present two degrées distant from the Ecliptique line towards the South , so it euer hath béene and euer shall be , and the like is to be said of all the rest of the fixed starres which do alwaies kéepe their Latitude , be it Northward or Southward , néere to the Ecliptique or farre from the same .

Of the Declination of the fixed starres . Chap. 29.

WHat is the declination of a starre ?

The declination is none other thing , but the distance of any fixed starre from the Equinoctiall , eyther Northward or Southward , which is mutable as well as the Longitude : for as the fixed starres do change their Longitudes , so also by little and little they decline either more or lesse from the Equinoctiall : As for example , the declination of the starre called Canicula , that is to say the lesser Dogge in the yeare of our Lord , 138. when Ptolomie liued , was 15. degrées 44′· and 38″· towardes the South . But in these dayes the declination of the said starre is but sixe degrées and 7′· towardes the South , and by reason that the fixed starres in processe of time doe change their Longitude and declination , they are not alwayes vnder one selfe signe , but doe flit out of one signe into another .

How is the Longitude , Latitude , and declination of anie starre to be knowne , and how are the starres themselues to be knowne in the firmament .

The Longitude , Latitude , and declination of any starre is to be knowne most truely by the Astronomicall Tables calculated of purpose , and you may know the same also without hauing regard to euery small minute , by helpe of the celestiall Globe , all the necessarie vses whereof I haue set downe in a little Treatise to be added hereafter to this Booke , and there also I shew you how to find out any starre in the firmament that is described in the globe , in which Globe are set downe as many stars as euer were known , a few excepted towards the South pole , which were founde out but of late dayes , of which starres I shall haue occasion to speake hereafter in my Treatise of Nauigation . In the meane time I will procéede to the ascention and descention of the starres both right , meane , and oblique .

Of the ascention and descention , that is the rising and setting of the starres , aswell according to the Astronomers , as according to the Poets . Chap. 30.

DOe the Astronomers and Poets differ touching this matter ?

Yea they differ greatly , aswell in name as in matter : for whereas of the Poets it is called ortus & occasus Signorum , that is to say , the rising and falling of the Signes , so of the Astronomers it is called Ascentio & descentio Signorum , that is to say , the ascention and descention of the Signes , againe they differ in matter , or rather in manner , for that the Astronomers do consider the rising and falling of the starres more exactly then the Poets , for the Astronomers do consider the degrées and minutes of the same , and also do ground their ascention and descention vpon more certain demonstrations then the Poets . Moreouer whereas the Poets by their manner of rising and falling , doe simply set downe the time of thinges done or to be done , the Astronomers doe the same a great deale more exactly , and by their manner of ascention and descention doe consider the increase and decrease of the dayes , of which Astronomicall ascention and descention , I mind here to treate first in generall and then in particular .

Of the Astronomicall ascention and descention in generall both right , meane , and oblique , and what a giuen Arke is . Chap. 31.

DEfine what the Astronomicall ascention and descention is .

Astronomicall ascention is that portion or Arke of the Equinoctiall line which riseth together with some giuen Arke of the Ecliptique line aboue the Horizon , and the descention is that portion or Arke of the Equinoctiall that goeth downe or setteth together with some giuen Arke of the Ecliptique line vnder the Horizon , according to the moouing of the worlde which is from East to West .

What meane you by a giuen Arke ?

A giuen Arke is as much to say as some supposed portion of the Ecliptique or of any other Circle , as if you woulde knowe the ascention of some supposed portion of the Ecliptique , contayning for example 25. or 30. degrées , here this portion of the Ecliptique contayning that number of degrées , is called the giuen Arke , of which Arkes some are called continuall , and some descrete or deuided , which I minde to English here whole and broken , for so I doe English quantitas , continua , & discreta in my Logicke , That Arke is sayde to bee continuall or whole which taketh his beginning from the first point of Aries , and so procéeding orderly , endeth at some other degrée of the saide Ecliptique . And that Arke is called discrete or broken , which doth not take his beginning from the first point of Aries , but beginneth at some other degrée of the Ecliptique , as for example , suppose that it beginneth at the fourtéenth of Taurus , and endeth at the fiftéenth of Gemini , this Arke is called a deuided or broken Arke , because it doth not beginne at the first point of Aries , and so procéede successiuely : moreouer you haue to vnderstande that the auncient Astronomers doe commonly make but two kindes of ascention and descention , that is , right and oblique , but there be in déede thrée kindes of ascention , that is to say , right , oblique , and meane ascention .

When is any ascention saide to bee right , oblique or meane ?

It is saide to be right , when that portion of the Equinoctiall which riseth or goeth downe together with the Ecliptique , is greater or more in circuit then that of the Ecliptique . And it is sayde to be oblique , when that portion of the Equinoctiall which riseth or falleth together with the Ecliptique , is lesser then that of the Ecliptique , Againe that is sayde to be meane ascention , when that portion of the Ecliptique which ascendeth , is neyther greater nor lesser then that of the Equinoctiall , for as in the right Spheare euery quarter of the Ecliptique hath a meane ascention , and equall to euery quarter of the Equinoctiall , beginning the quarters at any of the foure principall pointes , so if you take thrée signes in any other part of the Zodiaque , their ascentions will not agrée with a quarter of the Equinoctiall , sith there is no one signe that doth equally agrée with the like portion of the Equinoctiall , and all this matter dependeth vppon the knowledge of the vse of certaine Circles before defined .

Which be they ?

These thrée , the Zodiaque , the Equinoctiall , and the Horizon : for first the Zodiaque doth shewe the place of the Sunne , that is to say in what degrée it is of any signe together with the minutes of the same , and turning about euery day by the diurnall motion , doth both appeare aboue the Horizon , and also is hidden vnder the Horizon . Secondly the Equinoctiall with his equall rising and going downe , doth measure the time of the sunne whilest he maketh his abode vnequally and diuersely aboue the Horizon . Thirdly the Horizon deuideth the one Hemispheare from the other , on which Horizon is to be considered what Angle any signe or starre maketh therewith , in his ascention or descention , and according as any portion of the Ecliptique riseth or setteth rightly or obliquely , so in respect of the Angle which it maketh with the Horizon , it is called a right or oblique ascention or descention .

Why should the ascentions and descentions be measured rather by the equinoctiall line then by the Ecliptique , sith the course of the sunne measureth all times ?

The cause thereof is the obliquitie of the Zodiaque , hauing diuerse and variable situations , whereby the sunne abideth sometimes a great while aboue the Horizon , and sometimes but a little while , all which inequalitie is onely to be measured by the Equinoctiall , which is alwayes equally mooued vppon his Poles . Hitherto of the Astronomicall ascention and descention in generall , nowe of all thrée ascentions and descentions in particular .

Of the right , oblique , and meane ascention in particular , and of the chiefest causes of such diuersitie of ascentions . Chap. 32.

ANd for the better vnderstanding of the Astronomicall ascention and descention , wee will make this diuision , for either it is of some point or starre , or else of some portion of a Circle chiefely of the Ecliptique line . In the ascention of any point or starre , wee consider two thinges . First what Angle it maketh with the Horizon either right or oblique . Secondly the time from the rising of the first minute of Aries , which is the first beginning of the Longitude of any starre or Circle in heauen , and in respect of the Angle euery ascention is sayd to be right in a right Spheare , and oblique in an oblique Spheare : Againe the time of the ascention is to be measured by the degrées of the Equinoctiall from the first minute of Aries , vnto that degrée and minute of the Equinoctiall which ascendeth together with the starres , And note by the way that 15. degrées of the Equinoctiall do make an houre , and foure doe make one degrée of the same Equinoctiall , for foure times 15. doe make 60. minutes , which is an houre , Againe euery ascention considered , according to the time of his gate , is eyther right , oblique , or meane : if it be right , it is slowe : if it be oblique , it is quicke : if it be meane , it is equall . Now the ascention of any Arke or portion of a Circle is also eyther right , oblique , or meane : if it be right , it ascendeth slowely : if oblique , it ascendeth quickely : if meane , it ascendeth equally . And the better to vnderstand all these thrée kindes of ascentions , I will set downe these twelue rules heere following , whereof fiue doe belong to the right Spheare , and seuen to the oblique .

In the right Spheare all the foure quarters rising from the foure principall points , haue a meane ascention , and so hath all the foure points themselues .

In the right Spheare all those signes that be equally distant from the foure principall points haue equall ascentions .

3 In the right Spheare all starres or pointes that be in the Solsticiall Colure haue meane ascention .

4 In the right Spheare those signes that do ascend rightly , doe descend rightly , and those that doe ascend obliquely , doe descend obliquely .

5 In the right Spheare , Gemini , Cancer , Sagittarius , and Capricornus , doe ascende rightly , and all the rest obliquely .

6 In the oblique Spheare , the two Equinoctiall points haue meane ascention .

7 In the oblique Spheare ech halfe of the Spheare , beginning at eyther of the Equinoxes haue meane ascention : but this rule holdeth not , if that you beginne any other where .

8 In the oblique Sphere , those signes that do ascend rightly , do descend obliquely , & those which ascend obliquely do descend rightly .

9 In the oblique Spheare , the ascention of any supposed signe is equall to the descention of his opposite signe , and the descention of any supposed signe , is equall to the ascention of his opposite signe .

10 In the oblique Spheare , the ascention of any signe being added to his descention , is equall to the ascention and descention of the same signe being in the right Spheare .

11 In the oblique Sphere euery two signes equally distant from the two points of the Equinoctiall , haue equall ascentions and descentions .

12 In the oblique Sphere vnder the pole Arctique , al signes from Cancer to Capricorne , do ascend rightly , and all the rest obliquely , but contrariwise vnder the Pole Antarctique .

What is the chiefe cause of the diuersity of ascentions and descentions , aswell in the right as in the oblique Spheare ?

The chiefe cause is the diuersity of the Angles which the Zodiaque maketh with the Horizon , for the sharper that the Angles be the lesser portion of the Equinoctiall riseth together with the Ecliptique , and the righter that the Angles be , the greater portion of the Equinoctial riseth , but the Equinoctiall by reason of his vniformitie , maketh his Angles alwayes equall one to another , that is to say , in the right Spheare , it maketh right Angles , and in the oblique Spheare , though not right , yet in euery signe it maketh like Angles .

How to know the diuersities of the ascentions and descentions , as well in the right as oblique Spheare . Chap. 33.

THat is to be knowne most exactly by the Tables of ascentions calculated of purpose by Iohannes de Monte Regio , and by Reinholdus called in Latine Tabulae directionum , and you may knowe it also without hauing respect to euery minute by marking and obseruing the same in a materiall Spheare or Globe , that hath a standing foote with a firme Horizon , for if you will know the diuersities of ascentions in a right Spheare , then you must lay the Spheare or Globe so as the Horizon may passe through both the Poles , and in turning about with your hand the Equinoctiall , together with the Ecliptique from East to West , marke with what degrée of the Equinoctiall any signe beginneth to ascend , & marke that degrée of the Equinoctiall with a little peece of waxe , then turne the Globe or Spheare towardes the West , vntill the last degrée of the sayd signe doe appeare iust with the vpper edge of the Horizon , and then marke what degrée of the Equinoctiall is aunswerable to the said last degrée of the foresaid signe , and there set another péece of waxe , then count the degrées of the Equinoctiall contayned betwixt those two markes , and if it be more then 30. that signe is said to ascend rightly , if it be lesse then 30. then that signe ascendeth obliquely , if it be iust 30. then it hath a meane ascention , & by allowing 15. degrées of the Equinoctial to an houre , and 4′· to a degrée , you shall know in what time that signe riseth . As for example if you would know what ascention the whol signe Taurus hath in a right Spheare , and also in what time it riseth , doe thus , First lay both the Poles of the Spheare iust vpon the Horizon , so as the same Horizon may passe through both the poles , then bring the first point of Taurus to the East part of the Horizon , so as it may touch the vpper brimme or edge of the Horizon , and staying it there with your hande , looke what degrée of the Equinoctiall doth also touch the Horizon at that instant , which you shall finde to be 27. degrées 54′· and marke that degrée of the Equinoctiall with a little péece of waxe , or some other thing that may be easily put out or taken away , that done , put forwarde the foresaid signe Taurus still towardes the West , vntill the last degrée of the saide signe be ascended vp euen to the vpper edge of the Horizon , and there staying it with your hande , looke againe what degrée of the Equinoctiall doth rise withall , which you shall finde to be 57. degrées 48′· and there set another marke vppon the Equinoctiall , then by telling the degrées conteyned in the Equinoctiall betwixt the two markes , you shall finde the number of degrées to be 29. degrées 54′· and by allowing 15. degrées to one houre , and 4′· to a degrée , you shall finde that the whole signe Taurus spendeth in his rising one houre , 59′· 36″· But now sith the Meridian in any place ( as hath béene said before ) doth alwaies shew the right ascention of any starre , signe , arke , or point , because that cutteth both the Equinoctiall and the Horizon with right Angles : you may therefore find the right ascention of the sayd signe , or of any other signe or starre without remoouing the Spheare from your owne eleuation or Latitude in this manner following , bring the first degrée of Taurus close to the mooueable Meridian , and there staying it marke what degrée of the Equinoctiall the Meridian cutteth at that present , which you shall find to be 27. degrées 54′· which is the right ascention of the first point of Taurus , then hauing brought the last point of Taurus to the foresaide Meridian , marke what degrée of the Equinoctiall the sayd Meridian cutteth at that present , and you shall finde it to bée the 57. degrées 48′· now by counting vpon the Equinoctiall , the degrées conteyned betwixt those two markes , you shall finde the number to be 29. degrées 54′· and you may finde the selfe same number by subtracting the right ascention of the first point of Taurus , out of the right ascention of the last point of Taurus , & thereby you shal know the time of his rising to be the same that you found in the right Spheare . Now if you would know the ascention of any signe in an oblique Spheare , then hauing placed your Spheare according to your Latitude , which for example sake suppose to be 52. degrées , and that in such Latitude you would know what ascention the whole signe Taurus hath , and in what time he riseth , you must first bring the first degrée of Taurus to the East part of the Horizon , so as it may méete euen with the vpper edge of the Horizon , and there staying it , marke what degrée of the Equinoctiall riseth therewith , which you shall finde to be 12. degrées 48′· and hauing marked that degrée , put forward the foresaid signe Taurus towards the West vntill the last degrée thereof be ascended vp to the vpper edge of the Horizon , and then make another marke vpon the point of the Equinoctiall , which riseth at that instant with the last degrée of Taurus , which you shal finde to be 29. degrées 42′· and by counting the degrées contained in the Equinoctiall betwixt the two markes , or by taking the lesser ascention out of the greater , you shall finde the number of degrées to be 16. and 54′· whereby you may conclude that the ascention of Taurus in that Latitude is oblique , and that he spendeth in his rising one whole houre 7′· 36″· And looke what order is héere taught to finde out the ascention of any signe , the same order is to be obserued for the finding out of the descention of any signe , sauing that you must séeke for the descention of any signe in the West part of the Horizon of the Spheare or Globe , and not in the East part . As for example , if you would knowe what descention Taurus hath , and in what time he descendeth in the foresaide Latitude : here hauing brought the first degrée of Taurus to the West part of the Horizon , so as it may touch the vpper edge therof , and hauing also marked what point or degrée of the Equinoctiall toucheth the same Horizon at that instant , which you shal find to be 42. degrées 30′· cease not to turne the Spheare or Globe , vntill all the whole signe of Taurus be descended vnder the Horizon , and that the last degrée thereof doe méete iust with the vpper edge of the Horizon , and there stay it vntill you haue againe marked that point of the Equinoctiall which toucheth the Horizon at that instant , which you shall finde to be 84. degrées 54′· and by counting the degrées contained betwixt the two markes on the Equinoctiall , you shall find the number of degrées to be 42. degrées 24′· so as you may conclude that the descention of Taurus in that Latitude is right , and that he spendeth in his going downe two houres 48′·

How shall I knowe the right or oblique ascention of any of the fixed starres , and also at what houre of the day or night they rise & set , and how long they abide aboue the Horizon : finally when they are at the highest , and when they are at the lowest , called the depression or lowest Meridian Altitude of the starres ?

All these things are most truely knowne by Tables calculated of purpose , and also they are to be knowne by helpe of the celestiall Globe in such manner as shall be declared hereafter when we come to treate of the said Globe .

Of the ascentionall difference and vses thereof . Chap. 34.

WHat is the ascentionall difference ?

It is a portion of the Equinoctiall , wherby is knowne how much the right ascention and oblique ascention of any star , or portion of the Ecliptique line , or any other point in the firmament , doth differ one from another , As for example in that place where the Pole is eleuated 52. degrées , the right ascention of the first point of Taurus is 27. degrées and 54′· and the oblique ascention of the same point is 12. degrées 48′· here by taking the lesser out of the greater , that is 12. degrées 48′· out of 27. degrées and 54′· there will remaine 15. degrées and 6′· which is the ascentionall difference .

What vses hath the ascentionall difference ?

The ascentionall difference being knowne , all the oblique ascentions and descentions of the starres are easily knowne by the Tables of directions , againe by this difference is knowne the increase and decrease of the artificiall day in euery Latitude , and therefore it is called of some incrementum diêi . Moreouer it sheweth the semi-diurnall Arke of the artificiall day , for in euery oblique Spheare the artificiall day is alwayes either longer or shorter then the Equinoctiall day throughout the yeare , vnlesse the sunne be in either of the Equinoctiall points .

How is the increase or decrease of the day to be knowne by the ascentionall defference ?

That shall be declared hereafter in the 50. Chap. of this first booke , whereas wee treat of the artificiall day and night , in the meane time we will speake somewhat of the poeticall rising and setting of the starres .

Of the Poeticall rising and setting of the starres . Chap. 35.

DEfine what the Poeticall rising and setting is .

The Poeticall rising is the appearing of some starre aboue the Horizon , determined by the sunne , and the Poeticall setting , is either the going downe of some starre vnder the Horizon , or else the hiding thereof vnder the beames of the sunne .

How manifold is the Poeticall rising and setting ?

Thréefold , that is , Cosmicus , Acronicus , and Heliacus , the signification of which wordes shall appeare vnto you by the definitions of the foresaid thrée kindes here following : For ortus Cosmicus , called in Latine mundanus , which is asmuch to say here as the worldly or morning rising , is when any starre riseth in the morning aboue the Horizon , together with the sunne , or rather with that point of the Ecliptique line wherein the sunne is at that time . And the Cosmicall setting , called in Latine occasus Cosmicus , is when a starre goeth downe vnder the Horizon at such time as the sunne riseth , so as this kind of rising and setting is wholly to be referred to the rising of the sunne .

What is Ortus and occasus Acronicus ?

Ortus Acronicus which is as much to say as the Euening or temporall rising , is when any starre riseth aboue the Horizon in the Euening at the going down of the sunne : And occasus Acronicus , that is to say , the Euening setting is when any starre goeth downe vnder the Horizon , together with the sunne , and therfore this kind is alwaies to be referred to the going downe of the sunne , and not to his rising : And whatsoeuer signe or starre doth rise Acronicè , the same goeth downe Cosmicè , and whatsoeuer starre doth rise Cosmicè , the same goeth downe Acromicè . And generally all starres that rise in the day time , are saide to rise Cosmicè , and all those that rise in the Euening after the sunne set , are said to rise Acronicè .

What is Ortus & occasus Heliacus .

Ortus Heliacus , that is to say , the Solar rising , is when any star by departing from the beames of the Sun appeareth , & may be séen , which before being darkned by the sunne could not be séen , And occasus Heliacus , is when any starre by the nigh approching of the sunne ceaseth to bee séene , for by reason that the sunne by his yearely course & oblique motion of the Ecliptique , doth sometime approch to diuerse starres , and sometime by little and little retireth backe againe from the same , it falleth out that those stars to whom he approcheth , are by nighnesse of his great light , darkned and not séene , and by his departing from them , and especially when the sunne is in the East or West part of the firmament , they beginne againe to be séene . And therefore as in the other 2. kinds , the Horizon together with the rising and setting of the sunne , are to be considered as chiefe causes thereof , so in this last kinde the chiefe cause is to be referred to the highnesse or farrenesse of the sunne from the starre .

Whereto serueth the knowledge of this threefold Poeticall rising and setting of the starres ?

It serueth chiefely to vnderstand thereby those Poets and Histroriographers , which in shewing the time of any act done or to be done , doe not set downe the day of the moneth , but are wont to describe the time by the rising or setting of some notable starre , which they thinke most méete for their purpose , and thereby doe greatly adorne their stile , and specially being poeticall : And because that the times wherein such starres did rise or set , do greatly differ in these dayes from the auncient times . Many therefore of our moderne writers , as Garceus and others , haue made diuese Tables of purpose to finde out the difference , and thereby to come to the true knowledge of the times by the auncient men described , of which matter I leaue to speake , thinking it not méete to trouble young Saylers therewith , for whom I chiefely wrote this Treatise of the Spheare .

Yet some affirme that the auncient men did vse the foresaid poeticall rising and setting of certaine starres , and specially of the Pleiades , Hiades , Orion , Arcturus , Capella , and Lira , ( which starres were to them best knowne ) as a Kalender not onely to know thereby the difference of times , and seasons of the yeare : but also by their manner of rising , setting , hiding , and appearing to prognosticate and to fore-see tempests and stormes , yea and that in these daies wee also ( as some write ) might doe the like , though there were neither Kalender nor Ephemerides , and in that respect the knowledge hereof seemeth most necessarie for Mariners .

All such things are to be knowne more exactly by the Astronomicall ascention and descention , then by the Poeticall rising or setting of the starres : And you haue to vnderstande that the starres since those dayes haue changed their places , their longitudes , and declinations , and thereby in diuerse respects haue altered their Natures and qualities , yea and the very signes themselues : As for example , neither Taurus , Gemini , nor Cancer , is so hoat and dry now , as in times past , neither doth Scorpio cause so much thunder now , as in times past , some againe are more or lesse cold and moyst then they haue béene heretofore , the causes whereof I leaue to the discussing of the Astrologers , and so once againe ende with this matter .

Of time , what it is , and into what parts it is deuided . Chap. 36.

MOst men that write of the Spheare , after they haue spoken of the ascentions , doe immediatly treate of the diuersitie and inequalitie of dayes and nightes , but sith dayes , nights , and houres , are but parts of time , like as be wéekes , moneths , and yeares , I mind here therefore first briefly to treat of time , and then of all his chiefest parts in order , for if you will be instructed at large of these matters , then read the booke of Iohannes de sacro Busto de anni ratione , and also Iohannes Garceus his booke de tempore .

How define you time ?

Leauing to speake of time , without time , that is to say euerlasting and infinite , called of the Latines Eternitas , ascribed chiefely to God , & therefore not contained within the mooueable Spheares or heauens : I mind to speake here onely of that time which is a number measuring the moouing of the first mooueable , and of all other mutable thinges , which time had his beginning with the world , and shall ende with the same , and this time consisteth of two parts , that is first , and last , or rather before or after , successiuely following one another , and these two partes are knit together with a common bound called of the Latines Nunc , that is to say now , or at this present , which is the end of that which went before , and the beginning of that which followeth after , and therefore some doe deuide time into thrée parts , that is , time past , time present , and time to come , but the time present is a moment indiuisible , and is the beginning of time , euen as a point or pricke is the beginning of all Magnitudes , & yet least part therof it selfe : Againe time is deuided of some into greater and lesser parts , the greater are such as these : Kalendes , Nones , Ides , a wéeke , a month , a yeare , the space of fiue yeares , called of the Romaines Lustrum , and of the Gréeks Olympias , the Romaines did call it Lustrum a lustrando , that is to say , of going about , because that they vsed in the end of euery fiue yeares , with lights and torches of waxe to goe in precession round about the Citie , and did purge the same by sacrifising a Dogge , a Sowe , and an Oxe , and at that time also they did chuse their Dictator in a place called the fielde of Mars , but the space of fiue yeares called Olimpias , tooke his name of the high mount Olympus in Greece , whereas in the end of euery fiue yeares were celebrated all kind of martiall playes , as Fencing , Wrestling , Running , and such like in the honour of Iupiter Olympicus , also the space of 15. yeares called indictio , in which space those forraine Nations that dwelt farre off , and were tributary to the Romaine Empire , payd their tributes , that is to say , in the first fiue yeares they payde onely gold , in token of their obedience to the Empire : In the second fiue yéeres they paid siluer for Souldiers wages , and in the last fiue yeares they paide brasse towardes the reparation of armour and munition . Item the space of an hundred yeares , called in Latine seculum , and in English an age , wherof the playes that were celebrated in Rome euery hundred yeare , were called Ludi seculares , and last of all the space of a thousand yeares , called aeuum , contayning tenne ages , againe the lesser partes ( as Iohannes de sacro Busto saith ) are these fiue , the first is called in Latine quadrans , which is the fourth part of a day , that is sixe houres : The seconde punctus , which is the fourth part of an houre in the sunnes account , but in the Moones account the fift part of an houre : The third is called momentum , which is the tenth part of punctus : The fourth is called vncia , which is the twelfth part of momentum : The fift is called Atomus , which is the 48. part of vncia . But because in all the greater parts of time , there is no greater variation or difference , then in that which in Latine is called annus , and in English a yeare . I mind here therefore first to treat of a yeare , and then of monethes , wéekes , dayes , nights and houres .

Of the yeare , and of his diuerse kindes , and of the diuerse computations had thereof in diuerse ages , and amongst diuers Nations . Chap. 37.

BE there diuerse kindes of differences of yeares ?

Yea indéede , but I will speake here only of thrée kindes or differences , that is of the great yeare , the Solar yeare , and the Lunar yeare , whereof the two last are most necessarie for our purpose .

What is the great yeare ?

The great yeare is a space of time in the which not onely all the Planets , but also all the fixed starres that are in the firmament , hauing ended all their reuolutions do returne againe to the selfe same places in the heauens , which they had at the first beginning of the world : And therefore it is called of some the yeare of the world , and of some the great yeare of Plato , which contayneth according to Alphonsus , 49/000. yeares , whereof we haue spoken before , yet some affirme that the perfect yeare of the worlde containeth but 36000. yeares , whose reuolution is after one degrée in 100. yeares , but leauing this matter as not greatly profitable , we will speake now of the yeare Solar .

Of the sunnes yeare called in Latine annus solaris , and of the diuerse kinds thereof , and first of the Tropicall yeare , both equall and vnequall . Chap. 38.

WHat is the yeare Solar ?

It is that space of time in which the sunne departing from any point of the Ecliptique line , or from some fixed starre of the Zodiaque , goeth round about the Zodiaque by his owne proper moouing , which is from West to East , and so returneth againe to the selfe same point or starre from which he first departed , and the Astronomers doe make diuerse diuisions of the Solar yeare , for first they say that it is eyther Astronomical or Politicall : Secondly that the Astronomicall yeare is eyther Tropicall or Syderall . Thirdly that the Tropicall is either equall or vnequall , which vnequall yeare they otherwise call the apparant yeare , and true yeare , all which kindes haue in a manner one selfe definition , sauing that the Tropicall yeare taketh his beginning from the vernall Equinoctiall point , and the Syderall yeare from the former starre of the Rams horne , and doe differ chiefely in quantitie .

Shew then what quantitie , that is to say , how many daies , houres , and minutes euery such yeare conteyneth .

The equall Tropicall yeare being counted alwayes from the middle point of the vernall Equinoxe , contayneth 365. dayes , fiue houres , 49′· 15″· and 46‴· But the vnequall or apparant Tropicall yeare conteyneth sometime more and sometime lesse then the equall yeare , for sometime besides the 365. dayes and fiue houres , it amounteth to 56′· 53″· and 1‴· so as it is more then the equall Tropicall yeare by 7′· 37″· and 15‴· and sometime ouer and besides the foresaid 365. dayes and fiue houres , it onely contayneth 42′· 38″· and 27‴· which is lesse then the equall Tropicall yeare by 6′· and 3‴· which inequalitie chiefely chanceth by reason of the vnequall precession of the two Equinoctiall points before defined in the 26. Chapter .

Of the Syderall yeare , and how much it containeth . Chap. 39.

WHat is the Syderall yeare ?

The Syderal or starry yeare is that space of time wherein the sunne walking vnder the firmament , departeth from the first or foremost starre of the Rams horne , and returning to the same starre againe , which space of time alwayes and equally contayneth 365. dayes , sixe houres , 9′· and 39″· so as this yeare is alwayes greater then the Tropicall yeare , and by his equalitie doth alwayes rule and rectifie the inequalitie of the Tropicall yeare .

Of the Politicall yeare , and diuerse kinds thereof . Chap. 40.

WHat is the Politicall yeare ?

It is a yearely space of time which any people or nation attributeth to the course of the Sunne or of the Moone or of eyther of them which is diuerse and manifold , according to the diuerse customes of the Nations , of all which I meane not to speake at this present particularly , but of certaine speciall and necessary to be knowne , as of the Iulian yeare , of the Egyptian yeare , of the Iewes yeare , and of the Athenian yeare .

Of the Iulian yeare , and why it is so called . Chap. 41.

WHat is the Iulian yeare ?

The Iulian yeare is that which we vse at this present day which of all other yeares draweth nighest vnto the Tropicall yeare , for this yeare consisteth of 365. dayes and sixe houres , which sixe houres , if it should be reckoned euery yeare , it would make a great confusion , and therefore it is reckoned at the end of euery fourth yeare , which yeare consisteth of 366. dayes , for foure times sixe doth make 24. houres , which is one whole naturall day , whereof that yeare is called the leape yeare , And thereby the Iulian yeare is sayd to be two fold , that is common , contayning 365. dayes , and the other bissextile or leape yeare , contayning 366. dayes This worde bissextile is deriued of bis and sextus , because the sixth day next before the Kalends of March is twice repeated or reckned , which indeede is the 25. of Februarie , vpon which day the feast of Saint Mathias commonly falleth .

Why was it called the Iulian yeare ?

Because Iulius Caesar the first Monarch of the Romaine Empire caused the yeare ( according to the course of the Sunne ) to be reduced to the number of dayes and houres before expressed , who brought an excellent Astronomer with him at his comming from Egypt , aswell for that purpose as to teach the Mathematicall disciplines vnto the Romaines , yet you haue to consider that the Iulian yeare being greater then the Tropicall yeare , doth cause great diuersitie in that it maketh aswell the Equinoctiall & Solsticiall points , as also the entrance of the Sun into the other signes by little & little to anticipate or to runne before , for whereas in Iulius Caesars time the vernall Equinox was the 23. day of March , the same Equinox is now about the 11. day of March , which is sooner by 12. dayes .

Of the Egyptian yeare , and how many daies it containeth . Chap. 42.

THe Egyptian yeare contayneth the iust number of 365. dayes , by reason of which equalitie this yeare is very fit to serue the Astronomers turne in making their Astronomicall computations , but the Egyptian yeare hath no certaine place of beginning : For by omitting the sixe houres which is the Iulian yeare , it doth anticipate in the space of 4. yeares one whole day in such sort , as 1460. Iulian yeares doe make of the Egyptian yeares . 1461.

How many Moones the Iewes yeare , and the Athenians yeare doth containe . Chap. 43.

THe Iewes yeare contayneth for the most part twelue Moones , and sometimes thirtéene Moones , which kinde of yeares did agrée with the yeares of the Greekes , and of the Athenians , and also of the auncient Romaines before Iulius Caesars time : and the auncient Romaines did beginne their yeare from March , but the latter Romaines from the winter Solstice . Againe the Iewes did beginne their yeare at the first new Moone that followed next after the vernall Equinox : But the Athenians began their yeare at the new Moone that followed next after the Sommer Solstice : The most people of Asia began their yeare at the Autumnall Equinox : But the most part of those that dwel in these partes of the world , following the custome of the Romaine Church , doe beginne their yeare at the Kalends of Ianuary , which in old time was not much distant from the Winter Solstice , which Solstice at Christs birth was the 25. of December , but nowe the same Solstice is about the twelfth day of December , so as the Winter Solstice falleth sooner by thirtéene dayes then it did at that time . But we here in England do begin the yeare at the 25. of March .

Of the yeare Lunar , and of the kinds thereof . Chap. 44.

HOw many kindes of Lunar yeares be there , and which be they ?

Of Lunar yeares there be two kindes , wherof the one is ordinary called in Latine Annus communis , and the other extraordinarie or excessiue , called by a Gréeke name Embolismalis , The ordinary or common yeare , is the space of twelue Moones or changes , passing by course within the yeare Solar , and is called common because it hath twelue Moones Lunar , euen as the Solar yeare hath twelue monthes Solar , and consisteth of 354. dayes and a little more , so as the Solar yeare excéedeth the Lunar yeare by 11. dayes , for the yeare Solar conteyneth ( as hath béene saide before ) 365. dayes , in which account the Fractions in both yeares are omitted : And therefore if these two yeares should begin together at one selfe time , the Lunar yeare would end his course sooner by 11. dayes then the yeare Solar .

What is the extraordinarie Lunar yeare called Embolismalis ?

It is the space of thirtéene Moones or changes contayning 384 dayes , so as this yeare excéedeth the common Lunar yeare by 30. dayes , and is more then the yeare Solar by 19. dayes .

Of the diuers kinds of monthes , and into what parts euery Solar moneth is deuided according to the Romanes , that is , into Kalends , Nones , and Ides . Chap. 45.

HOw many kindes of moneths be there , and which be they ?

There be thrée kinds , that is , the month Solar , the moneth Lunar , and the moneth Vsuall .

The moneth Solar is that space of time which the Sunne spendeth in passing through any one of the 12. Signes .

The Lunar moneth is that space of time which the Moone spendeth whilest she departing from the Sun returneth to him againe .

The Vsuall moneth is that number of dayes which are set downe in our common Kalenders , whereof some containe thirtie dayes , some thirtie and one , and the moneth of Februarie hath but 28. dayes . But if you will readily know which containe more dayes and which lesse , kéepe alwaies in memorie these old English verses here following .

Thirtie dayes hath Nouember , Aprill , Iune , and September : Februarie hath 28. alone , And all the rest haue thirtie and one .

But when it is leape yeare , February hath 29. dayes , Againe the Romaines deuided the Solar moneth into Kalends , Nones , and Ides .

What be Kalends ?

The Kalends are the first day of euery moneth from which the Romaines counted the daies of the moneth procéeding backward , As for example the first day of Aprill they named the Kalends of Aprill , and the last day of March next before they called in Latin pridie Kalendas Aprilis , that is the day before the Kalends of Aprill , and the next day before that , the third Kalends of Aprill , and the next day before that the fourth Kalends , and so foorth vntill they come to the Ides .

Whereof sprange this name Kalends ?

Of the Gréeke verbe Calo , which is as much to say as call , for the first day of euery moneth the cryer standing in a high place made foure cals or more to signifie thereby to the people howe many dayes in that moneth the faires or markets called Nundinae should endure , and of Nundinae sprange this word Nonae , that is to say the dayes of the faires : For looke howe many Nones there were in euery moneth , so many faires there were , during which time the Romaines neuer worshipped any God because there was no holy day during that time , and therefore Ouid saith that Nonarum tutela Deo caret , that is to say , no God had tuition of the Nones .

What are Ides ?

They are those dayes by which the Nones are deuided from the rest , and these Ides doe deuide in a manner the whole moneth into two equall parts , for the first Ides most commonly falleth eyther on the 13. 14. or 15. day of the moneth .

How many Ides , Nones , and Kalendes doe belong to euery moneth ?

Of Ides euery moneth hath eight , but of Nones March , May , July , and October , haue sixe , and all the rest of the moneths haue but foure Nones , but they differ most in the number of Kalends , as you may perceiue by this Table following , which sheweth how many Kalendes , Ides , and Nones , doe belong to euery moneth . Thus farre of the moneth Solar , now I will speake of the moneth Lunar .

TheTable. Moneths . Kal. Ide . None . Ianuarie . 19. 8 4 Februarie . 16 8 4 March . 17 8 6 Aprill . 18 8 4 May . 17 8 6 Iune . 18 8 4 Iuly . 17 8 6 August . 19 8 4 September 18 8 4 October . 17 8 6 Nouember . 18 8 4 December . 19 8 4

Of the diuerse kindes of moneths Lunar . Chap. 46.

HOw many kindes be there , and which be they ?

Iohannes de Sacro Busto sayth that there be foure kindes , that is the moneth of paragration , the moneth of apparation , the moneth medicinall , and the moneth of consecution .

The moneth of Paragration , is that space of time in which the Moone departing from any one point of the Zodiaque , goeth by her proper moouing about the Zodiaque , and returneth againe to the said point from which she first departed which her reuolution is accomplished in 27. dayes , & 8. houres . And this reuolution of some is called a yeare , and by this account the Moone tarrieth in euery signe two dayes sixe houres and 29′·

The moneth of Apparation consisteth of 28. dayes , deuided commenly by foure wéekes , euery wéeke contayning seuen daies , for foure times seuen maketh 28. of which foure wéekes the first is counted from her first apparance vnto the ende of the seuenth day , and so foorth from wéeke to wéeke , so as the fourth wéeke endeth at the 28. day , in which account the odde houres during the Moones abode vnder the beames of the sunne , when as she is said to be combuste , are not reckned .

The moneth Medicinall contayneth but 26. dayes , and a halfe ( as Galen sayth ) and is deuided also into foure wéekes , the diuision being made by minutes .

The moneth of Consecution is that space of time wherein the Moone being in coniunction with the sunne , goeth about her Circle and returneth againe to the same point , and not finding the sunne there because he hath in that while passed through one whole signe , she hasteth after and in two dayes and foure houres 44′· and a little more she ouer taketh the sunne , and is againe with him in coniunction , of which her following and ouertaking , the sunne this moneth is called the moneth of Consecution , which moneth consisteth of 29. dayes and a halfe : during which time as the sunne by his owne proper course passeth through one signe or there abouts , so the Moone by her course in the selfe same time passeth through the whole Zodiaque and one signe more : And note by the way that the sunne in making his owne proper course doth not enter into any signe in the very beginning of any moneth , but rather about the midst of euery moneth , or at the least not much ouer or vnder that day .

Of a weeke . Chap. 47.

NOw because that euery moneth aswell Solar as Lunar is deuided into foure wéekes , I will speake somewhat of a wéeke .

What is a weeke ?

A wéeke called most commonly in Latine Septimana , which is as much to say as seuen mornings , is the space of seuen dayes , whereof the first is called Sunday , the second Munday and so foorth to Saturday , which names the Gentiles gaue to these seuen dayes in honour of the seuen Planets whom they worshipped as Gods , for they called the first day the day of the sunne , the second the day of the Moone , the third the day of Mars , the fourth the day of Mercury , the fifth the day of Iupiter , the sixth the day of Venus , the seuenth the day of Saturne .

Are there any more names belonging to this word weeke ?

Yea , it is called also by a Gréeke name Hebdoma , that is to say , contayning seuen dayes , and in the scripture it is called sometime Sabbatum , as when the Pharisie sayd that he fasted Bis in Sabbato that is to say , twice a wéeke , for the Iewes called Sunday the first of the Sabaoth , and Munday the second of the Sabaoth , and so foorth in order vntill Saturday , which in déede was their true Sabaoth or day of rest .

Why is it not still so counted amongst vs Christians , but changed into Sunday ?

For two causes , first to auoide the superstition of the Iewes , and partly it was done in the honour of Christ , whose day of birth , resurrection , and sending of the holy Ghost , was on the Sunday .

Of dayes and nights both naturall and artificiall . Chap. 48.

THe Astronomers doe deuide the dayes into two kindes , whereof the one is called naturall , and the other artificiall .

Which call you a naturall day ?

A naturall day is one entire reuolution of the Equinoctiall about the earth , whereunto must be added such portion of the Zodiaque , as the sunne in the meane while maketh by his proper motion , which is from West to East .

In what time doth the Equinoctiall euery day make his reuolution ?

In 24. houres , which space contayneth both day and night , according to which reuolution and number of houres , the most part of Horologies or clockes in the East countryes doe goe , and are set to shew the houres of the day , but yet diuersly , for some begin their naturall day at the rising of the sunne , as the Bohemians , and the Persians , and some at the going downe of the sunne , as the Italians , the Athenians , the Iewes , the Egyptians , and the Arabians , but the Astronomers reckon their naturall day from noonetide to noonetide .

If the Equinoctiall doth make his reuolution iust in 24. houres , then all naturall dayes are equall ?

That is true if ye onely consider the motion of the Equinoctiall , but if you adde thereunto ( as I saide before ) that portion which the sunne in the meane while maketh by his owne proper motion , you shall finde them to be vnequall , because that portion is sometime more , sometime lesse , according to the swift or slowe ascention of the signe wherein the sunne is .

What is the artificiall day ?

It is the distance or space that is betwixt the rising of the Sun , or going downe of the same .

The causes why the time betwixt the rising & going downe of the sunne is vnequall . Chap. 49.

IT must néedes be vnequall , because the abode of the sunne aboue the Horizon is variable , aswell for the vnequall ascention and descention of the signes , as also for the obliquitie of the Horizon and Zodiaque : and therefore the spaces of the artificiall dayes must néedes be vnequall : for the sun aswell in ascending from the beginning of Capricorne , to the beginning of Cancer , as also in descending from the beginning of Cancer to the beginning of Capricorne , describeth on ech part 182. Circles or Paralels , the middlemost whereof is the Equinoctiall . All which Paralels are deuided into two parts by the Horizon , and the Arkes which are aboue the Horizon are called the artificiall daies , and the Arkes beneath the Horizon are called the artificiall nights , which Arkes to those that dwell vnder the Equinoctiall in a right Spheare are alwaies equall , that is to say , the diurnall Arke is equall to the nocturnal , because their Horizon passeth through the Poles of the world , but to those that dwell in an oblique Spheare , aboue whose Horizon the Pole is any thing eleuated , be it neuer so little , the Arkes or Paralels are vnequall one to another , that is to say , eyther making short daies and long nights , or else long dayes and short nights , the Equinoctiall onely excepted , which aswell in the oblique Spheare as in the right , is alwayes deuided by the Horizon into two equall partes , and so maketh the dayes and nights equall in all places of the world . All which thinges you shall easily comprehend by these two figures following , whereof that on the left hande representeth the right Spheare , and the other on the right hande representeth the oblique Spheare .

A figure of the right Spheare .
A figure of the oblique Spheare .

And here I thinke it not amisse to shewe you howe to finde out the length of euery artificiall day and night throughout the yeare in euery Latitude by the materiall Spheare , and likewise howe to know at what part of the Horizon the sunne riseth and setteth euery day , and also how to finde out his Meridian Altitude , wherby you shall know how nigh or how farre from your Zenith hée is euery day .

How to find out by the materiall Spheare or Globe , and by helpe of the ascentionall difference before defined , the increase and decrease of euery day throughout the yeare in euery seuerall Latitude , and at what houre the sunne riseth and setteth . Chap. 50.

FIrst hauing set the Spheare at your Latitude , learne to knowe by some Table or instrument in what signe and degrée thereof the sunne is at such day of the moneth and yeare as you séeke , and bring that degrée close to the mooueable meridian , & there marke in what point or degrée the said Meridian cutteth the Equinoctiall , & what number it hath , for so shall you haue the right ascention of the degree of the sunne for that day . That done , bring the said degrée of the sunne to the East part of the Horizon , so as it may méete euen with the vpper edge thereof , and staying it there , marke what degrée of the Equinoctiall at that instant doth also touch the Horizon , and what number it hath , and that is the oblique ascention of the foresaid degrée of the sunne , then subtract the lesser number out of the greater , and that which remaineth shall be the ascentionall difference , as for example in this present yeare 1590. the xi . day of Aprill the sunne is in the first degrée of Taurus , whose right ascention in the Latitude 52. by doing as before is taught , you shall finde to be 27. degrées and 54′· and the oblique ascention thereof to be 12. degrées 48′· and the ascentionall difference to be 15. degrées 6′· which difference you must first double , and then conuert the same into houres and minutes by allowing 15. degrées to an houre , and 4′· to a degrée , that done , adde those houres & minutes to the Equinoctiall day , which is alwaies 12. houres , & the summe of that addition will shew you the length of the day , when the sunne is in the first point of Taurus , which is 14. houres & 12′· and that is the true length of that day . But you haue to note by the way that as you haue to adde the sum of the houres to the Equinoctial day , the sunne being in any of the 6. Northerne signes , so must you subtract the said houres from the Equinoctiall day , the sunne being in any of the sixe Northerne signes , As for example , suppose the sunne to be in the first point of Scorpio , the right ascention whereof is 207. degrées 54′· and the oblique ascention is 223. degrées , here by taking the lesser out of the greater you shall finde the ascentionall difference to be 15. degrées 6′· which being doubled , maketh 30. degrées and 12′· that is two houres and 12′· which if you subtract from 12. there will remain 9. houres 48′· which is the length of the artificiall day , when the sunne is in the first degrée of Scorpio , the one halfe whereof is called the semi-diurnall Arke of that artificiall day , which is 4. houres 54. minutes , whereby you may gather that the sunne at that time riseth foure minutes after seuen of the clocke , and setteth againe foure minutes before fiue : so likewise in the former example where the ascentionall difference was 15. degrées sixe minutes , which being doubled made 30. degrées and 12′· that was two houres and 12′· which being added to 12. houres made 14. houres and 12′· the halfe whereof is seuen houres and sixe minutes , whereby you may gather that the sunne did then rise 6′· before 5. of the clocke in the morning , and did set 6′· after 7. of the clocke in the Euening , and so you haue both the forenoone & afternoone of the day , which two times are best to be reckoned alwayes from 12. a clocke at noone , that is to say , the forenoone houres and minutes , from twelue backward , and the after moone houres and minutes from twelue forward , & by subtracting the whole length of the artificiall day from 24. houres , you shall haue the length of the artificiall night , as in the former example take 14. houres and 12′· from 24. houres , and there will remaine for the length of the artificiall night 9. houres and 48′·

How to know by the materiall Spheare or Globe , in what part of the Horizon the sunne riseth and setteth euery day , and thereby the length of the day . Also how to know the Meridian altitude of the sunne euery day throughout the yeare , & being at his Meridian altitude , to know how farre distant he is from the Zenith euery day . Chap. 51.

FIrst then hauing set the Spheare or Globe at your Latitude , which suppose to be 52. degr. bring the degrée of the sun for that day , to the East part of the Horizon , so as it may méete iust with the vpper edge therof , and there set a little péece of waxe vpon the Horizon , that doue turne the said degrée of the sunne to the West part of the Horizon vntill it méete againe with the vpper edge of the Horizon , and there set another péece of waxe vpon the Horizon , and those two markes will shewe you in what part of the Horizon the sunne riseth and setteth : As for example , I would know this present yeare 1594. in what part of the Horizon the sunne doth rise and set the twelfth of Iune , here finding by the Ephemerides that the sun is entred 16′· into Cancer at that day , I bring that point of the Ecliptique to the East part of the Horizon , so as it may méete iust with the vpper edge thereof , and there I set a little péece of waxe vpon the Horizon : that done I turne the said point of the Ecliptique to the West part of the Horizon , and whereas that point toucheth the vpper edge of the said Horizon , I set there another péece of waxe vpon the Horizon , then counting the degrées vppon the Horizon from the true East point thereof to the first péece of waxe Northward , I finde the number of the degrées to be 40. degrées or therabouts , which containeth thrée points and a halfe and somewhat more , of the Mariners Compasse , whereby I gather that the sunne riseth at that time very néere to the Northeast , and setteth néere to the Northwest , because the number of the degrées in both parts of the Horizon are like .

How shall I know how many points of the Mariners compasse are contained in any number of degrees , exceding the number of degrees and minutes contained in one point ?

The Mariners Compasse containeth 32. points , and euery point containeth 11. degrées and ¼ . of a degrée which is 15. minutes , wherefore whensoeuer you would know how many points of the Compasse are contained in any number of degrées be it great or small , multiply that number by foure , and deuide the product thereof by 45. and the quotient will shewe the number of the points , and if there be any remainder , then because the Mariner doth make euery point to haue foure quarters , multiply that remainder by foure , and deuide the product by 45. which is the common diuisor , and the quotient will shew the quarters , or if the remainder be but small , then multiply that remainder by eight , which are halfe quarters , and deuide the product thereof by 45. as before , and the quotient will shew the halfe quarters of a point : As in the former example in multiplying 40. degr. by 4. the product is 160. which if you deuide by 45. you shall finde in the quotient 3. whole points , & the remainder to be 25. which being multiplyed by 4. the product will be 100. which if you deuide by 45. you shall find in the quotient 2. quarters of a point , & the remainder to be 10/45. of a quarter , that is to say , if you can deuide one quarter into 45. parts , then you must take 10. of those parts , and adde them to the former summe , which being of smal importance is not to be regarded . But now to returne to my first matter , I say that by counting the degrées vppon the Horizon , from the first péece of waxe to the South point of the Horizon , I find the number of degrées to be 130. degr. & by allowing 15. degr. to an houre , I finde the halfe day to containe 8. houres 10′· which being doubled maketh 16. houres 20′· Then to know the Meridian altitude of the sunne , you must bring the degrée of the sunne right vnder the Meridian , and the number of the degrées contained in the Meridian betwixt the South point of the Horizon , and the said degrée of the sunne will shewe the Meridian Altitude of the sunne , so shall you find the Meridian Altitude of the sunne , being in the first point of Cancer to be 61. degrées , which once had , the distance from the Zenith is soone knowne : for if you subtract 61. degrées from 90. you shall find the distance of the sunne from the Zenith to be 29. degrées . Againe contrariwise the distance from the Zenith being subtracted from 90. the remainder will shew the Meridian Altitude , for those two numbers being added together , doe alwaies make a iust quarter of the great Circle , which is 90. degrées .

Of houres as well equall as vnequall , and into what partes they are deuided . Chap. 52.

AS of daies and nights there be 2. kinds , that is naturall and artificiall before defined , so likewise are there 2. sorts of houres , that is equal , and vnequall : An equall houre is the 24. part of a naturall day , and euery such houre containeth 15. degrées of the Equinoctiall , for 15. times 24. maketh 360. degrées , which is the whole circuit or Longitude of the Equinoctiall , which according to the diurnall moouing of the first mooueable maketh his reuolution in 24. houres as hath béene said before , and therefore this equall houre is also called an Equinoctiall or naturall houre . The vnequall houre is the twelfth part of an artificiall day or artificiall night , which daies and nights as they be sometimes long and sometimes short , according to the time of the yeare , so are the houres of the same . For both day and night , be it neuer so short , is deuided by the Astronomers into twelue houres , so as when here in some part of England towards the North , the artificiall day is 17. houres long , and the night but seuen houres , if you deuide either of these into twelue parts , you shal find that euery such twelfth part of the day shall containe more then the naturall or Equinoctiall houre by 25′· and the twelfth part of the said artificiall night to containe but the one halfe of an equall houre and fiue minutes , which notwithstanding being both added together , will make in all 24 , equall houres , which is a naturall day , so as by this meanes you may easily perceiue that a naturall day comprehendeth both kindes of houres , aswell vnequall as equall . Note also that the vnequall houres are called sometime artificiall and sometime temporall houres , artificiall , because they are dayly changed by the varietie of the artificiall dayes and nights , and are neuer equall but twice in the yeare , when the sunne is in eyther of the Equinoxes : and they are called temporall , because the auncient obseruers of time were woont to make diuerse clockes and Horologies to shewe these vnequall and temporall houres , of which clockes there are yet some to be séene at this day . Moreouer the houres both equall and vnequall are deuided not onely into quarters of houres , but also intominutes , for euery houre be it long or short , is deuided into 60. minutes , and euery minute into 60. secondes , and euery seconde into 60. thirdes , and so forth to fourths , fifts , sixthes , and into as many as you will , so as you make your diuision alwayes by 60. But you haue to note , that as the Astronomers doe deuide the artificiall day and artificiall night into houres both equall and vnequal , so the Iewes doe deuide ech of them into foure quarters , in manner and forme following .

How and in what manner the Iewes doe deuide the artificiall day and night , ech of them into foure quarters . Chap. 53.

THey deuide the artificiall day into foure quarters by allowing to euery quarter 3. houres , accounting the first houre of the first quarter at the rising of the sunne , and the third houre of the said quarter they called the third houre , and the third houre of the second quarter they called the sixth houre , which was midday or noonetide , againe they called the third houre of the third quarter , the ninth houre , and they called the second houre of the fourth quarter the eleuenth houre , and th●● called the last houre which was the twelfth houre of the day , euentide because then the sunne went downe .

Whereto serueth the knowledge hereof ?

By knowing this account you may the better vnderstande certaine places of the Scripture making mention of thinges done at certaine houres not like vnto our common houres , For whereas it is said in the Gospell of Saint Iohn , that Christ healed the Rulers sonne that was sicke of an Ague in Capernaum , and that the Ague left him the seuenth houre , is as much to say , as the Ague left him at one of the clocke in the afternoone : Againe where as mention is made in the Gospell of S. Matthew , of the labourers that came to worke in the Vineyard at the eleuenth houre is to be vnderstood at fiue of the clocke in the afternoone , or rather one houre before the sunne set , for you must thinke that the sunne riseth not nor yet goeth downe in Iewrie alwaies at sixe of the clocke , for then they should haue no artificiall day nor night , but all dayes and nights should be a like . Also they deuided the artificiall night into foure quarters , otherwise called by them the foure watches of the night , for the first thrée houres was the first watch , during which time all the Souldiers both young and old of anie fortified Towne were wont to watch , the second thrée houres they called the second watch , which was about the dead of the night , at which time the young Souldiers onely watched , and the third quarter of the night containing also thrée houres , and was called the third watch the Souldiers of middle age did watch , and the last thrée houres called the 4. watch which was about the breake of day , the old Souldiers onely watched . But nowe because the auncient Astronomers doe appoint the gouernment of the vnequal houres to the seuen Planets , it shall not be amisse to shewe you here what Planet raigneth euery houre both day and night .

How to knowe what Planet raigneth in euery houre of the day or night artificiall , as well by helpe of a table , as by a rule contained in one verse . Chap. 54.

BVt first I will describe vnto you the Table , and then briefly set downe the vse therof . In the first collum of this Table on the left hand are set downe the seuen dayes of the wéeke , whereof the first is Sunday , and the second Monday , and so foorth downeward to Saturday , against euery which day the Planets are placed towardes the right hand , euery one in course one after another , and in the first rowe of this Table which is the head or front thereof , are placed the houres of the day written in Arithmeticall figures , and in the next rowe of the saide front , are set downe the houres of the night written as you sée in common numerall letters .

TheTable. Houres of ●●e day . 1 2 3 4 5 6 7 8 9 10 11 12 Houres of ●●e night . iij iiij v vj vij viij ix x xj xij i ij Sunday . ☉ ♀ ☿ ☽ ♄ ♃ ♂ ☉ ♀ ☿ ☽ ♃ ♂ ●●nday ☽ ♄ ♃ ♂ ☉ ♀ ☿ ☽ ♄ ♃ ♂ ☉ ♀ ☿ ●●esday ♂ ☉ ♀ ☿ ☽ ♄ ♃ ♂ ♀ ☿ ☽ ♄ ♃ W●●nesday . ☿ ☽ ♄ ♃ ♂ ☉ ♀ ☿ ☽ ♄ ♃ ♂ ☉ ♀ ●hursday . ♃ ♂ ☉ ♀ ☿ ☽ ♄ ♃ ♂ ☉ ♀ ☿ ☽ ♄ Friday ♀ ☿ ☽ ♄ ♃ ♂ ☉ ♀ ☿ ☽ ♄ ♃ ♂ ☉ Saturday ♄ ♃ ♂ ☉ ♀ ☿ ☽ ♄ ♃ ♂ ☉ ♀ ☿ ☽

The vse of the Table .

Now the vse of the saide Table is thus : whensoeuer you would knowe in what houre of the day or night any Planet raigneth , you must first séeke out the houre of the day or night , and if it be of the day , then you shall finde it in the first rowe of the front , if of the night , in the second rowe of the front , as hath béene saide before : and from that houre descende with your finger to the common Angle standing right against the day which you séeke , and that will shewe you what Planet then raigneth . As for example , if you would know on Wednesday at 8. of the clocke of the day what Planet raigneth , then hauing found the number of 8. in the front , written in Arithmeticall figure , come straight downe from thence with your finger to the common Angle standing right against Wednesday , and you shall finde that Mercurie raigneth . And if you would knowe what Planet raigneth the same day at the eight houre of the night , then descende from the houre of the night downe to the common Angle , and you shall finde that the sunne raigneth , and so foorth of all the rest .

The rule contained in one verse , and the vse thereof .

The rule in verse is thus :

Sol , Ve , Mer , Luna , Saturnus , Iupiter , & Mars .

These vii . wordes ( the coniunction & being left out ) doe signifie the seuen Planets : For Sol is the Sunne , Ve standeth for Venus , Mer for Mercurius , Luna is the Moone , and the other thrée Planets following , as Saturnus , Iupiter , and Mars , doe make vp the number of seuen , which must alwayes followe one another , in such order as they are here set downe in the foresaide verse , and to haue the true vse of this rule , you must first apply euery Planet to his owne proper day , as Sol to Sunday , Luna to Munday , Mars to Tuesday , Mercurius to Wednesday , Iupiter to Thursday , Venus to Friday , and Saturnus to Saturday : for euery one of these Planets gouerneth the first houre of his owne proper day , and the Planet placed next to him in the verse , gouerneth the second houre of the same day , and so forth orderly , as for example , if you would knowe what Planet shall raigne on Sunday at the third houre of the day , you must first say that Sol doth raigne the first houre because that is his day , and Venus raigneth the second houre , and Mercurie the third houre according to your rule , & so by kéeping the order of the verse , you shall easily appoint to euery houre both of the day and of the night artificiall his owne gouernour : For though both day and night be deuided ech of them into 12. houres , making in all 24. houres , and that there be but seuen Planets , yet by appointing euery Planet to his owne proper day as gouernour of the first houre of the same day , and by obseruing the order of the verse in repeating the said Planets , you shall not faile to giue to euery houre his proper Planet . Thus hauing sufficiently spoken of the celestiall part , I will now procéede to the Elementall part of the worlde , contained in the second Booke of this Treatise .

The second part of the Spheare .
Of the Elementall part of the world . Chap. 1.

WHat doth the Elementall part containe ?

I told you before that as the celestiall part doth containe the eleuen heauens before described , so the Elementall part containeth the 4. Elements , that is to say , fire , Aire , Water , & Earth , which are of themselues pure substances , and the first & next beginnings whereof all mixt bodies are compound , and therefore not to be séene with our outward eyes : for as we our selues are bodies compound , so with our outwarde senses we can discerne nothing but that which is compound : and therefore the fire , aire , water or earth which we daily féele or sée , are not the Elements themselues , but things compounded of them . The natures and properties of which Elements I mind here but briefely to touch , sith the exact handling thereof belongeth rather to naturall Philosophers & to Physitians , then to Geographers , who haue to deale only with the situations of the earth with Zones , Paralels , Climes , Longitudes , Latitudes , distances & such like thinges belonging to the measure and description of this earth here which we inhabite .

Define these Elements .

Of the Fire and of his nature and motion . Chap. 2.

THe Fire is an Element most hoat and dry , pure , subtill , and so cleare as it doth not hinder our sight looking through the same towards the stars , and is placed next to the Spheare of the Moone , vnder the which it is turned about like a celestiall Spheare .

Of the Aire and into how many Regions it is deuided . Chap. 3.

NExt to the Fire is the Aire whichis an Element hotte and moyst , & also most fluxible , pure & cleare , notwithstanding it is farre thicker & grosser as some say , towards the Poles then elswhere , by reason that those parts are farthest from the sun : And this Element is deuided of the naturall Philosophers into thrée Regions , that is to say , the highest Region , the Middle Region , and the lowest Region , which highest Region being turned about by the fire , is thereby made the hotter , wherein all fierie impressions are bredde , as lightnings , fire drakes , blazing starres and such like .

The middle Region is extreame cold by contra opposition by reason that it is placed in the midst betwixt two hotte Regions , and therefore in this Region are bred all cold watry impressions , as frost , snow , ice , haile , and such like .

The lowest Region is hotte by the reflexe of the sunne , whose beames first striking the earth , doe rebound backe againe to that Region , wherein are bred cloudes , dewes , raynes , and such like moderate watry impressions , which thrée Regions of the aire with the rest of the Elements this figure doth plainely shew .

The highest region of the ayer Midle region Loest region Earth .

Of the Water , and whether it be round or not . Chap. 4.

NExt to the Aire , is the Water which is cold , moist , and fluxible , and being lighter then the Earth would of his owne nature surmount and couer the whole earth , had not God in the creation of the world deuided waters from waters ( as the Booke of Genes. saith ) and gathered together those waters that are vnder the firmament into certaine concauities of the earth , leauing other partes of the earth dry , and discouered , that man and beast might inhabite the same , & haue foode necessarie for their behoofe , so as now both water and earth doth make one entire and Sphericall bodie , which is enuironed with the Aire .

Is not the water a round bodie of it selfe without the earth ?

Many late writers doe deny the whole bodie to be round , affirming onely the Conuere superficies or vpper face of the water to be round , for ( say they ) the earth being not altogether round the Concaue superficies of the water cannot be round , notwithstanding the most part of the auncient writers do affirme the whol bodie of the water to be round , saying that the water hath the like shape in his whole , that it hath in his parts : For the parts which are drops , are round , ergo the whole is round . Againe they proue the water of the sea to be round by demonstration thus , suppose a ship to depart from the shore whereon some marke is set , which you may sée with a right leauelled line standing at the stearne of the said shippe , but saying further from the shore , you cannot sée the marke any more standing vpon the sterne , but shall be faine to goe vp to the toppe of the mast to sée it , by reason that the water being a round bodie riseth and swelleth in the midst , and so letteth your sight as this figure plainely sheweth .

Of the Earth and whether it be all round or not . Chap. 5.

NExt to the Water is the Element of the Earth , which of his nature is thicke , heauie , cold , dry , and not fluxible as is the water and aire , but is firme and apt to kéepe his place , and though some deny the earth to bee , round because of the high mountaines , and déepe dales and vallyes therin , which are nothing in comparison of the whole earth to alter that roundnes which it hath by nature , yet Aristotle affirmeth in his second booke de coelo & mundo , the fourtéenth Chapter , that the earth of her own nature is round , proouing the same as well for that the Moone when she is eclipsed in part , could not haue such horned shape as this figure representeth .

Vnlesse the earth were also round by the interposition whereof shee is eclipsed , either totally or in part as hath béen said before , Againe he prooueth the roundnes of the earth by the altering of the Horizon , for in going from North to South , our Horizon altereth in such sort as wee discouer those starres which we could not sée before , but were cleane hidden from our sight , some also deny that the earth is in the middest of the world , and some affirme that it is mooueable , as also Copernicus by way of supposition , and not for that he thought so in déede : who affirmeth that the earth turneth about , and that the sunne standeth still in the midst of the heauens , by helpe of which false supposition he hath made truer demonstrations of the motions & reuolutions of the celestiall Spheares , then euer were made before , as plainely appeareth by his booke de Reuolutionibus dedicated to Paulus Tertius the Pope , in the yeare of our Lord 1536. But Ptolomie , Aristotle , and all other olde writers affirme the earth to be in the middest , and to remaine vnmooueable and to be in the very Center of the world , proouing the same with many most strong reasons not néede full here to be rehearsed , because I thinke fewe or none do doubt thereof , and specially the holy Scripture affirming the foundations of the earth to be layd so sure , Psal. 104. that in neuer should mooue at any time : Againe you shall finde in the selfe same Psalme these words , Hee appointed the Moone for certaine seasons , and the Sunne knoweth his going downe , whereby it appeareth that the Sunne mooueth and not the earth . But leauing this matter , we will now speake of the compasse of the earth , and of the Longitude and Latitude thereof .

Of the compasse of the earth , and of the diuersitie of measures according to diuerse countryes . Chap. 6.

CAn the whole earth be measured ?

Yea very well , for sith the earth and the water ( as hath béene said before ) doe make together , one whole Sphericall or round bodie , and that euery great Circle as well thereof as of the heauens , containeth 360. degrées , there is no more then to be done , but to allow for euery such degrée 60. Italian miles , which differ not much from our English miles , so as in multiplying 360. degrées by 60. you shall finde the whole compasse of the earth to be 2160. miles , of which compasse if you would knowe the true Diameter , then hauing multiplyed the said compasse or circuit of 21600. miles by seuen , deuide the product thereof by 22. and the quotient together with the remainder , will shew the true Diameter which is 6872. miles , fiue furlongs , and 9/11 of a furlong , and the halfe of that is the semi-Diameter of the earth , which is 3436. miles , and 4/11 of a mile : and as the Italians and we English men doe measure great distances on the earth by miles , so the French , the Spanish , and the high Almaines , doe measure such distances by leagues both by lande and sea , and euery one differeth from other : for the French league containeth two of our miles , the Spanish league thrée , and the common league of Germany foure , and the great league of Germanie containeth fiue of our miles , yea in some places of Germany , as in Sueuia , the leagues are so long as a man shall skant ride thrée of them in a whole day . Againe the Grecians did measure the distances of the earth by furlongs , the Egyptians by signes , and the Persians by parasanges , all which measures doe greatly differ euen in the smallest partes , from whence all measures doe take their first originall : for as well amongest the auntient men , as amongest them of latter dayes , foure barley kernels couched close together side by side , and not end long , are saide to make a finger breadth , and thrée finger breadthes an inch , and foure finger breadthes a palme or hand breadth , and thrée palmes or nine inches a span , and foure palmes or hand breadthes a foote , and two foote and a halfe to make a common pace , and fiue foote to make a Geometrical pace , of which kinde of paces , 125. do make a furlong , and eight furlongs doe make an Italian mile , and foure such miles doe make a common Germaine league , as hath béene saide before , but by reason that the barley kernels be not in all Countryes of like bignesse , neither finger breadthes , inches , hand-breadthes , féete , nor any of the other measures are founde any where to be equall : for the French foote of Paris is longer then ours by an inch , and the Italian foote is longer by two inches and more , and yet their miles are somewhat shorter then ours : and the Germaine foote ( according to Stophlerus . ) is lesse then ours by two inches and a halfe . But to shew the diuersitie of measures would require a long discourse more intricable then profitable , and therefore I leane to talke any further thereof , wishing you when we speake of miles , furlongs , paces , or féete , to consider the measure thereof according to the inch or foote of our English standard .

Of the Longitude and Latitude of the earth . Chap. 7.

WHat meane you by the Longitude and Latitude of the earth ?

Longitude is as much to say as length , and Latitude signifieth breadth , for sith the earth is a bodie , it must néedes haue both length , breadth , and depth .

How define you such Longitude and Latitude , and how is it to be counted ?

The Longitude of the earth in generall is that space or vpper face of the earth which extendeth from West to East , and againe from East to West : And the Latitude in generall is that space which extendeth North & South euen from the one pole to thother . Now to know how such Longitude and Latitude is to be accounted , you must first vnderstand that the Equinoctiall Circle girding the earth in the very midst , is deuided into 360. degrées by reason of certaine Meridians which passing through the Poles of the world , doe cut ech halfe of the Equinoctiall in eightéene pointes , which being doubled doe make 36. spaces , euery space containing tenne degrées : and some doe deuide the Equinoctiall with 36. Meridiaans , cutting ech halfe thereof in 36. points , which being doubled doe make 72. spaces , euery space containing 5. degrées , which commeth all to one reckoning , for fiue times 72. doe make 360. as well as tenne times 36. of which Meridians , be there neuer so fewe or many ( for you may if you will make halfe as many Meridians as there be degrées in the Equinoctiall which amounteth to a 180. ) yet according to Ptolomie , that Meridian is saide to be first and furthest Westwarde which passeth through the Ilands called Insulae fortunatae , for the West Indies were not knowne nor discouered in his dayes , nor yet long time after , since the discouerie whereof , the late Cosmographers of these dayes doe make the first Meridian to passe through the Ilandes called Azores , which Ilandes , as appeareth by their Cardes are situated more Westwarde from the foresaid Insulae fortunatae , by fiue degrées , the reason that mooueth them so to do , is because the Mariners Compasse as they say , wil neuer encline to the true North pole , but when they sayle either by the Ile S. Mary or S. Michaell , affirming that in euery other place the Compasse doth varie from the true North , eyther by Northe asting or Northwesting . And by thus altering the auncient placing of the first Meridian , they must likewise alter all the Longitudes set downe heretofore by Ptolomie or any other auncient writer , notwithstanding the matter is easilie holpen , for by adding to euery Longitude Eastward fiue degrées , or by subtracting fiue degrées from euery Longitude Westward , you shall not greatly vary from those auncient Longitudes that be truely set downe . But to returne againe to my first purpose , I say tha● wheresoeuer this first Meridian cutteth the Equinoctiall , there beginneth the first degrée of Longitude , which procéedeth Eastward vntill you come to 180. degrées , which being the one halfe of the earth is as farre as you can goe Eastwarde , for then the earth being round , you must néedes turne againe Westward vntill you come to the 360. degrée , which is the last degrée of Longitude , and endeth where the first degrée beginneth : and therefore the Cosmographers measuring alwaies the Longitude by the degrées of the Equinoctiall , do define Longitude to be that portion of the Equinoctiall Circle , which is contained betwixt the first Meridian and the Meridian of any place supposed , but the distance betwixt any two supposed Meridians ( neither of them being the first Meridian ) is not called of them Longitude , but the difference of Longitude : For suppose the distance of the one Meridian to be twentie degrées distant from the first Meridian , and the other but tenne , these Longitudes you sée are not like but doe differ , and therfore the distance betwixt any such two places may be very well called the difference of Longitude , and not Longitude it selfe , which hath alwaies regard to the first Meridian and to none other .

Define once againe what Latitude is ?

Latitude is none other thing but the distance of any place from the Equinoctiall either towards the North pole or towards the South pole , so as there be two kinds of Latitudes , the one Northerne , and the other Southerne : And such Latitude is measured vpon the Meridian which passeth through any place supposed . For euery Meridian is also deuided into 360. degrées , and by reason that the Equinoctiall girdeth all such Meridians in the very midst , it deuideth them all into foure equall quarters euery quarter contayning 90. degrées , which is the greatest Latitude that any place can haue , as you may sée in this figure following , whereof the first Meridian on the left hand is put to signifie according to Ptolomie , that which passeth through the Fortunate Ilandes , or by the Azores according to the moderne Cosmographers ( if you will haue it so ) contayning the degrées of the Latitude both Northward and Southward , and through the midst of all the Meridians passeth the Equinoctiall contayning the degrées of the Longitude .

The figure of the longitude and latitude of the world .

What be the other manifold Circles in this figure deuiding the Meridians on ech side of the Equinoctiall , as well towards the North pole , as towards the South pole .

They be called Parallels , whereof we shall speake in the next Chapter , in the meane time marke well this figure that thereby you may the better conceiue what is the Longitude and Latitude of the earth .

Vnderstanding now the Longitude & Latitude of the whole earth , I am desirous to know how the Longitude and Latitude of euery seuerall place of the earth or sea is to be found out , & how farre any place is distant one from another .

How to know the Latitude of any place , aswell in the day as in the night . Chap. 8.

BEcause the Latitude of any place is more easie to be found ( as most men thinke ) then the Longitude , I will first treat of Latitude . The Latitude them is to be knowne by the Astrolabe , Quadrant , Crossestaffe , and by such like Mathematicall instruments , & that diuerse wayes whereof the most easie is thus : first with your Astrolabe or Quadrant , or any such like instrument , take the height of the sunne right at noone , when the sunne is in the first point of Aries or of Libra , which height if you subtract from 90. that which remaineth is the true Latitude of that place . But if you would knowe the Latitude at any other day or time of the yeare , then after that you haue taken the height of the sunne at noone , otherwise called the Meridian altitude , you must first learne to knowe the true degrée of the sunnes declination by the Table of the declinations before set downe , together with the vse thereof in the 13. Chap. of the first booke , or by some other Table more lately calculated , and if such declination be Northernly , then you must subtract that from the foresaid Altitude or height : but if the declination be Southernly , them you must adde the same vnto the foresaid height , and by such subtraction or addition , you shall haue the height of the Equinoctial aboue your Horizon , which being subtracted from 90. that which remaineth is the true Latitude of that place : and to be sure in taking the Meridian altitude , it shall be néedefull to take it diuerse times one after another , with some little pawse betwixt , to sée whether it increaseth or decreaseth , for if it doth increase , then it is not yet full noone , but if it decreaseth then it is past noone . This last way of finding out the Latitude , is , and hath béene most commonly taught , as well by the auncient , as moderne writers , as a most sure and readie way of finding the Latitude of any place .

What if the sunne doe not shine at noone , nor perhaps all that day ?

Then you must tarrie vntill night that some starre appeare , which you perfectly knowe , and such a one as both riseth and setteth . And hauing taken the Meridian altitude of that starre with your Astrolabe or Quadrant , you must learne what declination he hath , and whether it be Northerne or Southerne . For if the starre hath North declination , then you must subtract his declination from his Meridian altitude , and the remainder shall be the Altitude of the Equinoctiall , which being taken out of nintie , shall be the Latitude of the place , or eleuation of the pole : but if the declination of the starre be Southernly , then you must adde his declination to his Meridian altitude , and that summe shall be the Altitude of the Equinoctiall , which being taken out of 90. the remainder shall be the eleuation of the Pole . As for example , supposing that you knowe the starre called Arcturus , or Bubulcus , and that you finde his Meridian altitude by your Astrolabe or Quadrant to be 59. degrées 30. minutes and also that you haue learned by some Table , that his declination to the Northwarde is 21. degrées 30. minutes : here by taking his saide declination because it is Northernly , out of his Meridian altitude , you finde the remainder to be 38. degrées which is the Altitude of the Equinoctiall , which béeing taken from 90. the remainder wil be 52. degrées , which is the Latitude of the place whereas you made your obseruation , and this is a farre more readie waye then to waite all night to take the Meridian altitude , and also the depression of such a starre as neuer setteth , which is séeledome done in one selfe night . And therefore I would wishe all Mariners to acquaint themselues with manie starres that doe both rise and set , and so shall they be sure to finde one such starre or other , to be at his Meridian altitude at any houre of the night that they desire , if the starres doe shewe .

How to know the true Longitude of any place . Chap. 9.

THough the Longitude may be found out by diuers waies not easie for euery mans capacitie , yet because Gemma Frisius thinketh none so sure as to knowe the same by the eclipse of the Moone , ( which also as he saith ) may sometime faile by reason of the diuersitie of aspects and Latitude of the Moone , and for that cause hath inuented a more readie way to finde out at all times the Longitude of any place , I minde here therefore briefely to shewe you first the order of finding out the Longitude by the Eclipse of the Moone , and then how to finde out the same by that readie way which he hath inuented : the order then to knowe it by the Eclipse of the Moone is thus : First you must learne by some Ephemerides at what houre the Eclipse shall be in some place , where you knowe alreadie by some Table the Longitude , that done , you your selfe or some other for you , must the same day of the Eclipse obserue by the Astrolabe at what houre the Eclipse beginneth in that place , whereof you knowe not the Longitude : For if the Eclipse doe beginne in both places at one selfe houre , then assure your selfe that both places haue one selfe Longitude , but if it beginne sooner or later , then there is difference betwixt them , according to the varietie of the time , which difference is thus to be knowne : Take the lesser summe of houres out of the greater , and there shall remaine either houres , or minutes , or both , if houres , then multiply the same by fiftéene , if minutes , deuide the same by foure , ( for in this account fiftéene degrées doe make one houre , and foure minutes doe make one degrée ) and adde the difference so found to the Longitude , if the Eclipse doe appeare there sooner : if later then subtract the sayd difference from the knowne Longitude , and that which remaineth will shew the vnknowne longitude . But note by the way , that if there remaine any minutes after the diuision , you must multiply those minutes by 15. and so shall you haue the minutes of degrees .

Shew the vse of this rule by some example .

For example I finde by the tables of Ptolomey that the longitude of Paris in France is 23. degrées , and by some Almanacke or Ephemerides I finde that the Eclipse doth begin there at thrée houres after midnight , now by this I would know the longitude of Tubing a famous citie in Sueuia , which is a region of Germany , at which towne in the verie day of the Eclipse I cause to bee obserued by Astrolabe at what houre the Eclipse beginneth there , and I finde that it beginneth at 3. of the clocke and 24′· after midnight , then by subtracting the lesser number of time out of the greater , I find the remainder to bée 24′· which béeing deuided by 4′· which doe make one degrée , the quotient shall be 6. degrées , and that is the difference , which being added to the knowne longitude of Paris ( because the Eclipse is sooner there than at Tubing ) it maketh in all 9. degrées , whereby I gather that the longitude of Tubing is 29. degrées , by this meanes all the Tables of Cosmographers are most commonlie made , and yet manie times they greatly differ in their longitudes for lacke perhaps of vsing diligence in taking the right houre and moment of the Eclipse , and for not dulie considering the diuers aspects , and what latitude the Moone hath at that instant which may cause great error .

A readie way to finde out the longitude of any place , inuented by Gemma Frisius . Chap. 10.

THat way is done by the helpe of some true Horologie or watch apt to bee carried in iournying , which by an Astrolabe is to bée rectified and set iust at such houre as you depart from the place where you are , to goe to any other place whereof you are desirous to know the longitude , in which your going you must bee diligent to sée that your watch neuer cease going , and being arriued at that place whereof you séeke to know the longitude , you must tary vntill the Index do iustlie touch the pricke of some perfect houre , and also at that instant , to sée what houre it is by your Astrolabe , for if your Astrolabe and watch do both agrée in one , then assure your selfe that there is no difference of longitude , but that you haue trauelled still vnder one selfe Meridian eyther towards the North or South . But if they differ one houre or certaine minutes , then reduce them to degrées , or to the minutes of degrées in such order as is before taught , and thereby you shall finde the longitude which you desire to know . But to take the longitude of anie place vpon the sea by this manner of way , most men thinke it were a great deale better to doe it by the helpe of a great houre glasse , made to run 24. houres , which must be watched when it is ready to run out , that it may be immediately turned : for watches made of Iron or Stéele will soone rust vpon the sea . This way of taking the longitude and many others I doubt not but that M. Borrough hath tryed , & thereby is able to iudge which is best , wherfore in mine opinion hée may do his country great good by setting downe in writing that way which hee by experience knoweth to be best and readiest , for want whereof the Mariners cardes do make the sea men oft to erre in their voiages .

Another way taught by Appian to finde out the longitude of any place with the crosse staffe by knowing the distance betwixt the Moone and some knowne starre that is situated nigh vnto the Ecliptique line . Chap. 11.

FIrst séeke to knowe by the Astronomicall Tables the true moouing of the Moone according to the longitude at the time of your obseruation at some certaine place , for whose Meridian the rootes of those Tables are calculated and verefied . Also you must know the degrée of longitude of some fixed Starre nigh vnto the Ecliptique , going eyther next before or els next after the moouing of the Moone , then you must séeke out the distance of the moouing of the Moone and of the said Starre , which distance once had , apply the crosse staffe to your eye , and mooue the crosse vp and downe vntill you may sée the center of the bodie of the Moone with the one end of the crosse and the foresaid fixed Star with the other end of the crosse , so shall the crosse shew you by the degrées and minutes marked vpon the staffe , the distance of the Moone and of the foresaide Starre aunswerable to the place of your obseruation , which being set downe , set downe also the distance betwixt the Moone and the foresaid Starre that was first calculated , and then take the lesser out of the greater , so shall remaine the last difference , which may be rightly called the diuersitie of aspectes , which difference if you deuide by the moouing which the Moone maketh in one houre , you shall knowe thereby the time in which the Moone is or was ioyned with the first distance of the foresaid Starre , then hauing conuerted that time into degrées and minutes , adde or subtract the product thereof to or from that Meridian , vnto which the Tables ( whereby you first calculated the moouing of the Moone ) were verefied , that is to say , if the distance betwixt the Moone and the fixed Starre of your obseruation be lesser , then adde the degrées and minutes to the knowne longitude , so shall you find the place of your obseruation to be more Eastward , but if it bée greater , then subtract the degrées and minutes from the known longitude , and the place of your obseruation shall bee more westward . All which rules Gemma Frisius affirmeth to be true , so as the Moone be more westward than the fixed Starre : for if at the time of your obseruation the Moone bee more Eastwarde , then you must worke cleane contrarie , that is to say , if the distance betwixt the Moone and the fixed Starre bee lesser , you must subtract the degrées and minutes from the knowne longitude , so shall the place of your obseruation bee more Westward : but if it be greater , then you must adde the degrées and minutes vnto the knowne longitude , and you shall finde the place of your obseruation to be more Eastward .

How to know the distance of places , that is to say , how many miles one place is distant from another , and howe many wayes places are said to differ in distance one from another . Chap. 12.

THe distance may be knowne diuerse wayes , that is , either Arithmetically , Geometrically , or by the Tables of Sinus . But before I shew you the order of any of these wayes , you haue to vnderstand that any two places do differ in distance one from another one of these 3. maner of wayes , that is , either in latitude onely , or in longitude onely , or els in both : if two places hauing one selfe longitude do differ onely in latitude , then according to the Arithmetical way you must subtract the lesser latitude out of the greater , and the remainder shall be the difference , which being multiplied by 60. will shewe the number of miles as for example , London and Roan hauing in a manner one selfe longitude , doe differ onely in latitude , for the latitude of London is 51. degrées 32′· & the latitude of Roan is 49. degrées and 10′· which being the lesser latitude and therefore to bee taken out of 51. degrées 32′· there remaineth 2. degrées 22′· which 2. degrées being multiplied by 60. maketh 120. whereunto if for the 22′· annexed to the degrées you adde 22. miles ( for euery minute is a mile ) it shall make in all 142. miles , which by a right line is the true distance betwixt London and Roan . But you haue to note that the difference of 2. sundry latitudes is not to be knowne by subtracting the lesser out of the greater , vnles both the places be so situated , as both may haue either North latitude or South latitude , for if the one place haue North latitude and the other South latitude , the difference is to be known by addition , and not by subtraction : as for example , Naples in Italy hath 41. degrées of North latitude , and la Madalena in Aphricke not farre from Manicongo , hath 8. degrées of south latitude , both places hauing one self Meridian , here the difference of these two latitudes is to be knowne by addition , and not by subtracting the lesser out of the greater , for 8. and 41. being added together do make 49. and that is the true difference , which being multiplied by 60. maketh 2940. miles . So likewise the difference of the two longitudes is not alwayes knowne by subtracting the lesser out of the greater , vnlesse the two places haue both East longitude or els both West longitude : As for example Lisbona in Spaine hath in East . longitude 13. degrées and Cap de losslauos in the West Indies hath in west longitude 334. degrées : here the difference of these two longitudes is not to be known by taking the lesser out of the greater , but thus , first take 334. out of 360. and there will remaine 26. degrées , wherunto if you ad the East longitude for Lisbona , which is 13. degrées , it wil make in all 39. degrées , which is the true difference of the 2. longitudes , for if you should take 13. degrées out of 334. there would remaine 321. which is not the true difference . But to know the distance of two places differing in longitude , this Table here following is needfull .

TheTableofmilesaunswerabletoonedegreeofeueryseueralllatitude. 1 2 3 4 5 6 D M S D M S D M S D M S D M S D M S 1 59 59 16 57 41 31 51 26 46 41 41 61 29 5 76 14 31 2 59 58 17 57 23 32 50 53 47 40 55 62 28 01 77 13 30 3 59 55 18 57 4 33 50 19 48 40 9 63 27 14 78 12 28 4 59 51 19 56 44 34 49 45 49 39 22 64 26 18 79 11 27 5 59 46 20 56 23 35 49 9 50 38 34 65 25 21 80 10 25 6 59 40 21 56 1 36 48 32 51 37 46 66 24 24 81 9 23 7 59 33 22 55 38 37 47 55 52 36 56 97 23 27 82 8 21 8 59 25 23 55 14 38 47 17 53 36 7 68 22 29 83 7 19 9 59 16 24 54 49 39 46 38 54 35 16 69 21 30 84 6 16 10 59 5 25 54 23 40 45 58 55 34 25 70 20 31 85 5 14 11 58 54 26 53 56 41 45 17 56 33 33 71 19 32 86 4 11 12 58 41 27 53 28 42 44 35 57 32 41 72 18 32 87 3 8 13 58 28 28 52 59 43 43 53 58 31 48 73 17 33 88 2 5 14 58 13 29 52 29 44 43 10 59 30 54 74 16 32 89 1 ● 15 57 57 30 51 58 45 42 26 60 30 0 75 15 32 90 0 0

Describe this Table .

This Table is diuided into 6. collums , euerie collum containing , first the degrées of longitude , and then the miles and seconds of miles answerable to euerie degrée , for euerie degree of the verie Equinoctiall it selfe is in value 60. miles , but the further you goe from the Equinoctiall either Northward or Southward , euery degrée of latitude is lesser in value than other , and containeth fewer miles , as you may easilie sée by the said Table , procéeding from one degrée to 90. which is the greatest latitude that any place can haue .

How to know by the helpe of the foresaide Table the distance of two places differing onely in longitude . Chap. 13.

YOu must multiplie the difference of the longitudes by the number of miles answerable to the latitudes of the said places , omitting alwayes the seconds of miles set down in the said Table , because in this account they are of small importance : as for example , London and Antwerp , hauing both in a maner one selfe latitude , do differ onlie in longitude 6. degrées 42′· which difference being multiplyed by 37. miles answerable to 51. degrées of latitude , as you sée in the Table , doe make in all 247. miles and 54″· of a mile . But in making this multiplication , you must first multiply the 6. whole degrées by 37. and the product thereof will amount to 222. then by the rule of proportion you may find out the value of the minutes annexed to the degrées of the difference of longitude in saying thus , if 60′· which is one degrée , do require 37. miles , what shall 42′· require . And by working according to the rule of proportion , you shall find the fourth number which you séeke to bée 25. miles and 54″· which being added to the first product 222. maketh in all 247. miles and 54″· of a mile , which wanteth but 6″· to make vp another mile .

How to finde out the true distance of two places differing both in longitude and latitude by the Arithmeticall way . Chap. 14.

HOw is that done ?

First , take the difference of the longitudes and latitudes of both places by subtracting the lesser out of the greater , then conuert the same into miles by multiplying the difference of the two longitudes into the miles that be answerable to the latitude of each place , which miles you shall find in the Table aforesaid , and if there be any minutes annexed to the degrées of the difference of longitude , then reduce the same also to miles by the rule of proportion , as before is taught , and hauing added the two products together , halfe the summe , and set it by it selfe . Then multiply the difference of the latitudes into 60. miles , and adde thereunto the fraction of minutes annexed to the said difference if it hath anie fraction , allowing for euerie minute one mile , and set that number also by it selfe : that done , square the summes reserued , that is to say , multiplie ech one part by it selfe into it selfe , and hauing added the two products together into one summe , séeke out the square roote thereof , & that shal be the true distance of the two places . As for example , if you would knowe the true distance betwixt London and Venice , first you must knowe by some Table the longitude and latitude of both townes , wherefore finding the longitude of London to be 19. degrées , 54′· and the latitude thereof to be 51. degrées , 32′· And the longitude of Venice to be 35. degrées , 30′· and the latitude thereof to be 44. degrées , 45′· Now by subtracting the lesser longitude out of the greater , I finde the difference of longitude to bee 15. degrees , 36′· and by subtracting the lesser latitude out of the greater , I find the difference of latitude to be 6. degrées , 47′· Then knowing the latitude of London to be 51. degrées , I resort to the table of miles appointed for euerie degrée of latitude before set downe , and theer I find that to 51. degrées of latitude do answere 37. miles and certaine seconds , which being of small moment are not woont to be reckoned . Then in multiplying the difference of the longitudes which is 15. degrées 36′· by 37. miles I find the product of the 15. degrées so multiplied to be 555. and because there bée 36′· annexed to the foresaid 15. degrées , I séeke by the rule of proportion to knowe how many miles that fraction containeth , in saying thus , if 60. require 37. what shall 36. require ? and I find 22. miles , which being added to 555. maketh 577. then by séeking in the foresaid table how manie miles be answerable to the latitude of Venice , which is 44. degrées , I find the number of miles to be 43. by which number I multiply once againe the difference of longitude , which is 15. degrées . 36′· the product whereof , together with the fraction annexed thereunto being conuerted into miles by the rule of proportion , as before , doth amount to 670. which sum being added to the former conuerted longitude , which is 577. maketh in all 1247. the halfe whereof is 623. which halfe number I reserue by it selfe , that done I multiply the difference of the latitude , which is 6. degrées 47′· by 60. miles , in saying 6. times 60. maketh 360. whereunto I adde for the 47′· annexed 47. miles , & it maketh in al 407. which sum I reserue also by it self . Then I multiplie the first reserued number into it selfe , the product whereof is 388129. That done , I multiplie the seconde reserued number also into it selfe , the product whereof is 165649. which two last productes being added together , do make in all 553778. whereof the square roote being taken , is 744. miles , which is the true distance of Venice by a right line from London . And to the intent that the order and working herein may more plainly appeare vnto you , I haue set downe all the particular numbers of the same here by them selues , as it were in a Table .

longitude . latitude London . 19. degr. 54′· 51. degr. 32′· Venice . 35. degr. 30′· 44. degr. 45′· The difference of their longitudes and latitudes . 15. degr. 36′· 6. degr. 47′·

The difference of the longitudes conuerted into miles : for London is 577. For Venice . 670. The summe of the two conuerted longitudes added together , is 1247. The halfe whereof , which is the first reserued number , is , 623 The second reserued number , which is the difference of the latitudes , conuerted into miles , is 407 The summe of the first reserued number multiplied into it selfe , is 388129 The summe of the second reserued number multiplied into itselfe , is 165649 The summe or both added together is 553778 The square root whereof , which is the summe of the miles , is 744

How to finde out the distance betwixt two places , differing both in longitude and latitude by help only of a demi-circle deuided into 180. degrees without any Arithmetike . Chap. 14.

BVt now because the way before taught to finde out such distance by the Arithmeticall way may séeme perhaps to some folkes very busie and tedious , I haue thought good therefore to set downe this other way which was sent me not long since from my louing friend M. Wright of Cayes colledge in Cambridge who is wel learned in the Mathematicals , & is so apt therunto by nature , as he is like inogh to attain to such perfit knowledge therein as he may be able thereby hereafter greatly to perfect his country , if for want of sufficient exhibition he bée not forced to leaue so noble a studie , wherefore I wish with al my heart that all Gentlemen of abilitie were minded to shew their liberalitie towards him in that behalfe . But to returne to my matter , I say that the way to find out the foresaid distance is this here following : first hauing drawne a demi-circle vpon a right Diameter ( the larger that the demi-circle is the better , ) and diuided the same into 180. degrees , like vnto this hereafter described , and marked with the letters a b c d. whereof d. is the center and a. c. the Diameter . Then learne first by some Table to know the longitude the latitude of both places , and the difference of their longitudes , as you did before in séeking to know by the Arithmetical way the distance betwixt London and Venice , the difference of whose two longitudes is 15. degrées , and 36′· as you may sée in the former Table : for in working by this way , you haue chiefly to séeke out in the circumference of the demicircle but thrée things , that is , first , the difference of the two longitudes , secondlie , the lesser latitude , and last of all the greatest latitude . Knowing therefore the difference of the said two places in longitude to be 15. degrées , 36′· séeke out the same in the demicircle , beginning to count at A. and so proceed towards B. And at the end of those degrées and minutes set downe a pricke marked with the letter e. vnto which pricke drawe a right line by your ruler from D. the center of the demicircle . That done , séeke out the lesser latitude , which is 44. degrées , and 45′· in the forsaid demi-circle , beginning to account the same from the pricke e. and so procéede towards the letter B. and at the end of the said lesser latitude , set downe another pricke marked with the letter g. from which prick or point draw a perpendicular line which by help of your squire or compasses may fal with right angles vpon the former right line drawne from D. to e. and where it falleth , there set downe a pricke marked with the letter h. That done , séeke out the greater latitude , which is 51. degrées , and 32′· in the foresaid demicircle , beginning to account the same from A. towards B. and at the end of that latitude set downe another prick marked with the letter I. from whence draw another perpendicular line that may fal by help of your squire or compasses with right angles vpon the Diameter A. C. and there make a prick marked with the letter K. That done , take with your compasse the distance that is betwixt k. and h. which distance you must set downe vpon your said Diameter A. C. setting the one foote of your compasse vpon k. and the other towardes the center D. and there make a pricke marked with the letter L. Then take with your Compasse the length of the shorter perpendicular line g. h. and apply that widenesse vpon the longer perpendicular line I. K. setting the one foot of your Compasse at I. which is the end of the greater latitude , and extend the other foote towards K. and there make a pricke marked with the letter M. That done , take the distance betwixt L. and M. with your compasse , and apply the same to the demicircle , setting the one foote of your Compasse in A. and the other towards B. and there make a prick marked with the letter N. And the number of degrées contained betwixt A. & N. will shew the true distance of the two places , which you shall find to be 12. degrées and almost 24′· Now by allowing for euery degrée 60. miles , and for euery minute a mile , the summe of miles will agrée with the former distance found out by the Arithmetical way which was 744. miles . And thus you haue to deale to know the distance of any other two places whatsoeuer , differing both in longitude and latitude . But you haue to note by the way , that if the difference of the longitudes doth excéede the number of 180. then you must subtract that excéeding difference out of 360. and the remainder shall be the difference of the longitudes , and then work in all points as is before taught . And this way is as Geometricall , as that which Appian setteth downe in his booke of Geographie , to be done by the helpe of the terrestriall Globe , the order whereof here followeth .

Longitude . latitude . London . 19. deg. 54 ' . 52. deg. 32 ' . Venice . 15. deg. 30 ' . 44. deg. 45 ' . The difference of their longitudes and latitudes is 15. deg. 36 ' . 6. deg. 47 ' .

How to find out the distance of the places by the Geometricall way . Chap. 15.

HOw is that done ?

Most readily and easily by helpe of a terrestrial globe in this maner following . First take the distance of the 2. places by extending your compasse vpon the globe , from the one place to the other , which if you would know how many miles it comprehendeth , apply the same distance so taken vnto the Equinoctiall line , setting the first foot of your compasse vpon the first Meridian in that point , whereas it cutteth the Equinoctiall , then sée how many degrées of the Equinoctial are comprehended betwixt the two féet of your Compasse , and multiply those degrées by 60. & the product therof shall shewe you howe many Italian miles such distance is in length . But if either of the places or both , be wanting and not expressed in the Globe , then you must learne by the tables of Ptolomey or of some others , as of Appian , Gemma Frisius , Orontius or such like , the longitude and latitude of the said places , that done , hauing sought out the longitude of the first place in the Equinoctiall , turne the globe about with your hand vntill you haue brought the longitude right vnder the brazen Meridian , which being stayed there , séeke out in the said Meridian the latitude of the said place , and there set a marke vpon the globe , for there the place should stand , and doe in like maner to finde out the seconde place : Then by extending your Compasse from the one marke to the other , you shall haue the true distance , which distance if you apply to the Equinoctiall like as before is taught , the degrées thereof being multiplyed by 60. will shewe you how many miles those two places are distant one from another .

May not the distance of places be found out aswell by an vniuersall Map as by the globe terrestriall ?

Yes indéed and more readily by reason that for the most parte euery Map hath his proper skale , so as you néed to do no more but to take the distance of the 2. places with your Compasse , and to apply the same to the skale shewing the miles or leagues .

What if the Map haue no skale ?

Then you must séeke out the distance by such meanes as I do shew in my Treatise of the vse of vniuersall Mappes , and also in my description of Planctius his Map .

I pray you in the meane time proceed in shewing me the third way of finding out of the distance of places which you said was per tabulas Sinuum .

The order of finding out the distance of 2. places differing both in longitude and latitude per tabu●as Sinuum is plainly set down before in the ende of my Arithm●ticke , where as I doe make a plaine description of the said Tables , and do shew the vse thereof aswel by this as by diuers other examples , wherefore I wish you to resort to that treatise and you shal haue your desire . For hauing for this time sufficientlie spoken of the longitude , latitude and distance of places , and how the same is to bee found out , I thinke it méet nowe to treat of the 5. Zones , of Climes and Parallels , whereinto the amient Cosmographers thought good to deuide the earth , to the intent that euery part thereof night bée the better known how it is situated either Northward or Southward , whether it be hot or cold , or betwixt both , and of what length the day and night is in euery place , and what manner of shaddow the Sun yéeldeth euery where , and such like accidents , and first of the 5. Zones .

Of the 5. Zones . Chap. 16.

THe most ancient Cosmographers considering how the Sun by his oblique , and bariable course did warme with his beames one part of the earth more than another , gathered thereby that the earth had thrée temperatures that is to say , extreame hot , extreame cold , and a meane temperature , that is , neither too hotte , nor too colde : And therefore to shew vs vnto which of these temperatures , any part of the earth was subiect , they deuided the earth into 5. Zones , answerable to the 5. Zones of the firmament , by helpe of the Equinoctiall and the 4. lesser circles before described in the first part of this treatise , the 20. Chapter , and are there set foorth in figure which figure I thought good to set downe againe in this place , to the intent you might the better remember what was said there touching the Zones .

But other Cosmographers comming afterward , not satisfied with the 5. Zones , because they shewe nothing but the situation and the thrée temperatures of the earth , did deuide the earth into certaine Climes and Paraliels , to find out thereby ● the length of the day and night in euery place , and the true latitude thereof as Ptolomey and many others after him haue done , making such diuisions as we shal speake of hereafter . But trulie I think with Mercator , that the best and most exact way of deuiding the earth to serue all purposes is to bee made by degrées and minutes , wherein is lesse error than in Climes and Parallels , neither can Climes or Parallels be so well described when you drawe night to any of the Poles , for that the spaces as well betwixt the Parallels as betwixt the Meridians doe growe continually straighter & straighter , as you may sée in the figure of Parallels hereafter following .

I remember that in the 20. chapter wheras you described the 5. Zones , you said that the ancient men did greatly erre in affirming 3. of the Zones to be vnhabitable , that is , the two colde Zones and the hot Zone , I pray you therefore shew me here the cause why they erred ?

They erred for lacke of experience , because they had neuer traueled into those regions , but in these latter dayes men of diuers nations , specially the Spanish , English , French , & Flemmish haue trauelled very farre , some towards the North pole , and some towardes the South pole , and also through the burnt Zone , for those that sayle from the North parres towardes the South pole , or from the South partes towards the North pole , must néeds passe in their voiage through the burnt Zone : and these men doe affirme that they haue found all the 3. Zones , that is to say , the two cold and hotte Zones to be well inhabited . And our late Cosmographers doe not let to render cause why they should be habitable , for ( say they ) though the cold be very extream in those regions that lie next vnto the Poles , yet the Sun appearing and giuing shine vnto them both day and night , doth greatlie qualifie and moderate the extreame cold of those regions . But truly , in mine opinion , they haue small comfort of the Sun , sith it striketh almost round about their féet , without yéelding any warm reflexe from aboue , and especially to those that doe inhabite nigh to either of the Poles . Againe , they say that the burnt Zone is habitable , by reason that the night to them is continually as long as the day , the coolenesse whereof doth greatly refresh the extream heat of the day . But now let vs returne to our purpose , and speak somewhat of the Climes and Parallels , & because euery Clime consisteth of 2. Parallels , I think it best to speak first of Parallels .

Of Parallels . Chap. 17.

WHat be Parallels .

Parallels be lines either right or circular alwaies equallie distant one from another so as they can neuer méete . And of Parallels that are to be considered in the Spheare , some make 3. kinds , according to their thréefold signification , for some are called the Parallels of the sunne , who in departing from the Equinoctiall towards any of the Poles , maketh euery day throughout the yeare one Parallel , so as in going from the Equinoctiall to the Tropique of Cancer , he maketh 182 Parallels . And as many againe in going from the Equinoctiall to the Tropique of Capricorne . The second kinde of Parallels are called the Parallels of Latitude , And the third the Parallels of the longest day , which two last are in effect both one , for the further that any Parallel is situated from the Equinoctiall towards either of the Poles , the more Latitude it hath , & so by consequent maketh the day longer to those that dwell vnder that Parallel , of which Parallels the auncient Cosmographers do make in all but 21. procéeding proportionally either towards the North pole or South pole , as you may sée by this figure here following the middle line or Circle whereof is the Equinoctiall .

And euery Parallel procéeding from the said Equinoctiall , eyther Northward or Southward , doth lengthen the day by one quarter of an houre in such proportion as this Table here following sheweth , appointing to the first Parallel , and next vnto the Equinoctial 4. degr. & 15′· and to the second parallel 8. degr. 30′· and so foorth vntil you come to the 21. Parallel passing through the Iland , which Parallel is distant from the Equinoctial 63. degr. 16′· further then which Parallel Northward , the Tables of Ptolomie doe not extend , and Southward they extend no further then to that Parallel which hath lesse 20. degrées of Latitude .

TheTableofParallelsshewinghowmanydegreesandminuteseueryoneisdistantfromtheEquinoctiall,madeaccordingtotheruleofPtolomy. Parallels D M the first 4 15 the second 8 30 the third 12 45 the fourth 16 35 the fifth 20 30 the sixth 24 15 seuenth 27 30 the eight 30 45 the ninth 33 40 the tenth 36 24 eleuenth 39 0 twelfth 41 20 thirtenth 43 15 fourtenth 45 24 fifteenth 48 40 sixteenth 51 50 seuententh 54 30 eightenth 56 30 ninetenth 58 20 twentieth 61 10 the xxj . 63 16

Of Climes both old and new . Chap. 18.

WHat is a Clime ?

A Clime is a space of the earth comprehended betwixt two Parallels , in which space the longest day doth vary by halfe an houre .

How many Climes be there ?

The auncient Cosmographers deuided aswell that part of the earth which lyeth betwixt the Equinoctiall and the North pole , so much I say as they thought to be habitable , as also the habitable part which lyeth betwixt the Equinoctiall and the South pole , ech of them into 7. Climes , to euery of which Northerne Climes they gaue a seuerall proper name , according to the name of the place through which the midst of the said Clime did passe , for they called the first Clime Diameroes , Dia is a Gréeke preposition signifying in English By or Through , and Meroes is a Citie of Egypt situated in a certaine Ile enclosed with the flood Nilus , which Ile hath also the like name . The second Clime is called Dia Syenes , which is also a Citie in Egypt situated right vnder the Tropique of Cancer . The third Clime is called Dia Alexandrias , which is another Citie of Egypt situated vpon the West mouth of Nilus , falling into the sea of Egypt . The fourth Clime is called Dia Rhodou , Rhodos is the chiefest Citie of an Ile called Rhodos , standing in the sea called Mare Carpathium , washing the Southwest end of Natolia , sometime called Asia minor , which Ilande , together with the Citie Rhodos , Suliman the great Turke wan not many yeares since from the Christians . The people of this Ile in S. Paules time were called Colossians , to whom he wrote one Epistle , and they were so called of a great brazen Image called in Latine Colossus , containing in Altitude 105. féete , which was dedicated to the Sunne , or as some say to Iupiter . The first Clime is called Dia Rhomes , that is to say by Rome that famous Citie of Italy , and sometime head of all the world . The sixth is called of the old men Dia Boristhenes , notwithstanding the moderne writers thinke that it may be more rightly called Dia Pontou , Pontus is both a Sea and Country lying Eastwarde right against Constantinople . The seuenth Clime is called of the olde men Dia Ripheos , but of the moderne writers Dia Boristhenes which is a great flood of Scithia in the South part of Sarmatia , which falleth into the sea called Mare Euxinum . To these seuen Climes Appian addeth two others , so as in all he setteth downe nine Climes , making the eight Clime to passe through Ripheos which are mountaines enuironing Sarmatia on the North side , and the ninth through Denmarke .

What names did they giue to the Southerne Climes ?

The selfe same that the Northerne Climes haue , sauing that they put before euery such name this Gréeke word Anti , which is as much to say as Contrarie or Right against , as Anti Meroes , Anti Syenes , &c. as you may easily perceiue by this figure of Climes here following , notwithstanding none of those Southern Climes were knowne to Ptolomie more then Anti-meroes , and hardly all that , the furthest part whereof hath not twentie degrées of Latitude .

And note that euery such Clime is deuided into 3. parts , that is the beginning , the midst , & the end . And if you would know what degrées and minutes of Latitude euery such part hath , that is to say , how many degrées euery such part is distant from the Equinoctiall , then consider wel this Table following , which both briefly and plainely sheweth the same .

The beginning . The midst . The ende . Degrees and Minutes of houres . D M D M D M The first Clime . 12 45 16 35 20 30 The second Clime . 20 30 24 15 27 30 The third Clime . 27 30 30 45 33 40 The fourth Clime . 33 40 36 24 39 C The fift Clime . 39 0 41 20 43 30 The sixt Clime . 43 30 45 24 47 15 The seuenth Clime . 47 15 48 40 50 30 The eight Clime . 50 30 51 50 53 10 The ninth Clime . 53 10 55 30 56 30

May not the North and South part of the world ech of them be deuided into more then 9. Climes ?

Yes indéede as our latter writers affirme , for betwixt the Equinoctiall and the 66. degrée and 30′· of Latitude in which the longest day containeth full 24. houres without hauing any night , they make 48. Parallels , and thereby 24. Climes , for euery Clime containeth two Parallels , and euery Parallell maketh the day to increase by one quarter of an houre as hath béene saide before . But from thence forth , though you may continue the Parallels almoste to the very pole , yet you can make no more but 24. Climes by reason that the spaces of the Parallels towardes either of the Poles , do grow more narrow euery one then other , so as from 66. degrées and 30. minutes of Latitude vnder which the Circle Arctique passeth , the longest day is not to be counted any more by houres , but by whole dayes , wéekes , and monethes , in so much as they which dwell right vnder the North pole , whose Zenith is the pole it selfe , haue sixe monethes day whilst the Sunne abideth in the sixe Northerne signes , and 6. moneths night , the Sunne being in the sixe Southerne signes . And contrariwise they that dwell right vnder the South pole haue 6. months day , the Sunne being in the 6. Southerne signes , and 6. months night whilest the Sunne remaineth in the 6. Northerne signes . But because I thinke it a more readie way to account the length of the day by the degrées of Latitude , then by Climes or Parallels , I thought good here to set downe Orontius his Table made for that purpose , which from the 67. degrée of Latitude to the North pole agréeth in all points with the Table of Iohannes de Sacro Busto , in which Table from the saide 67. degrée is not onely set downe the longest day , but also what portion of the Zodiaque alwaies appeareth aboue the Horizon , which portion when it containeth one whole signe , then the day is one moneth long , and the night as much , if two signes , then the day is two months long and the night as much , and so foorth successiuely vntill you come to 6 signes , which is the one halfe of the Zodiaque , making both day and night ech of them sixe monethes long , as I haue saide before , and as you may plainely sée with your eye by placing the Spheare on his Horizon at euery such Latitude .

A Table shewing the longest day in euery degree of Laditude , proceeding orderly from the Equinoctiall to the North pole , by whole degrees without minutes from one degree to 90.

Degrees of latitude . The longest day . D M H M S 1 0 12 3 28 2 0 12 6 56 3 0 12 10 24 4 0 12 14 0 5 12 17 28 6 12 20 56 7 12 24 46 8 12 28 0 9 12 31 36 10 12 35 12 11 12 38 48 12 12 42 44 13 12 46 8 14 12 49 44 15 12 53 28 16 12 57 20 17 13 1 4 18 13 4 46 19 13 8 56 20 13 12 46 21 13 16 48 22 13 21 4 23 13 25 4 24 13 29 20 25 13 33 35 26 13 38 0 27 13 42 24 28 13 46 16 29 13 51 30 30 13 56 16 31 14 1 12 32 14 6 8 33 14 11 12 34 0 14 16 24 35 0 14 21 52 36 0 14 27 20 37 0 14 33 4 38 14 37 36 39 14 44 56 40 14 51 12 41 14 57 44 42 15 4 24 43 15 11 20 44 15 18 40 45 15 26 8 46 15 34 8 47 15 42 24 48 15 51 4 49 16 0 8 50 16 9 44 51 16 19 52 52 16 30 32 53 16 41 52 54 16 54 8 55 17 7 4 56 17 21 4 57 17 36 16 58 17 52 48 59 18 10 48 60 18 30 56 61 18 53 20 62 19 18 24 63 19 48 40 64 20 24 24 65 21 10 32 66 22 20 40

Degrees of latitude The longest day The arch of the Zodiaque alway appearing aboue the Horizon . D M Daies H M De. M 67 0 24 1 40 22 50 68 0 42 1 16 40 0 69 0 54 16 25 52 0 70 0 64 13 46 61 26 71 74 0 0 70 26 72 82 6 39 78 22 73 89 4 58 84 56 74 96 17 0 92 12 75 104 1 4 96 20 76 110 7 27 105 16 77 116 14 22 111 20 78 122 17 6 117 6 79 127 9 55 122 46 80 134 4 58 128 22 81 139 31 36 133 50 82 145 6 43 139 6 83 151 2 6 144 22 84 156 3 3 149 36 85 11 5 24 154 42 86 166 11 23 159 50 87 171 21 47 164 52 88 176 5 29 169 58 89 181 21 58 174 58 90 187 6 39 180 0

What things else do the Cosmographers teach to be considered in Climes and Parallels ?

Diuerse things , as first how many Italian miles euery clime hath in breadth and length , also what seasons of the yeare , and what shadowes the sunne yéeldeth to those that dwel vnder diuers Climes and Parallels .

What breadth and length do the auncient writers appoint to euery one of the seuen climes .

They appoint such breadth and length as this Table folowing sheweth , in which also is set downe the degrées and minutes of Latitude through which the middle Parallell for euery Clime passeth , and also the number of miles answerable to one degrée of euery such Latitude .

Climes . miles in bredth . miles in length . The degrees and minutes of Latitude through which the middle parallell of euerie clime passeth . The number of miles answerable to one degree of euery such Latitude . De. M. 1 465 20555 16 37 57 2 420 19453 24 15 54 3 370 18398 30 45 51 4 350 17299 36 24 48 5 270 16215 41 20 45 6 225 15136 45 24 42 7 195 14426 48 40 40

How the bredth & length of euery Clime is to be known .

The breadth is to be knowne by multiplying the degrées of difference contained betwixt the beginning & end of euery Clime by 60. miles : And if the degrées of difference haue any minutes annexed thereunto , then you must adde so many miles as there be minutes , to the product of the former multiplication . Nowe to haue the length of euery Clime , you must séeke to knowe by the Table of miles , how many miles be answerable to one degrée of that Parallell of Latitude which passeth through the midst of the Clime , and by that number of miles multiply 360. and the product thereof shall be then length . But if the degrées of that Latitude haue any minutes annexed thereunto , then you must finde out the miles of those minutes by the rule of proportion in saying thus : if 60. doe require so many miles as you founde in the Table of miles , what shall so many minutes require , and adde the quotient thereof to the product of the former multiplication , so shall you haue the true length of the Clime , all which thinges are obserued in the foresaid Table , which Table by obseruing like order , you may extend ( if you will ) to the number of 24. Climes , set downe by Orontius and by diuerse other moderne writers .

Of the diuerse seasons and shadowes incident to diuerse Climes and Parallels , and first what seasons and shadowes they haue that dwell right vnder the Equinoctiall . Chap. 19.

THose that dwel right vnder the first Clime , and specially right vnder the Equinoctial , which people at one instant may sée both the Poles , haue two sommers and two winters , for the sun hauing to passe right ouer their heads twice in the yeare , which is when he is in Aries , and againe in Libra , then they must néedes haue two sommers , because the sunne at both those times is nighest vnto them , but when he is in Cancer or in Capricorne , then he is furthest from them , and thereby maketh vnto them two Winters . But yet neither of them so cold as our winter is , whereby it appeareth that our two times of Spring and Autumne are to them 2. sommers , & our two times of sommer and winter are to them two winters .

What shadowes haue those inhabitants ?

They haue fiue sundry shadowes , for when the Sunne is in either of the Equinoxes , they cast their shadow in the morning when the sunne riseth towards the West . And at night when he goeth downe towards the East , and at noone day they haue no shadowe at all , but a perpendicular shadowe , which stréeketh right downe from head to foote , because the sunne being then in the Equinoctiall , must néedes at that time of the day be right ouer their heads , as you may plainely sée in euery material Spheare , hauing a foote with a firme Horizon , and dwely placed to shew a right Spheare , but when the sunne is in any of the Southerne signes , then the foresaid inhabitants doe cast their shadowe towardes the North , And when he is in the Northerne signes , they cast their shadowe towards the South , and because they dwell in a right Spheare , their daies and nights be alwaies equall .

Of the seasons and shadowes which they haue that dwell betwixt the equinoctiall and the Tropique of Cancer . Chap. 20.

THese haue also two sommers and two winters , and 5. shadowes like vnto the others because the sunne passeth twice in the yeare right ouer their heads , first in his declining from the Equinoctiall towards Cancer , and againe in his returning from Cancer towards the Equinoctiall , vnder which Clime Arabia foelix is said to be situated . For Lucan writeth that the Arabians comming to Rome to aide Pompey , meruailed much to sée that the trées did neuer cast their shadowes on the left hand , because in their countrey their shadowe is sometime on the right hande , and sometime on the left , sometime perpendicular , sometime orientall , and sometime occidentall , but in Rome and in all other places beyond the Tropique of Cancer the shadowe alwayes at noonetide tendeth Northward , the words of Lucan be thus , Ignotum vobis Arabes venistis in orbem vmbras mirati nemorum non ire sinistras .

How is Lucan to be vnderstood here in vsing this speech on the left hand ?

As all other Poets are , for they hauing regard to the West , do alwaies make the North part their right hande , and the South part the left . But the Astronomers hauing regard to the South , do make the West their right hand , and the East their left hand : againe the Geographers contrariwise hauing regarde to the North , doe make the East their right and the West their left hand , whereby you may sée that the right hand and left hand may be taken thrée manner of waies , that is according to the manner of the Poets , of the Astronomers , and of the Geographers .

Sith these inhabitants haue seasons and shadowes like to those that dwel right vnder the Equinoctiall , wherein then do they differ ?

They differ in that their dayes & nights be not alwaies equall , by reason that their Horizon declineth from the poles of the world , and by cutting the Parallels of the sunne with oblique Angles it deuideth those Parallels into vnequall parts , neither can their two sommers be so extremely hotte as the others , because the sunne for the most part is further from them .

Of the seasons and shadowes which they haue that dwell right vnder the Tropique of Cancer . Chap. 21.

THey haue but one sommer and one winter , by reason that the sun neuer passeth right ouer their heads but once in the yeare , and that is when he entreth into the first degrée of Cancer : In which time when the sunne riseth , the shadowe tendeth towardes the West , and at noone the shadow is perpendicular enclining on neither side , but falling right downe . And at night when the sunne goeth downe , the shadow tendeth towards the East : and at all other times of the yeare the shadow at noonetide falleth alwaies Northward , and as touching their daies and nights , they shorten and lengthen according as the sunne either approcheth towards Cancer , or retireth towards Capricorne .

Of the seasons and shadowes which they haue that dwell betwixt the Tropique of Cancer and the circle Arctique . Chap. 22.

THey haue also one sommer and one winter as we haue here in England , for the sunne neuer passeth right ouer their heads , by reason that so soone as he hath made his last and highest Parallell in Cancer , he returneth againe Southward , and therefore their shadow is neuer at noonetide perpendicular but shooteth Northward , and the dayes and nights do lengthen and shorten according as the sunne maketh his course either through the Northerne or Southerne signes . For whilest he passeth through the Northerne signes the dayes are longer than the nights , and in passing through the Southerne signes he maketh the nights longer then the dayes .

Of the seasons and shadowes which they haue that dwell right vnder the Circle Arctique , and how long their day is . Chap. 23.

THey haue but one Winter and one Sommer , and their shadow alwaies tendeth sidelong & Northward . And because their Zenith being in the circle Arctique , is at all times of the yeare all one with the Pole of the Zodiaque : the Ecliptique line therefore must néedes be all one with their Horizon , whereby the one halfe of the Zodiaque in a very moment doth rise aboue the Horizon , and the other halfe in the same instant goeth downe , and as the whole Tropique of Cancer appeareth alwayes aboue the Horizon : So the whole Tropique of Capricorne is alwaies hidden vnder the Horizon , so as when the sunne entreth into the first degrée of Cancer , their day is 24. houres long and their night but a moment , so contrariwise when the sunne entreth into the first degrée of Capricorne , their night is 24. houres long , and their day but a moment , as you may plainely sée by placing the Spheare at the 66. degrée and 30′ of Latitude .

What seasons , shadowes and length of day they haue that dwell betwixt the circle Arctique and the pole Arctique . Chap. 24.

THese haue like seasons and shadowes as those that dwell vnder the circle Arctique , sauing that their Winter is colder and longer , because the nigher they approch to the Pole , the further are they from the sunne , and also haue both longer dayes and nights , for the more that the Pole is eleuated aboue their Horizon , the greater portion of the Zodiaque doth alwaies appeare aboue the same , which portion if it containe one whole signe , then their day is a moneth long , and their night as much , if two whole signes , then their day is two moneths long , and their night as much , and so foorth as hath béene said before .

Of those that dwell right vnder the Pole . Chap. 25.

WHat seasons , shadowes , and length of day haue they that dwell right vnder the Pole , if there be anie such people ?

Truely you doe well to doubt thereof , for in mine opinion humane nature is not able to suffer the extreme cold that by reason must néedes be in those parts , neither do I thinke that euer any man either Christian or Heathen did euer sayle so farre as to discouer any land there . Notwithstanding if there by any such people their season is alwayes so extreme cold as no part thereof is worthy to be called a sommer , but rather a continuall winter , and as for their shadow sith the sun when he is at the nighest doth neuer mount aboue their Horizon more then 23. degrées 30′ at the most , their shadow must néedes go round about them nigher , for the most part to their féete then to their heads , for the pole is their Zenith , and the Equinoctiall their Horizon , whereby 6. signes which is the one halfe of the Zodiaque , doth alwaies appeare aboue their Horizon , and the other halfe is alwaies hidden vnder their Horizon , and thereby they haue 6. moneths day & 6. moneths night , day I say whilest the sunne is in the 6. Northerne signes , & night , whilest the sunne is in the 6. Southerne signes , and yet the night can not be so darke there as elsewhere , by reason that the sunne is neuer distant from their Horizon aboue 23. degrées 30′· which is only when he entreth into the first degrée of Capricorne . So likewise the day with them can neuer be so cleare as elswhere , by reason that the sunne mounting no higher aboue their Horizon then 23. degr. 30′· which onely is when he entreth into the first degrée of Cancer , hath no power to dissolue their grosse , thicke , cloudie and mistie aire , yet haue they some preheminence in that they may ( if their cloudie aire be not the let ) alwaies sée all the fixed starres that are placed in the sky betwixt the Equinoctiall and the Pole , because they neuer go down but are alwaies remaining aboue their Horizon : whereas in all other parts of the worlde , the said starres can not be séene all at once , for that they both rise and set , more or lesse in number according as the Zenith of the inhabitants of euery place is more or lesse distant from the Equinoctiall .

By what names certaine inhabitants of the earth are called , as well according to the diuersitie or likenesse of shadowes as of situation . Chap. 26.

YOu shall vnderstand that according to the diuersitie or likenesse of shadowes , the auncient Cosmographers haue giuen to the inhabitants certaine Gréeke names , whereof some are called Aphiscij , some Heteroscij , & some Periscij .

Shew what these names do dignifie .

Amphiscij be those that cast their shadowes both wayes , that is sometime towardes the North , and sometime towardes the South , as those that inhabite the burnt Zone .

Heteroscij be those that cast their shadowe onely one way , as those that dwell in either of the temperate Zones , for if they dwell in the North temperate Zone , they do cast their shadow alwaies at noonetide towards the North. And if they dwell in the South temperate Zone they cast their shadowe at noonetide towardes the South .

Periscij are those that cast their shadow round about them as those that dwell in either of the cold Zones , to whom the Pole is their Zenith : againe they giue them certaine names according to the diuersitie or likenes of their situation and of the seasons incident to those places , whereof some are called Antoeci , some Perioeci , and some Antipodes siue Antichtones .

What doe these names signifie ?

Antoeci be two sundry Nations , the one dwelling towardes the North pole , and the other towards the South pole , hauing one selfe Meridian and one selfe Latitude , that is to say be of like distance from the Equinoctiall , the one Southward and the other Northward , as the letters a. b. in the figure folowing do shew .

Perioeci be those that dwell in one selfe Parallell , how distant soeuer they be East and West as the letters b. c. in the said figure doe shew .

Antipodes be those that dwell féete , so as a right line being drawne from the one to the other , passeth through the Centre of the world as you may sée by the Letters a. c. in the figure following .

And the first of these called Antoeci haue contrarie seasons of the yeare , for when it is sommer to the one , it is winter to the other : Againe Perioeci , though they haue like seasons of the yeare , yet because they are so farre distant in length one from an other , it is therfore mid-day to the one when it is mid-night to the other . But Antipodes be contrary in seasons and in all other things , hauing nothing common , more then that they haue one selfe Horizon .

By what names certaine parts of the earth are called by reason of their diuerse shapes . Chap. 27.

NOw besides the foresaid names attributed to the inhabitants of the earth for such respect as is aboue said , the Cosmographers doe giue also to diuerse parts of the earth according to the diuers shapes therof diuers names also : for if any part of the earth be enuironed round with water either salt or fresh , it is called Insula , that is an Iland , as England , Ireland , & such like , but if the water goe round about it sauing in one part , then it is called peninsula , that is to say , almost an Iland , as Denmarke , Italy , Morea , and such like , and if it be a narrow straight enclosed with the sea on both sides , then it is called Isthmus as the narrow straight of Corinth lying betwixt Boetia and Achaia in Gréece which diuers Emperours of Rome haue in vaine attempted to cut , to th intent to make there through a nauigable passage : finally , when it is neither insula , peninsula , nor Isthmus , then it is called Continens , that is to say firme land , as Saxonie , Boemia , Sueuia , and such like , but these be speciall continents , for the Cosmographers of these daies do make but threé general continents , that is first so much as was knowne to Ptolomie , and to the rest of the auncient writers , secondly the West Indies , lately found out , and thirdly the South part of the world not yet wholly discouered : Againe the auncient men deuided that portion of the world , which was knowne in their daies into 3. parts , that is , Europe , Asia , and Afrique , whereunto the moderne writers haue added a fourth part called America , containing the West Indies . Now if you would knowe what Kingdomes , Regions , Cities , Townes , Seas with their Hauens , Ports , Bayes , and Capes , Iles , Floods , Marishes , and Mountaines , are contained in euery one of these foure parts , then peruse often the vniuersall Maps , and Terrestriall Globes , as well of the moderne as of the auncient Writers , and also the Tables of Ptolomie and of Ortelius , which I wish that they had béene made in such forme as the Tables of Ptolomie are : for hauing the North alwaies set in the front , it should be the readier to compare the shape or situation of any place or Region to the vniuersall Mappe , and by knowing the Longitude and Latitude of any place , it should be the easier to finde the same , as well in the speciall Table as in the vniuersall Mappe or Globe . The vse of which vniuersall Maps , I haue alreadie written in a seuerall Treatise by it selfe printed not long since . But now for so much as the knowledge of the windes hath béene alwaies thought a thing méete to be treated of by him that writeth of Cosmography , and specially of the Spheare , I will héere speake some what of them , neither doe I mind to make any great discourse thereof , as naturall Philosophers are woont to doe , but onely to define what the winde is , and to shew into how many parts the same is deuided , as well by the auncient as moderne writers , and by what names they are called , and somewhat of the qualities thereof .

Of the wind , what it is , what motion it hath , and of the diuers names and diuisions thereof . Chap. 29.

FIrst then you haue to vnderstand that Aristotle and the rest of his sect , doe define the wind to be an exhalation hotte and dry , engendred in the bowels of the earth , and being gotten out , is caried sidelong vpon the face of the earth .

Why is not his motion right vp & down , aswel as sidelong ?

Because that whilest by his heat he striueth to mount vp , and to passe through all the thrée Regions of the aire , the middle Region by his extreme cold , doth alwaies beate him backe , so as by such strife & by the méeting of other exhalations rising out of the earth , his motion is forced to be rather round then right .

What is the cause why he bloweth more sharpely at one time then another , and in one place more then another , and sometime not at all .

As the fumes that rise of new exhalations , & out of flouds and waters , may encrease his force , so lacke of heate and fumes doth diminish the same . Againe the roundnes of the earth is cause that he bloweth somtime more in one place then in another : also mountaines , hils , and great woods may hinder his force in some place of the earth whereas vpon the plain or vpon the broad sea he bloweth most sharpely , and as for his not blowing at all , it may chance diuerse waies , as either for lacke of sufficient heate to open the pores of the earth to let himselfe out , or for that some extreme frost and cold doth close the pores of the earth so straite as he cannot get foorth , or for that the sunne with his extreme heate consumeth the fumes and vapours that should maintaine him . But leauing to answere any more to these questions , I will shew you how many windes were obserued by saylers in old time , and how many are obserued at this present day , and how they were named . True it is that all Nations doe agrée in placing the foure principall winds , according to the foure quarters of the Horizon or Angles of the world , that is to say , East , West , North and South : but in the subdiuisions of the saide foure quarters they differ , for some deuide euery quarter of the Horizon into two , making onely eight winds , and some into thrée , and there by do make 12. winds which the auncient Gréekes and the Romaines did chiefely obserue , the names of the 8. windes are commonly expressed in the Italian tongue thus : Tramontano , North , Mezzodi , South , Lenante , East , Ponente , West , Griego , Northeast , Garbino , Southwest , Maistro , Northwest , Syroccho , Southeast , which names are often vsed by Christopher Columbus , Albertus Vesputius , and others that sayled first into the East and west Indies , and if you will know the names of the 12. windes vsed by the auncient Greekes and Romaines , then behold this figure here following , wherein you shal find them set downe in English , Gréeke , and Latine , that is to say , the English names without the Circle , and the Greeke and Latine names within the Circle , the Greeke vppon the right side of euery line pointing the winde , and the Latine name vpon the left side of euery such line .

Of the nature and qualities of the foresaid 12. windes . Chap. 29.

THe North wind called Septentrio or Aparctias is extremely colde and dry , prohibiting raine , it preserueth health by clensing the Aire of all pestiferous infections , but it causeth drie coldes , and hurteth the fruites and floures of the earth .

2 The Northeast and by North , called Aquilo or Boreas , is also colde and dry without raine , it hurteth the flowres and fruites of the earth , and specially the Vines when they bud .

3 The Northeast and by East , called Helispontus or Caecias , is hotte drying vp all things .

4 The East winde called Subsolanus , is hotte and dry , temperate , swéete , pure , subtill , and healthfull , and specially in the morning when the sunne riseth , by whom he is made more pure and subtill , causing no infection to mans bodie .

5 The East Southeast , called Eurus or Vulturnus , is also hotte and dry , he bloweth lowde , and therefore is called of Lucretius , Altitonans vulturnus .

6 The Southeast and by South called Euroauster or Euronotus , is hotte and moyst , and bréedeth clouds and sicknes .

7 The South wind called Auster or Notus , is hotte and moist bréeding thicke cloudes , great raines , and pestiferous aire .

8 The Southwest and by South , called Austro aphricus , is temperately hotte , & yet bréedeth sickens & raine as some write .

9 The Southwest and by West named Aphricus or Libs , is cold and moyst causing raine .

10 The West winde called Fauonius or Zephirus is temperately hotte and moyst , and wholsome in the Euening , it dissolueth frost , yce , and snow , and maketh flowres and grasse to spring , and some write that it causeth Thunder .

11 The West Northwest , called Corus or Syrus , is cold and moyst without any great rigour .

12 The Northwest and by North called Syrus or Trachias is cold and dry of earthly nature , bréeding snow and windes .

Of the moderne diuision of the windes . Chap. 30.

BVt the Mariners of these our latter daies , to be the better assured of their routes and courses on the sea , do deuide euery quarter of the Horizon into 8. seuerall windes , so as they make in all 32. windes , which of the Spaniards are called Rombes , which win des together with their names , you may sée plainely set foorth in this figure following representing the Mariners Compasse .

A figure of the 32. windes representing the Mariners Compasse .

And note that eight of these windes are called principall windes , that is North , South , East , West , Northeast , Southwest , Northwest , and Southeast , and all the rest are called Collaterall windes , but the first foure are chiefest , from whence the names of all the rest are deriued , neither doe the learned Pilots in their Tables , call the first foure windes , Rombes , but will say the first , second or third Rombe from North , from East , South , or West , vnto the number of seuen , for so many Rombes they make in euery quarter betwixt the foure chiefe and principall windes , of which matter I shall speake hereafter in my treatise of Nauigation . In the meane time I heartily pray all those that shall vouchsafe to reade this my treatise of the Spheare , to take my labour therein bestowed in good part , and where any fault is , friendly to correct the same without any skorne or disdaine .

A plaine description of Mercator his two Globes , that is to say , of the Terrestriall Globe and of the Celestiall Globe and of eyther of them : Together with the most necessary vses therof written by M. Blundeuill . Whereunto is added a briefe description of the two great Globes lately set foorth by M. Molinaxe : and of Sir Frances Drake his first voyage into the Indies .

He telleth the number of the starres and calleth them by their names . Psal. 147. depiction of astrolabe

Imprinted at London by Iohn Windet .

A plaine description of the two globes of Mercator , that is to say , of the Terrestriall Globe , and of the Celestiall Globe , and of either of them , together with the most necessary vses thereof , and first of the Terrestriall Globe , written by M. Blundeuill .

THe Terrestriall Globe is a round body couered with an vniuersall Map containing both sea and land , which is deuided by the later Cosmographers into foure principall parts , that is , Europe , Affrique , Asia , and America , the Longitude and Latitude of euery which part is alreadie set down in my description of vniuersal Maps and Cards . And in this Globe are also set downe certaine stars , some towards either of the poles , & some nigh to the Ecliptique line , whereof we shall speake hereafter when we come to treate of the Celestiall Globe . But you haue to vnderstand that the one end of this Globe is called the pole Arctique , that is to say , the North pole , and the other ende the pole Antarctique , that is to say , the South pole : vpon which two poles it is turned about . And this Globe is traced with certaine circular lines , whereof some be greater and some lesser , those greater which passe through both the poles , are called Meridians of this word Meridies , which is as much to say as noonetide , for when the Sunne toucheth anie of those Circles , it is noonetide to all those that dwell right vnder that Circle , of which Meridians though you may imagine to bée halfe so many as there be degrées in the Equinoctiall , which amounteth to the number of 180. yet there are set downe in this Globe no more but twelue which doe cut the Equinoctiall in 24. points , making thereby 24. spaces , euery space containing 15. degrées of the Equinoctiall , which fiftéene degrées doe make one houre , for the Equinoctiall is a great Circle which guirding the Globe in the very middest betwixt the two Poles , and representing the motion of the first mooueable , maketh his daily reuolution from East to West in foure and twentie houres . And this great Circle by deuiding the Globe into two equall parts , causeth the same to haue two Latitudes or breadthes , the one Northerne and the other Southerne , for that space of the halfe Globe which lyeth betwixt the Equinoctiall and the North pole , is called the North Latitude , and the other halfe of the Globe which lyeth betwixt the Equinoctiall and the South pole , is called the South Latitude , ech Latitude contayning euerie where from the Equinoctiall to either of the Poles nintie degrées . Againe this Circle is deuided into 360. degrées containing the whole Longitude of the earth , and euery degrée is 60. Italian miles , which being multiplied into 360. maketh in all 21600. miles , and such Longitude is to be counted from West to East , beginning the same at the first Meridian which passeth through the Iles of Canaria , otherwise called Insulae fortunatae , crossing the Equinoctiall and the Ecliptique line , in the first degrée of Aries , and also in the first degrée of Libra , which are the two Equinoctiall points , so as the one halfe of the Equinoctiall which goeth from West to East , containeth 180. degrées , and the other halfe returning againe from East to West contayneth also 180. degrées , which maketh in all 360. degrées , ending where the first degrée of Longitude did beginne : For though Mercator and others in their seuerall Mappes doe in these dayes make the first Meridian to passe through the Iles Azores which are fiue degrées more to the West . Yet Mercator in his great Terrestriall Globe dedicated first to the Lorde Granuella , and afterwardes to the Emperour Charles the fift , Anno Dom. 1541. placeth the first Meridian as before is said , both whose Globes my Worshipfull friend Sir Thomas Knyuet of Ashwelthorpe Knight , did courteously lend me I thanke him , who as he is very well learned in all the liberall Sciences himselfe , so is hee a great fauourer and furtherer of all such as delight in any learned or vertuous exercise : And by helpe of those Globes I wrote this Treatise . But to returne to my matter , there is another great slooping and ouerthwart Circle , called the Ecliptique line , vnder which the Sunne continually walketh , and this line is marked with the Characters of the twelue signes , euery signe containing in length thirtie such degrées as the Equinoctiall hath , so as in all , the Ecliptique line containeth 360. degrées , which is the Longitude of heauen , and the first degree of the Longitude of anie starre beginneth at the first point of Aries , and endeth at the same point . Nowe of the lesser Circles there be foure principall , that is , the two Tropiques , that is to say , the Tropique of Cancer towardes the North pole , and the Tropique of Capricorne towardes the South pole , betwixt which two Tropiques , the Sunne continually goeth right vnder the foresaide Ecliptique line , neuer mounting higher then the Tropique of Cancer , nor descending lower then the Tropique of Capricorne . Of the other two lesser Circles , the one is called the Circle Arctique enuironing the North Pole , and the other the Circle Antarctique enuironing the South Pole : all which foure Circles are called Parallels , that is to saye , equally distant one from another , and by reason of these foure Circles , the Globe is deuided into fiue Zones , that is to say , two colde Zones , two temperate , and one extremely hotte : of the cold Zones , the one lyeth betwixt the Circle Antarctique , and the South pole , and the other betwixt the Circle Arctique and the North pole .

And of the temperate Zones , the one lyeth betwixt the Tropique of Cancer and the Circle Arctique , and the other lyeth betwixt the Tropique of Capricorne and the Circle Antarctique . The hotte Zone is that which lyeth betwixt the two Tropiques , through the middest of which Zone passeth the foresayde Equinoctiall line , right vnder which , they that dwell , haue no Latitude at all , and therefore their dayes and nightes are alwayes equall , ( that is to saye ) twelue howres day , and twelue houres night . Also besides these foure foresayde lesser Circles or Parallels , the Globe is traced with eight other Parallel Circles on each side of the Equinoctiall towards eyther of the Poles , making on each side nine equall spaces , euery space contayning tenne degrees of the Equinoctiall , so as they doe make in all 90. degrées , which is a quarter of the great Circle . Moreouer there be certaine Mariners Compasses deuided into 32. circular lines , signifying the 32. winds , whereby Marmers doe saile , and do direct their ships from port to port , which lines doe diuersely crosse the other Circles before rehearsed . Besides these circular lines before described , there is fastned vnto the two poles of the Globe , a Meridian of brasse commonly called the mooueable Meridian within the which the Globe turneth about , & this Meridian is deuided into foure quarters euery quarter containing 90. degrées , so as the whole circuit thereof is 360. degrées , one of which quarters towardes the North pole is deuided with circular lines into thrée seuerall spaces , in the first and lowest whereof being next to the bodie of the Globe , are set downe the numbers of degrées of Latitude . In the next space aboue that , are grauen the Parallels and numbers of houres of the longest day in euery Latitude , and the spaces of euery houre are deuided with little stréeks into foure parts , signifying the foure quarters of an houre : and these houres doe encrease till you come to 24. houres , and from thence the day incereaseth by monethes , from one to sixe , ouer which is grauen this Latine word Menses , ( that is to say ) monethes , and the third and vppermost space containeth the Climes , beginning at one , and so procéedeth to 8. The like Climes , Parallels , and houres of the longest day , are to be accounted , also in the said Brazen meridian procéeding from the Equinoctial towards the South pole , though they be not there set downe . And vpon this Brazen meridian is placed at the North pole another little brazen Circle , together with his Index called the houre whéele , euery halfe wherof containeth twelue houres , the Index of which whéele being set vppon the North pole turneth about as the Globe turneth , and yet you must imagine the Pole it selfe to be immooueable . Also there is a quarter Circle of thinne brasse plate , deuided into 90. degrées fastened in such sort vppon the brazen Meridian , as you may remooue it too and fro , yea and also take it cleane off if you will , which quarter Circle hath a square head of brasse , signifying the Zenith or verticall point of any place , and this Circle is called the quarter of altitude , which quarter is deuided into 90. degrées , procéeding from the Horizon vpward to the Zenith , wherby may be described vpon the Globe the 90. Almicanterathes or Circles of Altitude . And this quarter serueth for diuerse purposes , as to finde out thereby the Altitude of any starre or point of the Ecliptique line , or of any other point in heauen , and the very edge thereof on the right hand , sheweth the Azimuth or verticall Circle of any place , and in what coast and part of heauen any star is . Also how any place of the earth beareth one from another as you shall more plainely perceiue hereafter by the examples of certaine propositions thereto belonging . Then there is another Circle of brasse plate somewhat thicker called the semi-Circle of position , which serueth chiefely for matters of Astronomie , as to finde out the twelue houses of heauen , and is to be fastned on the Horizon , so as it may be remooued and set vpon which side of the Globe you will . Also there belongeth to the Globe a little round squire of brasse , made with right Sphericall Angles , the head or stile whereof is to shewe the shadowe of the Sunne being set vpon the Globe , in stead whereof a néedle being set right vp , will sometimes serue the turne . Finally to the Globe belongeth another Circle called the Horizon , which is a broad Circle of wood , hauing a foote of wood , within which Horizon , the Globe together with the brazen Meridian is to be set at what Altitude you list . And this Horizon is deuided with diuers circular lines into seuen vnequall spaces , whereof the first and narrowest space next vnto the bodie of the Globe , containeth the degrées of the 12. signes of the Zodiaque , euery signe containing 30. degrées . The seconde space containeth the Characters and names of the said signs , and also the number of the degrées , as 10. 20. 30. The third the daies of euery moneth . The fourth the names of euery moneth , and of certaine Festiuall daies . The fift , the names of the 12. windes , which the ancient Gréekes and Romaines were woont to obserue , whereof I haue alreadie spoken in my Spheare . The sixth and seuenth do shew the 12. houses , together with their significations necessarie for the setting of figures and calculating of natiuities , and in the foote of the said Horizon is a little Compasse with a néedle to shew the North and South , according to which the Globe is to be placed . But yet not before the said néedle be truely rectified , according to the variation which it hath in that place where you are to vse the Globe : for otherwise the néedle may cause great error , whereof we shall treate in the next Chapter . Thus hauing briefely described the Terrestriall Globe , and all the parts therof , I thinke it good now to shew you how to place the Globe according to the foure quarters of the world .

This Treatise of the two Globes , containeth 50. Propositions as followeth .

HOw to place the Globe truely , according to the foure quarters of the world , and according to the Latitude of any Region . Proposition . 1.

To know vnder what Clime any place or Region is , and of how many houres the longest day is there , and also what Latitude any place described in the Mappe hath . Proposition . 2.

How to know what Longitude any place described in the Mappe hath . Prop. 3.

How to know the distance betwixt any two places described in the Mappe . Prop. 4.

To know how one place beareth from an other . prop. 5.

How to find out by the Globe the place of the Sunne , that is to say , the degree and minute of that signe wherein the sun is euery day throughout the yeare . prop. 6.

How to rectifie the Index of the houre-wheele for euery seuerall day throughout the yeare . prop. 7.

How to knowe euery day at what houre the sunne riseth or setteth . Proposition . 8.

How to know in what part of the Horizon the sunne riseth and setteth euery day . Prop. 9.

How to know the length of euery day and night throughout the yeare , aswell by helpe of the houre-wheele , as by counting the degrees vpon the Horizon . Prop. 10.

How to know by the Globe how much the sunne declineth euery day throughout the yeare from the Equinoctiall . Proposition . 11.

How to know by the Globe the Meridian altitude of the sunne , that is to say , his height at noone tid● euery day thoroughout the yeare , and how far he is then distant from your Zenith . Prop. 12

How to know the altitude of the sun at euery other houre of the day . Prop. 13.

How to know the houre of the day by the Globe . Proposition . 14.

To know how much the vnequall houres otherwise called the Planetary houres , doe differ from the artificiall houres throughout the yeare , and how many minutes euery vnequall houre containeth . Prop. 15.

How to know euery day when the dawning of the day , and the twilight of the night beginneth and endeth , and the time of their continuance . Prop. 16.

How to knowe the ascention of the sunne , both right and oblique . prop. 17.

The 18. proposition , containing the description of the Celestiall Globe , and shewing wherin it is like or differing from the terrestriall Globe .

The 19. proposition containing a particular description of the 48. Images of the fixed starres that are in the Celestiall Globe , together with their sundry names , and also the names of so many starres as are named in the Globe . prop. 19.

How to finde out in heauen any vnknowne starre described in the Globe two manner of wayes , that is , either by the helpe of some knowne starre , or else by knowing the true houre of the night . prop. 20.

How to knowe by the Globe the Meridian altitude which is the highest altitude of any starre , and also how high or low he is at any other time . prop. 21.

How to know by the Globe what starres are aboue the Horizon at any time of the day or night . prop. 22.

How to know by the Globe at what time any starre riseth aboue the Horizon , mounteth to the highest , and setteth , and with what degree of the Ecliptique he riseth , mounteth , and setteth : and also in what part of the Horizon he riseth and setteth . prop. 23.

How to know in what part of the firmament any starre is , and how many degrees it is distant from the Meridian at any houre , and being right vnder the meridian to know how farre it is distant from your Zenith . Proposition . 24.

How to finde the houre of the night by the Globe . pro. 25.

How to know the verticall stars in euery latitude . Prop. 26.

How to know the true place of any star , that is to say , in what signe and in what degree thereof any starre is . Prop. 27.

How to finde the place and Longitude of any starre by the Globe . Proposition . 28.

How to find out the Latitude of any starre . Prop. 29.

How to find out by the Globe , the declination of any star : Proposition . 30.

How to know the magnitude or greatnes of any starre , and his nature and qualitie by the Globe , and also the right ascention of the Arque of the Ecliptique , which accompanieth the right ascention of any starre . Prop. 31.

A Table of the fixed starres .

How to find out the right and oblique ascention of any star and also of the ascentionall difference . Prop. 32.

To know in what quantitie of time any whole signe or any other Arque of the Ecliptique doth rise or set . Prop. 33.

How to know by the Globe what stars doe rise or set euerie day Cosmically , Acronically , or Heliacally . Prop. 34.

To know in what time of the yeare any starre riseth or setteth , either Cosmically or Acronically . Prop. 35.

Of the Horoscope and the rest of the 12. houses . Prop. 36.

How to find out the Horoscope or ascendent at any time of the day or night by the Globe , and thereby to know the foure principall Angles of heauen . Prop. 37.

How to erect a figure by the Globe according to Regio Montanus his way , which is called the reasonable way , and is counted the best of all others . Prop. 38.

How to know the Latitude of any place or region by any of the fixed stars described in the Globe . Prop. 39.

Another way to find the eleuation of the pole . Prop. 40.

A third way how to find out by 2. stars the eleuation of the pole , not knowing their Meridian altitude . Prop. 41.

A fourth way to find out the Latitude of any Region by any known fixed starre or Planet that may be seene . Prop. 42.

A briefe description of the diurnal Table set down in Stadius his Ephemerides , together with the vse therof . prop. 43.

Howe to finde out the place of any Planet by the Ephemerides . prop. 44.

A briefe description of the Table of Stadius , set downe in the 112. page of his Ephemerides , to find out thereby the dayly latitude of the Moone be it North or South , together with the Cannon or rule thereof , plainely declared by example . prop. 45.

How to know the true place of the Sunne or Moone or of any other Planet euerie houre of the day throughout the yeare . prop. 46.

How to find out the place of the Moone by the Globe , when she is aboue the Horizon , without the helpe of any Ephemerides or other table whatsoeuer . prop. 47.

Another way to find out the place of the Moone without taking the altitude of any starre . prop. 48.

How to find out the longitude of any region . prop. 49.

Another way to find out the vnknowne longitude of any place by the Globe . prop. 50.

How to place the Globe trulie according to the 4. quarters of the world , and according to the latitude of any region . The first Proposition .

FIrst in bedding your Globe together with the brazen Meridian into the 2. nickes of the Horizon and also into the slit of the pinne which standeth in the middest of the foot , so that the North pole of your Globe be answerable to the North quarter or north wind of the world , described vpon the Horizon , and sée that that part of the brazen Meridian , wherein are described the Climes , Parallels & houres of the longest day , may stand aboue the Horizon , and also that the one halfe of the brazen Meridian may iustlie and euenlie appeare aboue the Horizon , and the other halfe vnder the Horizon : Againe , you must sée that the Equinoctiall line of the Globe doe méete iust with the middle point or stréeke of the brazen Meridian whereas the first degrée of latitude doth beginne , and also that the body of the globe doe not leane to the one side of the Horizon more than to the other , but to bee equally distant from the same in all places , and in any wise to sée that the Horizon stande alwaies leuell , to which ende some Globes haue a plummet of leade hanging by a little chaine or thréede , which because it will mooue with euery winde , I for my part do think it better for you to haue such a little leuell made of purpose , as you may set the same vpon anie place of the Horizon wher you list , and therby make the Horizon to stand leuel on euerie side as you will your self , and it may be made of a little péece of thicke boorde , like a Triangle , thus .

Then with your 2. handes laying holde of the 2. next pillers , turne the foot of the Globe vntil it stand right North and South , which is to be done thus . First find out the true Meridian of the place whereas you are to vse the Globe by such meanes as M. Burrough teacheth in his discourse of the variation of the Compasse , the 7. chapter . which vndoubtedlie is a most certaine way , by which you shall finde thrée thinges at one instant , that is , the true Meridian , the variation of the néedle , and the true latitude of any place . But if you haue not the instrument of variation , by helpe whereof this matter is to bée accomplished , then in some open place betwixt 8. and 9. of the clocke in the forenoon or sooner , vpon a smooth table or plancke , standing leuell , draw a good large circle with your Compasses , in the Center whereof must be fixed a round and straight pinne of Iron or Latton wyer , in length a good deal shorter than the semidiameter of that circle , and to the intent that the pinne or stile may stande right vp , without inclining on either side , it would be rectified by a true Squire : That done , waite diligently vntill the shadow of the pinne head doe iustlie touch the circumference of the circle , so as it neyther passe beyond the circle , nor come short of the same , and there make a pricke , and so let it stay vntill about thrée of the clocke in the afternoone , about which time the shaddow of the pin will beginne to approch nigh vnto the sayd circle , and so soone as it toucheth the same , make there another prick : that done , diuide the Ark or portion of the circle , contained betwixt those 2. pricks into two equall parts , and in the middest therof set another prick , them laying your ruler to the middle pricke , and to the Center of the circle , draw a right line through the Center & also through the middle pricke from the one side of the circle to the other , and beyond if you list : for you may make the said line of such length as you shall thinke most méete to serue your turne , and that line shall bee the true Meridian for that place , shewing the right North and South part of your Horizon , and by crossing the said liue with another right line in the very middest with right angles , you shall haue the true East and West , and to auoyd long wayting , you may draw diuerse circles one within another . Thus hauing found the true North and South , East , and West , place your Globe accordingly , so as the brazen Meridian of the Globe may aunswere the Meridian line alreadie drawne vpon the boorde or plancke , and to the intent that afterwarde you may knowe at all times both day and night , howe to place your Globe right North and South , it shall bée necessarie to drawe a right line vpon the foote of the Horizon , answerable to the foresaid Meridian first drawne vpon the boord or planck , and vponthat line to fasten a pretie handsome Compasse , hauing a needle of an inch long at the least , which is much more certaine , than such a little néedle as is woont to bee set in the foote of euerie Globe : And when the néedle standeth still , marke how much the North point thereof declineth eyther East or West from the true Meridian before fastned , and whereas you see the Needle to decline , bee it East or West , there set fast a little pinne of Latton to serue as a marke , whereto you may alwayes direct the North point of the needle when you would haue the Globe to stand right North and South , for there is no needle touched with the load stone , bee it neuer so good a stone , but it wil vary from the true Meridian line , either more or lesse , & therefore no trust is to be giuen to the néedle vntil you know the true variation therof , the finding of which variation , as I said before , is most trulie taught by M. Borough in his book before mentioned , wherunto I once againe refer you , and the rather for that it is written in our mother tongue . But Gemma Frisius teacheth to set the Globe right North and South thus , first goe into an open place whereas the Sunne shineth , and vpon some table standing leuel , and also the Globe standing leuell , the Pole being eleuated aboue the Horizon according to your latitude , fixe a right néedle in the degrée of the signe , wherein the Sunne is that day so as the nédle may stande right vp without inclining anie maner of way , and if it be in the forenoone , turne the East side of the Globe towardes the Sunne , moouing both the Globe and his seat to and fro , vntill you sée the néedle to cast no shaddow at all , for so shall the Globe stand right North and South . But if it bee in the afternoone , you must turne the West side of the Globe towards the Sun , and then worke as before , and by anie of these wayes before taught you shall not onely place the Globe answerable to the foure quarters of the world , but also you shall find that the Circles , Poles , and Axletrées of the Globe , are answerable to the Circles , Poles and Axletrees of the heauens . Now hauing placed your Globe answerable to the 4. quarters of the world , learne to know by some Table or moderne Map , or els by such wayes as are set downe both in my Spheare , and also in the latter ende of this Treatise , the latitude of the region wherein you dwell : As for example , the latitude of these partes here about Norwich is 52. degrées , then taking holde of the brazen Meridian aswell with your left hand aboue the Horizon , as with your right hand beneath the Horizon , turne the same vp and downe in the nickes of the Horizon , vntill the North pole bee eleuated aboue the Horizon 52. degrées , the last of which degrées must méete euen with the vpper brimme of the Horizon , that done , séeke for England in the Globe , not leauing to turne the bodie of the Globe to and fro vntil you haue brought that Meridian which passeth through England right vnder the brazen Meridian , and holding it stil at that stay , drawe the head of the brazen quarter of altitude right ouer the place , and ouer that Meridian vnder which you dwell , so shall the head of your brazen quarter stande for your Zenith in the verie middest of the Horizon and kéeping al things thus at a stay , you may sée how euery region or countrey is situated , and how it beareth from you , and which be vnder your Climate and which be not , and to be sure that the brazen Zenith may stand in his right place , it shall be néedfull to set it so as the left square side therof made with a long notch may touch the selfe same degrée of latitude in the brazen Meridian , which it hath of altitude , for looke how many degrées the Pole is eleuated aboue the Horizon , which is called the altitude , so many degrées must the brazen Zenith be distant from the Equinoctial of the globe , which is called the latitude , and is euer all one with the altitude of the Pole . Thus hauing shewed you howe to place the Globe according to the foure quarters of the world , I thinke good now to declare vnto you , first the vses of the terrestriall globe , whereof some are common also to the celestiall globe , as you shall perceiue hereafter by the propositions following , which propositions may be very well diuided into 3. sorts , whereof the first do properly belong to the description of the vniuersall Map , wherewith the terrestriall globe is couered , which are but few in number . The second kind of propositions do chiefly belong to the Sunne , and to his apparances , which are also common to the celestial globe , and may be found out by either of the globes . But the third kind do belong most properly to the fixed starres and to their apparances , and therefore I mind not to set downe them vntill I come to describe the celestiall globe , and to shewe the vses thereof .

The chiefest propositions belonging to the vniuersall Map , wherewith the terrestriall Globe is couered , are these following .

To know vnder what Clime any place or region is , and of how many houres the longest day is there , and also what latitude any place described in the map hath . The 2. Proposition .

HAuing set the Globe at your latitude , bring the place or region which you séek right vnder the brazen Meridian , and the vpper space of the said Meridian will shew the Clime . And the second or middle space will shewe the houres of the longest day . And the thirde space the latitude : As for example , if you would knowe vnder what Clime London is , then hauing found London , fixe a long néedle in the red spot next to the name of London ( for all townes for the most parte in the Globe are marked with red spottes ) that done , turne the Globe with your hand vntill the néedle do touch the brazen Meridian , and by staying the Globe there with your hand , you shall find London to be vnder the 8. Clime , and that the longest day in the yeare is there 16. houres and 20. minutes , and the third and nethermost space doth shew the latitude of the place , which is 51. degrées , 32′· Againe , by obseruing this order without changing your latitude , you shall finde Venice to be vnder the 6. Clime , and their longest day to be 15. houres and somewhat more , and the latitude of that Cittie to bee 44. degrées , 30′· And you shall finde Ierusalem to be vnder the third Clime , and the longest day there to be 14. houres , and the latitude thereof to be 31. degrées . 30′·

How to know what longitude anie place described in the Map hath . Proposition . 3.

THe Globe standing still at your owne latitude , pricke the néedle in the place whereof you séeke the longitude , and bring it as before , to the brazen Meridian , and staying it there , looke at what number of degrées the brazen Meridian cutteth the Equinoctiall , and that is the longitude of the place , counting the degrées from the first degrée of the Equinoctiall , which beginneth at the first point of Aries vnto the place of the section : by doing thus , you shall finde the longitude of London to bée 19. degrées , and the longitude of Venice to be 36. degrées , and the longitude of Ierusalem to be 67. degrées and 30. minutes .

How to know the distance betwixt any two places described in the Map . Proposition . 4.

OPen your Compasses so wide as you may set the one foote thereof iust in the one place , and the other foote in the other place , and applie that wydenesse to the Equinoctiall line , counting how many degrees of the Equinoctiall are contained betwixt the two féete of your compasses , & by allowing to euery degrée 60. miles , you shall haue the true distance of the places . Thus you shall finde the distance betwixt London and Venice to be 13. degrées , 30′· which being multiplied by 60. maketh 810. miles , and the distance betwixt London and Ierusalem to be 40. degrées . 30′· which being multiplied by 60. maketh 2430. miles . And it maketh no matter to what part soeuer of the Equinoctial you do apply the widenes of your compasses , so as you set the first foot at the very beginning of some one degrée , and let the other foot fall out as it will , either at a whole degrée at a half , or at a quarter of a degrée , which smal partes are to bee counted by minutes by coniecturall discretion . And note here that no 2. places can bee distant one from another East and West more than 180. degrées which is iust the one half the circuit of the Earth , beyond which half or on this side thereof the places must néeds be néerer together by means of the roundnesse of the Earth , and if either of the 2. places wherof you would know the distance be not expressed in the Globe , then learne to find out the longitude and latitude thereof by some table , whereof the first meridian is supposed to passe through the Ilands of Canariae and mark vpon the Globe where the said longitude and latitude do crosse , for there ought that place to stand which is missing , to which place direct the one foot of your compasses & then worke as before is taught . And by this means you may also know the distance betwixt any two starres contained in the Globe .

To know how one place beareth from another . Proposition . 5.

THough you may partly finde out this by marking the direction of the lines , procéeding from the marriners Compasse set downe in the Globe : yet in mine opinion it is readier to do it by applying the Flie described in my little treatise of vniuersal Maps , vnto the Globe by setting the center thereof vpon the first place from whence you goe , and by drawing a thred through that place whereto you would goe , in such order as is there taught , so you shall finde Venice to beare from London South East and by East , and Ierusalem to beare from London East , South East , and two quarters more towards the South . But Gemma Frisius teacheth to knowe by the Globe how one place beareth from another thus , hauing set the Globe and also the Zenith of the quarter of altitude at such latitude as the first place or region hath , from whence you would goe , bring that first place vnder the brazen Meridian , and there stay the Globe vntil you haue brought the quarter of altitude to the second place , and the nether end of the quarter of altitude will shew you vpon the Horizon amongst the windes how the second place beareth from the first , so shall you find Hispaniola to beare from Spaine right West , these be the chiefest propositions belonging to the vniuersall Map , wherewith the terrestrial Globe is c●uered , and therefore I will now set downe those that belong to the Sun , which may be done aswell by the one Globe as by the other , and first how to find out the place of the Sunne .

How to find out by the Globe the place of the Sunne , that is to say , the degree and minute of that signe wherein the Sun is euery day throughout the yeare . Proposition . 6.

THough the surest way be to find it out by the Ephemerides , which sheweth the verie minute , yet without hauing respect to the minutes , you may find it out by the Globe thus , séeke out the day of the moneth vpon the Horizon , and that will point you to the degrée of the signe wherin the sun is that that day : As for example , the 6. day of May pointeth right to the 25. degrée of Taurus the Globe standing leuel , and your rule being rightly laid vpon the Horizon , but during the leape yeare you must adde one degrée more than the Horizon sheweth euery day from the beginning of the leape yeare throughout al that yeare .

How to rectifie the Index of the houre wheele for euerie seuerall day throughout the yeare . Proposition . 7.

HAuing placed the Globe at your latitude and also found out the degrée of the Sun , as is before taught , bring that degrée of the Sunne to the brazen Meridian , and there staying it with the one hand , turne with your other hande the index of the houre whéele to the highest part of the said whéele , marked with the number of 12. setting the point of the index iust with the stréeke of the whéele made to shew the houre of 12. or neonetyde , and that will serue your turne for al that day , and thus must you doe euery day in which you haue to vse the helpe of the said houre whéele for any purpose .

How to knowe euery day at what houre the Sunne rifeth or setteth . Proposition . 8.

HAuing set the Globe at your latitude , and rectified the index of the houre whéele by the 7. proposition , turne the Globe to the East , so as the degrée of the Sun may touch the Horizon , and then the index of the houre whéele will shew you at what houre the Sunne riseth . Againe , if you bring the saide degrée of the Sunne vnto the West part of the Horizon , the index of the houre whéele will shew you at what houre he goeth downe . As for example , Anno 1590. the third of Iune , the Sun being in the 21. degrée 33′· of Gemini . I bring that point to the verie edge of the Horizon on the East part thereof , and there staying it , the index of the houre whéele sheweth that the Sunne riseth 8′· before 4. of the clocke in the morning : and the said point of the Ecliptique being turned to the West part of the Horizon , the index sheweth that he setteth 8′· after 8. of the clocke at night .

How to know in what part of the Horizon the Sunne riseth and setteth euerie day . Proposition . 9.

SEeke the degrée of the Sun in the Ecliptique liue , & turn it to the East part of the Horizon , then you shal sée whether it riseth iust East or not , and whether it inclineth towards the South , or towards the north , and likewise by bringing the saide degrée to the West part of the Horizon , you shall sée in what part of the said Horizon he goeth downe : As for example , in the last proposition the Sunne being in the 21. degrée , 33′· of Gemini , and brought to the East part of the Horizon , I finde that the Sunne did rise distant from the East towards the North 38. degrées , 30′· of the Horizon , which is thrée points and somewhat more of the Marriners Compasse from East , towardes the North , so as the Sunne riseth Northeast and by East , and a little more Northward , and the said place of the Sunne being brought to the West part of the Horizon , I find that hee setteth North West and by West , and somewhat more Northward .

How to know the length of euery day and night throughout the yeare , aswell by help of the houre wheele as by counting the degrees vpon the Horizon . Proposition . 10.

FIrst you must know when the Sunne riseth and setteth by the 8. proposition , then looke how many houres the index doth go from the Sun rising to the Sun setting , and that is the length of the day , which number if you take from 24. the night wil appeare : as for example , knowing by the 8. proposition , that the Sun being in the 21. degrée , 33′· of Gemini , riseth 8′· before 4. in the morning , and goeth downe 8′· after 8. of the clocke at night , I find by counting the houres which the index of the houre whéele hath run , that the length of the day is 16. houres and 16′· which you may know also by counting the degrées vpon the Horizon from the place of the Suns rising , vnto the South point of the said Horizon , which you shall find to be 128. degrées , which being doubled maketh 256. degrées , which if you diuide by 15. it will make 16. houres and 16′· as before .

How to know by the Globe how much the Sun declineth euery day throughout the yeare from the Equinoctiall . Proposition . 11.

HAuing found the place of the Sunne , bring the same to the brazen Meridian , and by counting how manie degrées are betwixt that place and the Equinoctiall , you shall know what declination the Sun hath that day : As for example , supposing the Sunne that day you séek to be in the first degree of Gemini , bring the said degree of the Ecliptique to the brazen Meridian , and you shall find vpon the same Meridian the Sun to be declined from the Equinoctiall Northward almost 20. degrees . Againe , supposing the Sun to be in the first degree of Aquarius , if you bring the same degree of the Ecliptique vnto the Meridian , you shal finde the declination of the Sun to be almost 20. degrees Southward .

How to know by the Globe the Meridian altitude of the Sun , that is to say , his height at noontyde euery day throughout the year , and how far he is then distant from your Zenith . Proposition . 12.

BRing the place of the Sun that day you seeke to the brazen Meridian , and staying it there , count vpon the said brazen Meridian how many degrees are contained betwixt the place of the Sun and the South point of the Horizon , and that is the Meridian altitude of the Sun for that day , which if you subtract from 90. the remainder wil shew how many degrees he is distant that day at noontide from your Zenith : as for example , supposing the Sunne to be that day you seeke in the first degree of Taurus , bring that degree of the Ecliptique to the brazen Meridian , and stay it there vntil you haue counted how many degrées of the said Meridian are contained betwixt the place of the Sunne , and the South point of the Horizon , and you shall find the number of degrees to be 50. which is the Meridian altitude of the Sunne for that day , which 50. degrées being taken out of 90. there remaineth 40. and so many degrées the Sun is that day at noontide distant from your Zenith . The like order is to be obserued in séeking to know the Meridian altitude of any Starre , or any other point in heauen .

How to know the altitude of the Sunne at any other houre of the day . Proposition . 13.

HAuing rectified the index of the houre whéele by the 7. proposition , if the houre which you séek be in the forenoone , turne the Globe so as the index of the houre whéele may touch that houre of the forenoone , at which you desire to knowe the altitude of the Sun , and there stay the Globe , vntill you haue brought the quarter of altitude on the Eastside of the globe vnto the place of the Sunne , so shall you find vpon the said quarter the altitude of the Sun at that houre . And if you desire to know the altitude of the Sunne at any houre in the afternoone , then turne the Globe so as the index of the houre whéele may touch the houre of the afternoone , and there stay the Globe vntill you haue brought the quarter of altitude on the West side of the Globe vnto the place of the Sun , and the said quarter will shew you the altitude of the Sun at that houre . As for example , I would know how high the Sun is at 8. of the clocke in the morning , the Sunne being in the first degrée of Taurus : here hauing rectified the index of the houre whéele , I turne the Globe so as the index of the houre whéele may lie vpon the 8. houre of the forenoone , and there I stay the globe vntill I haue brought the quarter of altitude on the East side of the Globe vnto the place of the Sunne , whereby I finde the altitude of the Sunne at that houre to be almost 28. degrées . Again , if you wil know how high the Sunne is being in the same degrée of the Ecliptique at 5. of the clocke in the afternoone , then turne the index of the houre whéele , so as it may touch that houre , and stay it there vntill you haue brought the quarter of altitude on the West side of the Globe vnto the place of the Sunne , so shall you find the altitude of the Sun to be at that houre 21. degrées .

How to know the houre of the day by the Globe . Proposition . 14.

THis is to be done two maner of wayes , the first is thus , set the Globe in some open place whilest the Sunne shineth , and you must sée that it stand both leuell and also right North and South , as is taught in the first proposition , and that the index of the houre whéele be rectified according to the degrée of the signe wherein the Sunne is that day by the 7. proposition , that done , fixe a néedle in the place of the Sunne , and turne the bodie of the Globe to and fro vntill the néedle cast no shaddow at all , and there staying the Globe , the index of the houre whéele will shew you the houre of the day . But if you séeke the houre in the forenoone , remember to turne the East side of the Globe towards the Sunne , if in the afternoone , then turne the West side of the Globe towards the Sunne . The second way is thus , hauing rectified the index of the houre whéele , take the altitude of the Sun with some Quadrant or Astrolabe , and hauing marked the same altitude vpon the quarter of altitude , applie it to the degrée of the Sun on the East side of the Globe if it bée in the forenoone : but if it be in the afternoone applie the quarter of altitude to the degrée of the Sun on the West side of the globe , and the index of the houre wheele will shewe you the houre which you séeke . As for example , the 6. of Iune 1590. the Sunne being in the 24. degrée , 24′· of Gemini , I found by my Astrolabe the altitude of the Sunne to be 48. degrées , which I marked vpon the quarter of altitude , and because I tooke the altitude of the Sunne in the forenoone , I brought the quarter of altitude marked with that degrée to the place of the Sunne on the East side of the Globe , and there staying the Globe I found that the index of the houre whéele did point to the 9. houre of the forenoone and somewhat past . The like is to be done to know any houre of the afternoone , so as you forget not to apply the quarter of altitude vnto the place of the Sunne on the West side of the Globe . And by taking the altitude of any knowne Starre , and working in like manner as before , you shall know the houre of the north , as shall bee taught hereafeer when wee come to treate of the Starres .

To knowe how much the vnequall houres otherwise called the planetarie houres do differ from the Artificial houres throughout the yeare , and how many minutes euery vnequal houre containeth . Proposition . 15.

FIrst you must knowe by the 10. proposition the length of the day , that is to say , how many houres it is long , and reduce those houres into minutes , and diuide the product by 12. and the quotient , together with the remainder ( if there bee any left after the diuision ) will shewe you the quantitie of the vnequall houre of the day , that is to say , howe many minutes it contayneth : The like is to bee done to knowe the vnequall houre of the night , for hauing the length of the artificiall night , worke as before , and you shall haue your desire : As for example , knowing by the 10. proposition the length of the day when the Sunne is in the 21. degrée . 33′· of Gemini to bee 16. houres 16′· here by reducing those houres into minutes , and by deuiding the product thereof by 12. you shall finde the vnequall houre of the daye to containe 81. minutes . and ⅓ of a minute or 20. seconds , which is more than one whole artificial houre of the day by 21. minutes and 20. secondes . Againe , knowing the length of the artificiall night by the said proposition to bée 7. houres , 44. minutes , if you do reduce the same into minutes , and diuide by 12. you shall finde thereby the vnequall houre of the night to containe no more but 38. minutes , and 40. seconds , which is lesse than the artificiall houre of the day by 21. minutes , and 40. seconds .

How to know euery day when the dawning of the day and the twilight of the night beginneth and endeth , and the time of their continuance . Proposition . 11.

HAuing rectified the index of the houre wheele by the 7. proposition , first finde out the opposite point to the degree of the Sunne , and turne the Globe together with that opposite point , and also together with the quarter of altitude towards the West , so as the opposite point may méete euen with the 18. degrée of the quarter of altitude , and staying the Globe there , the index of the houre whéele will shew at what houre the dawning beginneth . As for example , I would know at what houre the dawning of the day beginneth the 19. of Aprill 1590. when as the Sunne is in the 8. degree of Taurus , the opposite point whereof is the 8. degree of Scorpio , wherefore I turne the Globe together with the saide point opposite and also together with the quarter of altitude towards the West , so as the said point opposite may méete euen with the 18. degree of the quarter of altitude , and there staying the Globe , I finde by the index of the houre whéele that the dawning of the day beginneth at 2. of the clocke in the morning and 20. minutes after , which dawning alwayes endeth when the Sunne riseth , as in the former example , the Sunne being in the 8. degrée of Taurus doth rise 45. minutes after 4. of the clocke in the morning , so as the continuance of the dawning is two houres and 25. minutes , for by taking two houres and 20. minutes out of 4. houres and 45. minutes , there remaineth 2. houres and 25 minutes . Againe , the twilight beginneth when the Sun goeth downe , which in the former example is at 7. of the clocke and 15′· Nowe to know when the twilight endeth , you must doe thus , turne the Globe and the quarter of altitude towardes the East , so as the opposite point , which is the 8. degree of Scorpio may méete euen with the 18. degrée of the said quarter , and the index of the houre whéele will shewe you that the twilight endeth at 9. of the clocke and 45′· after , so as the continuance of the twilight is two houres , and 30′ for by taking 7. houres & 15′· out of 9. houres & 45′· there remaineth 2. houres and 30′· But you haue to vnderstand , that the dawning twilight is not alwaies to bee knowne throughout the yeare by the Globe , for from the 11. day of May to the 10. of Iulie , you shall find that the opposite point of the Sunne will not agrée iust with the 18. degrée of the quarter of altitude : because that no opposite point during that time will amount to aboue 16 or 17 degrees of the quarter of altitude at the most , because the Meridian altitude it selfe of any such opposite point is not aboue 17. degrees , for during all that time both dawning and twilight had néed in this our latitude to be accounted as night , vnlesse you will make no night at all .

Howe to knowe the ascention of the Sunne both right and oblique . Proposition . 17.

HAuing set the Globe at your latitude , bring the degree of the Sunne to the brazen Meridian , and there staying it , marke at what number of degrées the said Meridian cutteth the Equinoctiall , counting that number from the first point of the Vernall Equinoctiall point to that section , and that is the right ascention , As for example , the Sunne being in the first degrée of Gemini and brought to the Meridian , you shall finde that the meridian cutteth the Equinoctiall in the 58. degrée thereof , and that is his right ascention . Now if you would knowe his oblique ascention , being in the first degree of Gemini , bring that degree to the East part of the Horizon , so as it may touch the vpper edge thereof , and staying the Globe there , looke what degree of the Equinoctiall toucheth the Horizon at that instant , which you shall finde to bee the 30. degree of the Equinoctiall , and that is his oblique ascention . These are the chiefest propositions that belong to the Sunne , and are to be found by eyther of the Globes , wherefore I will now proceed to those propositions that are to bee knowne most properlie by the celestiall Globe , But first I will make a description of the saide celestiall Globe , whereby it shall plainelie appeare wherein the one Globe is like the other , and wherein the one differeth from the other .

The 18. Proposition , contayning the description of the celestiall Globe , and shewing wherein it is like or differing from the terrestriall Globe . Proposition . 18.

THe Celestial globe is like to the terrestrial globe , in that it is round , hauing both like Axletrées , Poles , hourewhéele with his index , brazen Meridian , quarter of altitude of brasse or Latton with his square head or Zenith , and a halfe circle of brasse or Latton , called the semicircle of position , also a standing foot with an Horizon of wood diuided into seuen seueral spaces , containing in a manner the self same things that are before described in the Horizon of the terrestriall Globe , also in the bodie of the celestiall globe are set downe certaine circles like vnto the terrestriall Globe , that is to say , the Equinoctiall and the Ecliptique line . Moreouer the foure lesser circles , that is to say , the two tropiques , the circle Arctique , and the circle antarctique . But the celestiall Globe differeth from the terrestriall Globe in these foure things following . First the celestiall Globe hath one thinne demicircle of brasse or Latton more than the terrestriall Globe hath , which Demicircle is diuided into two quarters , each quarter containing 90. degrées , made so at ech end as it may be fastened when néed is , vpon the 2. Poles of the Zodiaque , to find thereby the longitude and latitude of euery Star described in the Globe , and therefore may very well bee called the Semicircle of longitude and latitude . The second difference is that whereas the terrestrial Globe is traced with 12. Meridians , diuiding the Equinoctial into 24. spaces , euery space containing 15. degrées . The celestial Gobe is onely traced with 6. Meridians , diuiding the Equinoctiall into 12. spaces , euery space contayning 30. degrées . The thirde difference is that the celestiall Globe hath not those 8. Parallels of latitude wherewith the celestiall Globe is traced . The fourth difference is that whereas the terrestriall Globe is couered with an vniuersall Map contayning the foure principal diuisions of the earth , that is , Europa , Africa , Asia and America : the celestiall Globe is couered with a Map , wherein are painted al the fixed Starres that were knowne to the ancient Astronomers diuided into 48. Images , with which Images , to the intent you might bee the better acquainted , and that you might the more readily find out any Star described in the Globe , I thought good to set downe a particular description of the said 48. images as followeth .

The 19. Proposition , containing a particular description of the 48. Images of the fixed Starres that are in the celestiall Globe together with their sundrie names and also the names of so many starres as are named in the Globe , of which 48. Images 21. are ascribed to the North part of the firmament , 12. to the Zodiaque , and 15. to the South part of the firmament . Proposition . 19.

THis description is diuided into two parts according to the twofold declination of the fixed stars , that is to say , Northern and Southern , for those Stars are said to haue North Declination , which are situated betwixt the Equinoctiall and the North pole , and those to haue South declinatition which are situated betwixt the Equinoctiall , and the South pole , and because that sixe great Circles or Meridians , passing through the poles of the world , doe deuide the Equinoctiall into twelue equall spaces , euerie space containing 30. degrées , I will begin my description at the first point of Aries , which is the Uernal Equinoctiall point , and so procéed towards the right hand round about the Globe , setting downe all such Images , or partes of Images as are situated towards the north pole , and are contayned in euery seuerall space betwixt two Meridians , & hauing described al the north part , I wil vse like order in describing the South part . And you may behold all the Northern Images by turning the Globe about with your hand without taking the same out of his bed or seat , the pole being eleuated aboue the Horizon 50. or 60. degrées , but to view the Southerne Images , it shall be néedfull to take the Globe cleane out of his seat , and to hold it so as the north pole may stand right vp , so shal you sée euery Southern Image and Starre at your pleasure . And yet to know how the Starres are situated in heauen , you had néed to imagine your selfe to bee within the Globe , in the verie center thereof and not without the Globe for otherwise those stars that are situated in heauen on your right hand , if you haue regard to the outside of the Globe , will séeme to be on your left hand .

The Northern Images contained in the first space , intercepted betwixt the first Meridian and the second Meridian .

IN this space you shall first sée next vnto the Equinoctiall the following Fish of the signe Pisces together with the bond , both Southerne and Northerne , called in Latine Linum australe & Septentrionale , also the knot of the bonde , which is called Nodus , Syndesmon , and hipouraion , which is a faire Star of the third bignes .

Item the first part of Aries with the two Starres in his right horne from the former Starre , whereof the Astronomers do alwayes make their computation .

Item the whole Image of Andromeda , her head and right arme excepted , in whose girdle is a Starre of the third bignes called Mirach , and in her left foote a Starre of the third bignes called Alamac .

Item the Triangle called Triangulus and Deltoton with his foure starres .

Item the whole Image of Cassiopeia sauing her right arme , and the vpper parte of the back of her chaire , in whose breast is a starre of the third bignes called Shedar .

The Northern Images contained in the second space .

FIrst the head of the Whale called Cetus in whose snout is a starre of the third bignesse called Menkar .

Item all the hinder parts of Aries called in Gréeke Chrios , in English the Ram.

Item the right legge , necke , breast , right eare , and mozell of the Bull , in whose right thigh towardes the shoulder point is a star of the fourth bignesse called Alfon and in his breast a starre of the third bignesse called Alfo , and in his Mozel lying vpon his right legge , is another starre of the 3. bignesse called Alfon , and in his necke toward the Withers are 7. Starres of diuers bignes called by these diuers names , that is , Vigilie , Atlantides , Pleiades , and Athoratae , commonlie called the 7. Starres .

Item the whole head of Medusa , called Caput Medusae vel Gorgonis , and Ras Algol .

Item the whole image of Perseus otherwise called Chelube , his right hand , sword , and right foot excepted .

The Northern Images contayned in the 3. space .

FIrst the left leg of the Bull hauing 2. starres thereon : moreouer his head , hornes and most part of his right eare , on whose lefte eie is a Star of the first bignes , called Oculus Tauri , Palilicium , and Aldebaran , also in his face are certaine lesser starres called succulae and Hyades .

Item the vpper part of Orion otherwise called Alguze , holding a club in his right hand , & a Lions skin in his left hand on whose right shoulder is a Star of the first bignesse called Bed Alguze , and on his left shoulder a star of the second bignes called Bellatrix

Item the left foot of the former Gemini , contayning two stars of the fourth bignesse , whereof the one is called Propous .

Item the whole Image of Auriga , otherwise called Ericthonius and Heniochos , holding a raine and a whippe in his right hand and hauing a goat hanging on his backe , which hath two little Goates sucking her behind , which be two starres of the second bignes , called by diuerse names , as Hedi , heriphoi , and Sadateni , and in the flanke of the Goat is a starre of the first bignes called hircus , aix , holenie and Alhaiot .

The Northern Images contained in the 4. space .

FIrst the whole image of the little Dog , in whose left flanke is a Starre of the first bignes called Canis minor , Procion , Algomeisa , and Alsahere .

Item the whole image of the two twinnes called Gemini or Didimoi , the left foot of the former twin only excepted , which former twin is called Apollo , Castor , Anhelar , and the other is called Pollux , and Abrachaleus , in whose left eare is a star of the second bignes called Ras Alguze .

Item the tayle and halfe bodie of Cancer .

Item the Mozell of the great Beare , whereon is a Starre of the fourth bignesse .

The Northern Images contayned in the 5. space .

FIrst the head and necke of Hydrus .

Item the fore part of Cancer , called in Gréeke Carchinos , vpon whose right Clea is a Starre of the fourth bignes called Acubene , and betwixt his head and his right Clea is a starre called Presepe , Phatue , and Meelleph , and on his backe are two starres called Aselli and Onoi .

Item the fore part of Leo against whose heart is a star of the first bignesse , called with these names , Cor Leonis , Regulus , Basileus , and Calb alezet .

Item the fore part of the great Beare called Vrsa maior , Arctos Eliche , and Calisto .

The Northren Images contayned in the sixt space .

FIst the hinder parts of Leo otherwise called Alezet , in whose taile is a starre of the first bignesse called Cauda Leonis , and Deneb Alezet .

Item the head and shoulders of Virgo .

Item the darke starre of Bernices haire .

Item the hinder partes of the great Bear , his tayle excepted .

Item the hinder parts of the Dragons taile , containing two starres of the fourth bignes .

The northern Images contained in the seuenth space .

FIrst , the most part of Virgo , who is otherwise called Parthenos , Erigone , Preuindemiator , Protigiter , Almucedie , and Alaraph .

Item the left leg and left arme of Bubulcus , otherwise called Bootes vociferator , Arctophilax and Lanceator , betwixt whose legges is a starre of the first bignes called Arcturus , Asimech , and Alramech .

Item Bernices haire , called Cincinnus , Cesaries , plochamos , Berenices crinis , and Trica .

Item the tayle of the great Beare contayning 3. starres of the third bignes , wherein that next his rumpe is called Aliot , & that which is in the tippe of his tayle , is called Benenacz .

Item a part of the Dragons tayle contayning 2. Sarres standing nigh together nigh vnto the Circle Arctique .

The northern images contayned in the eight space .

FIrst the head and neck of the serpent called Anguis , serpens , Engchelis and Ophis .

Item the crowne of Ariadna , called Corona gnosia , Stephanos Ariadnis , and is commonly called Corona septentrionalis , that is , the Northern Crowne , in which is a Starre of the second bignes called Melfelcare , alpheta , and Muniir .

Item the most part of the Image of Bubulcus , hauing a club in his right hand , in whose left sholder is a Star of the fourth bignesse called Ceginus , and there is another of the fourth bignesse in his club , right against his face called Incalurus .

Item the fore part of the little Beare called Vrsa minor , arctos and Cinosura .

The northern images contayned in the 9. space .

FIrst the vpper part of Serpentarius , otherwise called Ophioucos , and alangue in the Crowne of whose head is a Starre of the third bignesse called iras alangue .

Item the whole Image of Hercules with the Lyons skin hanging on his left arme , otherwise called Engonasi , algethi , Nessus , and ignotum idolum ( his right hand holding the club , and his right leg excepted ) in whose head is a starre of the third bignesse , called Ras Algethi , & this Image lieth groueling with his heels towards the North pole , and his head towards the Equinoctial , which méeteth almost with the head of Serpentarius .

Item the head of the Dragon called Draco & Aben , in whose head is a Starre of the third bignes called Ras Aben.

Item a part of his tayle containing 6. starres .

Item the hinder part of the little Beare , contayning 2. stars .

The northern images contained in the tenth space .

FIrst , the vpper part of Antinous , hauing at each elboe a starre of the third bignes .

Item the last end of the Serpents taile , in the tip whereof is a starre of the fourth bignes .

Item the whole image of the Eagle called Aquila , Vultur volans , Aetos and Alcair .

Item the whole image of the Shaft called Sagitta , Telum , and Hoistos .

Item the whole image of the Harp called Lira and Alohore , that is to say , Vultur cadens , and Chelis , in the vpper part wherof towards the left hand is a faire Starre of the second bignes called Fidicula , Lira , Alangue , Vega , and Brineck .

Item the head , necke , and left wing of the Swan , called Auis , Cignus , Olor , Hornis , Adigege , and of some Gallina .

Item the neck , bodie and fore part of the Dragon contayning 11. Starres .

Item part of the little Beares taile , contayning one Starre next to his rump of the fourth bignes .

The Northerne Images contayned in the 11. space .

FIrst , the Crowne of Aquarius his head , contayning one Star of the fift bignes .

Item the little Horse called Equus and Hippos , whose neck is inclosed with a cloud , and in his head are foure little starres .

Item the head and two fore féet of the winged horse called Pegasus , on whose right nostrell is a Starre of the third bignes called Emph alpharaz .

Item the Dolphin called Delphinus , containing ten Stars , whereof one is of the third bignes .

Item the bodie , legs , and the right wing of the Swan , which lyeth on her back with her bellie vpward , in whose bodie towards the tayle , is a faire Starre of the second bignes called Deneb Adigege , and Arided .

Item the right arme and right leg of Cepheus , on whose right shoulder is a star called Alderaimim .

Item part of the little Beares taile contayning the middle starre of his tayle .

The Northern images contained in the 12. space .

FIrst , the most part of the former fish of the signe Piscis together with part of her band .

Item the necke , body , and wing of Pegasus , otherwise called Equus Gorgoneus , and Alpharaz , rising out of a cloud , in which cloud is the head of Andromeda , hauing on the right side thereof a faire starre of the second bignes , and in the right wing of Pegasus is a starre of the second bignes called Marcab Alpharaz and on his right shoulder another Starre of the second bignesse called Scheat Alpharaz .

Item the right arme and hand of Andromeda holding part of her chaine , in the ring whereof is a Star of the fourth bignes .

Item the left arme of Cepheus .

Item the tip of the little Beares taile , in which is the North starre called Alrucuba of the third bignesse .

The names of the Images contayned in the celestiall globe betwixt the Equinoctiall and the South pole together with so many starres as are named in that part of the globe , beginning as I did before in describing the northern Images at the Vernall Equinoctiall point , and so proceede from space to space contayned betwixt euery two Meridians towardes the right hand .

The Southern Images contained in the first space , beginning at the Vernall Equinoctiall point .

FIrst , the most part of the Whale called Cetus , Pistrix , and Balena ( his head and fore part of his bellie excepted ) in the mid bodie whereof towards the back , is a starre of the fourth bignes called Baten kaetos , and in the lower part of his taile is another starre of the third bignes , called Deneb Kaitos .

The Southern Images contayned in the second space .

THe fore part of the Whales belly and his ghilles , contayning fiue Starres .

Item the most part of the Floud Eridamus , called of some Nilus , and in Gréeke Potamos Eridanos , contayning 22. Starres , whereof one is called Angetenar , which is about the middest of the Floud nigh vnto the Whales bellie , and there is in the verie end of the Floud another starre of the first bignes , called Acarnar .

The Southerne Images contained in the third space .

FIrst , the nether part of Orion or Alguze from the middle of his backe downward , in whose girdle are thrée faire starres , whereof the middle starre is of the second bignesse called Orion or Alguze .

Item another part of the floud Eridamus which séemeth to come from the left foot of Orion , which starre in his left foote is called Algebar , Rigel , Alguze .

Item the whole Image of the Hare , called Lepus and Lagos containing 12. little starres .

Item the rest of Eridanus contayning foure Starres , whereof there is one called Beemum of the fift bignesse .

The Southerne Images contained in the fourth space .

FIrst , the whole image of the great Dog called Canis maior , and Syrios in whose mouth is a Starre of the first bignesse called Asceher and Alhabor .

Item the fore part of the great shippe Argos , with her two Oares hauing a scutchen with 4. Starres , the greatest whereof being of the third bignes is called Markeb , and vnder the vpper hatches in the fore part of the ship is a Starre of the fift bignesse called Alphard , and in the least oare towards the South pole , is a faire Starre of the first bignes , called Canopus and Suhel .

The Southern Images contained in the fift space .

FIrst , the mid part of the Serpent called Hydrus and Asuia , in the which is a faire Star of the 2. bignes called Alphard .

Item the hinder part , mast and toppe of the ship Argos , which séemeth to come out of a cloud , containing diuers Starres of diuers bignes without name .

The southern Images contained in the sixt space .

ITem another part of Hydrus whereupon standeth the image of the cuppe or boule called Crater , vas and patera , and also the Crowes head .

Item the hinder part of Centaurus , in euery part whereof are diuers Starres without name .

The southern Images contayned in the 7. space .

FIrst , the left wing of the signe Virgo and her left hand , holding an eare of wheat , whereon is a Starre of the first bignes , called Spica Virginis , stachis , Acimon , Alacel , and Azimech .

Item the Crowe called Coruus and Corax , his head & necke excepted , in whose left wing is a star of the fourth bignesse called Algorab .

Item the rest of Hydrus , whereon the Crow standeth , containing thrée starres without name .

Item the rest of Centaurus or Chiron with his borestaffe trimmed with boughs , his right hand and right foot excepted .

The southerne Images contained in the eight space .

THe whole image of Libra , the ring only excepted .

Item the fore part of Scorpio , whose fore cleas do lie vpon the two ballances , that is to say , his right clea vpon the North ballance , and his left clea vpon the South ballance , hauing vpon ech clea a star of the second bignes .

Item part of the Serpent called Anguis or Ophis , hauing one Star of the fourth bignes .

Item the left hand of Serpentarius , holding part of the Serpent , vpon which hand are two starres of the third bignes called Yedd .

Item vpon the head of Scorpio are 3. stars of the third bignes , standing all in a row and diuers others as well vpon his back as vpon his left little clea without name .

Item the whole Wolfe called Fera , Lupus and Therion .

Item the right hand of Centaurus holding the said Wolf by the belly in both which are diuers starres without name .

Item the right foot of Centaurus , in which is a faire starre of the first bignes , and is called by the name of Centaurus .

The southerne images contayned in the 9. space .

FIrst , the nether part of Serpentarius , that is to say , from his mid backe downward , hauing the serpent winding betwixt his legs and aboue his right arme , in both which are diuers Starres without name .

Item the hinder part of Scorpio from his mid bodie to the outermost end of his tayle , who hath diuers names , as Scorpius Nepa , Alatrab , in the midst of whose bodie is a faire Starre of the first bignesse called cor Scorpionis , Antares , and chalb Alatrab , and in the tip of his tayle are two Starres of the third bignesse , called Alascha and scomlec Alatrab .

Item the most part of the Bowe with the heade of the shaft of Sagittarius and his right foot , in all which parts are diuers stars without names

Item the Altar with the flame and smoake called Ara , thuribulum , lar , sacrarium , thimiaterion .

The southern images contained in the tenth space .

THe lower part of Antinous from the breast downward knéeling vpon an Altar contayning foure Starres without name .

Item the forepart of Capricornus his head .

Item the image of Sagittarius , otherwise called Crotus , and toxeuter ( his bow , the end of his arrow , and his right foot excepted ) hauing diuers Starres without name .

Item the Southerne Crowne , called Corona Australist , and Notios stephanos , and of Aratus , it is called Dinotos Cyclos , that is to say , the Southern circle which Crowne is placed betwixt the two fore legges of Sagittarius , and in the said Crowne are diuers Starres , amongst which there is one of the second bignes touching the left knée of Sagittarius called Corona Australis , who also hath on his left foote another Starre of the second bignes without name .

The southern Images contayned in the 11. space .

THe fore part of Aquarius otherwise called Ganimedes and Hydrochos , holding a handkercher in his left hand , wherein are thrée Starres , and he hath diuers Starres in his bodie without name .

Item the whole Image of Capricornus , otherwise called Pan Aigoceros , and Algedi the fore part of his head onely excepted in whose taile is a Starre of the fourth bignes called Deneb Aldegi .

Item the hinder part of the Southern Fish hauing diuers starres without name .

The Southerne Images contained in the twelfth space .

FIrst Aquarius his right hande holding the water pot called Vrna and Chalpi , out of the which he powreth the water down into the mouth of the Southern fish , which water is called Aqua and hydor in which are diuers Starres without name .

Item the lower bellie parte of the former fish of the signe Piscis wherein are two Starres without name .

Item both the thighes and legges of Aquarius , vpon the calfe of whose right leg is a Starre of the third bignesse called Scheat Aquarii , and Crus Aquarii .

Item the head of the southern fish called Piscis Meridionalis , and Iothis notios , in whose mouth is a faire Starre of the first bignes , called Fomahant .

But you haue to vnderstand that besides the 15. Southern Images before mentioned , ther ar lately found out by the Portugales and others that haue sailed into the East and West Indies 4. other images towards the south Pole , as the crosse or Crosier , the south triangle , Noahs doue or Pigion , & another image made like a Philosopher called Polophilax , all which are set downe in the celestial Globe , lately set forth , first at the great charges of M. Sanders , and now at the like charges of M. Molinax of Lambeth of whome I lately bought both the Globes , that is , the terrestriall and celestiall , and I wish that the longitude , latitude and declination of euery Star contained in the said 4. images were trulie set downe , for Plancius maketh some doubt thereof . Notwithstanding if you be desirous to know the longitude , latitude , and declination of the said Starres by help of the foresayd great Globe , then you must worke as I doe shewe you hereafter in séeking for any Starre contayned in Mercator his globe , so shall you haue your desire .

Moreouer , to most of the Stars described in the Globe are annexed the Characters of some of the 7. Planets , to shew the nature & qualitie of the Stars & some stars ar also marked with some one letter or other , the more readily to find out therby the foresaid characters , As for example to Cor Leonis are annexed the Characters of Iupiter and Venus , & vnder the sayd Star is set the letter m. to shew the Characters which are not alwayes set hard by the Starre , but sometime a good distance off , for where the characters are set nigh vnto the star , there néedeth no letter , as in the Star called spica Virginis , whereunto are annexed the characters of Mars and Venus , without any letter to signifie the same , and where diuers Starres be of one selfe qualitie , they are seuerally marked with letters of one selfe same shape , as about the Starre Spica Virginis you shall finde diuers little Starres , each one marked with the letter h. signifying their nature to bée all one , that is , to participate of Mars and Venus , to whose characters is also ioyned the letter h , signifying that they be of that nature and quality .

TheCharactersaretheseherefollowingsetouereueryPlanetshead. ♄ ♃ ♂ ☉ ♀ ☿ ☽ Saturne . Iupiter . Mars . Sol. Venus . Mercurie . Luna .

The nature of euery one of the Planets here followeth .

SAturne is colde and drie . Iupiter is temperately hot and moist . Mars is extreamly hot and drie . Sol hot and somewhat drie . Venus is temperately colde and moist . Mercurie is of changeable nature , and plyant to the nature , bee it good or bad of euery other Planet or fixed Starre whereto it is ioyned . Luna is cold and moist .

Besides the images and starres both Northern and Southern aboue mentioned , there is also set downe in the celestiall Globe a certaine impression called in Gréeke Galaxia , that is to say , the milke white way , the description whereof here followeth .

A briefe description of the milke white way , called in Greeke Galaxia , and in Latine Via lactea .

THis way as Garceus writeth , procéedeth from the signe Gemini , and so passeth through the legs and loynes of : Ericthonius , and from thence through the right side of Perseus , and then through the whole image almost of Cassiopeia , and from thence through the left wing of the Swan called Auis , Gallina and Cignus , and from thence through the Image called in Latine Telum in English a Dart , shaft , or quarell , and from thence through the flying Eagle called in Latine Vultur volans , and from thence through the greatest part of Sagittarius his bowe , & from thence through the Alter called Ara and Thuribulum , and from thence through the legges of Centaurus , and so to the ship called Argos , from whence rising againe , and passing through part of the great Dog called Canis maior , it returneth againe to Gemini .

Thus hauing described vnto you all the 48. Images , and shewed the names of as many Stars as are named in the Globe , and also the milke white way , I mind now to procéede to the propositions belonging to the fixed Starres described in the Globe , as followeth .

How to find out in heauen anie vnknowne Starre described in the Globe two manner of wayes , that is , either by the helpe of some knowne starre , or els by knowing the true houre of the night . Proposition . 20.

THe first and surest way is thus , take with your Quadrant or Astrolabe the altitude of the knowne Star , marking therewith in what part of the heauen the same is situated East or West , North or South , and then hauing set your Globe right North and South , and at the true latitude of the place wher you are , bring the quarter of altitude to the said Starre , and therewith turne the Globe vntill you sée that the said Starre hath the like place and the like altitude in the Globe that it had in heauen , then kéeping the globe stil at that stay , séek in the globe the star that you would find out in the firmament , & marke wel in what part of the Globe it is situated , and how it beareth from the knowne Star , either East , West , North or South , and bring the quarter of altitude to that Starre , that you may know the altitude therof by help of the sayd quarter , which altitude once taken , turne your eyes towardes that part of the firmament , and hauing placed the Diopter of your Astrolabe , at that altitude , look what Star in that part of the firmament doth answere to such altitude , and that is the Starre which you séeke , whose name for the most part is set downe in the Globe . The second way is thus , hauing learned the true houre of the night by some clocke or watch , bring the degrée of the Sun vnto the brazen Meridian , & holding it there , set the index of the houre whéele iust at the 12. houre , that done , turne the body of the Globe to or fro vntil the index of the houre whéel fal iust vpon the houre which you séeke , and kéeping the Globe at that stay , séek out the vnknown Starre in the Globe , and consider how it beareth from you in the Globe either East , West , North or South , then bring the quarter of altitude to the Star , that you may know thereby what altitude it hath in the Globe , which once found and hauing set the Diopter of your Astrolabe at that altitude , turne your eie towards that part of the firmament whereunto the place of the Starre found in the Globe directeth you , and that Star which answereth to that altitude is the Star which you séek . But this way is not so sure as the other way first taught , vnles you know the true houre indéed .

How to know by the Globe the Meridian altitude which is the highest altitude of any starre , and also how high or lowe he is at any other time . Proposition . 21.

HAuing set the Globe at your latitude , bring the star to the brazen Meridian , and there staying it , number vpon the brazen Meridian the degrées contayned betwixt the saide Star and the South or north point of the Horizon , according as the Starre is situated either Northward or Southward , for so in the latitude 52 you shall find the Meridian or highest altitude of the Star Arcturus or Bubulcus to be 60. degrées . But if you would know his altitude at any other time , then you must rectifie the index of the houre whéele by the 7. proposition , and hauing set the index at the houre wherein you séeke , stay the Globe there , vntill you haue brought the quarter of Altitude vnto the Star , and that wil shew the altitude of the Star at that houre if it be aboue the Horizon .

How to know by the Globe what Starres are aboue the Horizon at any time of the day or night . Proposition . 22.

OF Starres according to diuers latitudes some ar alwaies aboue the Horizon & some are alwaies vnder the Horizon , & some do both rise and set , if you would know what Starres he aboue the Horizon in the day time , then hauing rectified the Index of the houre whéele by the 7. proposition , take the altitude of the Sun with your Astrolabe or quadrant , and therewith consider whether the Sun be in the East part or in the West part of the firmament . Then bring the quarter of altitude on the East or West side of the Globe according as you saw the Sun at that present to be in the firmament , and make the degree of altitude , marked in the quarter of altitude to méete euen with the degrée of the Ecliptique line wherein the Sunne is that day , and there staying the Globe , you shall sée all the starres that be aboue the Horizon at that present , as well on the East side as on the west side of the Globe , and the Index will shew you at what houre you took the aforesad altitude . But if it be in the night season , and that the starres doe appeare , take with your Astrolabe the altitude of some known star , and by doing as is before taught , you shal haue your desire . But you must not forget first of all to rectifie the Index of the houre whéele by the 7. proposition .

How to know by the Globe at what time any star riseth aboue the Horizon , mounteth to the highest , and setteth , and with what degree of the Ecliptique he riseth , mounteth , and setteth , and also in what part of the Horizon he riseth and setteth . Proposition . 23.

HAuing rectified the Index of the houre whéele by the 7. proposition , bring the Star to the East part of the Horizon , so as it may touch the edge thereof , and the Index of the houre whéele will shew at what houre hée riseth and by looking at that instant to the Ecliptique line you shall sée what degrée of the Ecliptique riseth then with him . That done , bring the said starre to the brazen Meridian , and the Index of the houre whéele will shew at what houre hee is at the highest , and there staying the Globe , marke what degrée of the Ecliptique line doth fall right vnder the brazen Meridian at that instant , for that degrée is sayde to accompany him when hée is at his highest . Then bring the sayd Starre to the West part of the Horizon , and you shall finde by the Index of the houre whéele at what houre hee setteth , and what degree of the Ecliptique doth accompany him at his setting . As for example , I would knowe the sixtéenth day of Iune 1590. the sunne being in the fourth degrée of Cancer , when Arcturus otherwise called Babulcus and Lanceator doth rise , mounteth to the highest , and setteth , here hauing rectified the Index of the houre-wheele by the seuenth Proposition , I bring the said starre of the East part and very edge of the Horizon , and I finde that he riseth a little before twelue of the clocke at noone , and that the 28. degrée and 30′· of Virgo , riseth with him , and by looking amongst the windes vpon the Horizon right against the place of his rising , I find that he riseth Northeast and by East . Secondly by bringing the said star to the brazen Meridian , the Index of the houre-whéele sheweth that he is at his highest halfe an houre after seuen of the clocke at night , and is then plaine South , and that the 29. of Libra , doth then accompany him . Thirdly by bringing the saide starre to the West part of the Horizon , the Index of the houre-whéele sheweth that he setteth or goeth downe a quarter of an houre before foure in the morning , and that the fourth degrée of Capricornus doth accompany him at his setting : and by looking there vpon the Horizon , I find amongest the winds that the said star setteth Northwest and by West .

How to know in what part of the firmament any starre is , and how many degrees it is distant from the Meridian at any houre , and being right vnder the Meridian to knowe how far it is distant from your Zenith . The 24. Proposition .

HAuing rectified the Index of the houre-whéele by the seuenth Proposition , turne the Globe vntill the Index touch the houre wherein you séeke . And staying the Globe there , bring the quarter of Altitude to the starre be it on the East or West side of the Globe , and the nether end of the said quarter will shew vpon the Horizon among the winds , in what part of the firmament the star is : now if you would know how farre that starre is distant from the Meridian doe thus : looke what degrée of the Equator is at that instant vnder the Meridian , and there make a marke , & then turne the Globe vntill you haue brought the said starre vnder the brazen Meridian , and marke what degrée of the Equinoctiall the said Meridian cutteth at that instant , that done , count the degrées contained betwixt the two markes vpon the Equinoctiall , for so many degrées is the starre distant at that time from the Meridian eyther towards the East or West , and by allowing 15. degrées to an houre , and 4′· to a degrée , you shall know in what time the said starre hath to approch to the Meridian , or howe much he is past the Meridian , and hauing brought the said starre right vnder the Meridian , you shall knowe how farre it is distant from your Zenith , by counting the degrées that are contained in the Meridian betwixt the said star and your Zenith : As for example , the sixth of October 1591. ( the sunne being then in the 22. of Libra ) I finde the starre Cor Leonis at thrée of the clocke in the after noone to be West , Northwest , & there staying the Globe , I sée that the Meridian cutteth the Equinoctiall at the 246. degrée , whereas I make a marke , that done , I bring the star Cor Leonis to the said Meridian at which instant the Meridian cutteth the Equinoctial in the 146. degrée , which being taken out of 246. there remaineth 100. degrées , which is his distance from the Meridian , which being deuided by 15. I find in the quotient 6. houres and 40′· for so much as he is past the Meridian towards the West , and by bringing the foresaid starre to the Meridian , I finde him to be distant from our Zenith 39. degrées , and 30. minutes .

How to find the houre of the night by the Globe . The 25. Proposition .

HAuing set the Globe at your Latitude , rectifie the Index of the houre-whéele by the 7. Proposition , then hauing taken the Altitude of some starre that you knowe , and is in the Globe with your Astrolabe or Duadrant , bring the quarter of Altitude vnto the starre , be it in the East or West , according as you found the starre to be in the firmament , not leauing to turne the Globe vntill you haue made the starre to haue the like Altitude in the Globe vpon the quarter of Altitude , and also the like situation that you found it to haue in the firmament by your Astrolabe or Duadrant , & staying the Globe there , the Index of the houre-whéele wil shew the houre : As for example , in the yeare 1590. the first of Ianuarie the sunne being in the 22. of Aquarius , I hauing with my Astrolabe found that the starre called Canis maior , that is to say the greater dogge , was eleuated aboue the Horizon in the East part of the firmament 20. degrées , I brought the quarter of Altitude to the East side of the Globe , not leauing to turne the Globe vntill I had made the star to méete euen with the 20. degrée of the quarter of Altitude , and there staying the Globe , I found by the Index of the houre-whéele that it was 8. of the clocke at night , and a quarter past .

How to know the verticall starres in euery Latitude . The 26. Proposition .

THe Globe and the brazen Zenith being set according to the Latitude of the place where you are , turne the Globe from East to West , and as many starres as passe right vnder your Zenith are saide to be verticall as in the Latitude 52. you shall sée the taile of the great Beare , the head of Perseus , and diuerse others to passe through your Zenith in turning the Globe from East to West .

How to know the true place of any starre , that is to say , in what signe and in what degree there of any starre is . The 27. Proposition .

HAuing fastned the semi-Circle of Longitude and Latitude vppon the two Poles of the Zodiaque , the North pole whereof is in the Circle Arctique not farre from the right clawe of the Dragon , and the South pole therof is in the circle Antarctique right opposite to the other , then bring the saide semi-Circle to the starre , whose place you séeke , and marke therewith what point of the Ecliptique the saide semi-Circle cutteth , and that is the place of the starre . As for example , by bringing the said semi-Circle to the starre called Hircus , that is to say , the Goate , I finde it to be in the 15. degrée 30. minutes of Gemini , and that is his place .

How to find the Longitude of any starre by the Globe . The 28. Proposition

I Haue told you in my Spheare , that the Longitude of any starre is that arch or portion of the Ecliptique line , which is contained betwixt the first point of Aries , and that Circle which passeth through the Poles of the Zodiaque , and also through the bodie of the starre , which Circle , the semi-Circle of Longitude and Latitude here representeth : and by making the saide semi-Circle to passe through the starre called Hircus before mentioned , I finde that the number of degrées of the Ecliptique , contained betwixt the said Circle and the first point of Aries to be 75. degrées & 30′· which is the Longitude of the said starre , and thereby maketh his place to be in the 15. degrée 30′· of Gemini , as before is set downe in the last Proposition .

How to find out the Latitude of any starre . The 29. Proposition .

I Told you also in my Spheare , that the Latitude is none other thing , but the distance of any starre from the Ecliptique line , eyther towardes the North or South pole of the Zodiaque , which distance the semi-Circle of Longitude and Latitude made to passe through the bodie of the starre , and cutting the Ecliptique line doth alwaies shew , as in the former example , in making the foresaide semi-Circle to passe through the foresaide starre called hircus , I find by counting the degrées of the said semi-Circle contained betwixt the Ecliptique line and the bodie of the starre , that the Latitude of that starre is 22. degrées and 30. minutes towardes the North , likewise by bringing the said semi-Circle to the starre which is in the right shoulder of Orion called Bed Alguze , I find his Latitude to be 17. degrées towards the South .

How to find out by the Globe the declination of any starre . The 30. Proposition .

THe declination is none other thing but the distance of any starre from the Equinoctiall , either towards the North pole or South pole of the world , which is to be found thus : First hauing brought the starre right vnder the brazen Meridian , and there staying the Globe , count the degrées of the said Meridian contained betwixt the saide starre and the Equinoctiall point or stréeke of the said Meridian , and that shall be the declination of the starre : As for example , bring the starre Hircus vnto the Meridian , and you shall find the declination thereof to be 45. degrées towardes the North pole of the world , and that the starre which is in the foote of the Hare called Lepus , hath of South declination 23. degrées .

How to know the magnitude or greatnes of any starre , and his nature and qualitie by the Globe , and also the right ascention of the Arch of the Ecliptique which accompaneth the right ascention of any starre . The 31. Proposition .

MArcator describeth the sixe magnitudes of the starres by making sixe figures or shapes of stars placed not farre from the head of the great beare , whereof the first and greatest hath 16. pointes or beames , the second eight , the third sixe , the fourth fiue , the fift sixe , which indéede by that account would be but foure : and the sixth hath fiue , which would haue but thrée : but in mine opinion it had béene better to haue made the first magnitude with ten points , the second with nine , the third with eight , the fourth with seuen , the fift with sixe , and the sixt with fiue : so should the magnitude of euery starre described in the Globe haue béene the more easily knowne . But to the intent that you might exercise your selfe in finding out by the Globe , the place , Longitude , Latitude , and declination of any starre that is described in the Globe : I haue thought good to adde hereunto the Table of Garceus , shewing not only the Longitude , Latitude and declination of the most notable starres that are both Northward and Southward , but also the right ascention , magnitude or bignesse , the qualitie or nature of euery such starre , and also the Arch of the Ecliptique line , which accompanieth the right ascention of euery star , which Table though by the saide Garceus was calculated out of the Astronomicall Tables for the yeare of our Lord 1564. and not by the Globe , yet for your better exercise in matters of the Globe , I thought good to set downe such Longitude , Latitude , declination , magnitude , and right ascention , and all other things contained in the said Table , according as they are to be found out by the Celestiall Globe of Mercator , and not calculated by any of the Astronomicall Tables .

Though the Longitudes and declinations of the fixed starres set downe in this Table to serue Mercator his Globe , doe not altogether agrée with the great Celestiall Globe , lately set foorth by M. Sanderson and by M. Molinaxe , by reason that the Longitudes and declinations of the saide starres are méere lately calculated , that is to say , for the yeare of our Lord , 1592. yet it will serue to shewe you the way how to exercise your selfe in the said Globe , and thereby you may correct this Table , and make it to agrée in all points with the new great Globe , whereby you shall reape more pleasure then griefe or paine . For I had this Table ( as I haue saide before ) out of Garceus , and did here set it downe more for your exercise , and to acquaint you with the fixed starres that are described in the Celestiall Globe , then for any other purpose .

The vse of the Table next ensuing .

THis Table is deuided into eight collums , The first whereof on the left hande , containeth the names of the starres , the seconde the degrées and minutes of Longitude , together with the Characters of the 12. signes : the third the degrées and minutes of Latitude both Northerne and Southerne , the North Latitude being marked with the letter N. and the South Latitude with the letter S. the fourth containeth the degrées and minutes of declination both Northern and Southerne , marked with the letters N. and S. as before . The fift sheweth the magnitude or greatnesse of the starre , whether it be of the first , second or third bignesse , &c. The sixth containeth the degrées and minutes of the right ascention of the foresaide fixed starres . The seuenth containeth the Characters of the Planets , signifying the nature and qualitie of the starres . The eigth containeth the degrées and minutes of the right ascention of the Ecliptique line and the signes of the Zodiaque .

Here followeth the Table .

The names of the starres . The Longitude & place . The signes . The Latitude North & South . Declination Nor. & South Magnitude or greatnes The right Ascention Degr. Min. The nature of the stars shewed by the charrect● of the Planets The right ascention of the arch Ecliptique acco●anin●●●● starre ●he 〈◊〉 of the s●gnes . The first starre of the Rammes horne . 27 40 ♈ 7 40 Nor. 17 20 Nor. 3 23 0 ♂ ♄ 25 0 ♈ The right shoulder of Cepheus 8 0 ♈ 69 0 Nor. 61 0 Nor. 3 317 15 ♄ ♃ 15 0 ♒ The last starre of Eridanus . 22 30 ♈ 53 30 Sou. 41 0 Sou. 1. 43 30 ♃ ♀ 15 50 ♉ The shoulder of Andromeda . 16 20 ♈ 24 30 Nor. 28 26 Nor. 3 5 0 ♀ 5 0 ♈ The girdle of Andromeda . 24 50 ♈ 26 30 Nor. 33 0 Nor. 3 12 10 ♀ 13 0 ♈ The right side of Perseus 25 0 ♉ 30 0 Nor. 48 0 Nor. 2 43 0 ♄ ♃ 15 29 ♉ The head Algol or Meduza . 20 40 ♉ 23 0 Nor. 39 30 Nor. 2 40 20 ♄ ♃ 13 0 ♉ The South starre of Pleiades . 23 29 ♉ 4 20 Nor. 22 0 Nor. 5 50 10 ♂ ☽ 23 10 ♉ The North starre of Pleiades . 23 10 ♉ 5 0 Nor. 23 10 Nor. 5 50 0 ♂ ☽ 22 20 ♉ The Buls eye . 2 50 ♊ 5 0 Sou. 15 30 Nor. 1 63 0 ♂ 4 20 ♊ The right shoulder of Orion . 23 0 ♊ 17 0 Sou. 6 20 Nor. 1 83 20 ♂ ☿ 23 50 ♊ The left shoulder of Orion . 12 0 ♊ 17 0 Sou. 4 40 Nor. 2 72 0 ♂ ☿ 13 0 ♊ The right shoulder of Auriga 24 0 ♊ 20 0 Nor. 43 0 Nor. 2 82 0 ♂ ☿ 22 50 ♊ The Goate , or Hireus . 16 0 ♊ 22 30 Nor. 45 0 Nor. 1 71 30 ♂ ☿ 13 0 ♊ The former of the Kiddes 13 0 ♊ 18 0 Nor. 40 0 Nor. 4 68 50 ♂ ☿ 10 28 ♊ The latter of the Kiddes 13 30 ♊ 18 0 Nor. 39 40 Nor. 4 69 0 ♂ ☿ 11 0 ♊ The left foote of Orion . 10 10 ♊ 31 30 Sou. 9 0 Sou. 1 73 0 ♃ ♄ 14 20 ♊ The tayle of the little deare 21 10 ♊ 66 0 Nor. 86 0 Nor. 3 4 30 ♄ ♀ 4 30 ♈ The nuddle starre of Orions girdle . 18 0 ♊ 24 30 Sou. 1 50 Sou. 2 79 25 ♃ ♄ 20 20 ♊ Canopus in the shippe Argo . 8 30 ♋ 75 0 Sou. 52 0 Sou. 1 94 24 ♄ ♃ 3 30 ♋ The great dogge . 9 0 ♋ 39 10 Sou , 16 0 Sou. 1 97 0 ♂ ♃ 6 30 ♋ The little dogge . 20 30 ♋ 16 0 Sou. 6 10 Nor. 1 109 30 ♂ 18 0 ♋ The former head of Gemini . 14 20 ♋ 9 40 Nor. 32 0 Nor. 2 106 47 ☿ 15 28 ♋ The following head of Gemini 17 40 ♋ 6 30 Nor. 28 30 Nor. 2 110 5 ♂ 18 20 ♋ The North Asellus . 1 30 ♌ 3 0 Nor. 22 30 Nor. 4 124 0 ♂ ☉ 1 40 ♌ The South Asellus in Cancer . 2 20 ♌ 0 10 Sou. 19 20 Nor. 4 124 50 ♂ ☽ 2 0 ♌ Praesepe . the Manger . ●n Cancer . 1 20 ♌ 0 40 Nor. 20 0 Nor. neb. 124 0 ♂ ☽ 1 30 ♌ The shoulder of the great deare . 11 40 ♌ 49 0 Nor. 62 30 Nor. 2 161 0 ♂ 9 30 ♍ ●he bright starre of Hydrus . 21 0 ♌ 20 30 Sou. 5 0 Sou. 2 137 0 ♄ ♂ 14 30 ♌ The Lyons heart 23 30 ♌ 0 10 Nor. 13 30 Nor. 1 145 93 ♃ ♂ 23 30 ♌ The Lyons necke . 23 10 ♌ 8 30 Nor. 21 40 Nor. 2 148 40 ♄ 26 0 ♌ The Lyons tayle . 15 30 ♍ 12 0 Nor. 16 20 Nor. 1 171 10 ♄ ♀ 20 30 ♍ The middle starre of the great Beares tayle . 9 50 ♍ 56 0 Nor. 56 55 Nor. 2 196 0 ♂ 17 0 ♎ The right wing of Virgo . 3 10 ♎ 15 10 Nor. 12 30 Nor. 3 189 0 ♄ ☿ 9 40 ♎ The left shoulder of Bubulcus . 10 0 ♎ 49 30 Nor. 40 0 Nor. 3 213 0 ♄ ☿ 5 0 ♏ The Ranens bill . 6 30 ♎ 21 40 Sou. 22 30 Sou. 3 176 43 ♄ 26 20 ♍ Arcturus the great starre betwixt Bubulcus his legs 18 0 ♎ 32 0 Nor. 22 0 Nor. 1 209 0 ♃ ♂ 1 0 ♏ Spica virginis . 17 40 ♎ 12 0 Sou. 10 0 Sou. 1 195 10 ♂ ♀ 16 30 ♎ The middle starre in the front of Scorpio 26 40 ♏ 1 30 Sou. 21 0 Sou. 3 234 0 ♂ ♄ 26 30 ♏ The South ballance of Libra . 9 0 ♏ 0 40 Nor. 14 0 Sou. 2 217 0 ♄ ♂ 9 20 ♏ The North starre of the ballance Libra . 13 10 ♏ 9 0 Nor: 8 0 Sou. 2 223 10 ♃ ☿ 15 46 ♏ The left hand of Serpentarius . 27 0 ♏ 17 0 Nor. 3 30 Sou. 3 238 30 ♄ ♀ 30 0 ♏ The bright starre of Ariadnes crowne . 5 30 ♏ 44 30 Nor. 28 0 Nor. 2 229 0 ♀ ☿ 21 20 ♏ The heart of the Scorpion . 3 30 ♐ 3 30 Sou. 25 0 Sou. 2 241 0 ♃ ♂ 3 0 ♐ The head of Hercules . 8 40 ♐ 37 30 Nor. 15 0 Nor. 3 252 35 ☿ 14 0 ♐ The head of Serpentacius . 15 30 ♐ 36 0 Nor. 12 50 Nor. 3 258 30 ♂ ☿ 19 10 ♐ The head of the Dragon 20 20 ♐ 76 0 Nor. 52 0 Nor. 3 266 10 ♄ ♂ 26 30 ♐ The bright starre of Lyra 9 0 ♑ 62 0 Nor. 38 20 Nor. 1 275 20 ♀ ☿ 5 0 ♑ The Eagle , alias vultur volans . 24 50 ♑ 29 20 Nor. 7 30 Nor. 2 210 40 ♄ ♂ 20 4 ♑ The first starre of the taile of Capricorne . 16 0 ♒ 2 0 Sou. 18 0 Sou. 3 319 0 ♃ ♄ 17 0 ♒ The following starre thereof . 17 10 ♒ 2 0 Sou. 17 30 Sou. 3 320 10 ♃ ♄ 18 0 ♒ The taile of the Dolphin . 8 10 ♒ 29 30 Nor. 10 0 Nor. 3 304 0 ♄ ♂ 1 10 ♒ The Fomahant . 28 0 ♒ 23 0 Sou. 33 10 Sou. 1 339 0 ♀ ☿ 7 0 ♓ The point of the dart . 1 10 ♒ 39 30 Nor. 18 10 Nor. 4 295 0 ♂ ♀ 23 8 ♑ The tayle of the Swanne , 0 20 ♓ 60 0 Nor. 43 30 Nor. 2 307 10 ♂ ☿ 5 0 ♒ The right thygh of Pegasus . 22 50 ♓ 31 0 Nor. 25 0 Nor. 2 341 0 ♃ ♂ 9 30 ♓ The taile of the Whale . 27 0 ♓ 20 20 Sou. 19 50 Sou. 3 5 35 ♄ 6 0 ♈

How to finde out the right and oblique ascention of any starre , and also of the ascentionall difference . The 32. Proposition .

THe right ascention of any star is an arch or portion of the Equator , to be counted from the first point of Aries according to the succession of the signes , with which portion in a right Spheare any starre both riseth mounteth to the Meridian , and setteth : and in an oblique Spheare it is a portion of the Equator , wherewith the star is mounted to the Meridian : as for example , in a great Spheare the star called Cor Leonis , that is to say , the hart of the Lion , both riseth , mounteth and setteth with the 145. degr. 30′· of the Equinoctiall . But the right ascention of the said starre in an oblique Spheare , is to be found onely by bringing the said starre to the Meridian , and you shall find it to be all one with the right ascention in a right Spheare , for by bringing the starre called Cor Leonis to the Meridian in an oblique Spheare , you shall finde the right ascention thereof to be all one with that which it had in a right Spheare , that is 145. degrées 30′· of the Equinoctiall . But if you would know the oblique ascention of any star , then hauing set the Globe at your Latitude , bring the starre to the East part of the Horizon , & marke what degrée of the Equinoctial riseth therewith and that is the oblique ascention : As for example , the Globe standing at the Latitude 52. bring the starre , Cor Leonis to the East part of the Horizon , & by staying the Globe there , you shall find the 127. degrée of the Equinoctiall to rise with that starre . Now if you would know the ascentionall difference , that is to lay , the difference betwixt the right and oblique ascention , you haue no more to doe but to subtract the lesser out of the greater , and the remainder shall be the ascentionall difference , as in the former example , take 127 degrées out of 145. degrées and 30′ and there shall remaine 18. degrées 30′· and that is the ascentionall difference , by helpe wherof you may know the encrease and decrease of the artificiall day and night throughout the yeare in any Latitude , if you obserue that order which I haue already set downe in the first part of my Spheare , the 50. Chapter .

To know in what quantitie of time any whole signe or any other Arch of the Ecliptique doth rise or set . The 33. Proposition .

HAuing set the Globe at your Latitude , and rectified the Index of the houre whéele by the seuenth Proposition , bring the beginning of any Arch or signe to the East part of the Horizon , and marke what degrée of the Equinoctiall riseth therewith , that done , bring the end of the said signe or arch to the East part of the Horizon , and marke there also what degrée of the Equinoctiall toucheth the Horizon , and staying the Globe there , looke how many houres or parts of houres the Index of the houre-whéele hath runne betwixt the beginning and ending of the said signe or arch , and so you shall know the quantitie of that time , you may know it also by the number of degrées of the Equinoctiall contained betwixt the beginning and ending of the said signe or arch by allowing 15. degrées to an houre , and 4′· to a degrée . As for example , supposing the sunne to be in the first degrée of Taurus , you shall finde by working according to the rule before set downe , that the whole signe of Taurus doth spend in rising one houre 8′· Now if you will knowe howe much time he spendeth in descending , bring the first degrée of Taurus to the West part of the Horizon , marking what degrée of the Equinoctiall toucheth the Horizon at that instant , and also to what houre the Index pointeth , then turne the Globe still Westwarde vntill the last degrée of Taurus méeteth euen with the edge of the Horizon , and then marke againe as well the degrees of the Equinoctiall that toucheth the Horizon , as also to what houre the Index pointeth , and you shall find the number of degrées of the Equinoctist to be 42. degrees , which maketh two houres 48. minutes , which is answerable to the houre of the houre-whéele , and so much time the whole signe of Taurus spendeth in his descention or going downe .

How to know by the Globe what starres doe rise or set euery day Cosmically , Acronically , or Helically . The 34. Proposition .

AS for the thréefold Poeticall rising and setting of the stars , you shall find them plainly defined in the first part of my Spheare , Chap. 35. but to finde out the same by the Globe , you must doe thus . First hauing set your Globe at your Latitude , and sought out the place of the sunne for that day , bring the degrée of the sunne to the East part of the Horizon , and stay the Globe there , that you may sée what stars doe rise a little before the sunne , and which rise together with the sunne . For those that rise a little before the sunne , are saide to rise Helically , and those that rise together with the sunne , are sayde to rise Cosmically , and those starres that are in the very West part of the Horizon at the rising of the sunne , are saide to set Cosmically . Againe those starres that rise immediatly after the sunne , doe set Helically , that done , turne the degree of the sunne vnto the West part of the Horizon , and staying the Globe there marke what starres are readie to goe downe with him , for those are said to set Acronically , ans staying the Globe still there in the West , marke what starres at that present doe rise in the East part of the Horizon , for those are said to rise Acronically .

To know in what time of the yeare any starre riseth or setteth , either Cosmically , or Acronically . The 35. Proposition .

HEre hauing set the Globe at your Latitude , and knowing the degrée of the sunne , bring the starre to the East part of the Horizon , and therewith consider what degrée of the Ecliptique the Horizon cutteth at that present , that done , finde out the selfe same degrée vpon the Horizon in the narrow space of degrées next vnto the bodie of the Globe , and right against that degrée , you shall finde in what day and moneth that starre doth rise Cosmically , As for example , I would knowe at what time of the yeare Cor Leonis riseth Cosmically in the Latitude 52. I bring the starre Cor Leonis to the East part of the Horizon , and I finde that the Horizon cutteth the Ecliptique in the 23. degrée 30′· of Leo , which degrée being found againe vppon the Horizon , pointeth to the sixth day of August , so as I conclude that Cor Leonis doth rise that present day Cosmically , for then both he and the sunne are in a manner in one selfe degrée of the Ecliptique : nowe to know the Cosmicall setting of the saide starre , turne the same starre to the West part of the Horizon , and marke what degrée of the Ecliptique doth then rise in the East , and you shall finde the same to be the 23. degrée of Aquarius , which degrée béeing found againe vpon the Horizon in the narrow space of degrées next to the body of the Globe , containing the degrées of the Zodiaque , will point to the 31. day of Ianuarie , at which time the sunne is opposite to the said starre , and therefore it is saide to goe downe Cosmically , because it goeth downe when the sunne riseth . Now to know the Acronicall rising of any starre at any time , bring the starre to the East part of the Horizon , and marke therewith what degrée of the Ecliptique goeth downe in the West at that instant , for the sunne being in that degrée is opposite to the starre : As for example , by bringing the starre Cor Leonis to the East part of the Horizon , you shall find that the 23. degrée of Aquarius goeth downe at that instant , which degree being found againe vpon the Horizon , will shewe the day and moneth when the starre riseth Acronically , and so you shall find the starre Cor Leonis to rise Acronically the 31. day of Ianuary . Contrariwise if you woulde know when the said starre setteth Acronically , bring the said star Cor Leonis , to the West part of the Horizon , & therewith marke what degrée of the Ecliptique then setteth in the West , which degree being found againe vpon the Horizon , will shew that the said starre setteth Acronically the 6. of August .

Of the Horoscop and the rest of the twelue houses . The 36. Proposition .

THis word Horoscop doth not onely signifie the degree of the Ecliptique , otherwise called the ascendent which riseth aboue the Horizon in the beginning of any thing that is to be sought or knowne , but also sometimes the whole figure of heauen containing the 12. houses , and doth shew the very secrets of nature , so that there is nothing that chanceth to the inferiour bodies , but some cause thereof doth appeare by meane of the Horoscope in heauen , and therefore the Astrologians haue deuided the whole heauen into 12. houses , which are numbred from the Horoscope , which is the East Angle , and so foorth according to the succession of the signes , of which 12. houses the foure principall are foure points of the Zodiaque whereof two doe fall vpon the Horizon , and the other two vpon the Meridian , and are called principall points , poles , or Angles , that is the beginning of the first house , of the fourth house , of the seuenth house , and of the tenth house , and those that doe follow next any of these principall Angles , are called succéeding houses , in Latine Succedentes , as the second , the fifth , the eight , and the eleuenth house . And those that goe next before any of the foure principall Angles , are called falling houses , in Latine Cadentes , as the 12. the third , the sixth , and the ninth : and such houses as haue no familiaritie with the Horoscop or ascendent , as the second , the sixth , the eight , and the eleuenth houses are said to be slow and deiect , all which things this Table here following doth shew , containing the number and names of the houses , and also their significations .

The 12. houses . The names of the houses . The significations of the houses . 1 Angle East life 2 succeeding the lower gate gaine 3 falling the Goddesse brethren 4 Angle the bottom of heauen parents 5 succeeding good fortune children 6 falling euill fortune health 7 Angle the West wife 8 succeeding the higher gate death 9 falling God religion 10 Angle the middle of heauen kingdome 11 succeeding the good spirit benefactor 12 falling the euill spirit prison

A generall figure of the 12. houses of Heauen , according to the Iudicial of Astrology .

1 the horoscope or ascendent . ♈ 2 the house succedent ♉ 3 the house cadent . ♊ 4 the angle of the earth . ♋ 5 house succedent . ♌ 6 house cadent . ♍ 7 the angle of the occident . ♎ 8 house succedent . ♏ 9 house cadent . ♐ 10 the angle meridionall . ♑ 11 house succedent . ♒ 12 the house cadent . ♓

How to finde out the Horoscop or ascendent at anie time of the day or night by the globe , and thereby to knowe the 4. principall angles of heauen . Proposition . 37.

FIrst hauing set the Globe at your latitude , and rectified the index of the houre whéele according to the degrée of the signe where in the Sun is that day you séeke by the 7. proposition , if it be in the day time , take with your Astrolabe or Quadrant . the altitude of the Sun . But if it be in the night , take the altitude of some knowne Starre , that thereby you may knowe the houre of the day or night in which you séeke the ascendent . But if it happen that neither Sunne nor starre is to be séene that day or night , then learne by some true clocke or watch what houre it is , and hauing set the Index of the houre wheele at that houre , stay the Globe there , and therewith marke what degrée of the Ecliptique riseth in the East parte of the Globe aboue the horizon at that instant , and that degrée is the Horoscop or ascendent for that houre . As for example , you woulde find out the ascendent the 16. of Iune 1590. at 8. of the clock in the forenoone , at which day the Sunne is in the 4. degrée of Cancer , here hauing first set the Index of the houre whéele at that houre by staying the Globe there , you shal finde that the 21. degrée 30′· of Leo is the ascendent , which is the East angle or first house , whereby you may also at that instant find out the other three angles , that is , the west , South , and North angles , for the opposit point of the Zodiaque to the ascendent is the West angle or seuenth house . And that degrée of the Zodiaque , which is at that instant right vnder the Meridian aboue head in the South angle or 10. house , and the opposite point to that beneath , is the north angle or fourth house , for hauing found the ascendent , which is the East angle or first house to be the 21. degrée , 30. minutes of Leo , the West angle must needs be the 21. degrée . 30. minutes of A quarius . Againe ●he south angle or 10. house is the 8. degrée of Taurus , and the oposite point to that is the North Angle or fourth house which is ●he 7. of Scorpio .

How to erect a figure by the Globe according to Regio Montanus his way which is called the reasonable way , and is counted the best of all others . Proposition . 38.

FIrst you must find out the degrée of the ascendent as is taught in the last chapter , which is alwayes the first house , then staying the Globe with some pretie wedges of wood being thrust betwixt the Horizon and the bodie of the Globe , marke there what degrée of the equinoctial doth touch the Horizon at that instant , and number from thence vpward vpon the said Equinoctiall 30. degrées , to the ende of which 30. degrées , bring the semicircle of position being first fastened in his due place vpon the East side of the Horizon , and looke what degrée and signe of the Zodiaque the circle of position cutteth at that present , and that degrée shall be the twelfth house : then number againe other 30. degrées vpon the Equinoctiall vpwardes towards the brazen Meridian , and to that bring the semicircle of position , marking what degrée of the Zodiaque the said semicircle cutteth , and that shall be the 11. house , that done , looke what degrée of the Zodiaque is right vnder the brazen meridian aboue heade , and that shall be the 10. house . Then hauing set the semicircle of position vpon the West side of the Globe , number from the brazen Meridian westward vpon the Equinoctiall other 30. degrées , to the end whereof bring the semicircle of position , and marke what degrée of the Zodiaque the circle of position cutteth , and that shall be the 9. house , then from thence downward number vpon the Equinoctiall other 30. degrées , to the end whereof bring the semicircle of position , marking there what signe and degrée of the Zodiaque the saide semicircle cutteth , and tha● shall be the 8. house . Now hauing these sixe houses , the opposite points of the sayd sixe houses will shewe you the other sixe houses , and if you will know which houses , and also which signes are opposite one to another , marke well this Table following .

The houses opposite are these , 1 to 7 The Signes opposite are these , ♈ to ♎ 12 6 ♉ ♏ 11 5 ♊ ♐ 10 4 ♋ ♑ 9 3 ♌ ♒ 8 2 ♍ ♓

And to make al these things the more plain vnto you , I thought good to set downe this example following . Suppose that the 16. of Iune , Anno 1590. and at 8. of the clocke in the morning you would erect a figure to knowe how the 12. houses of heauen are situated at that present , first hauing drawne such a square figure as this here following , representing the 12. houses , learne by the last proposition , who is the ascendent at that instant , and you shall finde finde it to be the 21. degrée 30′· of Leo , which must bée set in the first house , and the 28. degrée of Cancer in the 12. house and the 22. degrée of Gemini in the 11. house , and the 9. degrée of Taurus in the 10. house , and the 5. degrée of Aries to be in the 9. house , and the 12. degrée of Pisces in the 8. house . Nowe the opposite house to the first house or ascendent is the 7. house which is the 21. degrée . 30′· of Aquarius , for the opposite signe must alwayes haue like number of degrées , then the opposit to the 12. house is the 6. house , which the 28. deg of Capricorne , and the opposit to the 11. is the 5. house , which is the 22. degrée of Sagittarius , and the opposite to the 10. house is the 4. house , which is the 9. degrée of Scorpio , and the opposite to the 9. house in the 3. house , which is the 5. degrée of Libra , and the opposite to the 8. house is the second house which is the 12. degrée of Virgo , all which things this figure here following doth plainely shewe . And if you would know what Planets should be placed in euerie house , you must learne that out of the Ephemerides , or out of some of the Astronomicall Tables .

16. Iunii , Anno 1590. the Sun being in the 4. degr. of Cancer , in the latitude 52.

Now because most men that doe shewe the vse of the Globes , doe also teach therein how to finde out the latitude and longitude of anie region , I thought good therefore to set downe here some of their manifolde waies touching the finding out of the same , notwithstanding that I haue alreadie written something thereof in the second part of my Spheare , in the 8. 9. 10. & 11. chapters .

How to know the latitude of any place or region by any of the fixed starres described in the Globe . Proposition . 39.

TAke in the night season with your Astrolabe the Meridian altitude of some knowne starre that is to be found in the Globe , then hauing brought the starre vnder the brazen Meridian , turne the Meridian vp and downe in the nickes of the horizon , vntill the same Starre haue the said altitude in the brazen Meridian , which you found it to haue in the firmament by your Astrolabe , that done , number the degrées of the Meridian contayned betwixt the Pole and the Horizon , and that is the latitude of that place .

Another way to find the eleuation of the Pole . Proposition . 40.

HAuing brought the Globe into an open place where the Sunne shineth at nonnetide , and placed the same right North and south as is taught in the first chapter , set a néedle in the degrée of the Sunne , and bring the same to the brazen Meridian , not leauing to turn the same Meridian vp and downe in the nicks of the Horizon , vntill the néedle cast no shadow at all , and there staying the Globe , looke how many degrées the pole is eleuated aboue the Horizon , for that is the latitude of that place .

A third way to finde out the latitude of any place without taking the Meridian altitude of any Starre . Proposition . 41.

TAke at any houre the altitude of 2 knowne stars , and such as are to be found in the Globe at one selfe instant whereof the one must be situated towards the East , and the other towards the West , then turne the Globe together with the Meridian vp and downe in the nicks of the Horizon , vntill you finde by helpe of the quarter of altitude each Starre to haue the selfe same altitude in the Globe , that it had in heauen , which being done , looke how many degrées the North pole is eleuated aboue the Horizon , and that is the latitude of that place . But if you would knowe the latitude of any place that is towardes the South Pole , then you must first place the South pole aboue the Horizon , and then worke as before .

A fourth way to find out the latitude of any region by any knowne fixed starre or Planet that may be seene . Proposition . 42.

FIrst take his Meridian altitude , and then learne to know eyther by the Globe or by some table his declination , which if it bee Northerne you must subtract the same from the Meridian altitude , if Southern adde it to the Meridian altitude so shal you haue the eleuation of the Equinoctiall , which being subtracted from 90. the remainder shal be the latitude of that place as for example . I find the Meridian altitude of Oculus Tauri to be 53. degrées , 50′· 32″· and his declination Northward to be 15. degrées , 50′· 32″· which declination being subtracted from the Meridian altitude , there remaineth 38. degrées , and that is the eleuation of the Equinoctiall , which being subtracted from 90. there remaineth 52. for the latitude of that place .

Hitherto I haue set down the chiefest propositions that are to be done by the Globe , touching the Sun and the fixed stars : & lastlie , shewed how to find out the latitude of any place , wherefore now I thinke good to shew you how to find out the place of the Moone , and of euery one of the rest of the Planets in the Globe by help of the Ephemerides , and thereby to know when euery planet riseth and setteth , and first of the Moone .

A briefe description of the diurnall table set downe in Stadius his Ephemerides , together with the vse thereof . Proposition . 43.

BVt for as much as the diurnall Table of the Ephemerides shewing the dayly motion of the Planets is very néedful to serue diuers turnes I think it not amisse here briefly to describe the same , and specially that of Iohannes Stadius whose diurnall table or Almanacke beginning at the 202. page of his booke , and at the yéere of our Lord 1583. continueth to the yeare 1606.

Of which table euery page on the left hand is diuided into 9. Collums . In the first collum wherof on the left hand are set down the dayes of the moneth , first the Gregorian dayes according to the Romane account , and next to that the dayes of the moneth according to our English account , then in the front of euery other collum are set downe the characters first of the Sunne , and then of other the sixe Planets that is to say , of the Moone , Saturne , Iupiter , Mars , Venus , Mercurie , and last of all , the head of the Dragon figured thus ♌ And right vnder these seuen Planets , and also vnder the head of the Dragon are set downe the signes and degrées wherein euery of these is euery day of the moneth throughout the yeare at noontyde , and in the foot of the sayd Table is set downe the latitude of euerie one of the 5. Planets , procéeding by the dayes of the moneth diuided into thrée partes . And in the margent of euery left page are set downe the chiefest feastes and Saints dayes that fall in euery moneth throughout the yeare . Moreouer , there is a Table on the right hand right against the left Table , in which are set downe first the dayes of the moneth , and then what coniunction or any other aspect the Moone hath with any of the other sixe Planets , that is , with the Sunne , with Saturne , with Iupiter , with Mars , with Venus and with Mercurie , which Planets are set downe in the front of the saide Table , and vnder them the characters of such aspectes as the Moone hath that day with any of the other Planets . The characters of which aspects are these here following .

☌ ☍ △ □ ⚹

Whereof the first signifieth a coniunction : the seconde an opposition , the third a trine aspect , the fourth a quadrat aspect , and the fift a sextile aspect .

Two Planets are said to be in a coniunction when they are both in one selfe signe . And to be in an opposition when they are in two seuerall signes opposite one to another . For then they be distant one from another 6. signes . And they are saide to bée in a trine aspect when they be distant one from another by four signes . And to be in a quadrat aspect when they are distant one from another by thrée signes . And to be in a sextile aspect when they are distant but two signes one from another .

How to finde out the place of any Planet by the Ephemerides . Proposition . 44.

NOw to find out the place of any Planet , or of the head of the Dragon by this diurnall Table , you must first séeke out the day of the moneth in the fist collum of the left table , right against that on your right hand in the sayd left Table , you shall finde in the common angle right vnder the Planet or Dragons head ( which so euer of them you séeke ) the signe and degrée wherein the said Planet or Dragons head is the said day at noontyde .

And to find out the aspects which the Moone hath with any of the Planets the same day , you must resort to the other table on the right hand , obseruing like order as before .

The example .

As for example the 21. of Aprill 1592. which is the first of May , according to the Romain account , I find by the table on the left hand , the Sunne to be in the 10. degrée 50′· of Taurus , the Moone to be in the 30. degrée , 47′· of Capricorne , Saturne , to bée in the 8. degrée , 10′· of Cancer , Iupiter to be in the 18. degrée . 28′· of Sagittarius , Mars to be in the 12. degrée , 6′· of Gemini , Venus to be in the 2. degrée , 0′· of Aries , and Mercurie to be in the 6. degrée , 20′· of Taurus , and the head of the Dragon to be in the 29. degrée , 45′· of Gemini ,

And right against this in the Table on the right hande you shall find the Moone to bee in a trine aspect with the Sunne , to bée in an opposition with Saturne , to bee in a trine aspect with Mercurie .

Thus hauing brieflie shewed you the vse of the diurnall Table , I will shew you now how to find out the latitude of the Moone as well North as South , by help of Stadius his Table , set down in the 112. page of his Ephemerides , and first I will briefly describe the said Table .

A briefe description of the Table of Stadius set downe in the 112. page of his Ephemerides to finde out thereby the dayly latitude of the Moone be it North or South together with the Canon or rule thereof plainly declared by example . Proposition . 45.

THis Table is diuided into 8. collums , wherof the first on the left hande contayneth the degrées of euery signe set downe in the front of the Table , which degrées are to be counted descending from one to 30. for so manie degrées there be in euery signe : and the last collum on the right hand containeth the like number of degrées belonging to the signes set downe in the base or foote of the sayd Table , and this number ascendeth vpward from 1. to 30. and for that purpose it woulde not haue bene amisse to haue set ouer each head of those 2. collums this word gradus , next vnder the word Signa . And of the other sixe collums the first thrée on the left hand doe containe the degrées , minutes and seconds of the North latitude , and the other thrée towards the right hand do containe the degrées , minutes , and seconds of South latitude . Moreouer , the 12. Signes are to bée numbred in the front from the third collum on the left hand from 1. to 5. forward toward the right hand , and at the foot from 6. to 11. backward towards the left hand , set downe in arithmeticall figures .

The rule or Canon together with a plaine example shewing the vse of the Table .

FIrst knowing the day of the moneth , resort vnto the diurnall Table of motion of the Planets in the Ephemerides , and hauing there found out the motion or place of the Moone , and also of the Dragons heade answerable to the day wherein you séeke , subtract the place of the Dragons head from the place of the Moone , which is easily done so often as the arke of the Moone is greater , that is to say , contayneth more signes and degrées than the arke of the Dragons heade , beginning your account in both arkes from the first point of Aries . But if the arke of the Moone be lesser than the ark of the Dragons head , so as you cannot make your subtraction , then you must add to the place of the Moone 12. signes , which is 360. degrées , and you must adde also thereunto the number of so many signes as are contayned betwixt the first point of Aries , and the first point of that signe wherin the Moone is at that present , which signe it selfe is not to bee numbred , and when you come to take out of that whole summe the place of the Dragons head , you must first adde to the said place of the Dragons head the number of so many signes as are contained betwixt the first point of Aries , and the first point of that signe wherein the Dragons head is at that instant , but not the signe wherein it is , and then hauing made your subtraction , remember alwayes to take out of that remainder 90. degrées , which is thrée signes , so oft as you haue neede to adde 12. signes to the place of the Moone , and not otherwise , and with that remainder you must resort to the foresaid table of the Moones latitude , as for example .

The Example .

Suppose that you would knowe what latitude the Moone had the first of Nouember 1590. here resorting to the diurnall Table of the Ephemerides , you find according to the day propounded , the place of the Moone to be in the 16. degrée 49′· of Taurus , and the place of the Dragons head to be in the 28. degrée 14′· of Cancer . Now according to the rule before giuen , you must take the place of the dragon here , which is 28. degrées , 14′· of Cancer out of the 16. deg. 49′· of Taurus , which is the place of the Moone , and because you cannot take the greater sum out of the lesser , you must adde to the lesser sum 12. signes , which make 360. degrées , and also one signe , for Aries going next before Taurus , in which signe the Moone is , so shall you make the whole sum to be 13. signes 16 degrées 49′· out of which summe you must subtract the Dragons head , which with the signes that goe next before Cancer , counting from the first point of Aries , do make 3. signes 28. degrées , 14′· which being subtracted out of 13. signes , 16. degrées , 49′ there remaineth 9. signes , 18 deg. 35′· out of which you must also subtract 90. deg. which is 3. whole signes , & so you find the remainder to be 6. signes , 18. degrées 35′· with which last remainder you haue to enter into the table of the Moones latitude , in the foot wherof you shal find 6. signes , and in the last collum on the right hand 18. & in the next collum towards the left hand , & in the common angle answerable aswel to the said 18. degrée . as also to the 6 signes , you shal find the latitude of the Moone to be 4. deg. 45′· & 17″· ( which seconds may be very wel omitted ) and her latitude to be south . But now because there are 35′· more annexed to the 18. degrées of the foresaid remainder , you must find out a proportionall part answerable to those minutes which is to be done thus .

Take out of the table the whole latitude answerable to 6 signes & 19. deg. which is one degrée more , so as now the latitude of the Moone is 4. deg. 43′· omitting the seconds . Then subtract 4. deg. 43′· out of 4. degrées 45′· & there remaineth 2′· Now to find out a proportionall part answerable to the former 35′· you must say thus . If 60′· require 2′· what shall 35′· require ? and the quotient yéeldeth 1′· 10″· which being subtracted out of 4. degrées 45′· there wil remaine 4. degrées . 44′· and so much was the south latitude of the Moone at that present day .

How to know the true place of the Sun or Moone , or of any other planet euery houre of the day throughout the yeare . Proposition . 46.

BVt now because that to finde out the true place of the Moone or of any other Planet in the Globe to work thereby certainly and truly , it is not inough to knowe their places in the Zodiaque at noonetyde , vnlesse you knowe the same at the verie houre in which you seeke . Stadius teacheth a briefe rule to find out the true place of any Planet at what houre so euer you desire by helpe of a short Table set downe in the 109. page of his Ephemerides , which rule and Table he borrowed of Reinholdus . And this Table consisteth of 3. collums , in euery front wherof are set first degrées , minutes , seconds , and thirds , and vnder them minutes , secondes , thirds , and fourths . And note that the first row of numbers on the left hand signifieth sometime degrées , and sometime minutes , eyther of which do extend in this Table to 60. & to 61. right against which rowe in euery collum is set down on the right hand the proportionall part for one houre , the rule is thus .

The Rule .

First finde out by the Ephemerides the place of the Planet wherein it is at noontyde the same day that you séeke , and also the place of the same Planet , wherein it is the next day following at noone , and subtract the lesser out of the greater , if the two places at noonetyde be still in one selfe signe , but if the two places bée in two seuerall signes , then you must count how many degrées of the Ecliptique the one place is distant from the other , and that shall be the difference , with which difference you must resort to the Table , and right against that on the right hande in the same collum wherein you found the difference , you shall also find the proportionall part for one houre , as by this example you shall more plainly vnderstand .

The Example .

Suppose then that you desire to knowe the true place of the Sunne or of the Moone , or of any other Planet at fiue of the clocke in the after noone the 30. of December 1591. at which day you find the Sunne at noontyde to be in the 18. degrée , and 7. minutes of Capricorne , and the next day at noone to be in the 19. degrée and 8′· of the same signe , the difference whereof you finde by subtraction to be one degrée and one minute , with which difference you must enter into the table , and séeking for one degrée in the first collum on the left hande , you finde next vnto that on the right hand 2′· 30″· then for one minute which before was named one degrée , you find next vnto it 2″· 30‴· which being added to the former summe last found , maketh 2′· 32″· 30‴· which is the proportionall part for one houre . Then hauing multiplyed that by 5. houres to serue 5. of the clocke in the afternoone , you shall find the product to be 12′· 42″· 30‴· which being added to the first Meridian place , which is 18. degrées and 7. minutes of Capricorne , maketh in all 18. degrées 19′· 42″· 30‴· And remember that if your difference be only minutes , then the bodie of the Table doth shew the proportionall part . But if the difference doth containe degrées , then the fourths must bee made thirdes , and the thirds secondes , the secondes minutes , and the minutes degrées . And this table serueth for the 7. Planets for euer . Notwithstanding Stadius setteth downe another Table to finde out thereby the proportional moouing of the Moone feruing as wel for houres as for minutes of houres , which table beginneth at the 144. page , and endeth at the 184. page of his booke .

In the front of which table in euery page are set downe the differences of the two places of the Planets which you haue found by subtracting the lesser out of the greater , if the planet be at both noonetydes in one selfe signe , but if she be at the two noontides in two seuerall signes , then such difference is to bee accounted vpon the Ecliptique line , to know howe many degrées the one place is distant from the other , and as for the minutes , if there be any , you may know the difference thereof by subtracting the lesser number out of the greater . And in the outermost collum on the left hande are set downe the houres and minutes marked in the foote with the letter H. signifying houres , and vnder that with the letter M signifying minutes , whereof the houres procéede but to 24. but the minutes extend to 60. which make one houre procéeding by euen numbers in this sorte . 1. 2. 4. 6. 8. &c. and therefore not finding that houre or minute which you séeke in the sayd collum , you must take the next number which is lesser by one , and so make it vp by adding the first and onlie odde one vnto it , and in each common angle you shall find the proportionall part that is to bee added to the first place of the Planet answering to the day of the moneth wherein you sought . All which thinges you shall more plainelie vnderstand by this one example .

The example .

Suppose then that you would knowe the true place of the Moone at 5. of the clocke . 17′· in the afternoone the 28. of December 1591. and looking in the Ephemerides you find the place of the Moone to be the same day at noonetide in the 26. degrée 12′· of Libra . and the next day at noontide to be in the 8. deg. 17′· of Scorpio . here by numbring how many degrées are contayned in the Ecliptique line betwixt those two places , and by subtracting the lesser number of minutes out of the greater , you shall finde the difference to be 12. degrées 5′· with which difference you must resort to the table of proportionall parts , and there hauing found the said difference in the front of the 13. page of the said table , look in the first collum on the left hand , for the houres and minutes according to the rule before set downe . As here in this example , because you cannot find 5. houres , you take foure , right against which on the right hand in the common angle you shal find 2. degrées 0′· 50″· then for one houre to make the 5. houres , you shall find in the common angle 30′· 12″· that done , séeke out the 17′· in the first collum , which number not being there you must take 16. against which in the common angle you shall find 8′· 3″· 20‴· and for the one minute which is to be added to 16. you shall find in the common angle against that one minute 30″· 12‴· and by adding all these summes together , you shall find the proportionall part to be 2. degrées , 39′· 35″· 32‴· which being added to the place of the Moone at noontyde first found in the Ephemerides , maketh in all 28. degrées , 51′· 35″· 32‴· of Libra , which is the true place of the Moone for that houre . But you haue to note that the denominations of the numbers contained in the common angles of proportional parts be not alwaies like , for when they are to answer houres then the first number on the left hand in euery angle signifieth degrées , and the rest minutes & seconds , but if they haue to answere minutes of houres , then the first numbers doe signifie minutes , and the rest seconds and thirds , as you may easilie perceiue by examining the former example . But nowe to returne to my first intention , which was to shewe you how to find out the true place of euerie Planet in the Globe , you haue to vnderstand that hauing found out the true place of any Planet at the day and houre wherin you séeke , by such meanes as is before taught , then resort to the celestiall Globe , and hauing set the same at your latitude , and also rectified the houre whéele according to the day wherein you séeke . Suppose that the 26. of May 1592. you would knowe in what part of the Globe the Moone is to be found at 5. of the clocke in the afternoone . Here first you must knowe her place at noone the same day , which is the 14. degrée , 28′· of Aries , and also her place the next day at noone , which is the 29. degrée 0′· of the same signe , Now by taking 14. degrées 28′· out of 29. degrées , you shal finde the remainder to bée 14. degrées , 32′· with which difference you must resorte to the table set downe in the 109. page . And by working as before is taught , you shal finde her place or longitude to be at fiue of the clocke in the 17. degrée 23′· 6″· 43‴· of Aries , which being counted vpon the Ecliptique line of the Globe , lay your semicircle of longitude and latitude to that point , and hauing learned her latitude by helpe of the Table of latitude before mentioned , by which you shall find her latitude to be at that time 4. deg. 46′· to the Southward , then hauing counted that latitude vpon the semicircle of longitude & latitude , make there a mark vpon the Globe , for that is her very place at that instant that is to say , at 5. of the clocke in the afternoone the 26. day of May 1592

Now if you would know when she riseth and setteth , you haue no more to do but to bring her place before marked on the Globe vnto the Horizon on the East part , and the index of the houre whéele being rectifyed as is before said , will shew the houre of her rising aboue the Horizon , and by bringing her place to the west part of the Horizon , the Index will shew the houre of her setting , and by bringing her place to the brazen Meridian , you shal know at what houre she is full South . The like Tables you shall finde also in Stadius to find out the true place of euery other Planet at anie houre , as of Saturne , Iupiter , Mars , Venus , and Mercurie , which Tables doe beginne at the 128. page , and ende at the 143. page of his booke , the order of working by which Tables is like in euerie respect vnto that of the Moone last taught . By which Tables you may finde out the true place of anie other Planet in the Globe at anie houre of the day or night , and thereby to knowe the houre of his rising and setting , if you rightly obserue the rules before taught touching the vse of the foresaid Tables . And note that you may finde out the place of the Moone by the Globe without the helpe of any Ephemerides , by such wayes as here doe followe .

How to finde out the place of the Moone by the Globe , when she is aboue the Horizon , without the helpe of any Ephemerides or other Table what soeuer . Proposition . 42.

HAuing set the Globe at your latitude , and placed it so as it may rightly answere the foure quarters of the world , take the altitude of the Moone with your Astrolabe or quadrant , and marke therewith whether she be at that present East , or West , that is to say , on this side of the Meridian , or beyond the Meridian , that done , take the altitude of some fixed Starre , which you know , and is to be found in the Globe , and also at that time is aboue the Horizon , marking therewith in what part of the firmament the said Starre is , and with your crosse staffe take the distance betwixt the moone and that star . Then hauing these thrée things , that is , the altitude of the Moone , the altitude of the Starre , and also the distance betwixt the Moone and the knowne starre . Mooue the globe together with the quarter of altitude to and fro vntill you haue made the Moone to haue the same altitude and like place in the Globe , that you founde it to haue in the firmament , and there make a marke vpon the Globe . Then bring the quarter of altitude towards the Starre , that the Star may haue the like altitude and like place in the Globe that it had in the firmament , and there hauing stayed the Globe that it may not mooue , take with your Compasse vpon the Equinoctiall the distance betwixt the Moone and the starre before found by the crosse staffe , and keeping your compasses at that wydenesse , put the firme foote of your Compasses in the fixed Star , and cease not to mooue the other foot together with the quarter of altitude towards the mark of the Moones altitude vntill you make them to méet , for there is the place of the moone .

Another way to find out the place of the Moone without taking the latitude of any without . Proposition . 48.

SEeke by your crosse staffe to knowe the distance betwixt the Moone and any two stars that you know & are to be found in the globe . That done , drawe vpon the globe on either starre an obscure circle according to the distance of the Moone from either of those two stars , which two circles will cut or crosse one another but in 2. points , the one whereof is the place of the Moon , and which of those places it should be , your eie will easilie tell you . And by this meanes it were no hard thing ( as Gemma Frisius satih ) for our sea men in these daies to finde out all the starres that be in the nether Hemispheare and were vnknowne to Ptolomie and to the other auncient Astronomers , and to cause them to be set downe in the Globe , yea and by this art the places of the rest of the Planets may be found out as well according to their Latitude as Longitude .

Moreouer Gemma Frisius sayth , that by knowing the true place of the Moone in the Zodiaque , you may also finde out therby the vnknowne Longitude of any Region .

How to finde out the Longitude of any Region . The 49. Proposition .

HAuing found out the place of the Moone in the Zodiaque , you must first know the very houre of her being in that place , and then learne by some Ephemerides or by the Tables of Alfonsus , at what houre the Moone doth enter into the selfe same degrée of the Zodiaque in some other Region or Towne whose Longitude you alreadie knowe , and hauing reduced the houres to 24. take the lesser number of houres out of the greater , the remainder whereof must be reduced out of houres and minutes into degrées thus : Multiply the houres by 15. and the minutes of houres by 4. so shall you haue the degrées of the Equator contained betwixt the two Meridians . And such distance so intercepted is called the difference of Longitude , which difference you must adde to the knowne Longitude if the houres in that place were more in number , but if the houres were lesse in number , then you must subtract the foresaide difference from the knowne Longitude , so shall you collect the vnknowne Longitude of that place or Region which you séeke , and how far it is distant from the fortunate Iles.

Another way to find out the vnknowne Longitude of any place by the Globe . The 50. Proposition .

HAuing set the Globe at your Latitude and rectified the houre-whéele by the seuenth Proposition , bring the place whose Longitude you know to the brazen Meridian , and direct the Index of the houre-whéele to that houre in which the Moone doth occupie the former defined place in that Region or Towne : that done , leaue not to turne the Globe vntill the Index of the houre-whéele come to that houre in which you sought the vnknowne place of the Moone , and the degrées of the Equator which the brazen Meridian cutteth , wil shew the vnknowne Longitude of the place which you séeke .

To the ende of this Treatise I haue thought good to adde a briefe description of the two great Globes lately set foorth first by M. Sanderson , and then by M. Molineux , and therewith to set down a briefe description of Sir Frances Drake his first voyage into the West and East Indies , and also the voyage of M. Tho. Candish both whose voyages and what course they helde are to be séene by helpe of two lines drawne on the terrestriall Globe , of which lines the one is redde shewing the voyage of Sir Frances both outward and homeward . And the other line is blew shewing in like manner the voyage of M. Candish .

A briefe description of the two great Globes lately set forth first by M. Sanderson , and then by M. Molineux .

THese Globes doe differ in a maner nothing at all from the Globes of Mercator touching the circles before described , but only in the Horizon , for the Horizon of these great Globes is deuided into 13. spaces as followeth .

Whereof the first narrow & innermost space next vnto the bodie of the Globe , containeth the degrées of the Zodiaque .

The second containeth the numbers of the said degrées procéeding from 10. to 30. in euery signe .

In the third space are set downe the names and the characters of the 12. signes , and also the characters of the Planets that gouerne the said signes .

In the fourth space are set downe the letters of the dayes of the wéeke .

In the 5. and 6. are set downe the numbers of the daies and names of the monethes according to the auncient Calender .

In the seuenth the festiuall daies .

In the 8. and 9. space the number of the daies and names of the monethes according to the new Romain Calender .

In the 10. & 11. are set down the numbers of the daies & names of the monethes according to the true Kalender lately calculated by a most excellent Mathematician and mine old acquaintance M. Dee of Mortlake , as I coniecture by the letters I. D. set downe vpon the said Horizon .

In the 12. are set downe the English names of the 32. Rombes or windes of the Mariners compasse .

In the 13. and outermost space are set downe the Latine names of the said 32. windes .

But the Mappe which couereth M. Molineux his Terrestriall Globe , differeth greatly from Mercator his terrestrial Globe by reason that there are found out diuers new places aswel towards the North pole , as in the East & West Indies which were vnknown to Mercator . They differ also greatly in the names , Longitudes , latitudes & distances of such places as haue bin heretofore set down not only in Mercators Globe which was made many yeres since , but also in diuers Maps more lately made , but who goeth nighest the truth I dare not iudge , because I was neuer in those places . But as touching the Map of stars which couereth the celestiall Globe of M. Molineux , I do not finde it greatly to differ from that of Mercator , sauing that M. Molineux hath added to his celestiall Globe certaine Southerne images , as the Crosse , the southerne Triangle , and certaine other stars , whereof some do signifie Noes Doue , & others do signifie the image called Polophilax , the images whereof are not here set downe , but you shal find them described in Plancius his Map , made in the yeare 1592. whose longitudes , latitudes , and declinations , how truely they are set downe in the said Globe or Mappe I am not able to iudge . He setteth downe also two cloudes nigh vnto the South pole , but not the vse thereof . Moreouer to this briefe description of M. Molineux his two Globes , I thought good to adde the first voyage of Sir Frances Drake , and of M. Thomas Candish , set foorth by two lines , the one redde , and the other blew described in the Terrestriall Globe of the said M. Molineux , and also how farre Sir Martin Furbosher sayled Northward as followeth .

The first voyage of Sir Frances Drake by sea vnto the West and East Indies both outward and homeward .

IN the great terrestriall Globe lately put foorth by M. Sanderson and by M. Molineux , the voyage aswel of Sir . Fr. Drake , as of M. Th. Candish is set downe , & shewed by helpe of two lines , the one red , & the other blew , wherof the red line procéeding first from Plymouth , doth shew what course Sir Frances obserued in all his voyage , aswell outward as homeward , and the blewe line procéeding also from Plymouth , sheweth in like maner the voyage of Master Candish , and in that Globe is also set downe how farre Sir Martin Furbosher discouered towards the North parts . But first I wil describe vnto you the voyage of Sir Fr. Drake that wrothie Knight & most Noble Neptune , according as that red line directeth in the said Globe .

First parting from Plymouth he sayled with a North Northeast winde to an Ile called Mogodore , vpon the coast of Maroccho , which place is not named in M. Molineux his Globe , and that place hauing in North Latitude 32. degrées , is distant from Plymouth according to that course , which the redde line sheweth 780. leagues . In this Ile he built a little Pinnis or shallop , and from thence he sayled to Cape Dalguere which is further Southward , and hauing in North Latitude 30. degrées , is distant from Mogodore about 40. leagues , and from thence hee sailed to the Iles Canariae , which are somewhat more Westward , and hauing 27. degrées in North Latitude , are distant from the Cape Dalguere about 100. leagues , and from thence he sayled to Capo Blanco , which is more Westernly , and hauing in North Latitude 21. degrées , is distant from the Canaries 120. leagues and somewhat more , and from thence he sayled to the Iles of Capo Verde , which hauing in North Latitude about 14. degrées , are distant from Capo Blanco about 140. leagues . And from thence to the great Cape of S. Augustine , which hauing in South Latitude about 8. deg. is distant from the Canaries 500. leagues , and from thence he sayled more Westernly vnto the mouth of the Riuer called Rio de Platta , which hauing in South Latitude 36. degrées , is distant from Cape Saint Augustine 740. leagues , and from thence to the Port Saint Iuliano , hauing in South Latitude about 50. degrées , whereas one Doughtie was executed for conspiracie , and this Port is distant from Rio de Platta 380. leagues . And from thence he sayled to the Cape Virgine Maria , which hauing in South Latitude 52. degrées 30′· is distant from Port S. Iulian 50. leagues , and from thence stréeking in betwixt the Ile , whose Northeast Cape is called the Ile of the name of Iesus , and the Port Famine , he entred into the straight Magellane , which hauing in South Latitude 53. degrées 30′· is distant from the Cape Virgine Maria 50. leagues , and from thence he passed through the Magellane straights to the Cape de Sancto spirito , which is a Cape of the South land , hauing in South Latitude 52. degrées 20′· and is distant from Cape Virgine Maria about 150. leagues , from thence he sailed somewhat Westernly about 20. leagues , & there fetching a turne about certain Ilands called Las Anegadas , he tooke his course Northward alongst the West coast of America vnto the Ile Lima , which hauing in South Latitude 12. degrées , is distant from the Ilands called Las anegadas 800. leagues , and from thence he sayled still Northward vnto Cape Guija , which hauing in South Latitude 1. degrée 30′· is distant from the Ile Lima 160. leagues , from thence still Northward he sailed to Cape S. Francesco , which hauing in North Latitude 1. degrée 30. minutes , is distant from Cape Guija 140. leagues , from thence he sayled stil Northernly to the Cape Mondecino , which is in the land called Quiuira , and this Cape hauing in North Latitude 40. degrées is distant by that course from S. Francesco 1740. leagues , from thence he sayled still Northward vnto a certaine Bay in the West part of Quiuira , which he named Noua Albion ( that is to say ) new Englande hauing in North Latitude 46. degrées . And this was the furthest part of his voyage outward , in which voyage hee sayled in all 6050. leagues , and from this Bay Sir Frances himselfe ( as I haue heard ) was of very good will to haue sailed still more Northward , hoping to find passage through the narrow sea Anian , which sea is not set downe by Master Molineux in his Terrestriall Globe as a straight , but rather as a maine Sea , bearing in bredth 400. leagues , and so from thence to haue taken his course Northeast , and so to returne by the Iles Crocklande and Groynlande into England , but his Mariners finding the coast of Noua Albion to be very cold , had no good will to sayle any further Northward , wherefore Sir Frances was faine to come backe againe Southward to Mondecino , which ( as hath béene said before ) is distant from the foresaide Bay of Noua Albion 140. leagues . From thence he sayled in a manner right Southeast to the Iles Moluccas and touched at the Iles Terenate , Tidori , Machian , and Motill , which are nigh vnto the Ile Gilolo , which is right vnder the Equinoctiall , amongest which Iles he remayned a certaine time , of which Ilands the Ile called Terenate hauing about one degrée of North Latitude , is distant by that course from the Cape Mondecino 1180. leagues , and from thence hee sayled Southwest vntill he came to the West end of the Ile Iaua maior , which hauing in South Latitude nine degrées 30. minutes , is distant from the Ile Terenate by that course 530. leagues , and from thence he sayled still Southwest to the Cape di Buona Speranza , which hauing in South Latitude 35. degrées , is distant from Iaua maior 1630. leagues , then from Capo di Buona Speranza he making his course Northwest , sailed to an Iland called Serra Liona , which is vpon the coast of Afrique , and hauing in North Latitude 7 degr. 30′· is distant from Capo di Buona Speranza 1090. leagues , then from thence he sailed towards the Iles of Capo verde vntil he came to the 12. degrée of North Latitude right vnder the first Meridian , which point is distant from the Iland Serra Liona according to that course 300. leagues , & from thence he sayled Northward nigh to the Iles called Azores on the West side therof , which hauing of North Latitude 40. degr. are distant by that course from the Iles of Capo verde 600. leagues from whence he directed his course Northeast to Plimouth , which hauing in North Latitude 51. degr. is distant by that course from the Azores 490. leagues , so as in his returne from Noua Albion to Plymouth he sayled in all 5960. leagues , which if you adde to the number of leagues of his outward voyage before set downe , which is 6050. leagues , you shall finde the totall summe of the leagues to be 12010. leagues , which is almost twice so much as the compasse of the whole world , which if you measure vpon the Globe by the Equinoctiall line containing 360. degrées , and doe allow for euery degrée thereof 60. Italian miles , you shall finde the number of such miles to amount to 21600. miles , which by allowing thrée miles to a league doe make no more but 7200. leagues . But if it might please Sir Frances to write a perfect Diarie of his whole voyage , shewing home much he sayled in a day , and what watring places he found , and where hee touched , and how long he rested in any place , and what good Ports and Hauens he found , and what anchorage good or badde , and what maner of people , what trade of liuing , and what kinde of building and gouernement they vsed , in what aire they liued , and whether the ground were fertile or barren , dry or well watered with floods and fountaines , what mountaines , what mines , what woods or forrests , what beastes , fowles , or fishes , fruites , hearbes , plants , or other commodities he founde therein , and in what manner of seas he sayled , and what windes and currents were most rife in euery place : Also what rockes , sandes , sholdes , and all other places of daunger and perill , and by what markes such places are to be shunned . And finally what Moone doth make a full sea in euery Port where he arriued , and what windes doe alter any tide or Current , and all other necessary accidents most méete for sea men to knowe . In thus doing the saide Sir Frances I say should greatly profite his countrie men , and thereby deserue immortall fame , of all which things , I doubt not but that he hath alreadie written , and will publish the same when he shall thinke most méete .

The voyage of M. Candish vnto the West and East Indies , described on the Terrestriall Globe by the blew line .

THis blew line as you sée taketh his beginning from Plymouth like as the redde line doth , whereby you may plainely sée , that M. Candish did not greatly differ in his course from that which Sir Frances helde , sauing that Master Candish hauing once passed the straight Magellane Westward , sayled not so farre Northward as Sir Frances did , for he sayled no further Northwarde but to the Port Saint Lucas , which is almost vnder the Tropique of Cancer , for that Port hath in North Latitude 22. degrées 30′· kéeping alwaies a nigher course vnto the maine , then Sir Frances did , and from the Port S. Lucas he sayled in a manner Southeast about 320. leagues , and then he directed his course Eastward towards the Moluccas , & before he came to the Moluccas , he fetched about the Ile Catana●● , which is more Northward then Sir Frances went in those partes , and from thence he sayled full South to the Moluccas , from whence he directed his course Southwest , and passed betwixt the Iles Iaua maior and Iaua minor , which two Iles in the said great Globe are made to haue one selfe Latitude ( that is to say about 10. degr. of South Latitude , counting from the Equinoctiall to the Parallel which passeth through both the Iles. But in all other Globes and Maps Iaua minor standeth 10. degrées more to the South , and is placed behinde the South promontorie called the land of Beach . Then hauing passed betwixt the foresaid two Iles , he held in a manner the like course that Sir Frances did in his returne to Plimouth , which the blew line doth shew so plainely as it néedeth none other description . Nowe as touching Sir Martin Furboshers voyage , because his owne description thereof is in print , nothing is set downe in this Globe , but only the outermost end of his voiage named here Furbushers straights , hauing in North Latitude about 63. degrées , in the mouth of which straights is a little Iland called Hales Iland on the right hand , and on the left hand another little long Iland called Lecesters point . Truely this knight for his valourous venter , aswell in this voyage as in diuers other his worthy seruices done vpon the sea deserueth great commendation , and I wish with all my heart that he and such like might be much made of , and rewarded according to their desert . And thus I leaue to speake any further of M. Molineux his Globes , the vse whereof is to be learned by those Propositions which I haue héeretofore set downe , shewing the vse of Mercator his two Globes , for the practise and manner of working both by his Globes and by these Globes is all one .

A plaine and full description of Petrus Plancius his vniuersall Map , seruing both for sea and land , and by him lately put foorth in the yeare of our Lord , 1592. In which Mappe are set downe many more places , aswell of both the Indies as of Afrique , together with their true Longitudes and Latitudes then are to be found either in Mercator his Mappe , or in any other moderne Map whatsoeuer , and this Map doth shewe what riches , power or commodities , as what kindes of beastes both wilde and tame , what plants , fruits , or mines any Region hath , and what kinds of marchandizes doe come from euery Region . Also the diuers qualities and maners of the people , and to whom they are subiect . Also who be the most Mightie and greatest Princes of the world : A Mappe meete to adorne the house of any Gentleman or Marchant that delighteth in Geographie , and therewith this Booke is also meete to be bought , for that it plainly expoundeth euery thing contained in the said Mappe , Written in our mother tongue by M. Blundeuill Anno Domini . 1594.

Imprinted at London by Iohn Windet .

IN DOCTISSIMI VEREQVE GENEROSISSIMI THO. BLVNDEVILI , IN PETRI PLANCII TABVLAM GEOGRAPHICAM ET HYDROGRAPHICAM ELVcidationes , Gualteri Hawghi 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . PLancius in tabula terras descripsit & vndas , Quaecunque & toto condidit orbe Deus : Multiplices horum partes , Regumque per illas Sceptra superba , simul totius orbis opes . Sed solis Gallis haec Plancius , atque Latinis Inter & hos , Doctis scripserat , haud alijs , Tu Generose tuis patrio haec Idiomate clarè Blundeuile refers , Cunctaque nota facis . Tu simul haec auges , lucemque hijs addis , & horum Vsum , multiplici non sine fruge doces . Nec solum doctis , sed & omnibus haec Idiotis Perspicua vt pateant , tu breuitate facis . Plancius haec alijs , tu nobis omnia tradis , Tuque tuis Anglis Plancius alter eris .
A plaine and full description of Plancius his vniuersall Mappe , set foorth in the yeare of our Lord , 1592. written in our mother tongue by M. Blundeuill .

IN this Mappe , as you sée are drawne with redde inke two lines or Diameters crossing one an other with right Angles in the verie midst of the Mappe , whereof the perpendicular Diameter passing through the Iles Azores , and also through the Iles of Capo verde , signifieth the first Meridian and Axletrée of the world , at the vpper end whereof is set the North pole , and at the neather end the South pole . And the other ouerthwart Diameter signifieth the Equinoctiall , & also the line of East & West , that is to say , East on the right hande , and West on the left hand . And in this line are set downe the degrées of Longitude , which are to be counted from the foresaid first Meridian vpon the saide Equinoctiall towards your right hande from one degrée to 180. set downe in Arithmeticall figures thus , 10. 20. 30. and so foorth vntill you come to 180. which is the East Longitude of the world , and the West Longitude beginneth on your left hand whereas is set downe 190. and then 200. and so foorth vntill you come to 360. degrées , which is the whole circuit of the Equinoctiall , and of the whole earth . And in the first Meridian are set downe the degrées of Latitude , which doe procéede from the Equinoctiall to eyther of the Poles from one degrée to 90. written in Arithmeticall figures thus , 5. 10. 15. and so foorth to 90. which degrées doe procéede with equall distances from the Equinoctiall to eyther Pole , and the like degrées of Latitude are also set downe vpon the two outermost Meridians , aswell on the right hande as on the left . And on ech side of the Equinoctiall are drawne with redde inke the two Tropiques , that is to saye , the Tropique of Cancer , and the Tropique of Capricorne , ech one being distant from the Equinoctiall 23. degrées 28′· which is the greatest declination of the sunne , and towardes ech Pole are also drawne with redde inke the two Circles both Arctique & Antarctique , whereof the circle Arctique passeth through the Northernly part of Island , hauing in Latitude 66. degrées 30′· and the Circle Antarctique passeth through the like degrées of Latitude towardes the South pole , and the distance of each of those Circles from eyther Pole is equall to the greatest declination of the sunne . And these foure Circles are Parallels to the Equinoctial , bounding the 5. Zones that is , the two cold , the two temperate , & the hotte Zone which lyeth in the midst of the world betwixt the two Tropiques . And on the left hand or West part of the Mappe are set downe from the Equinoctiall vpward towards the North pole , what number of miles and secondes of miles doe belong to euery degrée of the North Latitude , procéeding from the Equinoctiall towards the North pole : for though that euery degrée of the Equinoctiall being a great Circle containeth 60. miles , yet the further that you goe from the Equinoctiall towards either of the Poles , the lesser and lesser are your Parallel Circles in compasse , and therefore one degrée of euery such Parallel must néedes containe the fewer miles , but because the sea men doe commonly make their account on the sea by leagues , and not by miles : Plancius here right vnder the former Table of miles , setteth downe the number of leagues incident to euery degrée or Parallel , in descending from the Equinoctiall to the South pole , appointing thrée miles to a league , and therefore he setteth downe vppon the very Equinoctiall 20. leagues , which is 60. miles , and next to that he setteth downe 19. leagues 59′· and so procéedeth foorth diminishing still the quantitie of the leagues , euen till you come to the very South pole , and on the right hand of the Map hard by the outmost Meridian are set downe the nine Climes , and the longest day in euery degrée of Latitude , procéeding aswell from the Equinoctiall to the North pole , as also from the Equinoctall to the South pole , from 12. houres to 187. dayes and 7. houres , which maketh the longest day to those that dwell right vnder the North Pole to be halfe a yeare and the night asmuch . Besides the Circles and lines before mentioned , there are set downe in this Mappe certaine flyes of the Mariners Compasse , each one containing two and thirtie lines , which doe signifie the two and thirtie rombes or windes of the Mariners Compasse to knowe thereby howe one place beareth from another , and by what winde the Mariner hath to sayle to any place whereunto he would goe : In euery which fly the line of North and South may serue in stead of a Meridian , and the line of East and West may serue as a Parallel , by helpe whereof you may the more readily take with your Compasses the Longitude and Latitude of any place contained in the Mappe in such manner as is taught in my first Treatise of Vniuersall Mappes . Also in the very front of his Mappe he setteth downe the numbers of foure and twentie houres , euery houre containing fiftéene degrées of the Equinoctiall , which houres doe beginne on the right hande , and so procéede to the left , whereof the twelfth houre is placed at the ende of that Meridian which passeth through the Fortunate Iles. Nowe betwixt the 72. and 86. degrées of North Latitude hee setteth downe two long Ilandes extending from the West towardes the East somewhat beyonde the first Meridian , and from the saide Meridian more Eastwarde he setteth downe other two long Ilandes , affirming that the North Ocean sea breaking in betwixt these Ilandes with ninetéene gates or entrances , maketh foure straites and is continually carryed vnder the North Pole , and there is swallowed vp into the bowels of the earth , and he sayth further that right vnder the North pole there is a certaine blacke and most high rocke which hath in circuit thirtie and thrée leagues , which is ninetie and nine miles , and that the long Ilande next to the Pole on the West is the best and most healthfull of all the North partes . Next to the foresaid Ilandes more Southward he setteth downe the Ilandes of Crockland and Groynelande making them to haue a farre longer and more slender shape then all other mappes doe : At the East ende of the third long Ilande , is a straite hauing fiue gates or entrances , which by reason of their narrownesse and swift course of streame are neuer frozen , if this be true , I marueile howe any shippe durst enter through any of those straites to discouer the North sides of any of those Ilandes , and how and where it came out againe . Moreouer at the East ende of the last Ilande somewhat to the Southwarde , he placeth the Pole of the Lodestone which is called in Latine Magnes , euen as Mercator doth in his Mappe who supposing the first Meridian to passe through Saint Marie or Saint Machaell , which are two of the outermost Ilands of the Azores Eastwarde , placeth the Pole of the stone in the seuentie fiue degrée of Latitude , but supposing the first Meridian to passe through the Ile Coruo , which is the furthest Ile of the Azores Westwarde , hée placeth the Pole of the Lodestone in the seuentie seuen degrée of Latitude , I thought good to seuer the Ilands last before mentioned from the bodie of the Mappe as partes belonging rather to the North pole , then to Europe , Asia , Africa , or America : for if Virgill did not let to say that Englande was Penitus exclusa ab orbe , mee thinkes that I may much more rightly say the like of these Ilandes , notwithstanding Plancius maketh the first two long Ilandes , and also Groynelande and Crockelande to be part of Mexicana , which mée thinkes is not méete , sithe they be deuided by the North Sea , likewise hée maketh the lande which is vnder the South pole , and is not as yet discouered , to bee part of Magellanica , which ( in mine opinion ) ought not to bée so , sith it is not onely deuided from Magellanica , by the straight Magellanicum , but also from Afrique , and from the East Indies by the great Southerne Ocean , and I beléeue that when the South lande shall bee all discouered , it will containe twice so much lande as Magellanica doth , and then I doubt not but that the Geographers will giue it some other name , and make many diuisions thereof : In the meane time I will followe Plancius his owne diuision of the worlde , which greatly differeth from that of Mercator , and of all other moderne Geographers , for they doe deuide the whole earth but into foure partes , that is to say , Europe , Afrique , Asia , and America , but Plancius by deuiding America into thrée partes that is , into Mexicana , Peruana and Magellanica , deuideth the whole earth into sixe partes , that is to say , Europe , Asia , Afrique , and into the thrée partes of America last mentioned , according to which diuision , he describeth the earth in the french tongue in sixtéene pages , set downe at the foote of his Mappe , the foure last pages whereof doe onely containe the interpretation of the seuentie one little Tables or Inscriptions written in the Latine tongue , dispersed throughout the whole Mappe , expressing therein such thinges as hee thought most méete to bee noted in diuerse partes of the worlde , all which Tables or inscriptions , I haue héere also set downe in our mother tongue : And although that to the foresaide Tables or Inscriptions Plancius hath attributed certaine numbers for the more easie finding out of the sayde inscriptions , yet not easie inough by reason that one selfe number is set downe in diuerse Tables , and therefore to the intent that you might the more readily finde out euery Table that is proper to the matter whereof it maketh mention , I haue here following ioyned to the number of euery such Table , his proper Longitude and Latitude , which with your Compasse you may quickly finde out , and more certainely , then by his numbers , if you remember the order thereof set downe in my Treatise of vniuersall Mappes . But now I will declare the contentes of the foresaide sixtéene pages in order as followeth .

The title of the first page is thus . A briefe declaration of the diuision , forme or shape , and of the particularities of the world . The Contents of the first page .

THat the earth and the water doe make both together one rounde bodie , which the Cosmographers doe inuiron with fiue Circles , that is the Equinoctiall , the two Tropiques , and the two Polar Circles , and thereby doe deuide the worlde into fiue Zones , two colde , two temperate , and one extreme hotte : and though that the auncient Geographers doe affirme that thrée of those Zones were vnhabitable , the one for extremitie of heate , and the other two for extremitie of colde , yet within these hundred yeares last past it is knowne by good experience that those thrée Zones are well inhabited as the manifolde Countryes therein placed and greatly replenished with people of sundry languages doe well testifie , of all which thinges I haue written at large in my Spheare . And therefore I make this page the shorter .

The Contents of the second Page .

TO knowe the true situation of the Prouinces and places contained in this Mappe , it is necessarie first to knowe their Longitudes and Latitudes . The degrées of Latitude or of the eleuation of the Pole , which is all one thing , are counted from the Equinoctiall to eyther Pole , which is 90. degrées , and the degrées of Longitude are counted vpon the Equinoctiall from the Iles of Capo Verde towardes the East , and so round about the earth vntill you come to the number of 360. degrées . The Prouinces and Townes that are situated vnder one degrée of Longitude , haue at one selfe time like houres of the day , but those that are situate vnder diuerse degrées of Longitude , doe differ in number of houres , for when it is in one Towne noonetide , it is in an other Towne that is distant from thence towardes the East 30. degrées two of the clocke in the afternoone , and so consequently for euery 15. degrée of distance they differ one houre . Likewise they that dwell vnder one selfe degrée of Latitude haue equall quantitie of dayes and nightes , but yet so as they which dwell on the South side of the Equinoctiall haue the shortest day when wee haue the longest , and haue Winter when we haue Sommer . But those that are situate vnder diuerse degrées of Latitude , haue inequalitie of dayes and nights , for the nigher that any place is situate towardes any of the Poles the more houres the longest day of the yeare in that place containeth . But those that dwell right vnder the Equinoctiall haue alwayes their dayes and nightes of like quantitie , and I vnderstande here by the daye , the space betwixt the sunne rise and the sunne set , and you shall finde the quantitie of the longest day of the yeare in euery degrée of Latitude set downe in the Northeast part of this Carde . As for example to those that haue 30 degrées of latitude , the longest day is 13. houres 57′· and so the nigher that you go to the Pole the longer is the day , in so much as to those that dwell right vnder the Pole , the yeare is but a day and a night , that is to say , they haue 6. moneths day , & 6. months night . Moreouer the Geographers do diuide the earth into 9. climes for to distinguish therby the prouinces and regions by the quantity of the longest day , the middlemost parallel of euery clime increasing by halfe an houre , and you haue to consider that the degrées of latitude are in all places of like bignes euery degrée contayning 15. Almain leagues , or 60. Italian miles , but the degrées of longitude procéeding from the Equinoctiall towards any of the 2. Poles are vnequall , that is to say , euerie one containing fewer leagues or miles than other , but the degrées of the Equinoctiall it selfe are equall to the degrées of latitude , euerie one containing 15. Germaine leagues or 60. Italian miles , as you may plainly sée in the Table set downe in the Northwest part of this Map . And you haue to note that one Almaine league doth contain 4. Italian miles , and we haue described the degrées of longitude in the Southwest part of this Carde by the houres of the ships way , euery one decreasing lesse than other from the Equinoctiall to the Pole , whereby you may conceiue that two ships being right vnder the Equinoctiall 150. degrées distant one from another , and are to saile with like gate towards the North pole : when they shall come to the 60. degrée of latitude , their distance shall be no more but 75. leagues . And the further they goe towards the Pole the lesse distant they shall be one from another , in so much as when they be right vnder the Pole it self , they shall both méete , as you may sée in the 2. rounde figures contayning the description of the earth , and set downe in the 2. nether corners of the Map . This matter is to bee considered of the Mariners that they may thereby the better perceiue the imperfections of their sea Cardes . Moreouer , in the second page Plancius setteth downe the diuision of the earth as well according to the ancient as moderne Geographers , making first thrée generall continents or firme lands , whereof the first is so much as was knowne to Ptolomey and to the ancient Astronomers , as Enrope , Afrique , and Asia , the second conti●ent is called America , and the third continent is the South part of the world , not yet fullie discouered , called of Plancius Magellanica , and he diuideth the second continent called America into thrée parts , that is , Mexicana , Pervana , and Magellanica , and by adding those thrée parts to Europe , Afrique , and Asia , he diuideth the earth into sixe partes , and first he setteth downe the description of Europe together with her boundes or limittes , and then the commodities thereof , as followeth .

Of Europe .

EVrope is farre lesse than all the rest , and yet exceedeth all others in noblenesse , in magnificencie , in multitude of people , in might , puissance , and renowne , the which in times past hath commanded both Asia and Afrique as Quéene , by reason of the Monarchies of the Gréekes and of the Romans , and at this day is of great force by the power of the Turkes and Muscouites . Moreouer it commandeth many prouinces in Mexicana and Pervana by the power of the Spaniards and Portugals , and of other Christian Princes .

Europe is seuered from Asia and Afrique by the sea Mediterraneun● , and by the sea called Marmagior , and by the Marish or sea called Palus Meotis , and by the Flood Tanais and Dwina . The chiefest prouinces of Europe are these , Almanie , Italy France , Spaine , Denmarke , Norway , Swethland , Muscouia , Polonia , Hungaria , Sclavonia , and Greece . The chiefest Ilandes of Europe are these , England with Scotland , Ireland , Sardinia , Corsica , Sicilia , Candia , Nigro Ponte sometime called Euboia , and Stalimene sometime called Lemnos . And in this second page hée setteth downe also the discription of Almanie thus .

Almanie is reputed to bee the greatest prouince in all Europe , and is situated in the middest thereof , which is bounded on the East with Polonia and Hungaria , on the South with Dalmacia , and Italia , and on the West with France , and on the North with the North Sea , and with the Sea called Mare Balticum . The inhabitantes of this Countrey warre● in olde time with the Romanes for their libertie , and since manie hundred yeares past it hath holden the imperiall Scepter . About the time of Christ his birth , it was a rude countrey , as Cornelius Tacitus saith , full of Wood , bushes , and marishes , but at this day it is so adorned with great magnificent Townes and well fortified , and is furnished with such a number of Castles , and Villages , and with such a number of people , and with such pollitique gouernement , as it is to bee compared to any prouince whatsoeuer in all the worlde . The soyle thereof is verie fruitfull both for corne and Wine , and hath manie nauigable Floods stored with plentie of Fish . It hath most excellent Fountaines , and hotte bathes , greate mines of Golde , of Siluer , copper , Tinne , Lead , and Iron . The inhabitants doe exercise as well nowe as they haue done in times past the Art military , and it hath manie learned men verie skilful in all sciences , and in Mechanicall artes , they were the inuenters of Artillerie , of Gunpowder , and of the noble Arte of printing , and of making artificiall dials and horologies .

The chiefe merchandizes that are transported out of Almanie into other countreys are these , Golde , Siluer , Copper , Tinne , Leade , vitrioll , Allum , Quickesiluer , Collours of diuers sortes , Slates to couer houses , Wheat , Wine , Fish , woollen cloath , Linnen cloath , Bombasine , Fustian , Suile , Armour , all sortes of workes made of Iron , or brasse , and other merceries .

The Contentes of the third page .

IN the thirde Page hée describeth the 17. Prouinces of the lowe Countries , also Italie , France , Spaine and Denmarke , as followeth .

THe 17. Prouinces of the lowe Countries are counted a part of Almanie , by reason that the most part of the inhabitantes haue as well their originall as their language from the Almanes . But those of Artoys , of Henough Namures , and parte of the inhabitantes of Brabant , Flaunders , Lymburgh , and Liezenburgh doe speake the French tongue . These prouinces are situated partly within the ancient limits of Almany , that is to say , beyond the Orientall part of the Flood Rhenus , & partly in Gallia Belgica , as also are the prouinces of the 4. princes electors and of manie other prouinces of the Empire . In these prouinces are manie nauigable Floods and rich in fish , namelie the Rheane , Mosella , Mosa , and the Escaut , and there is great aboundance of all sorts of corne and cattell meete for mans vse , there is also a number of great Townes rich , mightie , and well peoplished , also of fortresses well fortified , faire Villages , but chieflie of braue and commodious portes and hauens , and an incredible number of shippes , and the continuall warres that they haue had and haue at this day do witnesse to all the world their great force , might , and riches . The inhabitantes in time of the Romaine Monarchie were and also are at this present greatlie renowmed for their skill in the arte militarie , and besides that , they are most excellent and industrious in all sciences and mechanicall artes , and they haue a great number of Mariners and Pilots well practized in the arte of Nauigation , and to these prouinces is attributed the inuention of the Mariners Compasse , according to the opinion of manie learned men , which trulie is one of the noblest inuentions that euer was found out since the world beganne . In the towne of Brughesse was inuented the Arte of Painting with Collours tempered with Oyle .

The Prouinces of Belgia doe sende vnto other Prouinces all sortes of Cloathes made of Woolle and Flaxe , linnen cloath of Cambria , Skarlettes interlaced with golde , Siluer , and Silke , Taffatas , Borattas , Grograines , single Buffin , Sayes of Leyden , and Howscot , Worsteds , and halfe Worsteds , Fustianapes of Vellures , and of Wool , Bayes , silk , parchment lace , Sarcenet and inkle , all manner of twisted thred silke ready drest , purified Sugar , Buffe , Shamoyes , stryped Marokines , painted Pictures , Bookes , Armour , Cables , Ropes , and other Munitions belonging to Shippes , Kniues , Pinnes , and all sortes of Mercerie or Haberdash ware , and Fish dried and salted .

ITalie being the mother of eloquence and of al Latine erudition doth extende it selfe like an arme towards the Southeast , lying betwixt the Tuscane sea and the gulfe of Venice , & is bounded on the west part with France , & on the north part with Almany being seperated from the sayd two prouinces by the Flood Varo and the Alpes , and al the rest is inuironed with the sea , at the time of the natiuitie of Christ and since , she most flourished , being adorned with the fourth Monarchie , and with the most mightie towne of Rome , which at that time was Quéene of many prouinces , of Europe , Asia , and Afrique , which citie in the time of the Emperour Vespasian , had in circuit 13. Italian miles and 200. paces , as Plinie writeth in his third booke and fift chapter , Flauius Vopiscus reciteth that this towne was inlarged by the Emperour Aurelius to 30. Italian miles , which is 10. houres of way or gate , allowing thrée mile for an houre . This Prouince hath brought foorth , as it doth at this present , inhabitants of great industrie and wit , and it contayneth many noble cities and of great renowme , as Rome , which hath bene sometime the head of the world , Venice the rich , Rauenna the ancient , Naples the gentle , Florence the beautifull , Genua the proud , and Milaine the great . In a town called Amalphe situate vpon the sea , betwixt Naples and Salerno , the Mariners Compasse was inuented , in the yeare 1300. according to the opinion of some , by one Iohn Goia , citizen of the same towne : notwithstanding Iohn Gorop Becanus doth attribute that inuention to Flaunders which séemeth the more likelie , for so much as al the Pilats & Mariners of France and Spaine and other places do name the 32. windes or rombes of the Compasse by the Belgicke names . The chiefe marchandizes that are sent out of Italie into other countries are these , Rice , Silke , Veluet , Satten , Taffatas , fine péeces of Linnen , Grograines , Rash , Stamin , Bombasins , Fustians , Feltes to make riding cloakes , plentie of rich armour , Wiar of golde and siluer , Allum , Galles , drinking glasses and looking glasses of Venice .

FRance hath bene alwayes estéemed to be the chiefest realme of all Europe , whose soyle is most fertile , and bringeth forth all kinde of Graine , and euerie other thing that is necessarie for mans sustenance , there is great store of wine , and great plentie thereof is distributed to other Prouinces nigh adioyning . The Prouince doeth abounde in oyle Oliue , and in Corrall , and in many other noble fruites . In France are many great townes well walled , as Paris , Roan , Amiens , Orliens , Tours , Nantes , Poicters , Burghes , Tholous , Lyon , Na●bona , & Marcelles . It hath 15. Archbishoprickes , and 108. Bishoprickes , and a great number of townes and villages , and 132000 parishes , it is greatlie peopled , and hath not such desertes or heathes as are in other prouinces of Europe , the French men haue bene , and are at this present renowmed in the Arte militare , and there be manie learned men in all faculties and sciences .

The chiefe Merchandizes that are caried out of France into other prouinces are these , Wheate , Rie , Beefs , Hogges Swine , and other cattell , Salt , wines , wilde Oliue , Chessnuts , Almonds , Prunes , Corrall , Diers wadde , Clothes , linnen , Canuas and Skinnes .

SPaine is inuironed round about with the Sea , sauing that on one side it is seperated from France by the mountaines Perenei . This countrey was sometime diuided into thrée Prouinces or kingdomes , that is , Taraconensis , Lusitania and Betica , but now it is subdiuided into many Realmes , that is to say , Castilia , Aragon , Portugall , Gallicea , Lyon , Navarra , Toledo , Valentia , Murcia , Granado , Cordoa , and Algarbia , the which Realmes if they had bene reduced to one bodie of a Realme , as France is , and as they be at this houre subiect to one only king and Lord , it should bee without doubt one of the most mightie and puissant kingdomes of Europe . The inhabitantes of Spaine haue bene and are at this present much renowmed in the art militare , and in feates of warre , and it hath brought forth in times past many great Clearkes , as Seneca , Quintilian , Lucan and Martial , and in our time it had Iohannes Lodovicus Viues , Iohannes Osorius , and Benedictus Arias Montanus . The prouinces of Spaine are become verie rich and mightie , by reason of their nauigation into America , Africa , Arabia , Persia , India , the Iles Moluecas and China , in which Prouinces ( China excepted ) the king of Spaine possesseth manie countries that be rich and of great power , and many townes and fortresses in a manner round about the earth .

The chiefe Merchandizes that grow in Spaine , and are carried into other Countries are these , wines , Oyles , Rice , all sorts of fruites of Spaine , Liquoras , Silke , great quantity of wolle , Lambe skinnes , Corke , Rosin , Stéele , Iron and Armour .

DEnmarke and Norway are verie great Regions , and are as large as the countries of Almany , bordering vpon Almany towardes the South , they extend towards the North to 71. degrée 30′· of North latitude , and towards the East they border vpon Swethland , and on the west and North side they are inuironed with the sea . These two Realmes are at this day vnder the gouernment of one only king , who also is lord of Island and of the Iles of Fero , Hitland , and Gothland . Iuthland was sometime the habitation of the Cimbres , who in times past made cruell warres against the Romanes .

The Merchandizes sent from the two forsaid realmes into other prouinces are these , Oxen , Barly , Malt , stockfish , tallowe , nuttes and Filberds , hydes of Oxen , and Bucke skinnes , Mastes for shippes , planckes , and the toppes of Wainscot , Soliues , and firewood to burne , pitch and tar , Sulphur and such other things .

The Contents of the 4. page , wherein he describeth Swethland . Polonia , Hungaria , Sclauonia , Greece , England , and Scotland as followeth .

SWethland is a great and mightie realme , bordering towards the East vpon Russia , and towardes the South vpon the East sea , called Mare Balticum , diuiding Swethland from Almany and Pomerania , and towards the west vpon Norway and Denmarke , and towards the North vpon Finmarke . Stockholme is the Metropolitane citie in this realme , wherein the king kéepeth his court . From this realme is transported into other prouinces these Merchandizes , that is to say , Copper , Iron , lead costlie Furres , hydes or skinnes of Elkes , of Oxen , of Buckes , of goats , tallow , tar , barly , malt , nuts and Filbirds , and such like .

The description of Muscouia which should follow next , is set downe in the third table or inscription , which standeth in the verie front of the table , written in Latine , the interpretation whereof hereafter followeth in his place .

THe kingdome of Polonia containeth Lituania , Podolia , the lesser Russia , Volhinia , Massouia , Samogitia , Prussia , and in a manner all Liuonia , which two last Prouinces did belong not long since to Almania . Polonia is bounded on the East with Muscouia and with the Tartaries , Perocopsiques , and on the South with Moldania , and Hungaria and towards the West with Almania , and towards the North with the sea Baltique and Muscouia .

The chiefe merchandizes that go out of this realme into other Prouinces are these , Wheate , Rie , and other graine , Spruse or Danske Beare , yellow Amber , Waxe honie , a certaine drinke made of Hony which we call Meade , hydes of Oxen dryed and salted , Flaxe , Hempe , Pitch , and Tarre , Ashes , Clauellees , wood Mazier , and of Cuvelier , and other such like merchandizes .

HVngarie is a verie fruitfull Realme , rich and mightie , and it is bounded on the East with Moldauia , and Valachia , and on the South with Bosnia and Croacia , & on the West with Almania , and on the North with Polonia , it hath manie nauigable riuers , wherein are great store of fish , that is to say , Danubius , Dravus , Savus , and Tibistus . The chiefe townes are these , Buda , Gran , Weissenburgh , Rab , Prezburgh , Agria , Colocza , and Belgrada . The inhabitants of this countrey are warlike and hardie , and haue bene long time heretofore a most faithfull Rampyre and Bulwarke to all Christendome , but in the end by reason of their ciuill warres , the better part of them haue bene subdued in our time , and are made most miserable slaues to the Turke .

The Merchandizes which goe out of Hungarie into other Prouinces are these , Golde , Siluer , Copper , and diuers sorts of Collours , Salt , Wine , Wheat , Beefs , and fresh fish of the riuer salted .

SClauonie is bounded on the East with Bulgaria , and Greece , and on the South side with the Gulfe of Venice , and on the West , with the North parte of Italie , and on the North side with Almanie and Hungarie . This Region containeth manie particular Prouinces as Liburnia , Croacia Bosnia , and Balmatia , the chiefe townes whereof are these , Raguza , Salona , Sabenica , and Zara. Sclauonie at this time is diuided into manie iurisdictions , for one great part thereof is subiect to the Turke , another part to the Emperour of Almany , and the rest situated vpon the sea coast is subiect to the Seniorie of Venice . At this day there is no tongue ( the Arabie tongue excepted ) that extendeth further than the Sclauonie tongue , for as it is the Vulgar tongue of Sclauonie , so is it familiar to them of Histria , Bohemia , Morauia , Sileucia , Polonia , and to the large Prouinces of the great Duke of Muscouia , Circassia , Perihoka , Georgiana , Mengrelia , Moldauia , Valachia , Bulgaria , Russia , Seruia , Albania , and to part of Hungarie , that is also familiar in the Court of the great Turke , and among his souldiers that serue in Asia and Afrique .

GReece sometime the mother of all science and erudition , is on the East , South and West side inuironed with the sea , but on the North side it is bounded with Seruia and Bulgaria , it hath in times past valiantly fought with and beaten the Monarchie of Persia for the libertie of their countrie , and finallie by Alexander the great hath triumphed ouer the same , and thereby erected the third Monarchie , by meanes whereof it came to passe that the Gréek tongue was made common throughout Asia , Syria , and Aegypt , vntill such time as the Saracens and the great Turke did corrupt and change the same . The Emperours did rule in Greece from the time of Constantine the great vnto the yeare 1542. in which yeare Mahomet the great Turke forced the towne of Constantinople , and abolished the Empire of Greece in such sort , as euer since this magnificent and strong imperiall towne of the Christians , hath bene the seat of the Emperour of Turkie , and all the countrey made slaues to the Mahometanes .

The chiefe merchandizes that come from this countrie to other Prouinces are these , Gold , Siluer , Copper , Vitrioll , diuers sorts of collours , wines , Oyle , Veluets , Damasks , Grograins , Turquesques and Wood.

ENgland together with Scotland making both but one Iland is the greatest and mightiest of all Europe . And England is enuironed on all sides with the sea , sauing on the North side , which bordereth vpon Scotland . The aire according to the situation is indifferent temperate , for though it be more Northward than Flaunders , yet it is not subiect to such hard frostes and cold winters . The soyle is verie fruitfull , bringing foorth great plenty of wheate and of other corne , it hath great plenty of fruit trées , and there be many large and faire woods , swéete fountaines , floods and riuers full of fish , and a number of good hauens , also it hath many ritch Mines , as of Golde , Siluer , Lead , Iron , and chiefly of fine Tinne , wherefore it may be worthily counted amongst the most puissant and richest Ilandes of the world . This Iland nourisheth also a great number of cattel méet for mans vse , and chiefly of shéepe , which yéeldeth fine and good wooll , in which partly consisteth the profite and riches of the countrey , in such sort , as the golden Fléece ought to haue bene sought for in this Iland , and not at Cholcos . The inhabitants most commonly are tall of stature , beautifull and white of visage , couragious and méet for the warre , also they are ingenious and studious in the Arte of nauigation , in so much as in these dayes they haue traffique into verie farre countreyes , as into Greece , Natolia , Syria , Aegypt , Barbarie , Muscouia , and into manie other prouinces . London being situated vpon the Thames is the Metropolitane and chiefe towne of this Realme , and the Staple of the trade of Merchandizes , and the Courte royall , but Cambridge and Oxford are Vniuersities .

The Merchandizes sent from England into other prouinces are these , broad Cloaths , Carsies , Stamines , Bayes , Sayes , Saffron , Tin , Leade , Wheate , Barley , Malte , Beare , red Hearing , sea Cole and wood .

SCotland is the North part of this Iland , and is likwise inuironed round about with the sea , sauing on that side with which it bordereth vpon England . This Countrey is not so fruitfull as England , notwithstanding it is sufficiently prouided of all things that is néedfull for mans nutriment , it is watred with diuers armes of the Sea , and is indued with many mountaines full of grasse , which serueth to féede their cattell . Edenburgh is the Metropolitane citie of this realme , wherein the kéepeth his court . The Scottishmen are good Souldiers , which can endure scarsitie and the iniuries of the aire , and are very desirous to win honour . The inhabitants of the South part thereof doe speake the English tongue : but those of the North , and those of the Iles Hebrides doe vse the Irish tongue , and those of the Orcades doe vse the Norway tongue .

The Merchandizes which Scotland sendeth to other countries are these , course clothes , Karsies , Stamins , Freeses , Wool , Barlie , Malt , Fish , Hydes , leaden Owre , and Smithes cole .

The contents of the fift page .

IN this page hée describeth Ireland , the Iles Azores , Corsica , Sardinia , Sicilia , Candia , nigro Ponte , Stalimene , all which Ilandes doe belong to Europe , and in the latter end of this page hée beginneth to describe Asia .

IReland is nigh vnto England and Scotland , and is very rich in meddow ground , and hath great plentie of cattell as well tame as wilde , and fish as well of the sea , as of fresh riuers , and greate quantity of foule and birdes , but it hath scarsitie of corne by reason of the great moistnesse of the ayre . This I le is frée from all venemous beastes , the inhabitantes are wilde people , great and strong , and swift in running , and by little and little they waxe euery day tamer than other , vnder the gouernement of the English men .

THe Iles of Azores are called of the Flemish Pilots & Marriners the Flemmish Iles , because those of Burghes were the first that discouered those Ilands , & albeit that at this present the inhabitants thereof are Portugales , there is yet a remnant of Flemmish families , as of the Bruines , of the Vltrickts & others . These Iles are fruitfull , and bee 9. in number , that is to say , the Ile of S. Marie , S. Michael , Tercera , Graciosa , S. George , Pico , Fayal , Flores , and Coruo .

Tercera amongst all the rest is the strongest , & bringeth foorth diers Wad . The I le of S. Michael bringeth foorth Sugar , and great aboundance of good Diers Wad .

COrsica is situate in the sea Mediterraneum , and bringeth foorth most excellent wines , rough Horses , and great hunting dogges : and this Ile is gouerned by the Genueses .

SArdinia is a verie fruitfull I le , and chiefly of Wheat , which is transported from thence into Italie and into Spaine , likewise it hath very good Wine , both red and white , and verie good Salt , it hath also certaine mines of siluer , but not of so profitable yéeld as in times past . The inhabitantes are strong , and able to indure great labour and trauell . In great townes they speake the Spanish tongue of Aragon , but in small townes they speake the vulgar tongue of the Ile .

SIcilia hath bene alwayes famous , and is called of Diodorus the Paragon of Iles , also the Gréekes and the Latines haue greatly celebrated this Ile in their writings . This Iland hath great aboundance of Wheat and of al other grain , also of wine , Sugar , waxe , Honey , Saffron , Silk , and of all things els appertaining to the vse of man . Wherefore this Ile , together with Aegypt was sometime called the Grange of the Romanes . In this Iland is the hil Aetna , which alwayes burneth , and in the sea of Sicill nigh vnto Drepano , as Plinie writeth in his 32. booke and second chapter , there groweth verie faire redde Corall , in shape like to such a trée or bush as is here figured , which while it is vnder the water is gréene and tender , but so soone as it commeth into the aire , it waxeth hard like a stone , and is red , there is found thereof also nigh vnto the sea coast of Prouince , also in Italie nigh vnto Monte Alto , and to Naples , likewise in the red sea , and in the Gulfe of Persia , and there be three sorts of Corral , that is , red , blacke , and white .

CAndia sometime called Creta , was in olde time enritched with the famous Labyrinth , and with a hundred cities , it had also a great number of good ships and expert Pilots , this Ile together with the others , as the Ile of Zante , Cephalonia , Corfue , and diuers others , be at this present gouerned by the Senate of Venice .

The Merchandizes transported out of Candia into other prouinces are these , noble wines , as Malmsey , Muskadine , Corrants graine of Scarlet , Sugar , Christall of the mountaine , Cotton , and Buckeskins .

NIgro Ponte , sometime called Euboia , is a verie fruitfull I le in Wheate , Oyle and Wine .

STalimene sometime called Lemnos , is an Ile which hath aboundance of wheate , and most excellent Wines . In this Ile they digge out in the moneth of August a certaine medicinable earth called of the Physitians Terra sigillata . There be manie other Iles besides these in Europe , as the Iles of Denmarke , the Iles of Zeland in Flanders , the Ile Frumentera , Iuica Maiorica and Minorica , and a number of Iles that are in Sclauonie and Greece .

ASia is seperated from Europe by the floods Tanais & Dwina & from Afrique by the narrow part of land , which is nigh to Aegypt , betwixt the Mediterrane sea , & the red sea . Asia far excéedeth in greatnes both Europe , Afrique , and Pervana , and also in riches , as in pearles of great price , and precious stones and spyces it excéedeth all the other countries of the world .

This region hath bene alwayes renowmed by the first and second Monarchie of the world , obtained by the Syrians & the Persians , as also it is at this day by the mightie Princes of China , and of Persia , and by the puissancie of the Tartarians .

In this part of the world man was created of God , placed in Paradice , seduced by Sathan , and redéemed by our Sauiour Iesus Christ , and in this region , were done in a maner all the histories and actes mentioned in the old Testament , and a great part of those of the newe Testament . The most celebrated prouinces of Asia are those that belong to the great Duke of Muscovia , also Tartaria and China , the rich prouince of India , as Guzarette , Corasan , Sigistan , Chirmania , Parthia , Persia , Media , Assyria , Armenia , Natolia Syria , and Arabia . The principall Iles of Asia are these , Iapan , Luconia , Mindanao , Borneo , Sumatra , Ceilan , and Cypres , for as for the Iles of Gilolo , Moluccas Banda & Celebes , they belong to that part of the world which is called Magellanica .

The most mightie Potentates of Asia are these , the king of China , the king of Persia , the great Turke , and the Emperour of Russia , otherwise called the great Duke of Muscouia , according to which Seniories all Asia is diuided into sixe partes , that is to say , the Asiaticall prouinces , belonging to the great Duke of Muscouia 1. Tartaria 2. China 3. the Indies 4. the Prouinces of the king of Persia 5. and those of the great Turk 6. And as touching the Asiatical Prouinces of the Emperour of Russia , and of the prouinces of Tartaria , we shall make mencion thereof hereafter , when wee come to translate the Tables or Inscriptions written in Latine , marked with the numbers 3. and 4.

The Contents of the 6. page , wherein he describeth China , and the plant of pepper there growing , with the shape thereof .

CHina or Sina is the thirde part of Asia , somtime called of Ptolomy Sinacum regio , which on the East side is enuironed with the sea called of the ancient Geographers Oceanns Sicicus , or the east Ocean , and on the West it is bounded with the Indies and with Brumas , & on the North with Tartaria . This countrey is for many causes estéemed to bée the most ample , the richest & most mightie Realme of all the world , for it extendeth from the 18. degrée to the 55. degrée of North latitude , and it containeth in longitude 450. leagues of Almanie , and it is diuided into 15. great prouinces or Realmes , that is , Quincii , otherwise called Paquin , Xanton , Xiancii , Sancii , Suchuan , Honao , Nanquii , Chequiam , Foquiem , Cantam , Quancii , Suinam , or Huinam , Quiecheu , Fuquam , or Hucquam , and Quiancii . This Realme is adorned with manie nauigable Floods , and full of Fish , it is verie fruitfull , and bringeth forth great aboundance of all kinde of graine and amongst the rest , of Rice , euery yeare thrée or foure times in a yeare . It hath goodly woods & forrests , wherein do kéepe a number of wilde Boares , Foxes , Hares , Conies , Sables , and Martins . The monntaines are full of grasse , seruing to féede infinite heardes or troopes of Cattell , both greate and small . There bée also manie mines of precious stones , of Golde , siluer , Copper , Stéele , and Iron , and a great number of pearles , but not verie round , and great aboundance of silke . The townes there are very great , fortified , and well peoplished , which is easilie knowne by the greatnesse of Cantan , which is one of the least Metropolitane cities of the Realme , and yet it contayneth in circuite 12. Italian miles , and 350. Geometricall paces , which is more than foure houres iourney , not reckoning the suburbs , which are verie large and full of people . The principall Metropolitane towne where the King kéepeth his court , is named Paquine , or Suntie , that is to say in their tongue , the celestiall or heauenlie citie , touching the greatnesse whereof , the Portugals and the Castilians doe write many incredible things , and according to the opinion of manie , that is the selfe same towne which Marcus Paulus Venetus calleth Quinzay , as that which hath diuers names in diuers languages . The like may be said of a towne in Flaunders , which the French men call Lile , the Fleminges Russill , and in Latine it is called Insulae . In these prouinces bée manie good Portes and Hauens vppon the Sea , and a greate number of shippes : by reason whereof the Inhabitantes are mooued to say , that amongst them , there are as manie that dwell in Shippes vpon the Sea , as bee of them that dwell in houses vppon the land , and that their King might easilie make a bridge to passe from China to the towne of Malaccha , which is distant from them 300. Almaine leagues . But aboue all there is one thing worthie of great admiration , and that is a wall which hath in length 400. Spanish leagues , which the King of China caused to be built , to defende the countrey against the inuasion of the Tartarians , of which thing if the ancient men had had any knowledge , they woulde haue counted this worke amongst the seuen woonders of the world . The inhabitantes are men of Spirite , and geuen to labour . There was also inuented by them such a kinde of writing , that euery man of what nation so euer hee were , béeing some what exercised therein might pronounce in his mother tongue , euen as it were ciphered : They inuented also certaine Charriots , wherein they might sayle by the Winde vppon plaine grounde , as they doe in shippes vppon the sea . There are also men amongst them that are well learned in all Sciences and especially in Architecture , wherein they excell all others , they are great louers of learning , and those that doe excell others therein are promoted to the most honourable estates : they haue good municipall lawes , and will suffer no Stewes , and they forbid that any man shall marrie any woman with whome he hath liued before in adulterie , and they gréeuouslie punish all offences , & do forbid idlenes as the mother of manie euils , yea they constrain blind men to get their liuing , by turning with their hands , milles made to grinde corne , or any other things , and in their warres against the Tartarians , they get the victorie more by fine pollicie and stratagems , and by multitude of people , than by prowesse or feates of Armes . The Portugals doe report that the King bringeth to the field 300/000. footmen , and 200/000. horsmen . Now as touching their religion they be Paynims and superstitious Idolaters , sauing that ther are in many places some Christians , as Marcus Paulus Venetus testifieth .

The chiefest Merchandizes transported out of China into other prouinces are these , Golde , precious stones , Pearles , Muske , Rubarbe , the medicinable root , China , Purslane , aboundance of silke , Sugar , Rice , and all sorts of graine . The plant of Pepper is sowne at the roots of other trées , but specially at the roote of that Indian trée , which is called Faufell , and at the roote of the Date trée , to the tops whereof it climeth vp , much like as Iuie doth vpon a trée , or like to that which is called in Latine Clematis in English Perwinkle , which will winde about euery hearbe that groweth nigh it , the root is but small , and his leaues thinlike vnto Citron leaues , but somewhat lesse , and sharpe pointed , gréene and biting in the taste , the graines doe grow nigh one to another , like the long grape , and are alwayes gréene vntil they be through ripe and drie , there be two kinds thereof , that is , white and blacke , but the plants of both are much like to our white and red Vines . Pepper groweth in places neere the sea side of Malacha , and in the Iles of Sunda and Cuda , situated nigh vnto the Ile Iaua maior , but the best kind of Pepper most plentifully groweth in the prouince of Malabar , betwixt the Cape Comori and the cape Canonar , but long Pepper is found in the realme of Bengala , and is another kind of plant , altogether vnlike to this .

The Contentes of the seuenth page : wherein he first describeth the red figge tree of India , and setteth downe the shape thereof , then he describeth the East Indies , and last of all hee sheweth the nature of the Elephant , whose shape he setteth downe in the same page .

THe Indian figge trée groweth round about Goa , the bodie thereof is high and great , and extending his braunches in a round forme , which like yellow or golden fillets do stoope down towards the earth , and so soone as it toucheth the earth , it bringeth foorth a newe generation of trées , which differeth nothing at all from the mother but onely in thicknesse , des Souches , and the branches of those doe bring foorth new trées in like manner , in so much as the mother with her off-spring will in short time spread as much grounde as containeth an Italian mile in circuit , the fruites are small Figges , and red as blood , as well without as within .

India tooke his name from the Flood Indus , which bordereth towards the East vpon the Realme of China , and towards the South vpon the great Ocean of India , and towardes the West vpon the sea of Arabia , and also vpon the Flood Indus , and towards the North vpon the sea Mare Euxinum , or Mar maior , and vpon Bramas .

This countrey is iudged at this day , as it hath bene long since to be the noblest and richest countrey in all the whole world , and it is diuided by the Flood Ganges into 2. partes , whereof the West part is called Indostan , or India intra Gangem , and the East part is called India extra Gangem , and it containeth many prouinces and Realms , as Cambaiar , Delli , Decan , Bisnagar , Malabar , Narsingar , Orixa , Bengala , Sanga , Mogores , Tipura , Gouros , Aua , Pegua , Aurea , Chersonesus , Sina , Camboia , and Campaa . These prouinces are watred with a number of goodly riuers , amongst the which Indus and Ganges are the most renowmed riuers of the world . Moreouer , these prouinces do abound in al things that may grow either within the earth or vpon the earth , except it be copper and lead , as Plinie affirmeth , also al manner of Plants that grow there are very great , braue and excellent good India exéedeth all other countries in precious stones and in spices , furnishing therewith almost all the worlde . It hath many rich mines of Gold , and great store of faire pearle , also greate multitude of all manner of cattell , horses onely excepted , which are brought thither out of Persia and Arabia . It is not long since that Callicute was the chiefe towne of Merchandize in India , but at this present Goa is the chiefe : there is also great trafique vsed at Dio , at Cananor , at Cochin , at Bengala , at Pegu , at Malacha , and at Sian .

Of the Elephant .

THe Elephant amongst all other foure footed beastes is the greatest saue the Dragon and the Crocodyle , he is very ingenious , in so much , as it is incredible that which the ancient men write of him , and also the moderne , which haue sought more diligently to know his nature and disposition , hée is of force incredible , and méete to drawe shippes and boats both out of the water and into the water , and to drawe artillery and ordinance , hée is also méete for the warre , his téeth that shoot out of his mouth are Iuorie , there is great number of them found in the Indies , and in Afrique , but the greatest and fittest for war , are found in the Ile of Ceiland , nigh to Calicute .

The Contents of the 8. page , in which he describeth the beast called Rhinoceros , and setteth downe his shape , and he describeth the Sinamon tree , shewing the shape of the trunke and of the leaf therof , and also the Muske Cat , with her shape , and in the latter end of the page he describeth the Realme of Persia .

THe beast called Rhinoceros , is as long and as large as the Elephant , but not so high , for his legges are shorter , hee is armed not like a Tortoise ( as Plancius saith ) for that is couered a● ouer with one shell , whereas this beast is armed with manifold strong , hard , & thick skales , whic● are yellow and spotted with purple , he hath a strong horne or bon● vpon his nose , whereof he taketh his name , and hee hath another little horne vpon his backe , and he is a great enemie by nature vnto the Elephant , he is found in the Realmes of Cambaia , and of Bengala .

The Sinamon tree .

THe Sinamon trée , is as big as the Oliue trée , the branches and griftes whereof are verie right , his leaues in collour are like to those of the Laurell trée , but in shape like to those of the Citron , his Flowers are white , and the fruites thereof are blacke and round , like a hazell Nut , the Sinamon it selfe is no other but the barke of the sayd trée , which groweth in the Prouince of Malabar , and in the Iles of Iava and Mindanao , but the best is found in the Ile of Ceiland .

The Muske Catte .

THe Muske Cat , is like in shape to a common Cat , but shée is greater than either Catte or Foxe , her muzzell is somewhat long and armed with sharpe téeth , and with harsh haire , which haires ( béeing angry ) she will set vp as a Swine doth his bristles , she is in collour like a Wolfe , but that she is spotted with blacke spottes , the nether part of her Muzzel and the haires of her beard are white , her féete are blacke , her flanckes are whitish , and doe waxe whiter and whiter towardes her bellie , and next to her genitories , shee hath a little bagge like to a bladder or purse , into the which doeth fall the precious greace or humour , which they call Ciuet and Zibeth , which Ciuet is gathered out from thence with a spoone , if shee bee in mans kéeping , but when she is abroad and at her owne libertie , her bagge being full , shee will voyd that Ciuet of her selfe , and it will yéelde such a swéete sauour , as all they that sayle by that coast may smell it a farre off , as I haue heard . These musk Cattes are brought from the Realmes of Pegu and Tarnassary .

The description of Persia .

AS those in Persia haue enioyed in times past the second Monarchie of the world , so at this present they be stil very mightie , for the king of Persia is one of the greatest Potentates in the whole world , as hée which commandeth all the great prouinces that doe border towards the East vpon the Flood Indus , and towards the South vpon the sea called Mare Caspium , and vpon the Flood Oxo , within which limittes , are comprehended all the greatest Realmes and landes , which the ancient Geographers were woont to call by these names , Assyria , Media , Susiana , Persia , Parthia , Hircania , Margiana , Bactriana , Pa●opanisa , Aria , Drangiana , Arachosia , Caramania , and part of Armenia maior , the which at this present are called by other names as you may sée in the Map .

The Persians are a hardie and warlike people , and thought to be the best ryders or horsinen in all the worlde , they haue verie hard warres with the Turks , they be of most frée and gentle nature , louers of ciuility , they make great account of learning and Sciences , they honour Nobilitie , wherein they greatly differ from the Turkes . Now as touching their religion , they be Mahometists , and yet in such sort , as both they and the Turkes doe count each one the other as Heretiques in that religion .

From the prouinces of Persia are transported into other parts of the world these Merchandizes , stones called Turquesses , very faire and excellent pearles , great quantity of silke , Veluet , Damaske armour , and a great number of most excellent horses .

The Contents of the 9. page .

IN this page he describeth and setteth downe the shape of the precious stone named Bezar , and the Dominions of the great Turk in Asia , and the citie Aden in Arabia , also hée describeth the beast called Cameleopardalis , and setteth downe his shape , also he setteth down the shape of thrée of the greatest Pyramides that are in Aegypt .

Of the stone Bezar .

THe stone Bezar or rather Pazar ( for that is his right name ) groweth in Persia in manner de Boucz , named Pazan , which are of diuers collours , but most commonlie redde .

This stone Bezar groweth in a concauitie in manner of a girdle about two handfull long and thrée inches broad , it is medicinable , and of great efficacie against all manner of poysons and venoms , and many other maladies , there is to be found of them in the entrie of Malacha and also in Pegu , but the best of them are in Persia .

Of the dominions which the great Turke hath in Asia .

THe great Turke doth possesse in Asia , Natolia , sometime called Asia minor , and almost all Armenia , Mesopotamia , called at this present Diarbech , or Diarbekir , Syria , and a great part of Arabia , the most notable Merchant townes of this countrey are these , Trapezunda , Alepp . and a porte vpon the sea called Tripoli , also Aman , Damasco , with his port Barutti and Mecha .

The Merchandizes that are sent from these Prouinces into other countries are these , great quantitie of silke , Veluet , Damask Turkie Carpets , Cotton , and grain of Skarlet .

The Citie Aden .

ADen is the chiefe Merchant towne of the vpper part of Arabia , which is gouerned by diuers Kinges , and this towne sendeth into other Prouinces of the fayrest Pearles , the true Baulme , Frankenscence , Mirrhe , and Horses .

The Beast called Camelopardalis .

THis beast is called of the Arabians , Gyraffa , but the name Camelopardalis is compounded of Camel and Pardale , which is a Leoparde , hée hath a verie long necke like vnto the Cammell , and is spotted with many spots , as is the Pardale or Leopard , he is a faire beast , and of gentle nature , as the shéepe , his head is like vnto the heade of a Hart or Stag , but greater , his hornes are small topped , and couered with haire , and are about a handfull and a halfe long , hee hath eares tongue and féete like to an Oxe , his forelegges are long and tall , and his hinder legges are short , whereby he séemeth alwaies to stand right vp , his head is somewhat higher than the Cammell , and this beast is to be found in Arabia , Aethiopia , and India .

The Pyramides .

IN Aegypt are manie Pyramides , whereof the two greatest are counted amongst the seuen woonders of the worlde , the greatest of them ( as witnesseth Peter Belon ) who most diligentlie viewed the same , is at the foote foure square , and euerie square containeth in length 324. paces , and in height 250. degrées or steppes , and euerie steppe hath in breadth 45. inches , which is thrée foote and 9. inches , but hée setteth not down what depth euerie steppe hath , which must not bée ouer déepe : for then how can anie man easilie mount vp to the toppe thereof , for hée saith it is plaine in the toppe , and so large as 50. persons may stand thereon . It is found by writing that 360/000 men wrought 20. yeares in building this Pyramides .

The second great Pyramides is somewhat lesse , and smooth on the outsides without any degrées or steppes , and the toppe thereof is sharpe pointed .

The contents of the 10. page .

IN this page hée describeth the Crocodyle , and setteth downe the shape thereof , secondlie , hee sheweth whereof the Mummie is bred . Thirdly , he describeth the Vnicorne , and setteth down his shape .

The Crocodyle .

THe Crocodyle is found in Aegypt , in the flood Nilus , and in India in the flood Ganges , and in the two Prouinces Mexicana and Pervana in manie riuers . This is a foure footed beast , which hath a horrible head , sharpe téeth , a verie small tongue , and a thicke tayle , and his skinne is hard and armed with hard scales , the nether parte of his mouth is immooueable , and the vpper part mooueable , contrarie to all beastes , hée doth deuoure both men and beastes , and doth kéepe more in the water than on the land , and that which is greatly to be woondred at , hée is ingendred of an Egge , as great as a Goose egge , and hee groweth by little and little vntill hee come to the length of 18. cubites , or as some say to 22. cubites , which maketh 33. foote .

Of Mummie .

MVmmie is made of bodies embaulmed , which they bring from Aegypt , whereas manie such embaulmed bodies were buried , about foure houres iourney beyond Cayre , whereas was sometime the great citie of Memphis , for before the natiuitie of Iesus Christ the Egyptians béeing Paynims did spare no cost to kéepe the bodies of their Parents from putrifaction , and therefore they are great palpable lyes , whereby fooles are perswaded , that the Mummie procéedeth of those bodies which do perish in the sands that be in the deserts of Arabia , as though it were possible that those bodies could bee preserued in those sands without stench or putrifaction .

Of the Vnicorne .

THe Vnicorne , as Lewes Vartiman testifieth , who saw two of them in the towne of Mecha , is of the height of a yoong horse or colt of 30. moneths old , which is two yeares and a halfe olde , hée hath the head of a Hart , and in his forehead he hath a sharpe pointed horne thrée cubites long , hée hath a long necke , and a mane hanging downe on the one side of his necke , his legges are slender , as the legges of a Goat , and his féete are clouen much like to the Goate , his hinder féete are hairy , and his haire in collour is like to a bay horse . This beast in countenance is cruell and wilde , and yet notwithstanding mixt with a certaine swéetnes or amiablenes . His horne is of a merueilous greate force and vertue against Venome and poyson . The Vnicorne is founde in Aethiopia , like as the Indian Asse is found in India , which hath likewise one onely horne in his forehead ,

The Contentes of the 11. Page , wherein hee first describeth Afrique , and then certaine fruites and spices , as Nutmegs , Mace , & cloues , & setteth downe the shape of them , then he sheweth which bee the mightiest Princes in Afrique , and thirdly he describeth Mexicana , which is the first north part of America .

AFrique being the third part of the World , is seperated from Europe and Asia by the sea Mediterraneum and the red Sea , and by the land straight which is betwixt Aegypt and Palestina . The chiefe Prouinces of Afrique are these , Aegypt , Barbarie , Biledulgarid , Sarra , Aethiopia , Nubia , the large Prouinces of the Abassines , falsly called the lande of Prester Iohn , and also Monomotopa . The most renowmed Iles belonging to Afrique are these , Socotora , Madagascar , S. Thomas , the Iles of Capo Verde , and the Iles of Canarie and Madera .

The Nutmegge tree .

THe Nutmeg trée groweth in the Ile of Bada , and differeth not much from the Peach trée , sauing that the leaues of the Peach trée are shorter and rounder : The fruit is couered with a thicke bark or huske , which when it is ripe cleaueth in sunder , and sheweth the Nut together with the shell , which is couered with Mace , the which at the first viewe is as red as Skarlet , and pleasant to behold , but when the Nut waxeth drie , the Mace do seuer from the Nut , and loosing by little and little their Skarlet collour , do waxe nigh vnto the collour of an Orange .

Of the Cloue tree .

THe Cloue trée groweth in the Iles of Moluccas , which in greatnes and shape is like vnto the Lawrell trée , sauing that the leafe thereof is somewhat narrower . It hath many branches , and a great number of flowers , first white , afterward gréene , and then red , but being dried , they become black . The cloues do grow vpon the outermost ends of the branches , one hard by another and whilest the flowers are gréene they excel all other flowers in swéet odour .

The chiefest Princes of Africa

THe most puissant Princes of Afrique are these , the Emperour of the Moores or Ethiopians , which of the Arabians and of the Mahometistes is called Aticlabassi , and of his owne subiects , he is called Acegue and Neguz of the Abassines , that is to say , Emperor and King of the Abassines and Mores . Then the king of Monomotapa , the king of Morocho , the king of Fez and Sus. The great Turke also possesseth many prouinces in Afrique .

The chiefest Merchandizes that come from Afrique into Europe are these , Gold , Iuory , wood of Ebony , Aloes , Baulme of Aegypt , Mummie , Mirrhe , Anil feathers , Sugar , Ginger , Dates and wines of Madera , and of the Ile of Canarie .

Mexicana .

MExicana which is the fourth part of the world , is on all sides enuironed with the sea , sauing that nigh vnto Nombre de Dios it is ioyned by a land strait to Peruana . The chiefe prouinces of Mexicana are these , the prouince of Mexico , otherwise called nova Hispania , terra Florida , Norum Bega , nova Francia , Estotiland , Saguenay , Chilaga , Toconteae , Marata , California , Tolm , Quivira , Agama , and Anian . The chiefest Iles lying on the North & Northeast part of Mexicana are these , Groynland , Crockland , Island , Freezland , Bacalaos , and Cuba .

The cheife Merchandizes that come from Mexicana into Europe are these , Gold , Siluer , Pearles , Cochenilles , to die with , Baulme , Salsaparillia , the root Mechoicana , Brimston , hydes of Oxen and Molue .

The contents of the 12. page .

IN this Page he first describeth the beast called in that tongue Aiotochli , in Spanish Armadillio . Then he describeth the 2. Prouinces Peruana , and Magellanica , then hee sheweth which hee the most mightie Princes of the world , and finally the diuers qualities of the people inhabiting the world .

THe beast Armadillio is found in the Realme of Mexico , and he is no bigger than a cat , hee is headed like a Swine , and hath the féet of a Herison , and a long tayle , he is armed with scales , whereof he taketh his name , he keepeth for the most part within the ground , and as some suppose , doth liue by the earth , by reason that he is neuer seene to eat abroad out of his den , the bones of his tayle are medicinable , and do remedie the paine and deaffnes of the eares .

Though Plancius saith that this beast is armed with scales , yet my countryman William Greenway , who is a proper seruiter both by sea and land , and hath bene in the West Indies , and hath eaten of this beast , affirmeth his flesh to be white & verie delicate , and that he hath no scales , but that his skin is white and smooth like to a pig new scalded , and that sometime hée will shrinke vp the skin vpon his backe into diuers plates , and specially towards his fore partes and hinder partes , in such sort , as hee will make them almost to méete , and the former plates do hang downe vpon his shoulders like vnto two Poldrons , and his haire is white and short , growing thin here and there one , and he is eared and tailed like a rat , euen as he is here portraide , sauing that he is throughout of one selfe collour , and without scales .

Peruana .

PErvana being the fift part of the world , is also enuironed on all sides with the sea , saue wheras the foresaid landstrait doth ioyne the same to Mexicana , and the chiefe prouinces which it containeth are these , Brasilia , Tisnada , Caribana , Carthagena , Peru , Charchas , Chili , Chica , & the land of the Patagones ▪ The most renowmed Iles are these , Hispaniola , otherwise called S. Domingo , Boriquen , & Margarita , which is the Ile of pearls .

The Merchandizes which are transported out of Pervana into Europe are these , gold , siluer , Emeralds , Pearls , the medicinable stone called Bezoar , Baulme , Ginger , Sugar , wood of Brasill , wood of Guaicum , called Lignum vitae , long Pepper , Pepper of Brasill , Cassia solutiva and hides of Oxen.

Magellanica .

THis is the sixt part of the worlde , which as yet is but little knowne , in such sort as we cannot write any thing touching the prouinces of the same , notwithstanding it is thought the prouince of Beach is verie rich , and hath aboundance of Golde , the chiefe Iles of Magellanica are these , Iava maior , and Iava minor , Timor , Banda , the Molucques , Romeros , the Iles of Salomon .

From the Ile Timor doth come into Europe , the white and pale medicinable simple called Sandalum .

From the Iles Banda doth come Nutmegs and Maces .

And from the Iles Molucques Cloues .

Which be the great Princes of the World ?

THe most mightie Princes of the world are these fiue , that is , the King of China , otherwise called the great Cham 1. the king of Persia 2. the great Turke 3. the Emperour of Aethiopia 4. the Emperor of Russia , otherwise called the great Duke of Muscovia 5. amongst which the king of China is a Pagan or Heathen : and the great Turke and the king of Persia are Mahometists : but the Emperour of Aethiopia and the great Duke of Muscovia do make profession of the Christian religion . Now as touching the king of Spaine , his puissance should bee much greater than it is , if his prouinces were not so seperated , and so farre distant on from another .

The qualities of diuers people in the World .

AS touching the qualities of peoples , though God almighty hath created al men of one selfe blood , and that all do take their beginning from the Arke of Noah , and that all men be of one selfe qualitie and shape of body , yet they differ in greatnesse , in proportion of members , and in collour : for the Patagones doe excéede all other creatures in greatnes . Againe , the men of China haue most commonly broad faces , little cies , flatte noses , and little beards , and those that haue smallest féete , are counted amongst them to be most beautifull , those of Africa haue grosser and thicker lippes than other people , the inhabitants of Agysimba , and of Guinea , and speciallie of the lands that be nigh vnto Cape de bona esperanza , are blacke , from whome the Orientall Indians do not much differ .

The Abassines or Moores of Aegypt , be of a duskish collour lyke to the Oliffe , the Inhabitants of Barbarie , be called white Moores , and those that dwell betwixt them and the Nigrites or blacke Moores be of a yellowish collour , the Spaniardes haue not found eyther in Mexicana , or in Pervana any Nigrites or Blackmoores , but onely in certaine villages nigh vnto Carque , the other nations vnder the hotte Zone , bee of collour browne bay , lyke a chessenut , and the nigher that they dwell to eyther of the Poles Arctique or Antarctique , the whiter most commonly they be , and as touching the rest all are like in qualities , shape , and fashion of bodie , as hath bene said before , wherfore they are méere lies that are woont to be told of the Pigmeans , in that they should bee but a foote and a halfe high , and likewise that which hath bene spoken of people , that shuld haue their heads their noses , their mouthes , and their eies in their breastes , or of those that are headed lyke a dog , or of those that haue but one eie , and that in their forehead , or of those that haue but one foote and that so great , as that it couereth and shadoweth all their bodie , or of those that haue greate eares hanging downe to the ground . All these are méere lyes , inuented by vaine men to bring fooles into admiration , for monsters are as well borne in Europe , as in other partes of the world .

NOwe in the foure pages following , hee setteth downe an interpretation of the Latine inscriptions dispersed throughout the Mappe , euerie which inscription hath his number added , which I doe also heare set downe in the same order , ioyning to euery number his longitude and latitude , to the intent you may the more easilie find out the said inscriptions , of which inscriptions there be in all 71. diuided into fiue partes , whereof the first parte conteineth 21. inscriptions , belonging partlie to Europe , but most to Asia , the second part containeth 12. inscriptions belonging to Afrique , the thirde part contayneth 11. inscriptions , belonging to Mexicana , the fourth parte contayneth 6. inscriptions belonging to Peruana , and the fift part containeth 21 inscriptions , belonging to Magellanica .

The 21. inscriptions belonging to Europe , but most to Asia .

The number of inscrip . Longitude Latitude North or South . 1 5. 85 . 30′· North. THis arme of the sea doth take his course through 3. places running continually towards the North pole , and is frozen 3. moneths of the yeare , and it containeth in breadth about 37. leagues . 2 38 0′· 86 . 30′· North. It is said that this Countrey is inhabited of Owarttes called in Latin Pigmei , being in height foure foot as those be of Groinland , which are called Serelinges . 3 81. 0′· 77. 0′· North. Muscouia is bounded on the North side with the sea Puzorique , called of the ancient Geographers Mare Glaciale , that is , the frozen sea , and towardes the East it bordereth vpon Tartaria , and towards the South vpon the sea called Mare Caspium and also vpon the Turkes , and Tartaries Perocopsiques , and towardes the West , it bordereth vpon Lituania , Livonia , and vpon the Realme of Swethland : as touching their religion they obserue the Faith and ceremonies of the Gréeke Church , all their Bishops called in their language Vladiques , and also their Metropolitane are vnder the obedience of the Patriarch of Constantinople . The people is wise and subtill , and yet louing seruitude more than libertie or fréedome , and they all confesse themselues to be the sernants and slaues of the great Duke , hauing seldome or neuer peace , for either they haue warres with those of Lituania , or with those of Livonia , or with those of Swethland , or with the Tartarians , or if they haue no warres , then they lie in garrison nigh vnto the 2. floods Tanais and Dwina , to defend their bounds from the depredations and inuasions of the Tartaries , they sharply punish robbers and stealers , and yet priuie theft and murder is seldome punished with death : their siluer money is not rounde , but hath the forme of an egge , their countrey is euery where ful of woods , and they haue great aboundance of rich furres , they send into all countries of Europe very good Flaxe and hempe to make Cables and ropes , and a great number of hides , as well of Oxen as of Elkes , great balles of Waxe , much salted Fish , and Whales grease . The great Duke , called of his seruantes the Emperour of Russia , is rightly accounted amongst the most mightie Monarches of the world . 4 98. 0′· 82. 0′· North. This arme of the sea hath 5. mouthes or entries , and by reason son that it is so straight and hath so violent a course , it is neuer frozen . 5. 123. 30′· 58. 0′· North. Vnder the name of Tartarie at this day are comprehended all the prouinces that border towards the East vpon the sea of China & towards the South , are limitted with the prouinces of China , of India , with the flood Oxio , with the sea Mare Caspium , & with the lake or marish called Palus Meotis , & towards the West they are bounded with the flood Boristenes , & with the limits of Muscouia : for the Tartarians haue conquered all the countries which they possesse at this present , so as Tartary comprehendeth all that countrey which the ancient men were woont to call Sarmasia , Asiatica , and also the Scythias , that is , intra Imaum and extra Imaum , they began first to bee renowmed in Europe , in the yeare of Christ 1212. The Tartarians are diuided into certaine commonalties , and Colonies , called of them Hordes , but for so much as they dwell in diuers prouinces , that do extend far , and be farre distant one from another , they differ also in their manners and trade of life , they be men of a square stature , hauing broade and grosse faces , their eies hollow sunke into their heads , and looking somewhat a squint , and thick beards , they be strong of bodie and hardy , they eat horses and all other beastes howe so euer they are slaine , sauing hogges , from which they abstaine , they are able to endure hunger , thirst , watch , and all discommodities , and when they are distressed in their voyages with hunger and thirst , they let their horses blood , and with that blood quench their hunger and thirst , which kind of meat they cal in their language Besermannen , they call theyr Emperour Cham , that is to say , Prince , and therefore Cambalu is interpreted to be the seat or towne of the prince . 6 125. 40′· 50. 0′· North. This round lake in the prouince of Sancii , tooke his first originall and beginning in the yeare 1557. by reason of an inundation or flood , which carried away 7. townes , besides villages and other places nigh adioyning , & a great multitude of people , wherof none were saued but onely one infant , sitting vpon a trée . 7 124 0′· 37. 0′· North. The inhabitantes of China , are of good spirit and ingenious , insomuch as they haue inuented certaine kinde of carts , wherein they may sayle vpon euen ground , hauing wind , and sayle as they doe in shippes vpon the sea . 8 171. 30′· 80. 30′· North.

Plancius in this inscription setteth downe the opinion of Mercator , touching the beginning of longitude , and touching the Adamant stone , otherwise called the Lodestone , in Latine Magnes .

Frances of Diep , a most skilfull Pilot , doth witnesse that the néedle of the Mariners compasse , doth turn directlie to the North Pole , being in the Ilands of Capo Verde , that is to say , the Ile of Sal , the Ile of Bonauista , and the Ile of Mayo , whereunto those doe agrée very nigh , which do say that the néedle doth the like in the Iles of Tercera , and of S. Marie , which are part of the Flemmish Iles , otherwise called the Azores : but some others do affirme , that the néedle sheweth the North pole best , being in the Ile of Corvo , which is the furthest Ile westward of the said Azores , and because the longitude of places by most liuely reasons , ought to take his beginning from the common Meridian of the world , and from the rocke or Pole of the Adamant stone , we here following the opinion of those that are most skilfull in this matter , haue set downe the first Meridian betwixt the Iles of Capo Verde , and the Azores , and because the néedle in al other places , declineth either more or lesse from the Pole of the world , ther must néeds be another Pole in some one place wherunto the néedle doth incline from all coastes of the worlde , and I haue found by the declination of the néedle obserued at Ratisbone , otherwise called Regensberg , that is the same place , which I haue set down in the Mappe , and I haue likewise marked in this Carde , the situation of the Pole of the stone , in respect of the Ile Corvo , to the intent that according to the outermost places limitted by the first Meridian , the outermost boundes , betwixt which , this Pole ought to bee found , might be knowne , vntill the diligent and curious consideration of the Pilots shall bring vs somthing of more certaintie . Thus farre Mercator .

9 162. 0′· 71. 0′· North. These are the plaine fieldes of Bargu , whereof the inhabitants are called Mecriti . 10 154. 30′· 62. 0′· North. The Mount Askai , in which are to bée séene the sepulchers of the Kings of Tartary . 11 171. 30′· 70. 0′· North. Vpon this Mountaine are set by the Tartarians two trumpetters of brasse , for a perpetuall memorie of their freedome gotten . 12. 175. 0′· 67. 0′· North. The Prouince Vng , which of our men is called Gogg . 13 170 . 30′· 65. 20′· North. The Realme of Tendus , which in the time of Marcus Paulus Venetus , which was in the yeare 1290. was gouerned by those Christians , which descended from the king Vncham . 14 165. 30′· 65. 0′· North. The Castell of King Vncham was builded in this place , against the inuasion of the Tartarians . 15 157. 0′· 65. 30′· North. The Prouince Mongul , which of our men is called Magogg . 16 159. 20′· 66. 0′· North. The desert Belgian , which is very great , all sandie and barraine . 17 159. 0′· 61. 30′· North. In this Countrie of Cergutha , is found the best sort of muske , which groweth like as an impostume or bagge , nigh vnto the nauell of a certaine beast . 18 165. 30′· 58. 0′· North. Campion is the Metropolitane Towne of Tanguth , wheras the inhabitants are partly Christians , partly Idolaters , and partly Mahometists . 19 158 30′· 53 0′· North. This wall hath in length 400. Spanish leagues , and was built betwixt the mountaines by the King of China , against the inuasions and excursions of the Tartarians . 20 176. 0′· 40. 0′· North. Iapan hath thrée Ilands much renowmed , separated one from another by a straite of the sea , whereof the first and greatest is deuided into 53. Prouinces or Realmes , of which Meaco is the Metropolitane citie , the second is called Xima , which hath nine Prouinces or Realmes , the third is called Xicoca which containeth but foure Prouinces or Realmes . 21 71 0′· 29. 0′· North. Medina Talnabi , is the towne wherein is to be séene the Sepulchre of Mahomet .

The 12. Inscriptions belonging to Afrique .

The number of inscrip . Longitude Latitude North or South . 1 67. 0′· 26. 0′· North. Coptos is a trim Marchant Towne , whereto are brought the Marchandises of India , and of Arabia , in which towne are dwelling many Christians . 2 58. 0′· 21. 0′· North. The flood Nilus , which by his inundations doth yearely water and fatte the Countrie of Egypt , and maketh it meruailous fruitefull . 3 55 0′· 17 0′· North. The flood Nubia , taketh his originall from the lake Nuba , as Ptolomie sayth . 4 50. 0′· 15. 0′· North. The flood Niger , here taketh his course , running vnder the ground 50. leagues . 5 62 0′· 5. 0′· North. In this place is the ample iurisdiction of the Emperour of Aethiopia , wrongfully called of those of Europe , the lande of Prester Iohn , the Arabs and the Moores doe call him Aticlabassi , but his owne subiects doe call him Acegue , and Neguz , that is asmuch to say , as Emperour and King . 6 58 30′· 2. 30′· North. The mount Amara , whereas are most carefully kept with continuall watch and ward of souldiers , the children and Grand-children of the Emperours of Aethiopia . 7 53. 0′· 7. 0′· South . It is said that this countrie is inhabited of Amazones , which are women that make warre . 8 55. 40′· 15. 30′· South . In this place the King of Monomotopa , hath his great and ample iurisdictions . 9 42. 30′· 15. 40′· South . This South part of Afrique , vnknowne to the auncient writers , is called by those of Persia and of Arabia , Zanzibar . 10 45. 30′· 1. 0′· South . Heere is digged out of the Mines , great aboundance of golde . 11 30. 20′· 20. 0′· North. Libia or inwarde Afrique , is at this day called Sarra , or the deserts . 12 19. 0′· 15. 30′· North. The people Azanagi , are of colour blacke graye , and they couer their mouthes as a member of shame , and doe neuer vncouer them but when they eate .

The 11. Inscriptions belonging to Mexicana .

The number of inscrip . Longitude . Latitude North or South . 1 301. 30′· 81. 0′· North. This I le is thought to be the best and most holesome of all the North parts . 2 301. 0′· 74. 0′· North. The I le of Crockland , the inhabitants whereof say that they had their originall from Swethland . 3 311. 0′· 61. 30′· North. In the yeare of our Lorde , 1500. one Gaspar Corteriale a Portingale , entred into these Regions , hoping to haue found some passage on the North towardes the Iles Molucques , but arriuing at the Riuer , which by meanes of the abundance of snowe there falling , is called Rio Neuado , which hath in North Latitude 62. degrées , he did leaue to sayle any further towardes the North , by reason of the great colde there , and turning to the South did fetch in all the sea coastes vntill hee came to Capo Razo , which hath in North Latitude about 48. degrées : and in the yeare 1504. the Britaines were the first that discouered all the sea costes of new France in America , nigh vnto the Golfe of Saint Laurence , which hath in North Latitude about 50. degrées and in the yeare 1524. Iohn Verazzan a Florentine , did part from the Port of Diep , the 17. of March , in the behalfe of Frances king of France , and sayled towardes the South sea coast of newe France , whereas he arriued at the 34. degrée of North Latitude , and from thence sayled towardes the East , viewing all the Sea costs vntill he came to Cape Britaine , which hath in North Latitude about 46. degrées . 3 311. 0′· 61. 30′· North. And in the yeare 1534. new France was againe visited by the Admirall Iaques Cartier , and in the yeare next following , it was conquered to the vse of the King of France , also in the yeare 1577. Martin Furbosher Englishman , arriued at the north straite which is betwixt Groinland & Estotiland , that place hauing by this Map in the North latitude 65. deg. séeking passage by the North vnto Cathay , wheras he found certain Iles , & a mine of gold , wherwith hauing loden his ships , he returned into England with great hope of profite , but his successe was not answerable to his hope . 4 275. 20′· 70. 0′· North. This daungerous beast is called Sucaratha , which being chased of Hunters , doth take her young ones vpon her backe , to saue both her selfe and them by flight . 5 271. 0′· 64. 30′· North. This is a great lake or sea of fresh water , the limits whereof are vnknowne as they of Canada doe say , and as they haue heard by relation of those of Saguenay . 6. 293. 30′· 52. 0′· North. Alongst this Riuer a man may saile very commodiously , towardes the countrie of Saguenay . 7 ▪ 287. 0′· 47. 40′· North. All those that dwell betwixt Terra Florida , & Terra de Laborador be called by one common name Canadois , but there be many diuers Nations as those of Hochelada , Honqueda , & Corterealia , they are all very courteous to strangers , they liue commonly by fish , & they are clothed with skins of wild beasts , as they be also that dwell further towards the North , this countrie is also called new France , because the Britaines which are French men did first discouer it in the yeere of our Lorde 1504. the conquest wherof was atchieued by the Admirall Iaques Cartier in the yeare of our lord 1535. to the behoofe & vse of the French king . In the mountaines towardes the South , doe dwell many and diuers Nations , which be cruell people , liuing without any law , & do séeke by continuall warre , to vexe and oppresse one an other , as the people Auanares , Albardi , Calicuares , Tag●●es , Apalcheni , Mocose , Capaschi , Chilage , and many others , amongst which there be some of such agilitie and swiftnesse , as they may contend with horses , who can run fastest , they eate as those of Florida , certaine kind of Spiders , Ants or Pismires , Leazards , Adders & other venemous beasts . The land of Baccalaos , is so called of the fish Baccalaos , which is there taken . Terra Florida is so called of Easter day , because Iohn Ponce of Lion did discouer it on Easter day , in the yeare 1512. the which day is called of the Spaniards Pasqua du flores : this is a fertile countrie and rich in gold . 8 271. 0′· 39. 0′· North. Marcus Nizza testifieth that this Prouince called the seuen Cities , is a very good countrey with whom Frances Vasker doth not agrée , for he saith they be places of no value , and like to Villages , and be vnder the iurisdiction of Ceuola , which at this present is called of the Spaniards Noua Granada . 9 211. 40′· 82. 0′· North.

The North Ocean sea , entring with 19. mouthes betwixt the Iles of the North , doth make foure straites of the sea , and running floods , which continually take their course towards the North , & there are swallowed into the bowels of the earth , like to springs of fountaines and floods .

In mine opinion this Inscription would haue béene one of the first of Mexicana , and not one of the last , sith it is no small leape , to turne so sodenly from the middest of America to the North pole .

10. 223. 0′· 55. 40′· North. This Countrie is desert and plaine , in which are many wilde Horses , and Oxen with high backes like Camels , and wild shéepe like vnto those which Boetius writeth ( in his description of Scotland ) to be found in one of the Iles of the Hebrides . 11 240. 20′· 40. 30′· North. New Spaine was brought by force of armes vnder the obedience of the Spaniard , in the yeare 1518. by their generall , Ferdinando Cortes who conquered the same with great losse of his souldiours , but with greater ruine to the inhabitantes , who fought for the libertie of their Countrie , the soile is very fertill , and the Countrie is rich in golde and siluer , for in the floodes are found great sands of gold , and in the mountaines is drawne out of the mines great quantitie of siluer , and alongest the sea side they take to their great profite an infinite number of Oysters , wherein are found very faire pearles . In this Prouince there are many sault lakes , the water whereof by force of the sunne is turned into sault : there groweth also great abundance of Cassia fistula , and an other kinde of fruite , which the inhabitants call in their tongue Cacao , it is like to an Almonde , they haue it in great price , for of it they make a certaine drinke , which they loue meruailous well . The sea and floods which doe wash this Prouince , doe furnish them with great plentie of fishe , and in those floodes are many Crocodiles , whose flesh the inhabitants doe eate , and this beast will grow to be twenty foote long and aboue . This Countrie is full of great mountaines and high rocks , there is great diuersitie of languages , in so much as one vnderstandeth not an other without an interpretor . Mexico is the Metropolitane and Royall towne , or rather the Quéene of all the principall Townes in the worlde , it is situated vpon the side of a lake or marish , yea the very foundation of it is a very marish , in such sort , as you can neither enter into it , nor come out of it , but by bridges , and it is aswell peoplished with inhabitants and Merchants , as any renowmed Merchant towne in Europe , the towne is very great , for it containeth in circuit 3. Spanish leagues .

The 6. Inscriptions belonging to Peruana .

The number of inscrip . Longitude . Latitude . North or South . 1 333. 0′· 3. 30′· South . This great flood Maragnon , is called of some Oreigliana , and also the flood of the Amazones , it was discouered by Vincent Iohn Pinsonio , in the yeare 1499. and was sayled in a manner from the head spring euen vnto the sea , by Francisco Oreigiano in the yeare 1542. the which voyage he perfourmed in eight monethes , hauing sayled 1660. leagues . This riuer kéepeth his water still fresh after it is entred certaine leagues into the sea , by reason that his course is so swift and violent . 2 330. 30′ 22. 0′· South .

Peruana is the South part of America , and is deuided by the Spaniardes into fiue goodly Prouinces , that is to saye , Castiliador , that is to say , the golden Castilia , Pompaiana , Peru , Chila , and Bresille : The Prouince Peru ( before the arriuall of the Spaniardes ) did extende a great deale further , when as their Countrie was yet vnder the gouernement of their naturall King , which then was called Ingas , as Giraua and others doe write . At this present it is limited on the North with the Towne Quito , and towardes the South with the Towne of Plata , that is to saye , the siluer Towne , and it was called Peruana as some write , of the flood and Port of the sea called Peru. It is at this present deuided ( according to the situation of the Countrie ) into thrée partes , that is to say , Sie ras , Andes , and the flat or plaine countrie .

The playne Countrie is that which lyeth alongest néere to the Sea-coast .

Sierras is that part which is full of mountaines .

And Andes is that part of the Mountaines which tendeth towardes the East . Of all the Prouinces in the worlde this is the richest in golde , and in Emeraldes , the Metropolitane Towne of this Prouince , is the Citie Lyma , otherwise called the Towne of ●inges . Castillelledor tooke his name of the great abundance of golde that is there . The Prouince Popaiana tooke his name of the renowmed Towne Popaian , Chila is a colde countrie by reason that it is so nigh vnto the South pole . The Prouince Brasilia tooke his name of the woode called Brasill , which groweth there in great abundance . To these Prouinces were méete to be ioyned these other Prouinces , that is to say , Caribana , Charcas , Chica , and the lande of the Patagones .

3 329. 0′· 28. 0′· South . This beast is called of some Haute , but of a certaine people of Brasill , it is called Haye , which beast was neuer séene to eate or drinke , as some write , and therefore some thinke that she liueth without meate or drinke , onely by the aire . 4 328. 0′· 35. 20′· South . This kind of beast is found in the Prouince of Parias , which in his fore parts is like to a Foxe , and behinde he is like to an Ape , sauing that he is footed like a man , and hee hath the eares of an Owle , and vnder her ordinary belly , she hath also another belly , which openeth and shutteth , wherein she lodgeth her young ones , vntill they are able to get their owne liuing , and she neuer suffreth them to goe out but onely to sucke , Gesner calleth this beast an Ape-foxe , or a Foxe-ape . 5 325. 30′· 36. 30′· South . This is the land of the Patagones , the inhabitants whereof are Giants nine or ten foote high , which doe paint their visages with diuerse colours , made of herbes . 6 335. 0′· 51. 30′· South . In the yeare 1582. the king of Spaine commaunded here to be built certaine fortresses , at the entry or mouth of the straight called Mare Magellanicum .

The 21. inscriptions belonging to Magellanica .

1 188. 30′· 17. 20. South . This land is new Guinea , so called of the Nauigants and Pilots , because the sea coast and situation thereof , is like to that of Guinea in Afrique , and it was called by Andrew Corsali the Florentine , the land of Piccinacoli , and perhaps it is that which Ptolomy calleth the Ile Labadia , if it may be called an Ile , for it is not yet knowne whether it be any part of the South firme lande or no. 2 209. 0′· 31. 30′· South . It is not vnknowne to all those that are exercised in Geography , that the degrées of Longitude do diminish and decrease from the Equinoctiall to either of the Poles , eyther North or South , whereby it falleth out , that the Prouinces which are next vnto the two Poles of the world , do differ greatly from that naturall shape which they haue by the roundnes of the earth , and for that cause we haue briefely drawne a description of the whole world in two round figures or Circles at the end of this Mappe , to the intent that euery man might sée their naturall situation so farre as may be shewed in Plano , that is to say , in flat forme . 3 231. 72. North. These Prouinces at this present are little knowne , yet it is sayd , that they are full of many kindes of wilde beastes . You shall finde this inscription in the round figure on the left hande nigh to the North pole , which indéede belongeth to Mexicana , and not to Magellanica . 4 247. 20′· 14. 0′· South . These two infortunate Ilandes were so called by Magellane himselfe , because he could find in them neither men , nor any thing else that was méete for mans sustenance . 5 283. 0′ 61. 20′· South . These Prouinces were discouered by a Spaniarde , who being separated by tempest of sea , from the fléete or armie , ranne wandring here and there , through this great Southerne sea . 6 350. 40′· 85. 20′· South . In the yeare 1493. when as the desire to saile into farre countryes increased more and more amongest the Castilians and Portugales , and that with great contention who should discouer most , Pope Alexander ordeined , that the Meridian which is 100. leagues distant towards the West , from euery one of the Ilands aswell of Capo Verde , as of those which they call Azores , should be the boundes and limits to eyther partie , for their nauigation , determining their rightes in such sort , as the Castilians should haue the West part of the worlde , to finde out vnknowne countries , and the Portingales the East parte , but there was such strife and contention betwixt them afterwarde touching the boundes of Nauigation , as this ordinance of Pope Alexander , pleased neither partie , and therefore in the yeare 1524. it was fully determined , that the Meridian which is distant 370. leagues towards the West from the Ile Saint Anthonio , being the most Westerne I le of all the Iles of Capo Verde , should by the common bound of Nauigation to both parties . 7 19 0′· 54. 30′· South . Here vnder the Latitude 42. degrées , & distant 450. leagues , from the Cape de bona speranza , and also 600. leagues distant from the Cape Saint Augustine , was found the Promontorie of the South lande , as Martin Ferdinando Denciso hath noted in the Epitome of some of his Geographie . 8 22. 0′· 65. 0′· South . In this our Geography , we haue in such manner described the circuit of the whole earth , as all the Countries are situated vnder their proper Meridians , which could not haue béene done , without extending them more or lesse from the West towardes the East , notwithstanding béeing desirous to satisfie those that are practised in the arte of Nauigation , wee haue described the North Prouinces of Europe in this our Geographie , in such sort as their situation wholly agréeth with the perticular Mariners Cardes . 9 57. 20′· 54. 0. South .

The 9. Inscription is intituled to the louers of Geographie .

We haue in this Inscription of the whole earth , employed all diligence to describe all the Seas and Prouinces , in such sort as euery place may haue his true Longitude and Latitude , in which matter , we haue spared neither labour nor cost , for we haue diligently conferred together the Sea Cardes , as well of the Castilians as Portingales , which they vse in their Nauigations to America and to India , and amongest others wee obtayned from Portingale a Mariners Carde , describing the whole earth very correctly , and besides that 14. other particular Cardes , in the which likewise all the Seas and Prouinces of the whole world , with their situation were comprised , all which Cardes being compared together , wee doe here nowe set foorth this newe description of all the Prouinces and Seas in the whole worlde , and that as correctly as may be , according to the consideration and obseruation which hath béen vsed by the most expert Geographers and Pilots , euen vnto this present houre : for in this Carde we doe describe all the sea coastes , Promontories , windinges in and out , Iles , Portes deapthes , sandes , showlds and rockes , also we haue added thereunto in place conuenient , the Mariners compasse , and the lines of the windes , which wee haue set downe as correctly as was possible for vs to doe , for the commoditie of Nauigation : and for so much as the true Longitude of the places can not be well obserued , without extending or enlarging too farre those Prouinces that are nigh vnto the North or South pole of the world , we haue therefore briefely comprehended the same in this our description of the world , in two rondles or circles , vnto the which we haue added another little Geographicall Carde , comprehending the north Prouinces of Europe , to the intent that in them euery man may sée with his eyes , the naturall situation of those Prouinces as well as it may be done in plano , as is more amply declared in other inscriptions of this Mappe . And this little Septentrionall Map last mentioned , is placed at the foote of this map , in the very midst thereof comprehending all the North parts that lie betwixt the 52. degrée , and 72′· of Latitude .

10 79. 30′· 56. 0′· South . Here is a very strong current of the sea , which ruuneth East & West , betwixt Madagascar and the Ile Romoros , in such sort as Nauigation there is very troublesome and laborious , as Marcus Paulus Venetus testifieth in his third booke & 40. chap. whereof it must néedes follow of necessitie , that the sea coasts of this countrie are not far distant from Madagascar , in such sort as the great Orientall sea doth ebbe and flowe through this straite with great violence into the West Ocean sea , whereunto agréeth the letter missiue of a Candiot , who was Embassadour for the Venetians to the king of Portingale . In which letter hee writeth that the men of this Countrie goe all naked . 11 91. 0′· 35. 0′· South . To this place arriueth a Portingale shippe called S. Paul . 12 212. 36 South .

The South firme lande , is called of some Magellanica , of Ferdinando Magellanus , who first discouered the same .

This Inscription , together with the foure next following , are to be found in the rondle on the right hand .

13 146. 46 South . Marcus Paulus Venetus , and Lewes Vartiman doe testifie in their bookes of perigrination , that here be very great and ample deserts . 14 75 52 South . Betwixt the Ile of Saint Laurence and the Iles Romeros , doe fall a most violent fluxe and refluxe of the Sea , East and West . 15 46 55 South . This is the lande of Popeniayes , so called of the Portingales , because those birdes in that Countrie are of incredible bignes . 16 15 46 South . This Promontory of the South land , is situate 450. Spanish leagues from the Cape de bona speranza , and 600. leagues from the Cape of Saint Augustine . 17 148. 0′· 34. 0′· South . As we haue in the first rondle on the left hande set downe the description of that part of the worlde , which extendeth from the Equinoctiall to the North pole : so in this other rondle , wee haue set downe a description of that part of the worlde which extendeth from the Equinoctiall to the South pole , in such sort as this rondle containeth all Magellanica , and almost all Peruana together with a great part of Afrique , and a great number of the most noble and renowmed Iles of the world , and herein you may plainely sée with your eye , the naturall situation of those Prouinces that are nigh vnto the South pole . 18 148. 30′ 20. 30′· South . The Realme of Maletur , which aboundeth in all manner of spices . 19 148 0′· 15. 40′· South . The Countrie of Beach is rich in golde , but little frequented by Marchants of other Countries , by reason of the crueltie of the people . 20 161. 0′· 20. 0′· South . Iaua minor , bringeth forth diuerse spices which haue not yet béene séene in Europe , as Marcus Paulus Venetus testifieth in his third booke 13. chapter . 21 170. 40′· 3. 0′· North. The Iles Moluccas are much renoumed for the great aboundance of spices , which are sent from thence into all countries of the world : The chiefest of those Iles are these , Ternarie , Tidoris , Motir , Machian , and Bachian , vnto which some doe adde Gilolo , Celebes , Burro , Amboino , and Bandar .

Besides all these Inscriptions , Plancius at the foure corners of his Mappe , setteth downe foure rondles , two aboue and other two beneath : and in that aboue on the left hand , representing the Northerne halfe of the celestiall Globe , he describeth al the North stars that are alredy , & in the other rondle on the right hand representing the southerne halfe of the celestial Globe , he doth not only set downe such Southerne stars as were knowne to the auncient Astronomers , but also such Southerne starres , as haue béen found out of latter dayes by those that haue trauelled into the East and West Indies , as the Crosse , the Southerne Triangle , Noes Doue or Pigeon , and an other in the shape of a man , called Polophilax , and certaine others , touching which starres hee setteth downe nigh vnto the foresaid rondle , a certaine inscription written in the Latine tongue , which I haue here interpreted word for word in our mother tongue as foloweth .

We haue here set downe the fixed starres in their true places answerable to the yeare 1592. and not 95. as the Printer hath made it .

Of the South pole and of the starres that are about the same .

LEast the South part of this Hemispheare or halfe Globe , should remaine voide and emptie , I haue taken these Southerne starres out of the obseruations of Andreas Corsalius Florentine , and haue diligently compared the same with the writings of Americus Vesputius , and of Petrus Medina , and haue reduced the saide starres into this forme or shape . But for so much as I haue séene nothing as yet to my satisfaction or contentment touching the Longitude , Latitude , Magnitude or nature of the said stars , I hartely pray all those that haue any more certaine knowledge of this matter then we , that they will enforme vs thereof , to the common good of all men : As touching the pole Antarctique , Corsalius writeth that there be two cloudie stars of a meane bignesse , which with a circular motion doe goe about another starre , that is distant from the Pole almost 11. degrées , and are sometime aboue and sometime beneath the said starre . Hitherto Plancius . But now to the intent that you may the better vnderstande all the foresaide foure rondles , I thinke it not amisse to describe the same vnto you , and to shew the vse thereof as followeth . You haue to note then , that the two vpper rondles : that which is on the left hand , signifieth the Northerne halfe of the celestiall Globe and the other rondle on the right hand , signifieth the Southerne halfe of the said Globe as hath béene said before , and each one of these rondles is traced with certaine circles and lines : The outtermost Circle whereof being deuided into 360. degrées , and containing the Characters of the 12. signes , signifieth the Zodiaque , or rather the very Ecliptique it selfe , the Centre of which Circle , is the pole of the Zodiaque , which by continuall turning about , describeth another lesser Circle hard by it , signifying in the North rondle the circle Arctique , and in the South rondle the circle Antarctique , the Centre of which lesser circle in eyther rondle , is the pole of the world , both which poles are distant from the pole of the Zodiaque 23. degrées 28′· which is the greatest declination of the sunne . Moreouer in either rondle are drawne vpon ech pole of the worlde , two other circles , the largest whereof signifieth in both rondles the Equinoctiall , and the lesser thereof in the North rondle , signifieth the Tropique of Cancer , and in the South rondle the Tropique of Capricorne : besides these circles ech rondle is traced with 12. right lines , signifying those sixe Meridians or lines which passing through both the poles of the Zodiaque , doe deuide the Zodiaque into 12. equall parts , euery part containing 30. degrées : for so many degrées doe belong to the Longitude of euery one of the 12. signes , whereby the Zodiaque hath in Longitude 360. degrées , which Longitude is to be counted from the first point of Aries and so foorth , according to the succession of the signes , and by helpe of these lines , you may know vnder what signe any fixed starre is : I shal not néede here to shew you , how the said fixed starres are situated , in either of the rondles , nor how they are named , because their images or shapes together with their names , are apparant to your eye . But if you would know the true place , the Longitude , Latitude , Magnitude and the nature of any fixed starre herein contained , then do thus : First to knowe the place and Longitude of any starre , lay a ruler or extend a thréede , so as it may passe through the Pole of the Zodiaque , and also through the bodie of the starre , whose place and Longitude you séeke , euen to the very Zodiaque , and somewhat beyond , and thereby you shall know in what signe , and in what degrée thereof that starre is , for that is his place , and you shall knowe his Longitude by counting from the first point of Aries , vnto that degrée , for that is his Longitude .

Now to know the Latitude of any starre , you haue to note that in each rondle there is a certaine right blacke line extending from the Zodiaque to the Pole , deuided by vnequall spaces into 90. degrées , which line is called the skale of the fixed stars Latitude , the vse whereof is thus : Set the firme foote of your Compasses in the very pole , and extend the other foote into the middest of the bodie of that starre whose Latitude you séeke , and turne that foote standing at that widenes to the skale , and the number of degrées written vpon the skale , if you count from the Zodiaque vpward towards the Pole , will shew you the Latitude of that starre . Againe , to knowe the magnitude of any starre , Plancius setteth down in the North rondle the selfe same meane which Mercator also vseth in his celestiall Globe , that is to say , by making certain shapes of starres representing the bignesse of euery starre , according to his greatnesse , that by marking & comparing those shapes together , you might finde out , or rather coniecture the greatnesse of the starre which you séeke . Lastly he sheweth the nature of any starre by setting downe nigh vnto the star , the characters of those Planets , of whose nature that starre doth participate , all which thinges you shall more plainely vnderstande by this example here following .

Suppose that you would know the place , longitude , latitude , magnitude and nature of the starre called Arcturus : here because this is a North starre , you must therfore resort to the North rondle , and there séeke out the image Bootes , betwixt whose legs is the star called Arcturus which you séeke . And by extending a thréed which may passe through the pole , and also through the bodie of that starre euen to the Zodiaque , and somewhat beyond , you shall find his place to be in the 19. degrée of the signe Libra , and his Longitude counting from the first point of Aries , vnto that degrée to be 199. degrées , and by obseruing the rule before giuen , touching the knowing of the Latitude of any starre , you shall with your Compasse finde the Latitude of this starre to be almost 32. degrées Northward , and by his shape you shall knowe that hée is of the first magnitude , and the characters of the two Planets Mars and Iupiter , placed hard by him , do shew that he is of their nature , that is to say , by participating of Mars he is extremely hotte and dry , and by participating of Iupiter he is hotte & moist , and looke what order is to be obserued in the North rondle , touching the North stars , the same is likewise to be vsed in the South rondle containing the Southerne starres : Amongest which you may sée the Images called the Crosse , whereby most Pilots in these dayes doe chiefely direct their course , being once past the Equinoctiall towardes the South pole , which Crosse , though Plancius doth here make to consist of fiue starres , yet I am sure that Martin Cortes and Peter Medina , and all other late writers do appoint thereunto but foure starres , the shape and vse whereof , I haue set downe in my Treatise of Nauigation , according to the direction of Peter Medina . And those that haue trauailed into the Indies , doe all affirme that to the Crosse there doe belong onely foure starres and no more , wherefore I maruaile much , that Plancius doth set downe fiue , whereunto perhaps he is induced by the relation of some Spaniarde that neuer sawe them . Thus hauing described vnto you the two vpper rondles , representing together the celestiall Globe , and also shewed the vse therof , I will now describe the two neather rondles , whereof that on the left hande representeth the North halfe of the Terrestriall Globe , & that on the right hand the other halfe of the same Globe , towardes the South .

You haue then to vnderstand that the Centre or middle point in each rondle , signifieth the Pole of the world , that is to say , the North pole in the North rondle , and the South pole in the South rondle , and vpon each pole are drawne certaine Circles , the outtermost whereof , and furthest distant from the Pole signifieth the Equinoctiall , which is deuided into 360. degrées , euery degrée containing 60. miles , which is the whole Longitude of the earth , from which circle at the end of euery tenth degrée , are drawn certaine right lines to the number of 18. which doe méete in the very Pole , and doe signifie halfe Meridians , whereof that which passeth through the Iles Azores , and also the Iles of Capo Verde , is the first Meridian , from whence the longitude of the earth taketh his beginning , and there also endeth : which Meridian in the rondle on the left hand , is deuided into 90. parts , procéeding from the Equinoctial to the Pole , signifying the North Latitude of the worlde , the like diuision and number of degrées of Latitude , hath also the first Meridian , in the rondle on the right hande , sauing that the saide Meridian tendeth vpwarde . Moreouer you haue to vnderstande , that in each of these rondles , are drawne nine Circles , equally distant one from another , called Parallels , which together with the Equinoctiall , doe make nine spaces , euery space contayning 10. degr. & besides these circles , there are drawne in ech rondle two other circles , the one greater , & the other lesser , the greater in the rondle on the left hand being distant from the Equinoctiall 23. degrées 30′· which is the greatest declination of the sunne , is called the Tropique of Cancer , and the lesser Circle being of like distance from the pole , is called the Circle Arctique , but the greater circle being of like distance from the Equinoctiall , in the rondle on the right hand , is called the tropique of Capricorne , and the lesser circle in the said rondle inuironing the Pole , is called the Circle Antarctique vpon which circles in each rondle you shall finde their names written . The chiefest vses of these two rondles are these : first to finde out the Longitude of any place , secondly the Latitude , and thirdly the distance betwixt any two places .

To finde out the Longitude of any place , you must doe thus : Extende a thréede , so as it may passe through the pole , and also through the place whose longitude you séeke , euen to the very Equinoctial , and somewhat beyond , and holding the thréed straight , the numbers of the degrées written vpon the Equinoctiall will shewe the longitude of the place .

And if you wil know the latitude of that place , or of any other , doe thus , Set the one foote of your compasse in the very pole , extending the other to the place whose Latitude you séeke , and kéeping your compasse at that widenesse , bring the mooueable foote to the first Meridian , whereon the degrées of latitude are marked , and there staying it , the number of the degrées , counting from the Equinoctiall vpward towards the pole , will shewe the latitude of the place . As for example , suppose that you would know the longitude and latitude of Lisbone , which is a famous towne in Portingale , here hauing first found out that towne in Spaine , which is nigh vnto the West Ocean , extende your thréede from the pole through the middest of that Towne to the Equinoctiall and somewhat beyond , and you shall finde that the thréed will cut the Equinoctiall in the 13. degrée , which is the Longitude of Lisbone . Nowe if you would knowe the Latitude of the same place , set the one foote of your compasse in the Pole , and extende the other foote to Lisbone , and kéeping your compasse at that widenesse , bring the mooueable foote to the skale of latitude , and so you shall find that Lisbone hath in North latitude 38. degrées and 30′· Now to knowe the distance betwixt any two places doe thus , Set the one foote of your compasse in the one place , and the other foote in the other place , and apply that widenesse to the Equinoctiall , and looke howe many degrées of the Equinoctiall that widenesse comprehendeth , and by allowing 60. Italian miles to euery degree , you shall haue the distance by a right line , betwixt those two places , for by doing thus , you shall finde the distance betwixt Lisbone & Compostella to be 120. miles . Thus I haue sufficiently ( I hope ) expounded euery thing contained in Plancius his Map , his general skale made for the same onely excepted , whereof I come now to speake . In this skale are set downe the miles of Russia , of Italy , of Englande , of Scotland , the French leagues , the houre leagues , the Spanish leagues , the Germaine and Garscoyne miles , which two are all one , the miles of Sweuia in Germanie , of Scandia and of Swethland , which last three are likewise all one , the vse of which skale is thus . Take with your compasse the distance betwixt any 2. places which you desire to know , and apply the same to the skale of such miles as you would know , & so many miles the two places shal be distant one from another as the number of the skale doth shew : but if the distance betwixt the two places be longer then the skale , then hauing first taken the whole length of the skale with your compasses , looke howe manie times that widenes of your compasse measuring by a right line is contained in the distance betwixt the two places , and if there be any odde space left , streighten your compasse to that odde space , & apply that to the beginning of the skale , and adde the number of miles which you there find to the first great number , so shall you haue the totall summe . And lo here for each an example : first suppose that you would know the distance betwixt Cape S. Marie , & Cape finis terrae , which are two Capes or headlands in the West side of Spaine , both hauing in a maner one selfe Longitude , & do differ only in latitude by 6. degrées , for the one hath 37. & the other 43. in latitude , which distance if you take with your compasse by setting the one foot in the one place , & the other foote in the other place , and applying that widenes to the skale of Italian miles , you shal find the distance to be 390. Italian miles , but if you measure the same distance according to the Geographicall maner , which is to allow for euery degree of latitude 60. miles , you shal find the distance to be no more but 360. Italian miles . Let your other example be thus : Suppose that you would know the distance betwixt Compostella in Spaine and Constantinople , which haue all one Latitude and doe differ onely in Longitude , heere because the distance betwixt these two places is longer then the skale , you must take with your compasse the whole length of the skale , and then to looke how many times that widenes is comprehended ( measuring by a right line ) betwixt the two said places , & you shall find that widenes to be comprehended in the distance betwixt those two places thrée times , wherefore if you multiply 840. by 3. it will make in all 2520. Italian miles . But if ( according to the Geographicall kind of measuring ) you doe multiply the difference of their Longitudes , which is 43. degrées , by the number of miles , which is also 43. belonging to the Latitude of both places , which Latitude is also very nigh 43. degr. you shal find that the distance betwixt those two townes is no more but 1870. Italian miles , which number of miles is not so great as that of the skale by 650. Italian miles . And therefore I can not thinke but that there is some error in the skale committed either by the Printer or else by the author through some negligence , and not for lacke of skill or knowledge how to make a true skale , being so excellent a Geographer as the Author by this & other his Maps heretofore made sheweth himselfe to be , or else there is some greater mistery therin , then I perhaps doe vnderstand , for in séeking to know the distance betwixt two places differing onely in Latitude , I finde the skale most times to agrée with the Geographicall kind of measuring , but if the two places do differ either in longitude onely , or else both in longitude and also in latitude , then I find the skale to differ very much from the Geographicall kinde of measuring , wherfore I thinke it good briefely here to set downe certaine readie waies of finding out the distance of two places , differing eyther in latitude onely , in longitude only , or in both , which I doe shew also at large in the second part of my Spheare , Chap. 14.

How to finde out the distance of two places differing onely in Latitude .

IF the two places haue both eyther North or South Latitude , then subtracte the lesser Latitude out of the greater , so shall you haue the difference : which difference , if you multiply by sixetie , the product shall be the number of miles , and if to the whole degrées of difference there be annexed any minutes , then you must adde to the product for euery minute one mile . But if one of the two places haue North Latitude , and the other South Latitude , then you shall finde their difference by addition , and not by subtraction . As for example , suppose that you would knowe the distance betwixt a towne called Pasquali , which is the outermost towne in Moroa vpon the sea towards the South , hauing in North latitude 35. degrées 30′· and a certaine towne in Afrique called Debsan standing nigh vnto the lake Zembre , which hath in South Latitude 12. degrées 30′· here by adding these two latitudes together , you shall find the summe to be 48. degrées , and that is the difference of their Latitudes , which difference if you multiply by 60. the product will be 2880. and that is their distance .

How to find out the distance of two places differing onely in Longitude .

IF both places haue either East Longitude or West Longitude , then subtract the lesser out of the greater , so shall you haue the difference , which difference you must multiply by the number of miles belonging to their Latitude , which you shall finde on the Northwest side of the Mappe , or by the Table of miles answerable to one degrée of euery Latitude set downe hereafter in the end of this Treatise : and the product thereof shall be the number of miles whereby the one place is distant from the other . As for example , I finde Compostella in Spaine and Constantinople , hauing both almost 43. degrées of North Latitude , to differ only in East longitude , for Compostella hath in East longitude 13. degrées 30′· & Constantinople hath in East longitude 56. degrées , the difference whereof by subtracting the lesser out of the greater you shall finde to be 42. degrées 30′· here if you multiply 42. by 43. miles belonging to one degrée of the foresaid Latitude 43. you shall finde the product to be 1806. then to finde out the value of miles for the fraction 30′ , by the rule of proportion , you must say thus : if 60. require 43. miles what shall 30. require , and you shall finde in the quotient 21. miles , which you must adde to the former summe 1806. and it will make in all 1827. miles and ½ . which is the true distance betwixt the two foresaid places . But if the one place haue east Longitude , and the other West Longitude , then you must finde the difference aswell by Addition as by Subtraction . As for example , suppose that you would knowe the distance betwixt S. Domingo in the Ile called Hispaniola , and a certaine place in Afrique called Septem montes , night vnto the Ocean sea , both places hauing 18. degrées of North latitude . And S. Domingo hath in West longitude 310. degrées 30′· and the place called Septem montes hath in East longitude seuen degrées . Here you must first subtract 310. degrées 30′· out of 360. degrées , and there will remaine 49. degrée 30′· whereunto you must adde the East longitude of Septem montes , which is seuen degrées , and it will make in all 56. degrées 30′· which is the difference of their Longitude . Nowe if you first multiply 56. degrées by 57. miles belonging to 18. degrées of latitude , you shal find the product to be 3192. miles , and to finde out the value of miles for the fraction 30′· you must say thus : If 60. require 57. miles , what shall 30′· require , and working by the common rule of thrée , you shall haue in the quotient 28. miles , and there remaineth 30/60. which is one halfe mile . Now by adding 28. miles and ½ . to 3192. it will make in all 3220. miles ½ . and that is the true distance betwixt S. Domingo and Septem montes , but by Plancius skale you shall finde the distance to be 3410. miles , which differeth from the other almost 200. miles .

〈◊〉 . 〈◊〉 . degrees . degrees . London . 22. 0′· 51. 32′· The greater 〈◊〉 . Hierusalem . 6● . 0. 32. 0. The lesser L●● . The difference of their Longitudes 47. 0.

How to finde out the distance of two places differing both in Longitude and Latitude by helpe of a semi-circle deuided into a 180. degrees , which I had from my louing friend Maister Wright of Cayes Colledge in Cambridge , of whom I make mention aswell in my treatise of the Spheare as in that of Nauigation .

FIrst drawe a Semicircle vppon a right Diameter , marked with the letters A. B. C. D. whereof let D. be the Centre like vnto this here described , and the greater that such semicircle is , the spaces of the degrées shall be the larger , and thereby the more easie to find out the minutes . Then hauing drawne your semicircle and deuided the same accordingly , suppose that by helpe therof you would find out the distance betwixt London and Hierusalem , which two townes doe differ both in Longitude and also in Latitude so much as is here set downe in the front of the figure according to such Longitude and Latitude as Plancius doth allow to eyther towne in his great Map . Now to find out the true distance of these two townes you must first take the lesser Longitude out of the greater , so shall you haue the difference of their Longitudes , which is 47. degrées , then count that difference vpon the Semicircle beginning at A. and so procéede to B. and at the ende of that difference , make a pricke marked with the letter E. vnto which pricke draw a right line by your ruler from D. the Centre of the Demicircle : That done séeke out the lesser Latitude which is 32. degrées 0′· in the foresaid Demicircle , beginning to account the same from the pricke E. and so procéede towards the letter B. and at the end of the said lesser latitude set down another pricke marked with the letter G. from which pricke or point drawe a perpendicular line , which by helpe of your squire or compasses , may fall with right Angles vpon the former right line drawne from D. to E. and where it falleth , there set downe a prick marked with the letter H. That done séeke out the greater Latitude which is 51. degrées and 32′· in the foresaid Demicircle , beginning to account the same from A. towards B. and at the end of that Latitude set downe another pricke , marked with the letter I. from whence drawe another perpendicular line that may fall by helpe of your squire or compasses , with right Angles vpon the Diameter A. C. and there make a pricke marked with the letter K. That done take with your compasse the distance that is betwixt K. and H. which distance you must set down vpon the said Diameter A. C. setting the one foote of your Compasse vpon K. and the other towards the Centre D. & there make a pricke marked with the letter L. then take with your Compasse the length of the shorter perpendicular line G. H. and apply that widenesse vppon the longer perpendicular line I. K. setting the one foote of your compasse at I. which is the end of the greater Latitude , and extend the other foote towardes K. and there make a pricke marked with the letter M. That done take the distance betwixt L. and M. with your compasse and apply the same to the Demicircle setting the one foote of your compasse in A. and the other towards B. and there make a prick marked with the letter N. And the number of degrées contained betwixt A. and N. will shew the true distance of the two places which you shall finde to be 39. degrées , which being multiplyed by 60. maketh in al 2340. miles , and whensoeuer you haue any minutes besides the whole degrées , remember to adde vnto the summe of degrées for euery minute one mile . By Plancius skale you shall find the distance betwixt London and Hierusalem to be 3040. miles which are 700. miles too many . But you haue to note by the way , that if the difference of the Longitudes doth excéede the number of 180. then you must subtract that excéeding difference out of 360. and the remainder shal be the difference of the Longitudes , and then worke in all points as is before taught .

By this rule and the other two rules first declared , you shal easily trie the skale of any Mappe whether it be true or not , so as you first haue the true Longitude and Latitude of the two places whose distance you séeke to know . And thus I end with Plancius Map , hoping not to offend him with any thing that I haue added thereunto for the better instruction of those that haue not béene exercised in such matters .

And yet I had almost forgotten one thing , which is this , I haue here before as you haue read , made mention of drawing certaine perpendicular lines in the former figure of the Demicircle by helpe of your Compasses . Wherefore I thinke it necessary here to set downe the order thereof .

How to make with you Compasses , a perpendicular line to fal from any point giuen vpon another right line , making therwith right angles without the helpe of any squire .

SEt the firme foot of your Compasse in the point giuen , and extend the other foote a little beyond the line right against the point giuen , & draw a secret Arch or portion of a Circle that may cut the said liue in two points , and deuide that part of the Arch which lieth betwixt the two sections into two equall parts , setting a pricke in the very midst therof : then hauing laid your ruler to that pricke , and also to the point giuen , draw a right line , & that line wil fall vpon the other line with right Angles , as you may sée by this figure .

TheTableofmilesanswerabletoonedegreeofeueryseuerallLatitude. D M S D M S D M S D M S D M S 1 59 59 19 56 44 37 47 55 55 34 25 73 17 33 2 59 58 20 56 23 38 47 17 56 33 33 74 16 32 3 59 55 21 56 1 39 46 38 57 32 41 75 15 32 4 59 51 22 55 38 40 45 58 58 31 48 76 14 31 5 59 46 23 55 14 41 45 17 59 30 54 77 13 30 6 59 40 24 54 49 42 44 35 60 30 0 78 12 28 7 59 33 25 54 23 43 43 53 61 29 5 79 11 27 8 59 25 26 53 56 44 43 10 62 28 10 80 10 25 9 59 16 27 53 28 45 42 26 63 27 14 81 9 23 10 59 5 28 52 59 46 41 41 64 26 18 82 8 21 11 58 54 29 52 29 47 40 55 65 25 21 83 7 19 12 58 41 30 51 58 48 40 9 66 24 24 84 6 16 13 58 28 31 51 26 49 39 22 67 23 27 85 5 1● 14 58 13 32 50 53 50 38 34 68 22 29 86 4 11 15 57 57 33 50 19 51 37 46 69 21 30 87 3 8 16 57 41 34 49 45 52 36 56 70 20 31 88 2 5 17 57 23 35 49 9 53 36 7 71 19 32 89 1 3 18 57 4 36 48 32 54 35 16 72 18 32 90 0 0

Though it be the common order of working to know by helpe of the former table , the distance of two places differing onely in Longitude , yet I thinke it a more sure way to find it out per Tabulas Sinuum , the rule whereof is thus .

First take the difference of the two Longitudes , by subtracting the lesser out of the greater , and the halfe of that shall be the Arch which you haue to séeke in the front of the Tables , then multiply the sine of that Arch by the sine of the complement of the common Latitude , and deuide the product thereof by the totall sine , the quotient whereof you must séeke out in the Tables amongest the sines , and the Arch of that sine is the one halfe of the distance , which being doubled shall be the whole distance contayning degrées of the great Circle , and euery such degrée contayneth of Italian miles 60. and of German miles 15. and by working thus you shall finde the distance betwixt Compostella and Constantinople to be 1846. Italian miles , supposing the common latitude to be 43. degrées , and the difference of their longitudes to be 42. degrées 30′· And by working by the common table you shall find the distance of those two places to be 1827. Italian miles as before , because the common Table hath no minutes of miles but onely seconds , which are not to be accounted of , & in working by Appian his Table hauing minutes of miles , you shal find the said distance to be 2184. Italian miles , and by Mercator his Map to be 1980. Italian miles , in whose Map the common Latitude of the said 2. places is 43. & the difference of their longitudes is 44. 0′· And by the skale set down in Plancius his Map , you shal find the distance to be 2520. Italian miles , in which Map the common Latitude of the two foresaid places is 42. deg. 30′· and the difference of their Longitudes is also 42. degrées 30′· Truely I must néedes confesse that it is not so easie to make a skale or trunke for a Mappe or a Carde drawne in plano , as for that which is drawne vpon a round bodie or Globe : and therefore it is no maruaile though the skales of Mappes drawne in plano , and likewise the trunkes set downe in the Mariners Cardes doe not alwayes shewe the true distance of places , which I beléeue is to bee done as truely and a great deale more readily by my friend Maister Wright his Semicircle before described , then by the rules of Gasparus Peucerus in his booke de dimensione terrae , which rules doe depend vpon the knowledge of the quantitie of the Angles and sides of Sphericall Triangles , which kinde of working is indéede more troublesome and tedious then readie or pleasant . But if Maister Wright would make his Demicircle an vniuersall instrument to find out thereby all the thrée kinds of distances as he promised me to doe , there were no way in mine opinion worthie to be compared vnto it , neither for the truenesse , easinesse , nor readinesse of working thereby .

A very brief and most plaine description of Maister Blagraue his Astrolabe , which he calleth the Mathematicall Iewell . Together with diuerse vses thereof , and most necessarie for sea men , written by Maister Blundeuill .

NON SOLO PANE VIVET HOMO : Luke 4

Verbum Dei manet in aeternum : IW

Imprinted at London by Iohn Windet .

To the Reader .

OF Astrolabes I haue neuer seene but three sorts , First that of Stofflerus , which for these hundred yeares past or there abouts , hath beene had in most price and estimation , as an instrument containing all the vses , or at the least the most part of all other Mathematicall Instruments , which because it requireth almost for euerie seuerall Latitude a seuerall Table : Gemma Frisius inuented since another kind of Astrolabe , hauing but one Table to serue for all Latitudes , and therefore he called it the Catholicon , that is to say , an vniuersall Astrolabe , which hath also been most esteemed & vsed many yeres . Since whose time , & of very late yeares , one of our owne countriemen a Gentleman of Reading besides London , called M. Blagraue , hath greatly augmented the saide Catholicon , and hath thereby as it were newly inuented a third kind of Astrolabe , which he calleth the Mathematicall Iewell , whereby are to be wrought more conclusions then by any other one instrument whatsoeuer , for which his most excellent inuention vsed therein , he deserueth great commendation . And to the intent that others which haue not beene exercised in such thinges , might the more easily attaine to the better vnderstanding of the said Iewell : I haue made a plaine description of all the partes belonging to the said Iewell , without committing any offence ( I hope ) to the Author thereof , to the intent that euery Gentleman might haue the perfect knowledge and vse of so worthie an instrument : But broade Astrolabes though they be thereby the truer , yet for that they are subiect to the force of the winde , and thereby euer moouing and vnstable , are nothing meete to take the altitude of any thing , and specially vppon the Sea , which thing to auoide , the Spaniards doe commonly make their Astrolabes or Rings narrow and waightie , which for the most part are not much aboue fiue inches broad , and yet doe waigh at the least foure pound , and to that end the lower part is made a great deale thicker then the vpper part towards the ring or handle : Notwithstanding most of our English Pilots that be skilfull , doe make their sea Astrolabes or ringes sixe or seuen inches broad , and therewith very massiue and heauie , not easie to be mooued with euery winde , in which the spaces of the degrees be the larger , and thereby the truer , of which kinde of Astrolabes or Rings , I shall speake hereafter in my Treatise of Nauigation , and also of the Mariners Crosse staffe . But whensoeuer you haue to take the altitude of the Sunne or of any other starre be in wandering or fixed , I would wish you to vse the Mariners heauie and massiue Astrolabe , which in mine opinion for that purpose , is the sittest and most assured instrument of all others and to finde out all other conclusions by helpe of Maister Blagraue his Iewell , or rather by helpe of the Celestiall Globe , which for Astronomicall matters is the perfectest instrument of all , in which are contained all the starres both southerne and Northerne that haue beene heretofore knowne , and those that are lately knowne or shall be knowne hereafter may be easily placed therein : which thing can not be so well perfourmed in any Astrolabe were it neuer so great . But because the Globe is combersome and not portable , and therewith costly , and especially if it bee of any greatnesse , ( for the greater the better ) and also if it chance to be broken in any part of the bodie thereof , it can neuer bee made againe perfectly whole , it is therefore no meete instrument for euery Mariner to haue , but onely for such as be of good habilitie , in steede whereof to finde out manye necessarie conclusions , Maister Blagraue his Mathematicall Iewell may serue verie well , and specially if it had on the backe parte the like mater and Rete or Nette , to serue the South Latitude of the worlde as it hath in the fore parte to serue the North latitude , for then I beleeue verilie , that the chiefest Sotherne starre , as the Crosse , the Southerne Triangle , Noahs Doue or Pigeon , and another called Polophilax , latelie found out by such as haue travelled by sea on the South side of the Equinoctiall , might easilie be placed by Maister Blagraue in that Rete , hauing once learned of the skilfull Seamen the true longitude , latitude , and declination of the saide Starres , so shoulde his Iewell in mine opinion bee much more serviceable to the sea men , then nowe it is , by reason that those fewe Southerne Starres that be contained in his first Net , are nothing so nigh vnto the South pole , as the Northerne Starres described in his first Net , are vnto the North pole , amongst which Northerne Starres , it were verie necessarie for the Sea men , that the seuen principall Starres as well of the great Beare as of the little Beare , were all dulie placed in his first Net . Trulie if Master Blagraue his affaires would suffer him to take paine herein , I beleeue that his Iewell should not be much inferiour to the celestiall Globe , which indeede it representeth , and in so doing he should greatly profite the sea men , and deserue thereby great good will and commendations at their handes .

What this word Astrolabe signifieth .

BEfore I beginne to describe vnto you the said Instrument and all other parts thereof it shall not bee amisse to shew you what this worde Astrolabe signifieth .

This worde Astrolabe is as much to say as the handle or instrument of the Starres , by helpe whereof the manifolde motions and apparences of the heauens and of the Starres therein contained are known , and it is called of some a planispheare , because it is both flat and rounde , representing the Globe or Spheare , hauing both his Poles flatte both together , the shape or figure of which Instrument I doe not here set downe because the Instrument it selfe is to be had for a small price in diuers places of London , which if you lay before you when you mind to reade this my description thereof here following , I doubt not but that you shall finde euerie part thereof so plainlie explaned as you shal need no other teache to instruct you therin , or to help you to vnderstand any of these Conclusions that are to be wrought by that Instrument , and speciallie those which I haue here set downe , such as the Table following sheweth , and as I thought most meet for sea men to know , and being throughlie exercised in them , you shall the more easilie vnderstand the manifold and necessarie conclusions set down by Master Blagnaue himselfe in his owne booke , which booke I would wish you to buy , and earnestlie to studie the same . But now I will first describe the saide Instrument , and then shew the vse thereof by so many conclusions as are contained in the table hereafter following .

The description of Master Blagraue his Astrolabe , otherwise called the Mathematicall Iewell .

TThis Astrolabe is diuided into two parts , whereof the one is called the forepart , and the other the backepart . The foreparte containeth two principall partes , that is , the Mater , which is vnmooueable , and the Rete , which is mooueable . Againe , the Mater is enuironed with a great circle , called the Meridian , passing through the 2. Poles of the world marked with the letters A.B. and through the two Solsticiall points of Cancer , and Capricorne marked with the characters belonging to those two Signes , and therfore may verie well bée called the Collure of the 2. Solstices . And this circle is diuided by two crosse Diameters iuto 4. quarters euery quarter containing 90. degrées , so as the whole circumference of the circle is 360. degrées , which degrées doe procéede from 90. to 90.

Nowe as touching the 2. crosse Diameters , the one passing through the Center , and also through the two Poles , The 2. crosse Diameters in the Mater . is the Axletrée of the world , signifying sometime the first Meridian , and sometime the right Horizon , and sometime the line of sixe houres as well for the Morning as Euening , and then it is the line of East and West , and sometime the line of North and South .

And the other crosse Diameter drawne with red incke , signifieth most commonly the Equinoctiall , The Equinoctial . and sometime the line of East and West , and sometime the line of North , and South , and especially when you haue to finde out the 12. houses of heauen , to which end are set down in the outermost Meridian these foure Latine words , first at the North pole marked with the letter A. Oriens , which is as much to say , as the East part : and at the south Pole marked with the letter B. Occidens , which signifieth the West part : and at the ende of the ouerthwart Diameter , drawne with red inke , and most commonly signifying the Equinoctiall , marked with the letter C. on the right hande is set downe Culmen Coeli , the highest part of heauen , which is the South , at which ende the Ringle or handle is fastened . And at the other ende of the saide Diameter , marked on the left hande with the Letter D. is set downe Imum Coeli , the lowest parte of heauen , which is the North. And by these two crosse Diameters the whole Mater is diuided into foure quarters , that is , Northeast , and Southeast , Northwest and Southwest . The Northeast quarter lyeth betwixt Imum Coeli , and the vpper ende of the Axletrée , whereon is written Oriens . And the Southeast quarter lieth betwixt the same point and Culmen Coeli , and the Southwest quarter lieth betwixt Culmen Coeli , and the lower end : of the Axletrée , whereon is written Occidens , and the Northwest quarter lyeth betwixt the point Occidens and Imum Coeli .

Besides these 2 crosse Diameters the Mater is traced with 180 Meridians which doe passe through both the Poles , The Meridians and houre lines in the Mater . whereofthe first passing through the center from Pole to Pole is the Collure of the Equinoxes , because it passeth through the first point of Aries , and the first point of Libra , and is otherwise called the Axletree , and signifieth sometime the right Horizon , and somtime the first Meridian from which vpon the Equinoctiall are counted the degrées of longitude both East and West . And al the foresaid Meridians are most commonly vsed as houre lines , for euery 15th . Meridan , counting from the Limbe , doeth signifie one houre , for 15. degrées do make an houre , and 4. minutes do make a degrée . And these hours at the end of euery 15th . Meridian are marked in the bodie of the Mater somewhat aboue the Tropique of Cancer with Arithmeticall Figures , procéeding from 1. to 12. forwarde towardes the right hand , and againe are marked somewhat beneath the Tropique of Capricorne with like Figures , proceeding backward towardes the left hand . Whereof the vpper numbers doe signifie the forenoone houres , and the lower numbers the afternoone houres , and are placed so as one selfe houre line doth passe through both numbers : for the line of the eleuenth houre in the forenoone serueth also to the first houre in the afternoone , and the forenoone houre of 10. serueth also to the houre of two in the afternoone , and the fore noone of 9. serueth to the houre of 3. in the afternoone : and the forenoone houre of 8. serueth to the houre of 4. in the after noone , and the forenoone houre of 7. serueth to the houre of 5. in the afternoone , and the forenoone houre of 6. serueth also to the 6. houre in the afternoone , and from thence foorth towardes the left hande , 5. serueth to 7. 4. to 8. 3. to 9. 2. to 10. and 1. to 11. as before .

But when soeuer you haue to finde out anie houre amongst the houre lines in the Mater , remember that the Axletrée which passeth through the Center and also through both the Poles is alwayes the sixt houre both of the morning and euening , from which you shall the more easilie finde out any houre that you séeke by allowing fiftéene Meridians to euery houre , as is before saide . And you haue to note that all these Meridians doe sometime signifie euerie one a seuerall oblique Horizon , sauing that the first Meridian or Axletrée signifieth the right horizon as hath bene said before . How to finde out the Horizon of euey seuerall latitude . And to find out in the Mater the horizon of euery latitude , reckon from the axletrée on both handes so many Meridians as may make vp the number of degrees of your latitude , and that Meridian shal be your Horizon . In counring of which Meridians you shall finde that euery fift Meridian is drawne with a greater blacke line then the rest . Nowe ouerthwart these Meridians are drawne 180. Parallels , The Parallels of the Mater . which proceede from 1. to 90. whose nombers are set downe in the dimbe of the mater thus , 10. 20. 30. 40. and so forth both vpwarde and downeward , vntill you come to 90. and these Paralle Is are otherwise called the circles of declination as well of the Sunne as of any star . Amongst which there be 2. Parallels painted red , The two Tropiques . where of that towardes the North pole signifieth the tropique of Cancer , and the other towards the South pole the tropique of Capricorne . Then there is another oblique ouerthwart redde lyne , which passing through the center from Tropique to Tropique , signifieth the Ecliptique line , The Ecliptick-line of the Mater night vnto which on each side are set downe the Characters of the twelue Signes , whereof Aries and Libra are placed at the center , and Cancer at the one end of the Ecliptique on your right hand , whereas it toucheth the said Tropique , and Capricorne at the other ende of the saide Ecliptique , on your left hand , whereas it toucheth the said Tropique of Capricorne , and the rest of the signes are orderly placed betwixt the said Tropiques , euery two Signes one against another that haue like declination , as Gemini and Leo , Taurus and Virgo , which are placed betwixt the center and Cancer , then Pisces and Scorpio , and Aquarius and Sagittarius are placed betwixt the Center and Capricorne . And euery Signe contained in the Ecliptique line is diuided with slaunting black stréeks into 15. spaces , euery space containing two degrées , which maketh 180. degrées , and béeing doubled , because the spaces are to bée counted doth forward and backward , do make in all 360. degrées , which is the whole circuit or longitude of the Ecliptique , and the degrées of the said Ecliptique are larger towards the Limb than towards the Center , whereby the spaces betwixt Signe and signe are not equall , but some larger than other some .

And note that vppon the outermost Limbe of the Mater are drawne thrée circles making two reasonable spaces , The Limbe of the Mater . both which spaces are diuided into 360. degr. both of them beginning at the North pole . And the degrées in the outermost space of the Limbe , beginning at the North pole do procéede from 15. to 15. downward towards your left hand , and so round about the instrument vntill you come to 360. which degrées do shew the houres of the Equinoctiall . And at the North pole , whereas the diuision beginneth , is set downe the figure of 6. signifying the sixt houre of the forenoone , then counting fiftéen degrées downwarde towards your left hand is set down in the foresaid space the figure of 5. then 4. 3. 2. and 1. And at the end of the Equinoctiall on the left hande is set downe 12. from which procéeding still downewards are set these numbers of houres , 11. 10. 9. 8. 7. and 6. which 6. standeth at the South pole , then from thence procéeding vpwardes are set downe these houres 5. 4. 3. 2. 1. and then 12. which standeth at the end of the Equinoctiall on the right hand , from whence procéeding still vpwards towards the North pole , are set downe these houres , 11. 10. 9. 8. & 7. and at the Pole it selfe is set downe the figure of 6. as before hath bene said , from whence you first began to count . And in the inner space of the same Limbe , the 360. degrées are to bee counted by 90. in such sort as the first 90. is placed at the North pole , the second 90. at the one ende of the Equinoctiall marked with C. The third 90. at the South pole , marked with B. And the fourth 90 at the other end of the Equinoctiall , marked with D. Euery which quarter is to be counted both vpward and downwarde , according as you sée the numbers written with Arithmeticall figures , and are placed one aboue another in the saide space , whereof the vpper figures are somewhat lesser than the nether figures , both of them procéeding from 1. to 90. the greater figures from the Equinoctiall vpward to the verie Pole , and the lesser Figures from the Pole downward to the Equinoctiall , and are thus set downe . 10. 20. 30. 40.

A Description of the Nette , called in Latine Rete .

THis parte is first inuironed rounde about with a great circle signifying most commonlie the Meridian or 90. Azimuth or Verticall circle , and sometime it signifieth the Equinoctiall , and especiallie when the Center is taken from the Pole , which Circle is diuided by two crosse Diameters into foure quarters , euerie quarter contayning 90. degrées , which degrées are to be numbred in the verie Limbe of the Rete both vpwarde and downwarde , and of those two crosse Diameters , the one ende marked with the letter A. signifieth the Zenith , and the other end marked with the letter B. signifieth the Nadir or point opposite , and the other crosse Diameter marked with the letters C. D. signifieth the Horizon , which for distinctions cause is otherwise called the Finitor , because the Meridians before described in the Mater are to be called and vsed sometimes as Horizons and this Finitor is a pretie broade Ruler , the verie edge whereof is diuided with small diuisions into 180. degrées , which being doubled by reckoning the same both forwarde and backwarde , ( beginning at the Center ) doe make vp thrée hundred and thrée score degrées .

This Finitor signifieth the Horizon of the Globe , The Finitor the verie edge whereof being diuided by little short stréekes into small portions or degrées , is alwayes to be applyed to any seuerall latitude when néede is . And the broader part thereof serueth onely to containe the numbers that are set therein both beneath and aboue to knowe thereby the number of euerie Azimuth hereafter described , which numbers doe procéede from the Center to the right hand thus , 10. 20. 30. 40. 50. and so foorth to 90. Then backwarde towardes the Center are set downe 100. 110. 120. 130. 140. and so forth vntil you come to 180. which is placed at the verie Center . Then from thence towards the left hande are set downe these numbers 190. 200. 210. and so foorth vntill you come to 270. and from thence turning againe towards the center are set downe 280. 290. and so foorth vntill you come to 360. placed at the Center .

And in the said Net are certaine circles , The Almican teraths . which are Parallels to the foresaide Finitor , procéeding towards the Zenith , and are in euerie respect like vnto the Parallels described in the Mater , and these Parallels are called Almicanteraths , that is to say , circles of Altitude , which beginning at the Finitor , doe procéede to the Zenith , marked with the letter A. from 1. to 90. And though there be out out in the Rete but 30. Almicanteraths , yet for so much as euery space contained betwixt euery 2. Almicanteraths do containe 3. degrées , they make in all 90. for 3. times 30. maketh 90. which degrées you may sée set downe in the Limbe of the Rete on both hands thus , 10. 20. 30. and so foorth , till you come to 90. which standeth at the very Zenith .

And these Almicanteraths are crossed with other circles called Azimuthes , The Azimuthes . that is to say , Verticall circles , which passing from the Finitor , do méet all in the Zenith , whereof though in this instrument there be set downe but 12. being 15. degrées distant one from another , yet you must imagine that there be 180. which the degrees set downe in the Finitor , in such manner as is before described do shew . And you haue to note that if the Sunne bée in the very beginning of the Azimuthes , which is at the center , then is hée full East , and if he be in the 90. Azimuth , then he is full South , and when he is in the 180. then he is full West , and when hée is in the 270. Azimuth , then hée is full North.

And if you would haue this account of the Azimuthes to aunswere the Mariners compasse , How to count the Azimuthes according to the Mariners Compasse . then diuide the number of the Azimuthes wherein the Sunne is by 11. degrées and ¼ which is 15. minutes , and the quotient will shew the rombe or winde of the Mariners Compasse , so as in your account you procéed from the first Azimuth towardes your right hand , that is , from East to South , and from South to West , and from West to the North , and so from thence againe to the East point whereas you first began . Also you haue to note by the way , that these Azimuthes doe sometime signifie the circles of position , the vse whereof you shall find set downe hereafter in the 31. position of this Treatise .

In the meane time I wil procéed in describing the Zodiaque , of the Rete , which in shape is like to an Egge , The description of the Zodiaque of the Rete called in Latine Figura ovalis , the one halfe whereof extendeth towardes the Zenith , and the other towards the Nadir or point opposite to the said Zenith , in which circle are placed the 12. Signes , whereof the 6. Northern signes , that is , Aries , Taurus , Gemini , Cancer , Leo , and Virgo are placed in the nether halfe towards the Nadir , and the other sixe Southerne Signes , that is , Libra , Scorpio , Sagittarius , Capricornus , Aquarius , and Pisces , are placed in the vpper halfe of the Zodiaque towards the Zenith , and euerie one of these Signes are diuided into 30. degrées , which are set downe with Arithmeticall Figures in the said Zodiaque thus , 10. 20. 30.

Moreouer , in the said Net are placed certaine fixed Starres to Of Starres 71 contayned in the Rete . the number of 71. whose names here do follow . Caput Ophiuci , that is , the head of Serpentarius , Aquila , the Eagle . Caput Engon , the head of Hercules . Cuspis Sagittarii , the shaft of Sagittarius . Palma Ophiuci , the hande of Serpentarius . Cauda Delphini , the tayle of the Dolphin . Romboides Delphini , which is a starre in the Dolphins backe . Cor Scorpionis , the hart of the Scorpion . Frons Borealis , Media , and Australis Scorpionis , the Northern , Southern , and middle front of Scorpio . Lucida Lyrae , the bright Star of Lyra , or Vultur cadens . Lanx Chaele Borealis & Australis , the North and South starre of the Balance . Corona Gnosiae , the crowne of Ariadna . Praecedens & sequens caudae Capricorni , that is , the former and follower in the tayle of Capricorne . Caput Draconis , the head of the Dragon Hastile Bootis , the Borespeare of Arcturus . Cauda Cigni , the taile of the Swanne . Fomahand , a star in the mouth of the Southerne Fish . Arcturus , a starre betwixt the legges of Bubulcus or Bootes . Humerus Bootis , the shoulder of Bubulcus . Dexter Humerus Cephei , the right shoulder of Cepheus . Crus Pegasi , the legge of the winged Horse . Spica Virginis , the Wheat eare in the hand of Virgo . Tres stellae in cauda Vrsae maioris , three starres in the tayle of the great Beare . Previndemiatrix , that is Virgo . Cauda Ceti , the taile of the Whale . Andromedae Scapulum , the shoulder blade of Andromeda . Cingulum Andiomedae , the girdle of Andromeda . Humerus Vrse maioris , the shoulder of the great Beare . Corvi rostrum , the beake of the Crowe . Corviala dextra , the right wing of the Crowe . Cauda Leonis , the tayle of the Lyon . Ceruix Leonis , the necke of the Lyon . Cor Leonis , the hart of the Lyon , otherwise called Regulus . Lucida Hydrae , the bright Starre of Hydra . Capita Geminorum , the heades of the two Twins named Apollo and Hercules . Cancer , the Crabbe . Canis minor , the little Dogge . Canis maior , the great Dog . Canopus , a faire starre in the left oare of the Shippe Argus . Humerus dexter Aurigae , the right shoulder of Auriga . Hircus , the Goate hanging at the backe of Auriga . Hedi , the two little Goats sucking behind at her paps . Humerus dexter Orionis , the right shoulder of Orion . Tres stellae in cingulo Orionis , thrée stars in the girdle of Orion . Humerus sinister Orionis , the left shoulder of Orion . Pes sinister Orionis , the left foot of Orion . Oculus Tauri , the Bulles eie . Pleiades , the seuen little starres in the Buls necke . Extremum Eridani , the last end of the flood Eridanus . Caput Medusae , the head of Medusa . Dextrum latus Persei , the right shoulder of Perseus . Cornu Arietis , the former Star of the Rams horne . Venter Ceti the bellie of the Whale . Iuba Ceti , the mane of the Whale .

And note that all these Starres in the Nette , How to know which stars be North or South . whose longest tippes or pointes do point from the center outward , towardes the Limbe , are Northern Starres hauing North declination , and those whose longest tippes do point inwarde from the limbe towards the Center , are Southerne Starres , hauing Southern declination . All which Starres are set downe in a Table in the beginning of his third booke , which doth not onely shew their names , but also their longitudes , latitudes , and declinations , their natures , their right ascentions and magnitudes or greatnesse . And whereas both in the Mater , and also in the Rete one selfe circle is made to haue diuers significations , the cause thereof shall plainely appeare vnto you by the vse of the Astrolabe , in séeking to finde out thereby the propositions of the foresaid booke .

And besides the partes before described , A description of the Labell . there is yet another part belonging to the forepart of the Instrument , called the Labell , the one end whereof is fastened to the Center of the Astrolabe , so as it may turne round about , and this Labell is diuided into 90. degrées twise set downe therein with Arithmeticall Figures to bée rekoned as well from the Center to the poynte of the Labell , as from the poynt thereof to the Center seruinge to dyuerse vses , yea the 90 degrées of the Label are sometimes to be repeated 4 times to make vp the number of 360. degrées , as shall hereafter plainlie appeare by the 12. proposition , shewing how to find out the right ascention of anie degrée or portion of the Ecliptique line .

And remember that the right line drawne from the Center of the Labell alongst the inward edge thereof is called the Fiduciall line , The fiducial line of the label . and is diuided into 90. small partes called degrées , which line or inward edge is alwayes to be vsed in any proposition , and not the outward edge or backe part of the Labell .

Thus much touching the fore part of M. Blagraue his Iewell with euerie particular part , with which forepart I wish you to be throughlie acquainted before you deale with the propositions here following , or with any other proposition contained in M. Blagraue his booke .

A briefe description of the backe parte of the saide Iewell .

IN the Limbe of the backe part is described the Theorique of the Sunne , to knowe therby in what Signe and degrée the Sunne is euerie day throughout the yeare , by laying the Diopter thereto , M. Blagraue calleth it a Ruler , which Diopter is made with two Pinules or square Tablets , each one pearced with two holes one greater than another , the lesser to take the height of the Sunne by his beame passing through the saide lesser holes , and the greater holes doe serue to take the altitude of the Sunne , béeing something darkened in the day time , so as hée casteth no beame , or els the altitude of any Starre in the night season by looking with your one eie , the other béeing shutte , through the two greater holes of the Pinules or Tablettes . And the middest of this Diopter is fastened with a pin to the center of the Astrolabe , so as the saide Diopter may turne round about , and the middle line of the said Diopter is called the fiducial line , because it rightlie directeth the sight of the eye to the foresaid holes , the degrées of the altitude of the Sunne or of anie Starre are set downe in the outermost space of the Limbe , diuided by 2. crosie Diameters into foure quarters , euerie quarter contayning 90. degrées , the number of which degrées are set downe in the outermost space of the Limbe of the said backe part with arithmeticall figures right ouer the heades of the saide degrées of altitude . And you haue to note , that the perpendicular Diameter signifieth the Meridian , that is , the line of South and North , that is to say , the South point at the ring or handle , and the North at the point opposite , and the other ouerthwart Diameter signifieth the right Horizon , that is to say , the line of East and West , the East being placed on the left hande , and the West on the right hand . And vnder the Theorique of the Sunne you may drawe as manie circles of such reasonable distance one from another , as in the spaces thereof may be set downe , the yeares of our Lord , the Dominicall letter for the leape yeare , and the Dominicall letter for the common yeares , the Prime or golden number , the Epact , and on what day of March or of Aprill Easter day euery yeare falleth , in such order as M. Blagraue hath himselfe described in a Table made of purpose in the seconde booke of his Iewell the 11. chapter .

Thus hauing described euery particular thing as well in the forepart as in the backe part of the saide instrument , I will now shew you how to vse the same , and howe to finde out thereby all the conclusions contayned in the Table here next following .

The Table contayning 32. necessarie conclusions to be wrought by this Astrolabe .

FIrst , howe to finde out the place of the Sunne ( that is to say ) in what signe and degree thereof the Sunne is euerie day throughout the yeare , being not leape yeare , and also the opposite point of that degree . Proposition . 1.

How to know the place of the Sunne in the leap yeare , and how to finde the leape yeare . Prop. 2.

How to take the altitude of the Sun or of any star . Prop. 3.

How to take the Meridian altitude , that is to say , the highest or greatest altitude of the Sunne or of any starre . Prop. 4.

How to know the altitude of the Sunne at any houre without seeing the Sunne . Prop. 5.

How to know the Meridian altitude of the Sunne or of any starre in the Net without seeing them . Prop. 6.

How to know the declination of the Sun , or of any starre contayned in the Net . Prop. 7.

How to find out the latitude of anie Region diuers waies . Prop. 8.

How to know the houre of the day by the Sunne , and also in what part or coast of heauen he is at that instant . Prop. 9.

How to finde the rising and setting of the Sunne euerie day in euerie latitude , and thereby the length of the day , and also in what coast or part of the Horizon he riseth and setteth . Prop. 10.

How to knowe euery day at what houre the Moone riseth and setteth , and how long she continueth aboue the Horizon and also when she is full South . Prop. 11.

How to find out the right ascension of the Sunne , or of any degree or portion of the Ecliptique . Prop. 12.

Another more readie way to finde out the right ascention of any degree or portion of the Ecliptique by the Rete . Prop. 13.

How to finde out the ascentionall difference of the Sunne or of any degree or point of the Ecliptique . Prop. 14.

How to find out the oblique ascention of the Sunne , or of any point of the Ecliptique . Prop. 15.

How to find out the right ascention of any Arke or portion of the Ecliptique , and therwith to know what time it spendeth in rising in a right Spheare . Prop. 16.

Howe to finde out the oblique ascention of any Arke of the Ecliptique in any latitude , and what time it spendeth in his rising . Prop. 17.

How to finde out the oblique descention of any point of the Ecliptique in any latitude . Prop. 18.

How to find out the oblique descention of any Arke giuen of the Ecliptique , and therewith to know the time which it spendeth in his setting . Prop. 19.

How to knowe the height of any starre at any houre without seeing the starre , and thereby to finde out in the firmament all the starres that be described in the Net , and are to be seene with the eie ▪ Prop. 20.

How to finde out the ascentionall difference of any starre , Prop. 21.

How to know the oblique ascention of any star . Prop. 22.

How to knowe what starres do neuer rise nor set in any latitude . Prop. 23.

How to know at what houre of the day or night any starre riseth or setteth . Prop. 24.

How to know howe long any starre continueth aboue the horizon in euery latitude , Prop. 25.

How to find out the starres houre , and therby to know the houre of the night . Prop. 26.

How to finde out the distance betwixt any two starres contayned in the Net . prop. 27.

Another way to know the distance of any two stars , theri longitudes and latitudes being first knowne , and also by that meanes to finde out the distance betwixt any two places vpon the earth . prop. 28.

How to find out the degree of Medium Coeli at any houre of the day , that is to say , the degree of the Zodiaque that is in the Meridian at any houre that you seeke , and also the degree called Imum Coeli . prop. 29.

How to find out the horoscope or ascendent at any time of the day or night , and thereby to haue the foure principall angles of heauen . Prop. 30.

How to find the circles of position , and to know how much the pole is elevated aboue euerie such circle in any latitude , without the knowledge whereof you cannot find out the 12. houses by this Astrolabe . prop. 31.

How to find out all the 12. houses of heauen , and therby to erect a figure at any houre of the day or night . prop. 32.

The vses of the Astrolabe , and first howe to finde out the place of the Sunne ( that is to say ) in what signe and degree thereof the Sunne is euerie day throughout the yeare , being not leape yeare , and also the opposite point of that degree . The first . Proposition .

LAy the Diopter which is on the backside of the Astrolabe vpon the day of the moneth and that end of the Diopter with his fiduciall line will shew you in what signe and degrée thereof the Sunne is that day , and the other ende of the Diopter will shewe you the opposite point to that degrée , as for example : I would know the place of the Sunne the 17. of Iuly , here by laying the Diopter vpon that day , I find the Sunne to be in the fourth degrée of Leo , and the opposite point thereof to bee the fourth degrée of Aquarius .

How to know the place of the Sunne in the Leape yeare and how to finde the leape yeare . Proposition . 2.

WHen it is leape yeare , you must alwayes adde one degree more euerie day during that yeare vnto the place of the Sunne found by the first proposition , which leape yeare you shall know by diuiding the yeare of the Lord by 4. for if there be no remander left , then that yeare is leape yeare , so shall you finde the yeare of our Lord 1596. to be leape yeare .

How to take the altitude of the Sunne , or of any starre . Proposition . 3.

TO take the altitude in any time of the day when the Sunne shineth , you must turne your face and also the left Tablet or Pinule of the Diopter towards the Sunne , holding the Astrolabe by the ring with your right forefinger or middle finger being the somewhat bowed , in such sort as the Astrolabe may hang plumbe , and then with your left hand lift the Diopter vp and downe vntill the Sun with his beame due iustlie stréeke through both the holes of each Pinule of the Diopter , so as you may sée the shadow of the two holes of the vpper Pinule to play vpon the two holes of the nether Pinule , then marke vpon what degrée of altitude the thinnest edge or fiduciall line of the Diopter falleth in the outermost skirt or border of the backe of the Astrolabe , for that is the Sunnes altitude for that present . But if the Sun be couered with a cloud , so as it shineth not cleare mough to cast any shadow , and yet so as it may be séene with the eie , then hang the Astrolabe by the ring vpon your right thomb , and turning your face towards the Sun ▪ lift vp your hand with the Astrolabe so high , as by moouing the Diopter with your left hande vp and downe , you may with your right eie ( the other béeing shut ) sée the Sunne through the greater holes of both the Pinules of the Diopter , and marke vppon what degrée of altitude the vpper ende of the Diopter falleth , and that is the altitude of the sunne at the time , and in this maner you must also take in the night season , the altitude of any starre .

How to take the Meridian altitude , that is to say , the highest or greatest altitude of the Sunne or of any starre . The 4. Proposition .

GOe into some open place whereas the sunne shineth somewhat before noonetide , and there hanging the Astrolabe vpon your right fore finger or middle finger , take the altitude of the sunne in such manner as is before taught , at diuerse times with some pawse betwixt euery time to know therby whether such altitude increaseth or decreaseth , for if it increaseth , then the sunne is not yet at the Meridian , but if it decreaseth , then it is past the Meridian , and therefore you must watch diligently to take him when hee is at the highest . And you must do the like to take in the night season , the Meridian altitude of any knowne starre , sauing that then you must hange your Astrolabe vpon your thumbe before your right eye , and to doe as is taught in the last Proposition .

How to know the altitude of the sun , or of any starre at any houre of the day without seeing the sunne or starre . The 5. Proposition .

SEt the Finitor at your Latitude , and séeke out amongst the houre lines or Meridians of the Mater in what point the Parallel or Circle of declination of the sun crosseth the houre line which you séeke , and the Almicanterath and Azimuth passing through that point do shew both the altitude and also the coast or part of heauen , wherein the sunne or starre is at that instant . This is a very necessary Proposition , for by knowing the height of the sunne at euery houre of the day in whatsoeuer signe the sunne is you may make Tables for particular dials , as Cylinders , houre Quadrants , and such like to serue any latitude , yea rather this Instrument as M. Blagraue rightly sayeth is a Table of it selfe readie made to serue such purposes . Let the example of this Proposition be thus , Suppose that at 8. of the clocke in the morning the 21. of Aprill 1592. the sunne being then in the tenth degrée of Taurus , and his declination 15. degrées Northward , you would knowe the altitude of the sunne at that houre , which by working as the rule teacheth , you shall finde to be almost 30. degrées , and that hee is about 13. degrées distant from the East towards the South .

How to know the Meridian altitude of the sunne , or of any starre euery day throughout the yeare , without seeing eyther sunne or starre . The 6. Proposition .

FIrst séeke to know the declination of the sunne or star eyther by the seuenth Proposition next following , or else by some Table , & whether it be North or South . and knowing his declination , bring the Finitor to your Latitude , and staying it there séeke out in the limbe of the mater on your right hande the said declination , and there marke what Almicanterath toucheth that point , for that Almicanterath being counted vppon the limbe of the Rete from the Finitor , doth by and by shew the Meridian altitude of the sunne or starre for that day . As for example , I would know the Meridian altitude of the sunne the first of Iuly 1592. at which time his Parallel or declination is 22. degrées and certaine minutes Northward . Here hauing layde the Finitor to the Latitude 52. I finde that the 60. Almicanterath toucheth the Parallel , and that is the Meridian altitude of the sunne that day . Againe suppose that I would know the Meridian altitude of the starre Oculus Tauri , in the Latitude 52. Here hauing brought the Finitor to the said latitude , I finde that the 53. Almicanterath toucheth his Parallel or declination , which is 15. degrées 49′· Northward , so as I find his Meridian altitude in that latitude to be 53. degrées . But you haue to note that if the sun or star haue south declination , then you must count his parallel from the Equinoctial downward towards the south Pole , so shall you find the Meridian altitude of the great dogge called Canis maior , whose Southerne declination is 16. degrées to be 22. degr. & 30′· And by knowing the Meridian altitude of any star , you may also knowe how farre he is distant from the Meridian or South line , if you subtract from his Meridian altitude his altitude taken at any other time of the same night , for the remainder will shew his distance from the Meridian , & if the starre at the time of taking his altitude , be in the East part of the firmament , then he is so much short of the Meridian , and if he be in the West , then he is so much past the Meridian . As for example , knowing the Meridian altitude of Oculus Tauri to be 53 degrées , and his other altitude newly taken to be 30. deg. I find his distance from the Meridian to be 23. deg. aswel by subtraction , as by counting vpon the limbe of the Rete the degrées contained betwixt the two foresaid Almicanteraths . And because the said starre was in the East part of the firmament when I tooke his altitude , I conclude that he wanted 25. degrées of arriuing to the Meridian , which maketh one houre and a halfe , and a little more .

How to find the declination of the sunne , or of any starre described in the net . The 7. Proposition .

THis may be done two maner of waies , first knowing the place of the sunne , séeke his place in the Ecliptique line of the mater , and looke what Parallel of the mater passeth through that degrée , and the number of that Parallel will shew the declination ( that is to say ) how farre the sunne is distant from the Equinoctiall , counting from the Equinoctiall vpon the outermost Meridian or inner limbe of the mater , and according as the signe wherein the sunne is be it Northerne or Southerne , so is the declination of the sunne , and must bee counted eyther vpwarde or downewarde accordingly . The seconde way is thus , bring the Fiduciall liue of the labell to the degrée of the sunne in the Zodiaque of the net , and the number of degrées counted vpon the labell betwixt the limbe of the net , and the place of the sunne which the labell toucheth vpon the outward edge of the sayde Zodiaque will shewe his declination . By either of these two wayes you shall finde the declination of the Sunne , being in the tenth degrée of Taurus to be 15. degrées Northward . And by this last way you may know the declination of any starre contained in the net thus . Hauing found in the net the starre whose declination you séeke , lay the Fiduciall line of the labell to the longest tippe of that starre , then count vpon the labell how many degrées are contained betwixt the limbe of the Rete , and that point whereas the labell toucheth the longest tippe of the saide starre , and that shall be his declination either Northerne or Southern according as the said tippe point either outward or inward : for if outward from the Centre , then it is Northerne , if inwarde towardes the Centre , then it is Southerne , as hath béene sayde before in the description of the Net , and by doing thus you shall find that the starre called Canis maior , that is the greater dogge , hath in South declination 15. degrées 55′· Againe you shall finde the first starre of the Rams horne called Cornu Arietis to haue in North declination 18. degrées .

How to find out the Latitude of any Region . The 8. Proposition .

FIrst you must know the place of the sunne , and also his declination , and hauing taken his Meridian altitude , recken the same amongst the Almicanterathes from the Finitor vpwardes , and turne about the Rete from the Pole arctique towards the Equinoctiall on your right hande , vntill it toucheth the Parallel of the sunne , for then looke on your left hand and you shall finde the Finitor to stand at that Latitude which you séeke . As for example , the 12. of Aprill 1591. the sunne being in the first degrée 20. minutes of Taurus , and his declination being then 12. degrées Northward , I finde his Meridian altitude to be 50. degrées , which I count vpon the limbe of the Rete procéeding from the Finitor vpwardes towards the Zenith , and then I turne the Rete vntill I haue brought that Almicanterath to the Parallel of the Sunne , which is 12. degrées , counting the same from the Equinoctiall on the right hande of the mater towardes the North Pole , and there staying the Rete , I finde that on the left hande the Finitor lyeth vpon the 52. degrée of Latitude counting from the North pole downe towardes the Equinoctiall . The common way of finding the Latitude is thus , if it be in the day time , then take the Meridian altitude of the Sunne , and if the Sunne be in any of the sixe Northerne signes , then subtract the declination of the sunne out of his Meridian altitude , and the remainder shall be the altitude of the Equinoctiall aboue your Horizon , which being taken out of 90. the remainder will shew the altitude of the Pole , but if the sunne be in any of the sixe Southerne signes , then you must adde his declination to his Meridian altitude , and the summe thereof shall be the altitude of the Equinoctiall , which being taken out of 90. the Remamder will shew the Latitude or eleuation of the Pole . But to know the latitude of any place in the night season , you must take the Meridian altitude of some knowne starre which both riseth and setteth , then after that you haue taken his Meridian altitude with your Astrolabe , you must learne to knowe his declination , and whether it be Northerne or Southerne , for if the starre haue North declination , then you must subtract his declination from his Meridian altitude , and the remainder shall be the Altitude of the Equinoctiall , which being taken out of 90. shall be the latitude or eleuation of the Pole , but if the declination of the star be Southernly , then you must adde his declination to his Meridian altitude , and that summe shal be the altitude of the Equinoctiall , which béeing taken out of 90. the remainder shall be the eleuation of the Pole . And there be diuers other wayes of finding out the latitude of any place , which I haue partly set downe in my Treatise of the two Globes about the latter ende thereof , and partly in my Treatise of Nauigation , whereas I speake of the North starre and of his Guardes .

How to know the houre of the day by the Sunne , and also in what part of heauen he is at that instant . The 9. Proposition .

TAke the altitude of the Sunne , and knowing the latitude of the place where you are , bring the Finitor of the Rete to that Latitude , and hauing stayed it there , looke in what point the Almicanterath or altitude of the Sunne crosseth the Suns Parallell or Circle of declination in the mater , and the houre line passing through that point will shew the true houre , and at that instant you may also know in what Azimuth ( that is to say ) in what part of heauen , or as the Mariners terme it in what r●mbe or winde , as East , West , North , or South &c. the Sunne is at that instant , for that Azimuth which passeth through the foresaide point , is the Azimuth of the Sunne . As for example , the 21. of Aprill 1592. the sunne being in the tenth degrée of Taurus , and his declination being then 15. degrées Northward , I find by the Astrolabe , Quadrant , or Crosse-staffe , the altitude of the Sunne in the forenoone to be 30. degrées . Wherefore I hauing laide the Finitor to my Latitude which is 52. and stayed it there , I marke in what point the Almicanterath cutteth the sunnes foresaid Parallell in the mater , and I find that the houre line of eight in the forenoone cutteth that point , which sheweth that it was then eight of the clocke in the morning , and that the sunne was about 14. degrées distant from the East towards the South .

How to find the rising and setting of the sunne euery day in euery Latitude , and thereby the length of the day , and in what coast or part of the Horizon be riseth and setteth . The 10. Proposition .

BRing the Finitor to your Latitude , and staying it there , looke in the mater where the Parallel of the Sunne doth cut the Finitor , and the houre line which crosseth that point , will shewe the houre of his rising and setting , and the number of houres betwixt his rising and setting is the length of the day , and the number of the Azimuthes betwixt that point of the Finitor , and the Centre or first Azimuth will shew you in what part or coast of the Horizon he both riseth and setteth . As for example , séeking by this rule to knowe at what houre the sunne riseth the 19. of Iune 1592. he being then in the sixth degrée 39. minutes of Cancer , and his declination then 23. degrées 19′· I find that he riseth 12. minutes before foure , and goeth downe 12. minutes after 8. and thereby I find the length of the day counting from sunne rise to sunne set to be 16. houres and 24. minutes , and that he riseth from the East towardes North 40. degrées , which according to the Mariners reckoning is Northeast and by East two quarters and somewhat more towards the North.

How to knowe euery day at what houre the Moone riseth and setteth , and how long she continueth aboue the Horizon , and also when she is full South . The 11. Proposition .

FIrst you must learne by some Almanacke or Ephemerides in what signe and degrée the Moone is , and whether it be a Northerne signe or a Southern signe , for if she be in a Northerne signe , then bring her place to the Horizon of your latitude in the Northeast part of the Astrolabe , but if she be in a Southerne signe bring her place to the said Horizon in the Southeast part of the Astrolabe , & there hauing stayed the Rete bring the labell to the place of the Sun for that day , & the labell will point to the houre of the Moones rising in the limbe of the mater : but because the Almanack or Ephemerides do not set downe the true place of the Moone but only at noone , you must therfore consider whether it be in the forenoone or in the afternoone that you séeke , for if it be in the afternoone you had néed to know how many houres are run from noone , and then for euery houre to adde halfe a degrée to the place of the Moone which you found at noonetide , but if it be in the forenoone , then you must subtract from her place at noone , for euery houre halfe a degrée , so shal you go very nigh to finds her true place in the Zodiaque for that houre , though you know not her latitude which is but 5. deg. at the most , & therfore can cause no great error in this matter . Now to knowe when she setteth you must do thus , if the Moone be in any Northern signe , them you must bring her place to the foresaid Horizon in the Northwest part of the Astrolabe , & by laying the labell to the place of the sunne , it will point to the houre of her setting , but if she be in any Southerne signe , you must bring her place to the Horizon in the Southwesth part of the Astrolabe , and the labell being laide to the place of the Sunne will point to the houre of her setting . Now if you would knowe how long time she is aboue the Horizon , and also at what houre she is full South , then count the houres betwixt her rising & setting , and that shall be the time of her continuance aboue the Horizon , and the very middest of that is the true houre that she is full South : As for example , the fourth day of September 1592. in the latitude 52. at nine of the clocke at night the Sunne being in the 21. degrées 47′· of Virgo or there abouts , & the Moone being in the 2. degrée 30′· of Capricorne , I am desirous to know when the Moone did rise that day , and by working according to the rule before set downe , I founde that she did rise about thrée of the clocke in the afternoone , and that she went downe at tenne of the clocke at night and halfe an houre past , and that she was full South or at the Meridian a little before seuen of the clocke in the afternoone . And as by this rule you may finde out the time of the rising and setting of the Moone , so may you finde the time of the rising and setting of the other fiue Planets , that is Saturne , Iupiter , Mars , Venus , and Mercury , any day throughout the yeare , so as you know their places in the Zodiaque , which the Ephemerides of Stadius doth shewe , not onely at noonetide , but also at any other houre of the day by helpe of certaine Tables made of purpose , the vse of which Tables I haue set downe in the latter end of my Treatise of the two globes .

How to find out the right ascention of the sunne , or of any degree or portion of the Ecliptique . The 12. Proposition .

TAke the Rete and the label cleane from the Astrolabe , and séeke out in the Ecliptique line of the mater the signe and degrée whose right ascention you would know , and marke what Meridian cutteth that point : that done , place the labell vpon the pinne which standeth in the very Centre of the Iewell , and make the fiduciall line thereof to lie right vppon , and alongest the Equinoctiall line eyther towardes your right hand or towards your left , according as the signe & degrée whose ascention you séeke is placed in the mater . Then marke where the foresaid Meridian cutteth through the labell , and also through the Equinoctiall line , and the number of degrées contained in the labell , betwixt the Centre and that point of the Equinoctial is the right ascention of that degrée of the Ecliptique which you séeke , which number of degrées you must count vpon the labell in this maner . For if that signe and degrée be contained betwixt the first point of Aries and the first point of Cancer , then you must begin to count vpon the labell at the Centre , and so procéede forwarde towards Cancer , the right ascention of whose first point is 90. degrées , but if the signe and degrée which you séeke , be betwixt the beginning of Cancer and the beginning of Libra , which is at the very Centre right against Aries , then you must count vpon the labell backward from 90. to 180. by adding to euery tenth space of the labell 10. degrées , so as the first number procéeding towards your left hand shall be 100. and next to that 110. and so foorth vntill you come to 130. which is the right ascention of the first point of Libra , and from Libra you must count towards Capricorne 190. then 200. and so foorth till you come to 270. which is the right ascention of the first point of Capricorne : and from thence you must count 280. 290. then 300. and so foorth towards the Centre vntill you come to 360. which is the right ascention of the last point of Pisces , so as though there be set downe in the labell but 90 degrées both forward and backwarde , yet 90. being foure times repeated , doe make in all 360. which is the whole Longitude of the Equinoctiall . As for example , you would know perhaps the right ascention of the tenth degrée of Sagittarius , here by séeking in the mater you shall find that the Meridian passing through that degrée will cut both the labell being layde towards your left hande , and also the Equinoctiall in the 248. degrées 21. minutes , which is the right ascention of the tenth degrée of Sagittarius , in counting whereof remember to beginne from 180. that is from the Centre , and this ascention agréeth with the Table of right ascentions set downe in Stadius his Ephemerides the 44. page of his booke . Againe in séeking to know the right ascention of the 10. degrée of Taurus , if you beginne to count vpon the labell from the Centre , which is the first point of Aries , you shall finde that the Meridian which passeth through the 10. degrées of Taurus will cut the labell , and the Equinoctiall line in the 37. degrée 35′·

Another morereadie way to finde out the right ascention of any degree or portion of the Ecliptique by the Rete . The 13. Proposition .

LAy the Finitor eauen with the Axletrée , signifying here the right Horizon , so as the first point of Aries may méete with the East point of the saide Horizon , and lay the labell right vpon and alongest the Equinoctiall line , eyther towardes Cancer or towardes Capricorne , according as the signe and degrée which you séeke is placed in the Ecliptique of the Mater , and marke therewith what Meridian passeth through that degrée , and follow the same vntill it cutteth the Equinoctiall , and also the labell lying thereon , and by counting vpon the labell the number of degrées to that point in such order as is before taught , you shall haue your desire . As for example , if you séeke to knowe the right ascention of the tenth degrée of Leo , here hauing placed the Finitor as before is taught , séeke out in the Mater what Meridian passing through that degrée cutteth the said Equinoctiall , and in what point , and lay the labell to the said Equinoctial according as the signe requireth , be it towards Cancer or Capricorne , and by counting the degrées vpon the labell as is before taught , that is from 90. procéeding from the limbe backeward towardes the Centre you shall finde the right ascention of the 10. degrée of Leo to be 132. degrées , which differeth from the foresaid Table but 27′· a thing of small moment considering the narrowe spaces of the Meridians in the Mater .

How to finde out the ascentionall difference of the sunne , or of any degree or point of the Ecliptique . The 14. Proposition .

BRing the Finitor to your Latitude , and there staying it looke amongst the Meridians in the Mater in what point the Parallel of the Sunne cutteth the Finitor , and the number of the Meridians contayned betwixt the Axletree , and that point shall be the ascentionall difference . As for example , in the Latitude 52. I would knowe the ascentionall difference of the Sunne being in the fourth degrée of Cancer , at which time his Parallel of declination is 23. degrees Northward , here by working according to the rule I finde the ascentionall difference to be 33. degrées 30′·

How to find out the oblique ascention of the sunne , or of any point of the Ecliptique . The 15. Proposition .

HAuing found the right ascention and also the ascentionall difference by the former Propositions , consider whether the declination of the sunne or of any other point of the Ecliptique be North or South , for if it be North , then subtract the ascentionall difference out of the right ascention , and the remainder shall be the oblique ascention . But if the declination be South , then adde the ascentionall difference to the right ascention , and the summe thereof shall be the oblique ascention . As in the former example knowing the declination of the sunne to be 23. degrees Northwarde , I subtract the ascentionall difference which was 33. degrees 30. minutes , out of the right ascention of the sunne , which was 94. degrées , and there remaineth 60. degrees 30′· which is the oblique ascention of the sunne being in the fourth degree of Cancer , and his declination being 23. degrees Northward as is before supposed .

How to find out the right ascention of any arke or portion of the Ecliptique , and therewith to knowe what time it spendeth in rising in a right Spheare . The 16. Proposition .

BRing the ende of the giuen arke founde out in the Zodiaque of the Rete , vnto the left ende of the Equinoctiall marked with D. and there staying it , lay the Fiduciall line of the labell vppon the beginning of the saide Arke , and the number of degrées in the innermost limbe of the mater contained betwixt the left end of the Equinoctiall , and the Fiduciall line of the labell shall be the right ascention of that Arke , and the houre set downe in the outermost limbe wherunto the labell pointeth , shall be the time which the said arke spendeth in his rising , so shall you finde the arke of the whole signe Taurus to be 30. degrées , and to spend in his rising two houres .

How to finde out the oblique ascention of any arke of the Ecliptique in any Latitude , and what time it spendeth in his rising . The 17. Proposition .

FInd ▪ the oblique ascention both of the beginning and ending of the arke by the 15. Proposition , then subtract the said ascention of the beginning of the arke out of the ascention of the ending of the said arke , alwayes remembring that if the oblique ascention of the beginning be greater then the other , then to adde to the lesser 360. and out of that summe make your subtraction , for the remainder shall be the oblique ascention of the whole arke , then count the number of that ascention in the innermost limbe of the mater beginning at the left end of the Equinoctiall marked with D ▪ and so procéede vpwarde towardes your right hand , and where the said ascention endeth , there laye the labell , & the houre whereunto the labell pointeth sheweth the time which the saide arke spendeth in his rising in that Latitude . As for example , I would know the oblique ascention of the whole arke of the signe of Taurus , and what time that arke spendeth in his rising : first here hauing found the oblique ascention of the first point of Taurus to be about 13. degrées in the Latitude 52. and the oblique ascention of the last point of that signe to be about 30. here by taking 13. out of 30. there remaineth 17. degrées , which is the oblique ascention of the whole arke of the signe Taurus , which 17. degrées , I count in the limbe of the mater from the end of the Equinoctiall marked with D. vpward , and there laying the labell , I finde that it pointeth to one houre and two degrées , which maketh 8′· so as I conclude thereby that the whole arke of the signe Taurus , spendeth in his oblique ascention one houre and eight minutes .

How to finde out the oblique descention of any point of the Ecliptique in any Latitude . The 18. Proposition .

HAuing found out the oblique ascention of the poynt opposite to the point giuen , adde thereunto 180. degrées , and the summe thereof shall be the oblique descention of the point giuen , alwayes remembring if the summe of addition doe excéede 360 to subtract out of that summe 360. and the remainder shall be the descention of the point giuen . As for example , you would knowe the oblique descention of the first point of Taurus , whose point opposite is the first point of Scorpio , and his oblique ascention according to Reynholdus his Tables is 222. degrées 36. minutes , whereunto if you adde 130. you shall make the totall summe to be 402. degrées 36. minutes , from which summe if you subtract 360. degrées according to the rule before giuen , there will remaine 42. degrée 36. minutes , which is the oblique descention of the first point of Taurus .

How to finde out the oblique descention of any arke giuen of the Ecliptique , and therewith to knowe the time which it spendeth in his setting . The 19. Proposition .

HAuing found the oblique descention of the beginning , and also of the ending of the giuen arke by the last Proposition subtract the descention of the beginning out of the descention of the ending , and the remainder shall be the oblique descention of the giuen arke , alwayes remembring if the subtraction can not be made to adde thereunto 360. that done , deuide that by 15. and the quotient will shewe the number of houres which the giuen arke spendeth in his setting : and remember that if in making that diuision there be any remainder left , to multiply that remainder by foure and so you shall haue the minutes . By working thus you shall finde the oblique descention of the whole Arke of Gemini to be 36. degrées 49. minutes , and in the Latitude 52. to spende in his going downe two houres and 24. minutes .

How to know the height of any starre at any houre , without seeing the starre , and thereby to find out in the firmament all the starres that be described in the net , and are to bee seene with the eye . The 20. Proposition .

LAy the labell to the houre supposed vpon the limbe of the mater , and then bring the place of the sunne for that day to the Fiduciall line of the labell , and there hauing stayed the Rete , bring the Fiduciall line of the labell to the tippe of the starre whose altitude you séeke , and the labell will shew you vpon the limbe of the Mater how many degrées that starre is distant from the South . Againe by counting vpon the label , the degrées contained betwixt the point of the labell , and the tippe of the said star you shall haue his declination , which is North or South according as the starre is Northerne or Southerne . Then bearing in mind , aswell the stars distance from the South , as also his declination , work thus , Bring the Finitor to your Latitude , and vpon the stars Parallell or Circle of declination in the Mater count from the limbe of the Mater , the distance of the starre before found , and marke what Almicanterath crosseth that point , for there is the altitude of the star at that instant , and the Azimuth which cutteth that point , sheweth in what part or coast of heauen the starre is at that present . And remember to séeke the Parallell of the starre on that side of the Mater either on your right or left hand , so as it may fall amongst the Almicanterathes , for otherwise you shall not finde that you séeke . As for example the 26. of October 1591. at nine of the clocke at night , you would know the height of the first star of the Rams horne called Cornu Arietis the Sunne being then in the 12. degrées and 12. minutes of Scorpio . Here after that you haue laid the labell to the houre supposed , and brought the degrée of the Sunne for that day to the Fiduciall line of the labell , and stayed the Rete there , bring the labell to the tippe of the starre in the Rete , and the labell will shewe in the limbe of the Mater how many degrées the starre is distant from the South , which you shall finde to be 28. degrées and 30. minutes , and the declination of the sayde starre being counted vpon the saide labell , to be 18. degrées , then kéeping those two numbers in minde , bring the Finitor to your Latitude supposing the same to be 52. degrées , and there staying the Rete , séeke for the Parallell of the starre which is 18. degrées on your right hande , and vppon that Parallel count by the Meridians from the limbe inwarde , the stars distance from the South , which was 28. degrées 30. minutes , and you shall finde that the 49. Almicanterath cutteth that point , so as you may conclude that the altitude of the starre called Cornu Arietis , was at that houre 49. degrées , and that the 47. Azimuth doth also passe through that point which sheweth that the starre is 47. degrées from the East towardes the South , which according to the Mariners account is Southeast , and somewhat more to the Southward . Now to finde out the saide starre in the firmament , if it be a starre light , you haue no more to doe but to lay the Diopter of your Astrolabe at that altitude , and to turn your face towardes that coast ( that is to say ) Southeast and somewhat more to the Southwarde , and the next bright starre which answereth in that coast to that altitude is the starre which you seeke .

How to finde out the ascentionall difference of any starre . The 21. Proposition .

BRing the Finitor to your Latitude , and staying it there , looke in what point the Parallell of the starres declination cutteth the Finitor , and the number of the Meridians contained betwixt that point and the Centre , is the ascentionall difference , so shall you find the ascentional difference of the Buls eye called Oculus Tauri , whose declination is 15. degrees 48′· to be 21. degrees .

How to know the oblique ascention of any starre . The 22. Proposition .

YF the declination of the starre be Northern , subtract the ascentionall difference out of the right ascention , and the remainder shall be the oblique ascention of the starre , but if his declination be Southerne you must adde the ascentionall difference to the right ascention , and that shal be the oblique ascention of the starre . As for example , because Oculus Tauri is a North starre , subtract his ascentional difference which is 21. degrées , out of his right ascention which as the foresaide Table sheweth is 62. degrées 30′· and there will remaine 41. degrées 30′· which is the oblique ascention of the sayd starre . But if it were a South starre as Spica Virginis , whose declination is almost 9. degrées Southwarde , and her ascentionall difference is 11. degrées 30′· then you must adde the ascentionall difference to her right ascention , which is 195. degrées 51′· so shall you finde her oblique ascention to be 207. degrées 21′·

How to knowe what starres doe neuer rise nor set in anie Latitude . The 23. Proposition .

YF the starre be a North starre hauing a greater declination be it neuer so little , then is the altitude of the Equinoctiall answerable to your latitude : As for example , suppose your latitude to be 52. which if you take out of 90. then the remainder is 38. degrées , which is the altitude of the Equinoctiall answerable to that Latitude , otherwise called the complement , then that starre neuer setteth in that Latitude : Againe if it be a South starre hauing greater declination neuer so little then the complement of your Latitude , then that star neuer riseth aboue your Horizon . As for example , in the latitude 52. the starre called Hircus , that is the Goate , being a North starre neuer setteth in that latitude because his North declination is 45. degrées , which is greater then the complement of your Latitude by 7. degr. Also the star called Lyra , whose declination is 38. degr. like vnto the complement of your Latitude doth neuer set , but only toucheth your Horizon aswell at his rising as setting , so contrariwise the star called Canopus , being a South star hauing 51. degr. 38′· of South declination , neuer riseth aboue your Horizon in the foresaid latitude 52. And by this rule you may iudge in like maner of all the rest of the fixed starres both North and South .

How to know at what houre of the day or night any starre riseth or setteth . The 24. Proposition .

FIrst marke by the outwarde or inward shooting of the longest tippe of the starre , whether it bee North or South : for if it be North , then count amongest the Meridians in the Mater so many Meridians as your Latitude amounteth to , beginning at the Axletrée , and so procéeding towardes the Northeast , which is betwixt the North pole and the Equinoctiall on your left hande , but if the starre be South , then count your Latitude procéeding from the saide Axletrée towards the Southeast , which is betwixt the ringle and the North pole on the right hande , and bring the tippe of the starre to that Meridian which now signifieth your Horizon , and there staying the Rete , bring the labell to the place or degrée of the sunne in the Zodiaque of the Rete , in which the sunne is that day you séeke , and the houre in the limbe whereto the labell pointeth , is the houre at which the starre riseth that day or night . Now to knowe when the same starre setteth , you haue no more to doe but to worke with the starre in the Northwest part in such order as you obserued before in the Northeast part of the Astrolabe . As for example , I would knowe at what houre the Buls eye called Oculus Tauri , doeth rise the last day of Iune the Sunne being then in the 17. degrée and 40. minutes of Cancer . Here because this starre is a North starre , I bring the longest tippe thereof to the 52. Meridian which is our Latitude counting from the Axletrée towardes the Northeast part of the Astrolabe which is on my left hand , for that Meridian is alwayes the Horizon seruing the Latitude 52. and there staying the Rete , I bring the Fiduciall line of the labell to the place of the sunne which at that day is the 17. degrée 40′· of Cancer as I said before , and I find that the label pointeth to one of the clocke 30′· after midnight : wherefore I conclude that Oculus Tauri riseth that day at that present houre . Now to knowe at what houre that starre goeth downe the same day , I bring his longest tippe to the said Horizon towards the Northwest , and staying the Rete there , I lay the labell to the 17. degrée 40′· of Cancer as before , so that the labell pointeth to foure houres and 20. minutes in the afternoone , at which time he goeth downe , so as he continueth at that time aboue the Horizon in the Latitude 52. 14. houres 48. minutes . And to knowe the abode of any starre aboue the Horizon , the next Proposition doth also shew .

How to knowe how long any starre continueth aboue the Horizon in euery Latitude . The 25. Proposition .

BRing the Finitor to your Latitude , and looke in what point the Parallel or declination of the starre cutteth the Finitor , and the number of the Meridians in the Mater contained betwixt the limbe ▪ and that point do shew the halfe time of his abode aboue the Horizon , which being doubled is the whole time of his abode aboue the Horizon . And in numbring the said Meridians whereof 15. do make an houre , remember to beginne to count from the right side of the Mater , procéeding towardes your left hand , and remember also that the middle Meridian or Axletrée , signifieth alwayes the sixth houre , so shall you not erre in your account . As for example , hauing brought the Finitor to the Latitude 52. looke in what point the Parallel of the foresaid starre Oculus Tauri cutteth the Finitor , & by numbring the Meridians procéeding from the limbe on your right hand towardes the left , you shall finde the Parallell of Oculus Tauri , being 15. degrées 49′· to cut the eleuenth Meridian , which being doubled and then the summe thereof deuided by 15. maketh 14. houres and 48′· as before .

How to finde out the starres houre , and thereby to knowe the houre of the night . The 26. Proposition .

FIrst hauing taken the starres altitude , set the Finitor to your Latitude , and looke at what houre-line in the Mater that starres Almicanterath and the Parallell of his declination do méete , & hauing sought out the same houre in the limbe right against that point , bring the labell thereunto , for that is the starres houre , and there staying with your fingers end the very point of the labell , bring the longest tippe of the starre to the Fiduciall line of the labell , and staying there the Rete , turne the labell to the degrée of the Zodiaque of the said Rete , wherein the sunne is that day , and the labell will point to the true houre of the night set downe in the outwarde limbe of the Mater . As for example , I suppose that the seuenth of October 1591. the sunne being then in the 23. degrée 15′· of Libra , I tooke the altitude of the starre Hircus , that is the Goate , which I found to be 20. degrées , here hauing set the Finitor to my Latitude 52. I looke in the Mater at what houre line that Almicanterath , and the Parallell of the same starre which is 45. degrées Northwarde doe méete or crosse one an other , and I find that they méete iust vpon the fourth houre line of the forenoone , wherefore I séeke right against that point the same houre in the outermost limbe of the Mater , for that is the starres houre , and hauing placed the labell at that houre , I stay it there with my finger vntill I haue brought the longest tippe of the starre Hircus vnto the Fiduciall line of the labell , and there staying the Rete , I turne the labell to the degrée of the sunne which is 23. degrées 15. minutes of Libra , and I sée that the labell pointeth to the seuenth houre of the night set downe in the limbe of the Mater and halfe an houre past . Maister Blagraue sayth that the sooner you take the altitude of the starre whereby you séeke to knowe the houre of the night , you shall haue the houre more truely .

How to finde out the distance betwixt any two starres contained in the Net . The 27. Proposition .

FIrst consider whether the declination of both stars be eyther South or North , or that the one bee South and the other North , for if both their declinations be South or North , then you must not leaue to turne the Rete to and fro vntill you haue brought the longest tippes of both starres to one selfe Meridian in the Mater , that done count vpon the said Meridian howe many degrées are contained betwixt the tippes of the said starres , for that is the distance betwixt them . But if the one starre haue North declination and the other South then turne the Rete to and fro vntill both starres doe lie vpon two such seuerall Meridians as ech of them is equally distant on ech hand from the Axletrée , then count how many degrées or Meridians are contayned betwixt the Axletrée and either of those stars ( which you will ) and that is their distance , so shall you finde the distance betwixt Oculus Tauri and Canis minor being both North stars to be 46. degrées 20′· and the distance betwixt Oculus Tauri and Canis maior , whereof the first is a North starre , and the other a South starre to be iust 46. degrées .

Another way to knowe the distance of any two stars , their Longitudes and Latitudes being first knowne , and also by that meanes to finde out the distance betwixt any two places vppon the earth . The 28. Proposition .

FIrst séeke to knowe the difference of their Longitudes by subtracting the lesser Longitude out of the greater , then count that difference from the outermost Meridian of the Mater towardes the Center , and marke well that Meridian at which your account endeth , then number vpon the limbe of the Mater from the Equinoctiall the greater Latitude either Northward or Southward according as the Latitude is , and to that point bring the Zenith of the Rete , then vpon the selfe same Meridian before marked count from the Equinoctiall the lesser Latitude , and looke what Azimuth passeth through that point , for the degrées which are contayned betwixt that point and the Zenith shal be the distance : and thus doing you shal find the distance betwixt Oculus Tauri & Canis maior to be 46. degrées and 15′· and the distance betwixt London and Venice to be 12. degrées and 20′· which 12. degrées being multiplyed by 60. maketh 720. miles , whereto if you adde for the 20′· 20. miles , it will make in all 740. miles .

How to find out the degree of Medium coeli at any houre , of the day ( that is to say ) the degree of the Zodiaque that is in the Meridian at any houre that you seeke , and also the degree called Imum caeli . The 29. Proposition .

FIrst séeke out the place of the sunne for that day in the Zodiaque of the Rete , and hauing laid the labell to the houre supposed vpon the limb of the Mater , bring the place of the sunne to the Fiduciall line of the label , and there staying the Rete , looke what degrée of the said Zodiaque cutteth the noone line or Meridian at noonetide , which you shall easily find by laying the labell to the houre of 12. at noone , for the Fiduciall line of the label crossing the Zodiaque , will shew the degrée of mid heauen at that houre . As for example , the 26. of Iune 1592. I would know the degrée of Medium coeli , at eight of the clocke in the morning , the sunne being that day in the 14. of Cancer . Here by laying the labell to the said houre in the limbe of the Mater , I bring the place of the sunne to the Fiducial line of the labell , and there hauing stayde the Rete , I bring the labell to the twelfth houre at noone , and I find that the labell cutteth the Zodiaque of the Rete in the 18. of Taurus , which at that houre is the degrée of Medium coeli , whose point opposite is the 18. degrée of Scorpio , and that is at that houre Imum coeli ▪ which the labell being laid to the 12. houre of midnight will shew .

How to finde out the Horoscope or ascendent at any time of the day or night , and thereby to haue the foure principall angles of heauen . The 30. Proposition .

HAuing laid the labell to the houre giuen vppon the limbe of the Mater , stay it there with your finger vntill you haue brought the place of the sunne for that day vnto the Fiduciall line of the labell , and there staying the Rete , looke what degrée of the Zodiaque therof toucheth or crosseth the Horizon answerable to your Latitude , for that is the ascendent at that present houre . As for example , I would knowe the ascendent at eight of the clocke at night the twelfth of October 1580. the sunne being then in the 28. degrée 14′· of Libra . Here hauing layde the labell vpon the sayd houre , I bring the 23. degrée 10′· of Libra to the Fiduciall line of the labell , and there staying the Rete , I find by helpe of the labell that my Horizon which is the 51. 40′· Meridian , counting on both hands from the Axletrée , that the first degrée of Cancer , doth crosse my Horizon in the Northeast quarter , wherefore I affirme that to be the ascendent or first house , whose point opposite being the first of Capricorne , is the descendent or seuenth house , then by bringing the Fiduciall line of the labell to the South end of the Equinoctiall at which the ringle hangeth , I finde by helpe of the labell that the 24. degrée 40′· of Aquarius cutteth the Equinoctiall , which is the tenth house or Culmen coeli , whose point opposite being the 24. degrée 40′· of Leo , is the fourth house , otherwise called Imum coeli , and thus you haue all the foure principall houses of heauen for that houre , as you may sée in this figure here following .

A figure of the 4. Angles of heauen , made the 12. of October 1580. the sunne being then in the 28. degree 10′· of Libra , for the latitude 51. 40′·

But the difficultie of finding out the true ascendent consisteth in knowing whether it is to be sought in the Northeast quarter , or in the Southeast quarter of the Iewell . The Northeast quarter is that which lyeth betwixt Imum coeli , and the North pole or East end of the Axletrée , because in this case the Axletrée signifieth the line of East and West , and the Equinoctiall signifieth the line of South and North , at the South end whereof is fastened the ringle or handle . But in the former example you may plainely sée that the North part of the Zodiaque of the Rete doth cut the Horizon , as well in the Northeast quarter as in the Southeast quarter with two seueral degrées and signes , for in the Northeast quarter the Zodiaque cutteth the Horizon with the first of Cancer which is the ascendent , and in the Southeast quarter it cutteth the Horizon with the eight of Taurus , which is not the ascendent : for you haue to vnderstand that euery degrée of the Zodiaque doth both rise and set either towards the North or towards the South , the first point of Aries and of Libra onely excepted , both which do rise right East , and goe downe right West , euen as the Equinoctiall doth , whereof M. Blagraue doth gather a rule how to find out by his Iewell when the ascendent is to be sought either in the northeast quarter of the Mater , or in the southeast quarter , which is thus : The ascention of any of the sixe Northerne signes is to be sought for in the Northeast part of the Iewell , & the ascention of any of the sixe Southerne signes is to be sought for in the southeast part of the Iewell . For although that the North part of the Zodiaque of the Rete containing the 6. Northerne signes cutteth the Horizon answerable to your Latitude , aswel in the Northeast as in the Southeast part of the Iewell , yet you must séeke the ascendent in the Northeast part , and not in the Southeast part of the Iewell , because that euery degrée of any of the 6. Northerne signes riseth Northernly : so contrariwise if the South part of the Zodiaque containing the sixe Southerne signes doe cut the Horizon , aswell in the Northeast part as in the Southeast part of the Iewel , yet you must séeke the ascendent in the Southeast part , and not in the Northeast part of the Iewell . As for example , the second day of August 1592. the sun being in the 20. of Leo , I would know the ascendent at foure of the clocke in the afternoone : here hauing laid the labell to that houre , & brought the place of the sun to the Fiduciall line thereof , I find that the 13. degrée 30′· of Aquarius doth cut the Horizon seruing to your latitude 52. in the Northeast part of the Iewell , and that the 19. of Sagittarius cutteth the said Horizon in the Southeast part of the Iewell , which must be the ascendent , because that euery degrée of any of the southerne signes riseth Southernly , and not Northernly .

How to find the Circles of position , and to know how much the pole is eleuated aboue euery such Circle in any Latitude , without the knowledge whereof you can not finde out the twelue houses by this Astrolabe . The 31. Proposition .

FIrst bring the Zenith of the Rete to your Latitude , so shall the Azimuthes become circles of position , then vpon the Parallel of declination of the point giuen , according as the declination thereof is north or south : count from the limbe on your right hand , the number of the houres giuen amongst the Meridians in the Mater , & the Azimuth which the Meridian of that houre cutteth shal be the Circle of position , which had , you shall find the eleuation of the pole aboue the Circle of position thus : Count your Latitude amongst the Almicanteraths from the Zenith vpon the Circle of position found , & to that point whereas your Latitude endeth , bring the Fiducial line of the label , & reckon vpon the label how many degrées are contained betwixt the limbe of the Rete & that point , for the number of those degrées is the eleuation of the Pole aboue the Circle of position . As for example vsed by M. Blagraue himselfe , the 12. of October 1580. the sunne being then in the 29. degrée 10′· of Libra , and his South declination 11. deg. 10′· I would know at 8. of the clock at night in what Circle of position the sunne at that time was , here I bring the Zenith to the Latitude of Reading , which is 51. deg. 40′· then for 8. of the clocke I count 8. houres amongst the Meridians in the Mater , from the South part of the Iewell vppon the Parallell of the sunne being then 11. deg. 30′· and by attributing to euery houre 15. Meridians , I find that the Meridian whereas the 8. houre endeth doth crosse the 28. & ½ . Azimuth , counting the Azimuths from the Zenith line , and that is the Circle of position wherin the sun was then vnder the earth in the Northeast quarter . Now to knowe how much the Pole is eleuated aboue that circle of position I do count from the Zenith the foresaid Latitude 51. degrées , 40′· vpon the said Circle of position , and to the point where that Latitude endeth I bring the labell , & counting thereon from the limbe of the Rete the degrées contayned betwixt the said limbe , and the foresaid point , I find the number of them to be 43. degrées 30′· wherfore I conclude that the Pole is eleuated aboue the said Circle of position 43. degrées 30′·

How to find out all the 12. houses of heauen , and thereby to erect a figure at any houre of the day or night . The 32. Proposition .

YOu shall vnderstande the order of this better by this one example giuen by M. Blagraue himselfe , then by manifold rules which be also so plainely expressed and obserued in this example , as you néede none other instruction to erect the like figure in any Latitude , and at any houre of the day or night . Suppose then that you would erect a figure for one that was borne the twelfth of October 1580. at eight of the clocke at night in the Latitude 51. 40′· the sunne being then in the 28. degrée 10′· of Libra , and his declination being at that time 11. degrées 10′· Southward : Here first set downe in your figure alreadie drawne the foure Angles before found by the 30. Proposition answerable to the said day & houre , then to find out the rest of the houses do thus , bring the Zenith of the Rete to the latitude presupposed , which is 51. degrées 40′· & staying it there , marke what number of Azimuthes or Circles of position doth cut euery 30. degrée of the Equinoctiall , counting the degrées of the Equinoctiall by helpe of the Meridians from the limbe towardes the Centre , but you must count the number of the Azimuthes or Circles of position from the Zenith line towardes the limbe , of which Circles of position there be no more but sixe at the most ( that is to say ) thrée betwixt the limbe and the Centre on the one side , and as many on the other side of the Centre , the eleuation or rather depression of euery one whereof you must séeke to knowe by the last Proposition . But M. Blagraue sayth that you néede to knowe the eleuation or rather depression , but only of two Circles of position , that is of that which sheweth the eleuenth and third house , and of that which sheweth the twelfth & second house , and so sayth Stadius also in the beginning of his Ephemerides , treating of the 12. houses , for hauing them you haue all the rest . The order of working according to M. Blagraue his rule is thus : hauing brought the Zenith to the foresaid Latitude , looke what Circle of position cutteth the Equinoctiall in the first 30. degrées next to the limbe , and you shall finde that the 47. Azimuth or Circle of position ( counting from the Zenith line ) cutteth that point of the Equinoctiall , and therewith serueth to the eleuenth house , and also to the third house , and the eleuation of this Circle is 32. deg. then from thence tell vppon the Equinoctiall 30. degrées more , which do make in all 60. degrées , and through that point you shal finde the 19. Azimuth or Circle of position to passe , whose eleuation is 47. degrées and this Circle serueth to the twelfth house & to the second house . Now kéeping well in mind those two last eleuations , that is , 47. and 32. worke thus : Lay your labell to the eight houre of the morning , which is 30. degrées distant from the right Horizon or Axletrée , and bring to the Fiduciall line thereof the Culmen coeli or 10. house first set downe in your figure which is the 24. of Aquarius , and staying it there , looke what degrée of the Ecliptique cutteth the 32. Horizon which serueth to the 11. house counting from the right Horizon towards your left hande . And by following that Horizon vp towards the North pole , you shall finde that the 25. degrée 30′· of Pisces cutteth the same Horizon very nigh vnto the Pole , but by my instrument I finde it to be the 27. degrée of Pisces , which perhaps is not truely made , and therefore set downe in the eleuenth house of your figure 25. degrées 30′· of Pisces . Then bring the foresaid Culmen coeli together with the labell to the tenth houre of the morning , which is 30 degrées further towards the South , and staying it there , looke what degrée of the Ecliptique cutteth the 47. Horizon , seruing to the twelfth house , and you shall find that the 20. degrée of Taurus cutteth that Horizon , wherefore set downe the 20. of Taurus in the twelfth house of your figure , then bring the Culmen coeli together with the labell to the 12. houre at noone , and you shall sée the ascendent which is the first of Cancer to cut the oblique Horizon , which is 51. degrées 40′· and from thence bring the label and culmen coeli to two of the clocke in the after noone , and staying it there you shall sée the 21. degrée of Cancer to cut the 47. Horizon which sheweth the second house and also the twelfth house as before , wherefore set downe 21. of Cancer in the second house of your figure , that done , remooue the labell together with Culmen coeli to foure of the clocke in the afternoone , and there staying it , you shall sée the seuenth of Leo to cut the 32. Horizon which sheweth both the third house and the eleuenth house as before , wherefore set the seuenth of Leo in the third house of your figure , so haue you eight of the houses , that is to say , the first , the second , the third , the fourth , the seuenth , the tenth called Culmen coeli , the eleuenth and the twelfth , so as there want onely foure , that is , the fifth , the sixth , the eight and the ninth house . The fifth is opposite to the eleuenth , the sixth to the twelfth , the eight to the second , and the ninth to the third , whose opposite signes ech one hauing like number of degrées are to be placed in the foure houses of your figure that be wanting . As in the fifth houses the 25. degrée 30′· of Virgo , in the sixth house the 20. degrée of Scorpio in the eight house the 21. degrée 30′· of Capricorne , and in the ninth house the 7. degrée 10′· of Aquarius , so haue you all the twelue houses . And by this meanes M. Blagraue sayth that you may make the places of the twelue houses to serue for euer in any Latitude , so as you doe distinguish from the rest with some colour those two Horizons whereof the one doth shew the eleuenth and the third house , and the other Horizon sheweth the twelfth and second house , yea and by this meanes you may ( as he sayth ) make as true Tables to finde out the twelue houses in euery Latitude as those that be calculated of purpose .

The figure of the foresaid twelue houses .

The figure of the heauens the twelfth day of October 1580. at eight of the clocke in the afternoone , for the Latitude 51. 40′·

A new and necessarie Treatise of Nauigation containing all the chiefest principles of that Arte. Lately collected out of the best Moderne writers thereof by M. Blundiuile , and by him reduced into such a plaine and orderly forme of teaching as euery man of a meane capacitie may easily vnderstand the same .

They that goe downe to the Sea in ships , and occupie their busines in great waters : These men see the workes of the Lord and his wonders in the deepe . Psalme . 107.
Of Nauigation , what it is , and with what order the principles thereof are here taught by Master Blundeuill , according to the rules of the best moderne writers of that Arte.

NAVIGATION is an Arte which teacheth by true and infallible rules , howe to gouerne and direct a ship from one Port to another safely , rightly , and in shortest time : I say here safely , so farre as it lieth in mans power to performe . And in saying rightly , I meane not by a right line , but by the shortest and most commodious waye that may be found : And by saying in shortest time , I meane thereby according as the ship is good of sayle , and according as both winde and tide shall serue .

The meanes to attaine to this Arte as to all other Artes are two parts , whereof euery Arte consisteth , that is Methode and Practise , which is as much to say heere as instruction and experience : For of the chéefe points of this Arte , some are to be learned by instruction , and some onely by experience : for what instruction I pray you wil serue to make a good coaster , that is to say , to know any Cape or Cliffe when he séeth it , vnlesse he hath first séene it & hath before taken good markes thereof . Also to know the currents in all places , the deapth and qualities of waters , sandes , flats , or shoulds , and such like , the perfect knowledge whereof consisteth chiefely in experience . Of which experience I minde not here to treate , because I knowe that a Dutchman called Wagoner or Aurigarius hath lately written of these things notably wel , & especially for these our seas and the Northeast seas , whose booke I doubt not but that some of our learned sea men will set foorth in our mother tongue , with some augmentation of their owne experience for the East and West Indies , for the profite of their countrie and to their owne praise & commendation . In the meane time I will procéede with instruction , the chiefe pointes whereof are these here following : first to know the vses of such instrumentes as euery skilfull sea man ought to haue with him that mindeth to sayle any long voyage , which are these here following : First a perfect Kalender or Ephemerides , the Mariners ring or Astrolabe , the Crosse-staffe called of the Spaniardes Balla stella , the two Globes both Celestiall and Terrestriall : Of all which things I haue in a manner alreadie written seuerall Treatises . Item an vniuersall Horologe , to knowe thereby the houre of the day in euery Latitude , and a Nocturnlabe to know thereby the houre of the night . And it were necessarie for him that sayleth long voiages to carry with him a Topographicall instrument to describe therby those straunge Coasts & Countries wherby he sayleth , the vse of which instrument is plainely taught by William Borne in his booke called the Treasure of Trauailers , and by Maister Digges in his booke called the Pantometria , wherefore I shall not néede to speake any further thereof : Also the Mariners compasse , whereof I do not only make a plaine description and shew the vses thereof , but also how to finde out the variation of the same , according to the precepts of the best moderne writers therof . And lastly the Mariners Carde , whereof whilest I speake , I doe not onely shewe the making thereof , and how to drawe the Parallels in better sort then they are vsed to be drawne in the common Cardes , but I set downe therewith the chiefest vses of the Card , amongst which is taught how to know by help of the Card and certaine Tables made of purpose , what way your shippe hath made in sayling by any rombe or wind , yea in sailing right East & West , which heretofore ( as Cogniet sayth ) hath béene thought a thing vnpossible to be knowne , but only by coniecture : And being out of your way how to know in what place you are , & to marke the same in your Card that you may the more readily direct your ship againe to the place whereunto you would go . But all these instruments serue to little purpose , vnlesse you know also the North star with his guards , & diuerse other stars situated aswel towards the North as towardes the South pole , together with their Longitudes , Latitudes , declinations , and greatnesse , to knowe thereby the Latitude of any place , & the houre of the night : also the course of the sunne and his declination , by helpe whereof you may know the Latitude of any place , the times and seasons of the yeare , the houre of the day , and the length of the day and night in euerie latitude . And finallie you must know the course of the Moone , whereon dependeth the knowledge of the tydes in all places : Of all which thinges I mind here to treat in such order as the Chapters hereafter following doe shew .

A generall Kalender or Almanacke for euer , containing these 9. Chapters next following .

HOw to find out the golden number euery yeare . Chap. 1.

How to finde out the Epact in euery yeare . Chap. 2.

How to know the Epact by the Mariners rule vpon your thombe . Chap. 3.

How to knowe the age of the Moone in everie moneth throughout the yeare . Chap. 4.

How to know the change , ful , and 4 quarters of the Moon , euerie Moneth throughout the yeare . Chap. 5.

How to know in what signe and degree the moone is euery day throughout the yeare . Chap. 6.

How to find out the moouable feastes everie yeare , onelie by knowing the day of coniunction in the moneth of Februarie . Chap. 7.

How to finde out the circle of the Sunne , called in Latine Cyclus Solaris , and thereby the Dominicall Letter in euery yeare . Chap. 8.

How to find out the number of indiction . Chap. 9.

A briefe description together with the vse of the diurnall Table or Almanacke of Iohannes Stadius . Chap. 10.

How to find out the place of any Planet by the Ephemerides .

Chap. 11.

Of the Mariners Ring or Astrolabe , and of his crosse staffe Chap. 12.

A briefe description of M. Hood his crosse staffe , and of all the partes thereof . Chap. 13.

How to set the partes of M. Hoods staffe together to serue such Astronomicall vses as do chiefly belong to the Mariner . Chap. 14.

The shape or figure of the foresaide staffe , hauing all his partes set together to serue for Astronomicall vses . Chap. 15.

How to take the altitude of the Sunne at any houre that he is to be seene with the eie by M. Hoods staffe . Chap. 16.

Howe to take the altitude of any Starre with M. Hoods staffe . Chap. 17.

How to take the distance betwixt two stars with M. Hoods staffe . Chap. 18

Of the Mariners Astrolabe . Chap. 19.

A briefe description of the Mariners Astrolabe , and the vse thereof . Chap. 20.

A brief description of the Mariners crosse staffe . Chap. 21

The vses of the Mariners crosse staffe . Chap. 22.

Of the Wind , what it is , and of the divers kindes and names thereof . Chap. 23.

A briefe description of the Mariners Compasse , and vse thereof . Chap. 24.

Of the Lodestone , and of the variation of the Compasse in Northeasting and Northwesting . Chap. 25.

How to finde out the variation of the Compasse in everie latitude . Chap. 26.

Of the Mariners Card , and of the making therof . Chap. 27 ▪

The shape and figure of the first liniaments of the Mariners Carde drawne after the olde maner , and how to set downe the places of the land or sea therein . Chap. 28

A Table to drawe thereby the Parallels in the Mariners carde , together with the vse thereof in truer sort than they haue bene drawne heretofore . Chap. 29.

The draught of the Meridians and Parallels of the Mariners Carde or nauticall Planispheare according to the former table . Chap. 30.

The foure chiefest vses of the Mariners card . Chap. 31

How to know the way of your ship , and how many leagues are to bee accounted for one degree of latitude in euerie rombe whereby you saile . Chap. 32.

How to account the leagues in sayling directlie East or west without changing latitude or altitude of the Pole . chap. 33.

A Table to helpe you to know what way your shippe hath made in sayling right East or West without chaunging your latitude together with a brief description & vse thereof . chap. 34.

An example of counting the way of your shippe in sayling right West . chap. 35.

An other example of counting the way of your ship in sayling right East . chap. 36.

To know how much you goe out of your way in sayling by one wrong rombe or by more . chap. 37.

Of the North Starre , otherwise called the Lodestarre , and of his guards , and how to know the same . Chap. 38.

The vses of the North starre and of his guards . chap. 39.

To knowe by helpe of a little Table made according to the Mariners rule touching the 8. principall rombes , shewing how much and when the Lodestarre is either aboue or beneath the Pole , that you may know thereby the true altitude of the Pole in taking the height of the Lodestarre with your Astrolabe or crosse staffe . chap. 40.

How to make an Instrument which wil shew at any houre of the night how much the Lodestar is either aboue or beneath the Pole in everie other rombe as well as in the 8. principall rombes , and also the true houre of the night . chap. 41.

How to know by the foresaid twofold instrument as wel the mounting and descending of the North star as the true houre of the night both at one instant and also the elevation of the Pole , Chap. 42.

What stars are to be observed by those that sayle beyond the Equinoctiall vnder the South pole . chap. 43

Of the Sunne , and of his motion , and of the chiefest apparances belonging to him . chap. 44

A Table shewing the declination of the Sunne euerie day throughout the yeare , and the vse thereof . Chap. 45.

Of the foure seasons of the yeare , that is , Spring time , Summer , fall of the lease , called otherwise Autumne , and winter . Chap. 46.

How to knowe when the Sunne riseth and setteth in everie latitude , and thereby the length of the day and night , and also in what rombe or wind he riseth and setteth , and how much he declineth everie day from the Equinoctiall either Northward or Southward . Also howe to know the elevation of the Pole , otherwise called the latitude of any place , by knowing the Meridian altitude of the Sun , & his declination . Chap. 47.

Of the shaddow of the Sunne , and how to know therby the houre of the day in any latitude by help of an vniversall diall . Chap. 48.

Of the Moone and of all her diuers motions . Chap. 49.

How to know in what signe the point Auge of the Moone is in any yeare . Chap. 50.

When the Moone is saide to be in Coniunction with the Sunne , or to be at the full , and what her greatest latitude is aswell from the Ecliptique line , as from the Equinoctial . Chap. 51.

How to know in what part of the Zodiaque the head of the Dragon is euerie yeare . Chap. 52.

How to know the tydes in any place by the Moone . Chap. 53.

How to know by helpe of an instrument the tydes at any place . Chap. 54.

How a generall Rutter shewing the tydes in al places shuld be made . Chap. 55 ▪

A general Kalender or Almanacke for euer .

WHat this word Kalends signifieth , & from whence it is deriued is before set downe in the firste booke of my Treatise of the Spheare the 45. chapter . But because there be generall rules to knowe the coniunction of the Moone with the Sunne , her full , and all her foure quarters , and also the mooueable feasts and Dominicall letter , and such like things easilie to be learned without the help of any particular Kalender , I thought good first to set downe this general Ralender , contayning 9. propositions , as followeth .

How to find out the Golden number euery yeare . Chapter . 1.

THe Golden number is the number of 19. procéeding from 1. to 19. and so to begin again at 1. And it is so called because it was sent in golden letters from Alexandria in Aegypt to Rome : For in 19. yeares the Moon doth make all her sundry motions & changes , and returneth again to the place where she first began . And to finde the foresaid number the way is thus . Adde 1. to the yeare of the Lord wherof you inquire , and diuide the same by 19. and the remainder shall be the Golden number for that yeare , as for example , being desirous this present yeare 1590. to knowe the Golden number , I adde 1 to the said yeare , and so make it 1591. which being diuided by 19. there remaineth 14. which is the Golden number of this present yeare .

But when there is no remainder , then 19 is the Golden number , and remember that the Golden number beginneth alwaies at the first of Ianuary , and the Epact the first of March .

How to finde out the Epact in everie yeare . Chapter . 2.

THe Epact is a number not excéeding 30. because the Moone betwixt change and change , neuer passeth 30. dayes , and thereby the common Lunar yeares consisting of twelue Moones is lesser than the Solar yeare by 11. dayes , for to euerie Moon are attributed no more but 29. dayes and a halfe , which make in all 354. dayes , so as the common Solar yeare consisting of 365. dayes excéedeth the Lunar yeare 11. dayes , from whence the Epact taketh his originall , which Epact is found thus . Multiply the golden number of the yeare by 11. the product whereof if it be vnder 30. then it is the Epact. But if the product be aboue 30. then diuide the product by 30. and the remainder shall be the Epact. As for example , to know the Epact in the yeare 1590. the golden number béeing 14. as before : here hauing multiplyed the same by 11. and diuided the product thereof by 30. I find the remainder to be 4. which is the Epact of the said yeare . Also by knowing one former Epact you shal haue it euer after , by adding thereunto 11. and if the number doe excéede 30. then you must diuide the same by 30. and the remainder shall be the Epact , as by adding 11. to 4. I know the Epact shall be the next yeare , which is 1591. the number of 15. and by adding 11. to 15. the Epact in the yeare of our Lord 1592. shall be 26. and so forth .

How to knowe the Epact by the Mariners rule vpon your thumbe . Chapter . 3.

FIrst , you must suppose the inside of your left thumbe to be diuided into thrée spaces , and the nethermost space to containe 10. the middle space 20. and the highest space towards your thumbes end , to containe 30. and knowing first the golden number , begin to tell the same at the nether space saying there 1. at the middle space 2. at the thirde space 3. then beginne againe at the lowest space , and there say 4 , and so continue your account still after that manner , vntil you haue the full summe of the Golden number , and marke vpon what space the full summe falleth , for the Golden number being added to the number of that space , doeth shewe the Epact , so as the totall summe doth not excéede 30. for then you must subtract 30. and the remainder shall bée the Epact : as for example , in the yeare 1591. the Golden number is 15. which béeing counted vppon your thombe in such order as is before taught , it will fall vpon the highest space , which is 30. to which if you adde the Golden number , which is 15. it will make in all 45. from which summe if you subtract 30. there will remaine 15. which shall bee the Epact for that yeare , so as the Epact and Golden number in that yeare are like numbers . For euery thrée yeares they are alwayes like , as when the Golden number is either 3. 6. 9. 12. 15. or 18. the Epact hath also like number .

Howe to knowe the age of the Moone in everie moneth throughout the yeare . Chapter . 4.

ADde to the Epact the number of monethes from the beginning of March , together with the moneth wherein you séeke , and also the number of the dayes past of that moneth , wherein you séeke , and the summe of such addition will shewe you the age of the Moone , as for example , I would know the age of the Moone the sixt day of December in the yeare 1590. Here knowing the Epact of that yeare to bee 4. I adde thereunto the number of the moneths from the beginning of March , which are 10. moneths , and also the number of daies of December , which are sixe , and the summe thereof is the age of the Moone , if the sum bee lesse than 30. But if the summe of such addition do excéede 30. then you must subtract 30. and the remainder shall bée the age of the Moone , so as the moneth wherein you séeke haue 31. dayes , for if it hath lesse than 31. dayes , then you must subtract but 29. and the remainder shall bee the age of the Moone , as for example , suppose you séeke the age of the Moone the 22. of Nouember in the yeare 1589. in which yeare the Epact was 23. here by adding to the Epact first 9. Moneths , and then twentie two dayes , it maketh in all 54. out of which by subtracting 29. because Nouember hath but 30. daies , the remainder is 25. which was the age of the Moone in that moneth , day , and yeare .

How to know the change , the full , and foure quarters of the Moone euerie moneth throughout the yeare . Chapter . 5.

MArtin Cortes in his booke called the Art of Nauigation teacheth a rule to find the day of the change , in euerie moneth by knowing the age of the moone that day you séeke , and then by reckoning from that day backward the number of the dayes that went next before , or els by taking the age of the Moone out of the dayes of the moneth that went next before , as for example , the last of October 1592. I finde by the fourth proposition the age of the Moone to be 5. which being taken out of 31. ( for so many daies October hath ) there remaineth 26 , which day by this meanes should be the day of the change , but the very day indéed was the 25. of the said moneth , for the former rule to know the age of the Moone is not so true it selfe , but that sometime it will fall a day either ouer or vnder . But Gemma Frisius teacheth to find out the day of the change in euery moneth thus . Adde to the Epact the number of the moneths from the beginning of March whereof that Moneth wherein you séeke the change , must be counted as one , and then subtract the product or sum of that addition from 30. and the remainder will shew the day of the change , which rule I finde to be true so long as the summe of the addition doth not excéed 30. but when the sum of the addition is more than 30. which will commonlie chance when the Epact is a great number , as 26. or 29. he teacheth no rule for that , wherefore I thinke it best to take such summe out of 59. and the remainder shall be the day of change , or at the most but one day ouer or vnder , and hauing the day of the change , you shall haue the day of the full by adding 15. dayes more to the day of change : And hauing the change and the full , you shall easily haue all her foure quarters by adding or subtracting 7. dayes .

Howe to know in what signe and degree the Moone is euerie day throughout the yeare . Chapter . 6.

SOme doe set downe rules to know in what signe and degrée the Moone is euerie day , but such as are not true . And to say the truth the Moone hath so many diuers motions as it cannot be done but by speciall Tables calculated of purpose . And by what rule soeuer you work , you must first know the place of the Sunne , which the Ephemerides most trulie sheweth , and in looking for that , you shall also find hard by it the place of the Moone , that is to say , in what signe and degrée shée is euery day , and so orderly the place of any other Planet , and therefore I leaue to speake any further thereof .

How to find out the moouable Feastes euerie yeare , onely by knowing the day of coniunction in the moneth of Februarie . Chapter . 7.

THe way to finde out the Coniunction in euerie moneth is before taught in the fift proposition . And hauing the day of the coniunction in Februarie , you may assure your selfe that the next tuesday following is alwayes Shrouetuesday , for though the coniunction it selfe doe fall vppon a Tuesday , yet the next Tuesday after that , shall bee Shroue-tuesday , and the next Sunday after that , is called Quadragesima , which is the first Sunday in Lent , and sixe weekes next af-that is Easter day , whereunto if you adde fiue wéekes more , that is to say , 35. dayes , you shall haue Rogation Sunday , and 4. dayes next after that is Ascention day , and tenne dayes next after Ascention day is Pentecoste or Whitsunday , and seuen dayes next after that is Trinitie Sunday , and foure dayes next after that is the Feastiuall day called Corpus Christi . And you haue to note that the Sunday called Adventus Domini , which wee call the first Sunday in Aduent , is alwayes the fourth Sunday before Christmasse day , and the Sunday called Septuagesima is the third Sunday before Quadragesima , otherwise called the first Sunday in Lent , and betwixt Septuagesima and Quadragesima there are other two Sundayes , whereof that which is next before Quadragesima is called Quinquagesima , and the next Sunday before that is called Sexagesima .

How to finde out the circle of the Sunne , called in Latine by a Greekish name Cyclus Solaris , and thereby the Dominicall Letter in euery yeare . Chapter . 8.

THis Circle was inuented more to finde thereby the Dominicall Letter than to shew any greate chaunges of the Sunnes motions therein : And because there bee seuen dayes in the weeke , commonlie signified by seuen letters , A. B. C. D. E. F. G. This Circle therefore is made to containe twentie eight yeares , for foure times seuen doe make twentie eight , by helpe whereof is knowne the true order of the letters , whereof A. signifieth alwayes the first day of euerie yeare , and for euery leape yeare are appointed two Dominicall letters , whereof the one continueth from the beginning of that yeare vntill Saint Mathias euen , and the other from thence to that yeares ende , as you shall more plainlie perceiue by that which followeth . And first I will shew you how to finde out the iust number of the circle of the Sunne euery yeare , which is done thus .

Adde to the yeare of the Lord giuen or supposed 9. and diuide that product by 28. and the remainder shall bée the number of the foresaide Circle : As for example , I would know the number of the saide Circle in the yeare 1590. whereunto by adding 9. I make the summe to bee 1599. which béeing diuided by 28. there remayneth 3. Nowe to finde out the Dominicall letter by the number of the sayd Circle , you must resorte to the figure following , consisting of thrée Circles , making two spaces , in the vpper space whereof are set downe the aforesaide seuen letters , and in the nether space the numbers of the Sunnes Circle , in such sorte as euerie number hath his proper Dominicall letter standing right ouer his heade , and for euerie leape yeare there are set down two Dominicall Letters . Al which letters are to be counted backward , & not forward as they are placed in the Almanacks . As for example , hauing fouud the number of the Sunnes circle for the yeare 1590. to bée 3. in the lower space , you shall finde the letter D. standing ouer his head , which is the Dominicall letter for that yeare . And note that when 28. is the number of the Sunnes circle A. is alwayes the Dominicall Letter , from which number you must beginne to count againe at 1. and so procéed backward to 28.

How to find out the number of Indiction . Chapter . 9.

THis number consisteth of 15. yeares , and is commenly set down in al the Charters of the Bishops of Rome , and in the instruments and writings of their pronotaries , and therefore is called Indictio Romana , wherein they séeme to followe the auncient Romanes , which vsed the like indiction of yeares , but to other purposes , whereof I make mention in my Spheare : The way to find out this number euerie yeare is thus . Adde to the yeare of the Lorde giuen 3. and deuide the product thereof by 15. and the remainder shall be the number of the said indiction . But you haue to vnderstand that this Indiction is to be counted from September , and not from March as is the Epact.

But the surest , and therewith the most generall Kalender to serue in al places is the Ephemerides or dayly Almanack and especially that of Iohannes Stadius , which wil serue for 14. years yet to come : And before those yeares be all expired , I doubt not but that the like will be set forth by some one or other of our learned Astronomers , amongst whome for many years past the Germanes haue bene most famous . I chose the Ephemerides of Stadius because he is more portable , and of lesse price than that great Ephemerides of Leovitius . The chiefest vses whereof , and most méete for Mariners , though I haue already set downe in the latter end of my Treatise of the Globes , to the intent you might by the helpe thereof find out in the Globe the true place of the Moone and of euery one of the other 5. Planets that hath both longitude and latitude , yet I thinke it good once againe here in this place , most properlie requiring the same , briefly to shewe the vse of the saide Ephemerides , and speciallie of the diurnall , Table which beginneth at the yeare of our Lord 1583. endeth with the yeare 1606.

A briefe description together with the vse of the diurnall Table or Almanacke of Iohannes Stadius . Chapter . 10.

THis Table beginneth at the 202. page of his booke called the Ephemerides or dayly Almanacke . And euerie page of the said Table that is on the left hand is diuided into 9. collums . In the first collum whereof on the left hand are set down the daies of the moneth . First , the Gregorian dayes according to the Romane account , and next to that the dayes of the moneth according to our English account , and then in the front of euerie other collum are set downe the characters , first of the Sunne , and then of the other sixe Planets , that is to say , of the Moone , Saturne , Iupiter , Mars , Venus , Mercurie , and last of all , the head of the Dragon figured thus ♌ . And right vnder these seuen Planets , and also vnder the heade of the Dragon are set downe the signes and degrées whererein euerie of these is euery day of the moneth throughout the yeare at noonetyde . And in the foote of the saide Table is set downe the latitude of euery one of the fiue planets , procéeding by the dayes of the moneth diuided into thrée parts . And in the margent of euerie left page are set downe the chiefest feasts and Saints dayes that fall in euery moneth throughout the yeare .

And moreouer , in the Table on the right hande right against the left Table are set downe first the dayes of the moneth , and then what coniunction or any other aspect the Moone hath with anie of the other sixe Planets , that is , with the Sunne , with Saturne , with Iupiter , with Mars , with Venus , and with Mercurie , which Planets are set downe in the front of the said Table , and vnder them the Characters of such aspectes as the Moone hath that day with any of the other Planets . The characters of which aspects are these here following .

☌ ☍ △ □ ⚹

Whereof the first signifieth a Coniunction , the second an opposition , the third a trine aspect , the fourth a quadrat aspect , and the fift a sextile aspect .

Two Planets are saide to bee in a coniunction when they are both in one selfe signe .

And to be in an opposition when they are in two seueral signes distant one from another , that is to say , 6 signes distant one from another .

And to be a trine aspect when they are distant one from another by foure signes :

And to be in a quadrat aspect when they are distant one from another by thrée signes .

And to be in a sextile aspect when they are distant but 2 signes one from another .

How to find out the place of any planet by the Ephemerides . Chapter . 11.

NOw to find out the place of any Planet , or of the head of the Dragon by this diurnall Table , you must first séeke out the day of the moneth in the first Collum of the left Table , and right against that on your right hand in the said left Table , you shall find in the common angle right vnder the Planet or Dragons head , which so euer of them you séeke , the signe and degrée wherein the saide Planet or Dragons head is the said day at noontyde .

And to find out the aspectes which the Moone hath with anie of the Planets the same day , you must resort to the other Table on the right hand , obseruing like order as before .

The Example .

As for example , the 21 of Aprill 1592 which is the first of May according to the Romane accompt , I find by the Table on the left hand , the Sunne to be in the 10. degrée , 15. minutes of Taurus , the Moone to be in the third degrée 47 minutes of Capricorne , Saturne to be in the 8. degrée 10 minutes of Cancer , Iupiter to be in the 18. degrée 28 minutes of Sagittarius , Mars to be in the 12 deg. 6 minutes of Gemini , Venus to be in the 2 degrée 0 minutes of Aries , and Mercurie to be in the 6. degrée 20 minutes of Taurus , and the head of the Dragon to be in the 29 degrée 45 minutes of Gemini . And right against this in the Table on the right hand , you shall find the Moone to bee in a trine aspect with the Sun , to be in an opposition with Saturne , to be in a trine aspect with Mercurie .

Thus much touching general Kalenders or Almanacks . Now as for particular Kalenders I take that which Robert Norman hath set downe in his new Attractiue to be the fittest for our countrie men , which containeth many necessarie things , as the contents thereof together with the Table next following the same doth shew , which Kalender I think it superfluous to be set down againe here , and the rather for that I wish euerie Mariner not to be without that booke , and especially contayning besides the Kalender many other good precepts touching nauigation , to which booke is also ioyned a verie learned and Mathematicall discourse touching the variation of the Mariners Compasse made by Maister William Borough controller of her Maiesties Nauie , who in mine opinion is one of the skifullest men in the Art of nauigation that is in this Realme .

Of the Mariners Ring or Astrolabe , and of his crosse staffe . Chapter . 12

NOw according to the order set down in the beginning of this Treatise , I must néeds speak somewhat of the common Marmers Ring or Astrolabe , and of his crosse staffe which serue not to so many purposes as that of Stofflerus or of M. Blagraue before described , but onlie to take the altitude of the Sunne , or of any Starre or Planet , neither are so many conclusions to bee wrought by their common crosse staffe as by that of Gemma Frisius , or by that which Master Hood hath latelie inuented . Sith our Marmers for the most part doe vse their common crosse staffe to none other ende but eyther to take the altitude of the Sun or of any fixed star or planet , or els to take the distance betwixt two stars , the making of which staffe is plainly set downe by Martin Cortes in his arte of Nauigation , and also the making of their Astrolabe .

But Cogniet and Wagoner do set downe a new kind of Crosse staffe , hauing 3. transoms or crosses euery one longer & shorter than another by the one halfe , affirming that so many conclusions may be wrought therby as by that of Gemma Frisius , but in mine opinion they are not to be compared in al respects to that of Gemma Frisius which though by reason that the yard therof is of so great a length as it is not maniable in a shippe , yet vpon the land it is most serviceable , nor yet to that of Maister Hoods inuention , which is most maniable , and therwith verie light of carriage , by whose staffe I thinke verilie that as many things may be wrought as by anie other kind of crosse staffe whatsoeuer , and with lesse trouble , yea , and in matters of Astronomie more trulie by reason that the yard and transome be of one selfe length , and thereby the degrées bée larger throughout the whole quadrant than they be in the common crosse staues , for in them the degrées from 50. vpwards to 90. are verie small and haue narrower spaces than M. Hoods staffe hath . Againe , whereas in vsing the Mariners crosse staffe in such latitude , as the Sunnes beames be of great force , they are faine to haue glasses made of purpose to saue their sight , and in some places all too little . But in vsing M. Hoods staffe they shall not néed to behold the Sunne it selfe at all , but onely to marke vpon what degrée of the yard the shadow of the Vane stréeketh . Moreouer , when the Sunne or starre is 50. or 60. degrées high , they are fain to vse their Astrolabe and not their staffe , which Astrolabe in mine opinion , as I haue said before , is the best instrument of all others to take the altitude of the Sunne in the day , or of any starre in the night , and because I haue here commended vnto you M. Hoods staffe , I will first set downe a plaine description thereof together with those few Astronomicall vses which do chieflie belong to the Mariner , without committing anie offence I hope to the author thereof , and then I will describe vnto you the Mariners ring and his common staffe together with the vses of the same .

But as I was about to describe vnto you M. Hoods staffe , a friende of mine comming in the meane time desired mée that I woulde first set downe the making and vse of the crosse staffe with thrée transomes , which Wagoner and Michaell Cogniet doe so much commend , and as I heare , is vsed of manie sea men in these daies , whose request I could not well denie , and therfore loe here followeth both the making and vse thereof .

Of the crosse staffe hauing three Transames or Cursours , commonly vsed in these dayes , the vse and making whereof doth hereafter followe according to the description of Wagoner and Michaell Cogniet .

FIrst prepare a right staffe of firme woode that is square euery way , bearing in thicknes thrée quarters of an inch , and in length thrée or foure foote , for which you must also prepare thrée Transames or Cursours euery one shorter then another by the one halfe , for the longest would containe in length twelue inches , the seconde sixe inches , and the shortest or least thrée inches . And euery one of these Transames or Cursours must be cut with a square hole in the very middest , so as they may be made to runne iust vpon the staffe to and fro , these things being prepared , you must deuide thrée sides of the staffe into certaine degrées to serue the thrée seuerall cursours as followeth .

First you must vpon a smooth square Table somewhat longer euery way then the staffe , and for want of one such Table you may ioyne two Tables that may stand eauen together , drawe by helpe of a true long squire a right Triangle marked with the letters A. B. C. and let A. be the right Angle , and the Centre as you sée in the figure following , and let both the sides of the saide Triangle be of like length to your staffe , then putting the one foote of your compasse in the Centre A. and the other foote in B. or C. drawe a quarter of a circle from C. to B. and hauing deuided that quarter into two equall partes , make a pricke in the middest marked with the letter D. and laying your ruler to that pricke and to the Centre A. drawe a right line which shall be A. D. then deuide the halfe Quadrant D. B. into nine equall partes making 90. equall degrées procéeding from B. to D. so as the first degrée may be at B. and the 90 at D. that done , take with your compasse the iust halfe of the longest Transame , and kéeping your compasse at that widenesse , set the one foote in the Centre A. and the other foote in the line A. C. so farre as that widenesse will extende , and there make a pricke marked with the letter E. then from that point drawe a right line that may be a Parallell to the line A. B. which line shall be E. F. then take with your compasse the halfe of the middle Transame , and kéeping your compasse at that widenesse , set the one foote in A. and the other foote in the line A. C. so farre as that widenesse doth extend and there make a pricke marked with the letter G. and from that pricke drawe a right line that may be a Parallell to A. B which line shall be G. H. Thirdly take the one halfe of the least Transame with your compasse and transferre that widenesse to the line A. C. setting the one foote of your compasse in A. and make a pricke with the other foote in the line A. C. as before , marked with the letter I. and from that pricke drawe another right line that may be a Parallell to the line A. B. which shall be the line I. K. Now to graduate the first side of your staffe to serue the longest Transame , you must lay the ruler to the centre A. and drawe right lines from thence to euery degrée of the circumference contained betwixt 90. and 30. and those lines shall deuide the line E. F. into so many vnequall spaces as doe belong to the first side of the staffe , for you must lay the first side of the staffe to that line to be marked according to the diuision of that line , the neather section whereof towardes the lower ende of the staffe must be marked with 90. and the vpper section with 30. so is the first side of your staffe truely deuided to serue the longest Transame , now to serue the middle Transame you must deuide the line G. H. by drawing right lines from the Centre A. to euerie degrée of the circumference contained betwixt 30. and 10. which lines will deuide the line G. H. into vnequall spaces , of which spaces the lowest must be marked with 30. and the highest with tenne , according to which diuisions you must marke the seconde side of your staffe by laying the side close to the line G. H. so shall that side be marked to serue the middle Transame , then lay your ruler againe to the Centre A. and drawe right lines to euery degrée of the circumference contained betwixt 10. and the first or second degrée next to B. and those lines shall deuide the line I. K. into vnequall spaces , the lowest whereof is to be marked with 10. and the highest with 2. or 5. according as your instrument will beare , and according to those sections you must deuide the thirde side of your staffe by laying the same close to that saide line I. K. and remember to make your lines of diuision so finely as is possible , so shall your staffe be the more truely graduated . And noce that in graduating your staffe , you shall not néede to drawe right lines from the Centre to euery degrée of the circumference contayned betwixt the 90. degrée and the 60. degrée , for when the sunne or any starre is higher then 60. degrées , you must vse your Astrolabe and not your staffe which will not serue you to looke so right vp to take so great an altitude eyther of sunne or starre .

A figure of the foresaide Triangle .

The shape of the staffe with his three Transames , together with the vse thereof .

Whensoeuer you would take the Altitude of the Sunne or of anie starre , you haue first to consider whether the sunne or starre be 30. degrées or more high : for then you must place the longest Transame vppon your Staffe . And set the lower end of your staffe marked with ninetie to your eye , which is alwayes to be done howe high or lowe so euer the sunne or starre be , and you must mooue the Transame eyther forwarde or backewarde vntill you may sée by the vpper ende of the Transame the bodie or middest of the sunne or starre , and with the neather ende of the Transame the Horizon , and then looke in what degrée the Transame cutteth the staffe , for that is the Altitude of the Sunne and starre at that present , but if the sunne be not 30. degrées high , then you must put on the middle Transame , and if hee bée lesse then tenne degrées high , you must put on the shortest Transame , and then do as before . Thus much touching the Crosse staffe with 3. Transames , nowe I will describe vnto you Maister Hood his staffe , and shew you the vse thereof .

A briefe description of Maister Hoode his crosse staffe , and of all the partes thereof . Chap. 13.
The figure or shape of euery part of the said staffe .

The transome .

The yearde .

THis instrument as you sée consisteth of foure parts .

The crosse sockett

The Vane .

First of 2. square rulers ech one bearing in thicknesse thrée quarters of an inch or there abouts euerye way square , and in length I would wish ech ruler not to containe aboue one yarde , for then in some vses they would waxe toppe heauie .

Of which two rulers the one marked with the letters A. B. is called the Transame , which is deuided on the one side into 45. degrées beginning at one and so foorth to 45. And euery degrée is deuided againe into sixe lesser parts making 60. minutes , for sixe times ten maketh 60. which serueth for Astronomicall vses .

And on the opposite side to that , the said Transame is deuided into 1000. equal parts beginning at 25. and so increaseth by 25. vntill you come to 1000. Euerie which 25. partes is deuided into fiue lesser partes , and euery one of those againe into fiue partes , which maketh in all 25. for fiue times fiue is 25 and this serueth for Geometricall vses .

The other square ruler called the yarde & marked with the letters C. D. is deuided into degrées like to those of the Transame procéeding from 45. to 90. which together with those degrées that be in the first side of the Transame doe make vppe the number of a iust Quadrant to serue Astronomicall vses . And the side opposite to that is deuided into 1000. degrées procéeding from 25. to 25. like in all respectes vnto the opposite side of the Transame to serue Geometricall vses .

The third part belonging to this instrument is a double socket of brasse marked with this letter E. ioyned together with right Angles , and standing crosse one to another . And ech socket hath a skrewe to kéepe the same fast to his staffe at any degrée that you list to set the same : And also a long notch to the intent it may be layde close to any degrée of the ruler whereto it belongeth .

The fourth part belonging to this instrument is a vane of brasse marked with the letter F. the vpper edge whereof is pearced with a little round hole for the beame of the sunne to passe thorough the same . And this vane is made with a socket , and with a skrew to hold it fast to that ruler whereon it is set . And for certaine Geometricall vses it were necessarie ( as some thinke ) that there were two such vanes .

Thus hauing described all the partes of the foresaide instrument , I wil now shew you how to set those parts together : which to serue Astronomicall vses is thus done .

How to set the partes of Maister Hoodes staffe together to serue such Astronomicall vses as doe chiefely belong to the Mariner . Chap. 14.

FIrst put the neather ende of the Transame marked with the 45. degrée into the double socket , so as the Transame may stand right vppe , and that the notch of the socket may méete euen with the sayde 45. degrée , and turne the skrewe that the socket may stand fast at that degrée , that done put the yarde into the crosse socket , so as the notch of the sayd crosse socket may lie iust vppon the 45. degrée of the yarde , and there make it fast by turning the skrewe : then put on the vane , which to serue Astronomicall vses must be set at the highest ende of the Transame , in such sort as the vpper edge of the vane pearced with the hole may stande eauen with the first point or stréeke of the saide Transame , and when the Transame standeth on the left hande of the yarde , then the vane must be placed on the right side of the Transame : But if the Transame doe stande on the right hande of the yard , then the vane must be placed vpon the left side of the Transame : And the yard and Transame would be so set together as the degrées of the Transame procéeding downeward from one to 45. may looke towards you , that is to say , may stand right before your face : And the degrées of the yarde procéeding from 45. to 90. would lie vpwarde , so as the ende of the yarde marked with 90. may point to your breast or be set to your eye as occasion shall require , for such Astronomicall vses as heere doe followe .

The shape or figure of the foresaide staffe , hauing all his partes set together to serue for Astronomicall vses . Chap. 15.

How to take the altitude of the funne at any houre that he is to be seene with the eye by M. Hoods staffe . Chap. 16.

THe Staffe being set in such order as is before taught , goe into some open place whereas you may sée the sunne , and turning the ende of the yarde marked with ninetie towardes your brest , holde the yarde so leuell as you can , that it may be a iust Parallell to the Horizon , and turne both your face , and also the vane of the transame towardes the sunne . Then you haue to consider whether the sunne at that present be eyther iust 45. degrées high , or more , or els lesse then 45. degrées , which you shall easily knowe thus : For if he be iust 45. degrées high , then the shadowe of the vpper edge of the vane will stréeke iust vpon the 90. degrée of the yarde lying at your brest : But if he be more then 45. degrées high , then the shadowe of the vane will stréeke short of the 90. degrée : And if it be lesse it will cast no shadowe at all vppon the yarde , but stréeke cleane beyond it ouer your shoulder . Nowe knowing by this meanes whether the altitude of the sunne be more or lesse then 45. degrées , you shall take his true altitude thus :

If his altitude be lesse then 45. degrées , then make the Transame to sinke downe through his socket vntill the vpper edge of the vane standing vppon the Transame doe caste his shadowe iust vppon the nintieth degrée of the yarde . Then looke at what degrée the notch of the Socket wherein the Transome standeth , doth cut the saide Transame , for that is the true altitude of the sunne at that instant .

But if the altitude of the sunne be more then 45. degrées , then drawe the double socket vpon the yarde nigher towardes your brest , vntill you sée the shadowe of the vane to fall iust vppon the ninetieth degrée of the yard , that done , looke vpon what degrée of the yarde the notch of the double socket cutteth , and that is the true altitude of the sunne at that instant .

How to take the altitude of anie starre with Master Hoodes staffe . Chap. 17.

THere is no difference betwixt the order of the taking the altitude of the sunne and of anie starre , but onely that in taking the altitude of any starre , you must set the ende of the yarde marked with the 90. degrée to your eye , staying the same vppon the vpper bone of your chéeke , and that in such sort as you may sée the starre by a right line passing from the ende of the yarde , lying at your eye to the vpper edge of the vane at one instant , being alwayes sure that the yarde lie leuell , making a iust Parallell to the Horizon , which Horizon is more easie to be séene vpon the sea , then vpon the lande , for on the sea there be neither hils nor trées to hinder the sight thereof .

How to take the distance betwixt two starres with Maister Hoodes staffe . Chap. 18.

THe instrument being set in such order as was first taught : Set the ende of the yarde marked with 90. at your eye : and let the other end of the yarde directly point to one of those starres whose distance you séeke , and turne the edge of the vane to the other star be it on the right hande or on the left , according as you thinke good your selfe : Then you must consider whether the distance of the two starres be iust 45. degrées or more or else lesse , for if it be iust 45. degrées , then the outward edge of the vane , and the ende of the yarde marked with 90. will be answerable in your sight to that distance without any more adoo . But if the distance be lesse then 45. degrées , which you shall easily perceiue by your eye ( for then the starre that is on your right or left hand , will appeare within the vane and fall short thereof ) then you must thrust down the Transame in his socket vntill you may sée the two starres by a right line passing from the ende of the yarde marked with 90. and lying at your eye , right foorth to the outwarde edge of the vane , that done , looke in what degrée the double socket cutteth the Transame , and that shall be the true distance of the two starres . But if the distance of the two starres be more then 45. degrées , which you may knowe also by your eye , because the starre which is on your left or right hande , will appeare cleane without the vane : then drawe the double socket together with the Transame towardes your eye , vntill you may sée the two starres by a right line as before , and the degrées which the notch of the double socket cutteth on the yarde , shall be the true distance of the two starres .

Thus much touching the Astronomicall vses of M. Hoodes staffe meete for the Mariner . Now as for setting of the said staffe to serue Geometricall vses , and to finde out the manifolde conclusions to be done thereby , I wholly referre you to M. Hoodes owne booke , thinking it good now to speake somewhat of the Mariners Astrolabe or Ringe , and of his vsuall Crosse-staffe .

Of the Mariners Astrolabe . Chap. 19.

MIchaell Cogniet setteth downe the making of a new kinde of Astrolabe , which he calleth an vniuersall Astrolabe , onely because the declination of the sunne euery day is added in manner of a Table to the one ende of the Diopter of the saide Astrolabe , which is of no great importance sith the sunnes declination doth alter in the space of 30. yeares and lesse , which being expired , the declination is to be new calculated : In the meane time that Table which Norman hath set downe in his Attractiue , and also that which is more lately calculated set downe by my selfe in the first part of my Spheare , and likewise in this Treatise whereas I treat of the motion of the sunne and of his declination , may serue these 20. yeares and more without any greater error : Wherfore leauing to speake any further of Cogniet his vniuersall Astrolabe , I will describe here the common Astrolabe and Crosse-staffe vsed by most Mariners , the shape or figure of both which here foloweth .

A briefe description of the Mariners Astrolabe , and the vses thereof . Chap. 20.

THe limbe of the Mariners Astrolabe is traced as you sée with thrée Circles , making two spaces to containe therein the degrées and numbers of altitude : and these Circles are deuided by two crosse lines called Diameters , and cutting one another in the very Centre of the Astrolabe into 4. equall parts , of which quarters the vppermost on the left hand towards the Ringle is only marked with degrées and numbers , as you may sée in the figure : In which figure the perpendicular Diameter signifieth the Zenith , or the line of South and North , at the vpper end whereof is fastened the Ringle or handle . And the other ouerthwart Diameter signifieth the Horizon the one end whereof on the left hande signifieth the East point , and the other end on the right hande the West point . And to this Astrolabe ( as to all other ) doth belong a ruler or Diopter , which as you sée hath at ech end a square tablet pearced with two holes , the one greater and the other lesser , the greater to looke through with your eye , to take the altitude of any starre or of the sunne being so darkned by some cloud as though it casteth no shadowe , yet it may be séene with the eye : And the lesser hole is for the sunnes beame to passe thorough when he shineth cleare , and some Astrolabes are deuided in like maner of the Zenith on both sides , and haue two Diopters , whereof the one is pearced with a great hole , and the other with a smaller hole to serue to such purposes as is aboue saide .

The vse of this Astrolabe is onely to take the altitude of the sunne at any time of the day , or of any fixed starre or Planet in the night , in such sort as is before taught in my description of Maister Blagraue his Astrolabe , in the third and fourth Proposition thereof .

A briefe description of the Mariners Crosse-staffe . Chap. 21.

THis staffe consisteth of two foure square rulers of woode , whereof the one is called the yarde , and the other the Transame or Crosse : And the yard containing in length most commonly thrée quarters of a yarde is the longer péece , and is deuided into 90. vnequall degrées : for from 10. to 90. the degrées doe grow lesser and lesser , and the Transame containing for the most part but one third part of the yard , is cut with a square hole in the middest , so as it may runne vp and downe vpon the yard , & some Transames haue two running vanes to be set at what widenes you list , to take thereby the distance of any two starres .

The vses of the Mariners crosse-staffe . Chap. 22.

THe vse of the Mariners staffe chiefly consisteth in two points , that is , to know therby the altitude of the sun in the day time , or of any starre in the night season , and the other is to know thereby the distance betwixt any two starres .

The first is thus done , hauing put the staffe or yarde through the square hole of the Transame , set the end of the yard which is marked with 90. to your eye , laying that end vpon the vpper point of your chéeke bone kéeping your legs close together , and hauing directed the other end of the yard towards the sunne or star whose altitude you séeke ▪ mooue the Transame to and fro or vp & downe , vntill you may sée with the one eye ( winking with the other ) the one end of the Transame to méete ▪ iust with the Centre or middest of the sunne or starre , and the other end to touch the Horizon both at one instant , and that degrée of the yard which the square edge of the Transame cutteth , wil shew how high the sunne or starre is at that present : And in taking the altitude of the sunne or of any star , remember alwaies to holde the yarde so leuell as it may be a iust Parallel to the Horizon , for so shall you take the altitude the more truely . Now as touching the second point , which is to knowe the true distance betwixt any two starres you must doe thus .

Set the end of the yarde marked with ninetie to your eye as before , and mooue the Transame to and fro vntill you may sée the one end of the Transame to answere the one starre , and the other end thereof to answere the other starre , for that degrée of the yard which the Transame cutteth will shewe the distance betwixt the two starres : And if the two starres be so nigh together , as they wil not conueniently answere the two ends of the Transame , then you may vse the two mooueable vanes , which ( as I saide before ) you may set at what widenesse you list , but yet so as they may both stand equally distant from the Centre of the Transame or from both ends thereof .

Thus hauing shewed the vse of the Mariners Ring and crosse staffe , I will now procéede to the other instruments , whereof the two Globes are part , which Globes I haue alreadie described and shewed the chiefest vses thereof in a Treatise by it selfe , and as for the vniuersall Horologe to know thereby the equall houre of the day in euery Latitude , and also the Nocturnlabe to knowe the houre of the night , I shall speake of the one when I come to speake of the sunne , & of the other whereas I treate of the North starre and of his guards , for there I will set downe the figure and shape of ech instrument , and shewe how to vse the same : But the two chiefest instruments belonging to a Mariner , are his compasse and his Carde , whereof I come now to speake .

But sith the Fly of the Compasse representeth the 32. windes , I thinke good first briefely to define what the wind is , and to shew howe many windes the auncient Mariners did vse , I say here briefely because I haue alreadie spoken sufficiently thereof in the latter end of my Spheare .

Of the winde , what it is , and of the diuers kindes and names thereof . Chap. 23.

THe winde according to Aristotle , is an exhalation hotte and dry , engendred in the bowels of the earth , and being gotten out is carried sidelong vpon the face of the earth . And of windes there be foure principall which doe take their names from the foure quarters of the earth from whence they blowe , that is North , South , East and West : And though the Gréekes and Latines deuide euery quarter into thrée partes , and thereby make in all but twelue windes , whose names both Gréeke and Latine are set downe in the latter ende of my Spheare , yet our latter Sea men to be the more assured of their Rowtes and Courses , doe deuide euery quarter of the Horizon into eight windes , so as in all they make 32. windes , giuing them such names as are set downe in this figure heere following representing the Flye of the Mariners compasse .

A briefe description of the Mariners compasse , and the vse thereof . Chap. 24.

THe Mariners Compasse may be very well deuided into two essentiall partes , that is , the Fly , and the wyars touched with the Lodestarre called in Latine Magnes . And first you haue to vnderstand , that the flie is a round white Carde traced with 32. lines all passing thorough the Centre of the Circle , which lines doe signifie the 32. windes in such sort as the figure before set downe doth shew , of which lines that which is marked with the Flower-deluce signifieth the north , whose point opposite is the South , and that which is marked with a Crosse signifieth the East , whose point opposite is the West , and the outtermost Circle of the said Fly signifieth the Horizon , which circle is also deuided into 360. degrées like vnto those of the Equinoctiall , so as euery space bétwixt point and point cóntaineth 11. degrées and 15. minutes , which is the fourth part of a degrée . Moreouer , this Circle is deuided in 24. houres by allowing to euery point thrée quarters of an houre , which is 45. minutes , for an houre containeth 60.m m and halfe an houre containeth 30.m m and the quarter 15 m. Moreouer , the common Mariners doe deuide euery point of the Compasse into foure quarters to make the more exact account of their Routes or Rombes . But now as touching the other essentiall part of the Mariners Compasse which is the

wyars , you shall vnderstand that they are of yron or stéele , and made in this forme , and being touched at the one end with the Lodestone , they are fastened to the backe side of the Fly , eyther right vnder the line marked with the Flowre-deluce , or else somewhat distant from the same , either towardes the East or else towardes the West , for such cause as shall be declared in the next Chapter , for those wyars being thus touched , doe make that part of the Fly that is marked with the Flowre-deluce , to stand alwayes towardes the North , and to that ende the Fly hauing in the Centre a latten socket , is put into a turned boxe , in the middest whereof is a sharpe pointed Latten pinne , vpon which the Flie turneth about , and that turned boxe is couered with glasse , partlie to kéepe the Flie cleane , but chieflie that the winde should not mooue it to and fro . And this turned boxe is hanged by two rounde narrowe plates of Latten in another square boxe of thinne wainscot boord , so as it may alwayes hang leuell , howsoeuer the shippe swayeth or inclineth on either side , and though of the Mariners Compasse there be diuers vses , as you shall perceiue hereafter , yet the chiefest end and vse thereof is to shewe the North part of the world , whereby the Shipmaister knowing what course hée hath to holde and how the port or place to which hee goeth , beareth from the place from which he departeth , may by looking alwayes to his Compasse , know how to direct his ship accordinglie .

Of the Lodestone , and of the variation of the Compasse in Northeasting and Northwesting . Chapter . 25

THe Lodestone or Adamant , called in Latine Magnes , & in Italian Calamita , hath two marueilous great and secret properteies or vertues , the one to drawe stéele or Iron vnto it , and the other to shew the North and South parte of the world , of which stones some bée of more force than others according to the place from whence they come . For those are counted best which are found in the East Indies vpon the coaste of China and Bengala , which is no shell but a whole stone of sanguine collour like to Iron , and is firme , massie , and heauie , and will drawe or lift vp the iust waight of it selfe in Iron or stéele . And as Norman saith in his newe Attractiue , such stones are commonlie solde for their waight in Siluer .

Next to this , Norman commendeth the redde stone of Arabia , hee commendeth also the stone of Almaine , which is in collour like to Iron , but spongeous , and thereby lighter than the other . And as for the blacke and white stone of Elba , which is an Ilande , not farre from Piombino , hée saith that the vertue thereof is but of small force , and of no long continuance . But the worste are those that come from Norway , whose collour is mixt with graie , I haue hearde that there bee of them here in England , but howe good I knowe not , it is maruell that Norman maketh no mention thereof , whose Treatise called the newe Attractiue , together with Maister Borough his addition I would wish all that be studious in this Arte to reade most diligentlie , for truelie in mine opinion the secretes of this Stone and variation of the Compasse , was neuer better disciphered , nor by more experiences tried than it hath bene by these two men last named . The booke is most necessarie for nauigation and of an easie price , méete for euerie poore mans purse , though perhaps in some pointes not méete for euerie mans vnderstanding . Now whensoeuer the wyers are to be touched with this stone , they must be made very cleane and void of all rust , to the intent that the Iron may the more firmelie receiue the vertue of the stone . And it is well knowne by good experience , that by vertue of the Lodestone the North point of the Compasse declineth alwayes from the true North , either to the East , or West more or lesse according to the latitude of the place wherein you are vnlesse you be right vnder the Meridian of the Azores . And therefore , most men in these partes of the worlde doe vse to set the North point of the wiers not right vnder the Floure de Luce signifying the North part of the world , but rather somewhat inclining towarde the East halfe a point or thereabout to auoide the Northeasting and Northwesting of the Compasse , the cause of which declination by diuers learned men hath bene diuersly taught , but not rightlie as Michaell Cogniet saith , vntill Mercator found out the true cause , who first learned by the experience of one Francis of Deepe , an excellent Pilot , that in sayling vnder the Meridian which passeth thorough the Iles called the Azores , the néedle doth decline neither East nor West , whereupon Mercator by calculating the variation of the Compasse at Ratisbone , found that the Pole of the Lodestone ought to be put in that Meridian which passeth through the foresaid Iles so as it may be distant from the Pole of the world sixtéene degrées and ½ or rather as Maister Borough saith , sixtéene degrées and twentie two minutes , and by that calculation Cogniet hath found the variation of the Compasse at the towne of Antwerp to be 9. degrées . And Maister Borough by the helpe of the newe instrument of variation first made by Robert Norman , and afterward perfected by himselfe , hath found the variation for London to be 11. degrées and fiftéene minutes , which is a whole point from North to East . But whensoeuer you depart from the foresaid Meridian of the Azores , bée it neuer so little , either towardes the East or West , the néedle will varie and decline accordinglie . And his greatest declination is when you come to a full quarter of that Parallell wherein you saile , for from thence it declineth lesse and lesse vntill you come againe vnder the foresaid Meridian , which thing Cogniet doeth plainelie demonstrate by this Figure here made of purpose .

In which Figure the letter A. which is in the verie center of the Circle representeth the north Pole , and the right line , marked with the letters C. B. signifieth the Meridian which passeth through the Iles Azores , in which line is a point or pricke marked with the letter D. signifying the Pole of the Lodestone , distant from the North Pole sixtéene degrées and a halfe . Nowe , if your shippe bee in the point B. then your needle declineth on neither side , but pointeth right to the North Pole , and also to the Pole of the Lodestone . But if you sayle Eastwarde and arriue to the point E. then the right line of the North is E. A. but your néedle declineth on the right hande towardes his owne Pole D. which is to the East , so much as the angle A. E. D. doth shewe . Likewise if you saile from B. Westwarde to the point F. the right line of the North shall bée F. A. but your needle will decline to his owne Pole D. towardes the West , so much as the angle A. F. D. doth shew . And to bée short , the néedle doeth neuer shewe the right North , but onelie in C. or B. to eyther of which pointes the nigher that you approch the lesse your néedle declineth , and the more that you goe from anie of these two pointes the more your néedle declineth either East or West , but the greatest declination thereof is in H. or G. for then you are a iust quarter from the foresaide Meridian . Also by this Figure he plainelie sheweth that of two sundrie townes hauing one selfe Meridian , that which is néerest to the Pole of the worlde hath greatest declination . As for example , suppose the letter I. in the foresaide Figure to signifie that towne which is nigher to the Pole , and the letter E. to signifie that towne which is more distant from the Pole , both townes béeing vnder one selfe Meridian E. A. Here you sée that the Néedle being in E. doth shewe the Pole of the Lodestone by the line E. D. and béeing in I. it sheweth the Pole of the Lodestone by the line I. D. Nowe according to the doctrine of Euclyde , the angle A. I. D. is greater than the angle A. E. D. whereby it followeth that the néedle declineth more in I. than in E. But whereas Mercator affirmeth that there should bée a mine or great rocke of Adamant , whereunto all other lesser rockes or néedles touched with the Lodestone doe incline as to their chiefe fountaine , that opinion séemeth to mée verie strange , for truelie I rather beléeue with Robert Norman , that the properties of the stone , as well in drawing stéele , as in shewing the north Pole , are secrete vertues giuen of God to that stone for mans necessarie vse and behoofe , of which secrete vertues no man is able to shewe the true cause .

How to find out the variation of the compasse in every latitude . Chapter . 26.

WIlliam Boorne in his Regiment of the Sea teacheth diuers wayes how to finde out the variation of the Compasse , as well by the Sun in the day time , as by the North starre in the night season . First , thus mark at what point of the Compasse the Sunne riseth and setteth , for if hee riseth at the East point of the Compasse , and goeth down at the West Northwest point , then the Compasse is varied one whole point , that is to say , the North point of the Compasse standeth North and by East . But because the aire is seldom cleare at the rising and setting of the Sunne , you may do thus , take with your Astrolabe or crosse staffe the altitude of the Sun in the forenoone , the sooner the better , at some iust point of the compasse , & take again his altitude in the afternoone , when he is in like degrée of altitude , and mark therwith at what point of the Compasse the Sunne hath such altitude , and by the difference thereof you shall know the variation of the Compasse . As for example , you finde by your Astrolabe or crosse staffe , that the Sunne is 20. degrées high at the Southeast point of the Compasse , and obseruing the same againe in the afternoone you finde his height to bée 20. degrées at the West Southwest point of the Compas , wherby you sée that the compasse is varied one whole point , that is to say , that the North point standeth North and by East , and the South point is South and by West . He teacheth it also another way , which is thus .

Take with your crosse staffe or Astrolabe the height of the Sun at noontyde , which is called the Meridian altitude of the Sun , and thereby you shall haue the true Meridian of the place where you are , with which Meridian if the South point of your Compasse doth agrée , then your Compas hath no variation at al , but if the south point therof do swarue or incline on either side from the said Meridian , marke how much it differeth , and that difference will shewe you the variation . Nowe to knowe the variation of the Compasse by the North starre , doe thus , set your Compasse with the North starre , and if you finde them to agrée , then there is no variation , but note that this is to be done when the two guardes or pointers of Charles waine are right ouer or right vnder the North starre , for if these two starres be West from the North star , then the North star is a third part of a point vnto the Eastward of the North Pole , and if the said two starres be right East from the North starre , then the North starre is a thirde part of a point vnto the Westward of the north Pole .

The compasse ( as Bourne saith ) doth varie most in sayling long voyages East and West , and though it varieth two or thrée points , yet you may know what course to holde without alteration of the Wiers any maner of way . As for example , suppose the Northeast point to stand right North , and your course is to goe right West , here in this case you may vse the Southwest point in stead of the West point , whereby you may perceiue that it maketh no great matter which point standeth due North , so as you take that point of the Compasse for the North which directly pointeth to the North. But Robert Norman and Master Borough by the help of their new inuented Instrument of variation , doe shewe how to finde out the variation of the Compasse much more exactlie than euer it hath bene heretofore taught , which instrument together with the booke , I would wish all sea men to haue , and therby to learne the perfect vse of the instrument , which vse that booke teacheth both plainlie and learnedlie , in which booke is also prooued by diuers demonstrations that there is no such attractiue point as some haue dreamed , but rather a respectiue point whereunto the néedle of your Compasse wil alwayes turne in what part of the world soeuer you saile . But of the place where that respectiue point should be , diuers learned Pilottes haue had diuers opinions , for some haue imagined it to be in the heauens , and some aboue the heauens , if it were in the heauens , then the néedle would dayly turne about , and alter according to the motion of that heauen , wherein the said point is , which is nothing so . And to bée aboue the heauens , it is contrarie to the old rule of Philosophie , which saith that Extra Coelum non est locus , and therefore Norman and Maister Borough haue great reason to say that it is in the bodie of the earth beneath the Horizon , for they haue tried by diuers liuely experiments that the North parte of the néedle of his owne accorde and nature would alwaies decline downwarde if it be not otherwise counterpoised or letted , & by their demonstrations supposing the Meridian of the Azores to be the first or common Meridian , and also knowing the altitude of the Pole at London to be 51. degrées , 32. minutes , they find that according to that latitude the Pole of the Lodestone , béeing in the vpper face of the earth , and right vnder the foresaid common Meridian is 25. degrées 44. minutes distant from the Pole of the worlde . And the point respectiue to bee distant in a right line from vnder the Horizon of London 71. degrées , 50 minutes and the variation of the Compasse to be as hath bene said before , 11. degrées and 15. minutes , but to find out the true place of the respectiue point in euerie latitude , they say that no certaine rules can as yet be set downe , by reason that the compasse doeth vary more sodenly in one place than in another , for in some place it will varie more in sailing 200. leagues than in another place in sayling 400. leagues . Againe , it will sometimes be retrograde , for M. Borough in sayling betwixt the North Cape and Vaigates , towards the Northeast , and looking by his computation that the néedle should haue increased his variation towards the East , hée found that it was sodainly turned backward towarde the West , notwithstanding both Norman and M. Borough doe affirme that though the Compasse hath in seuerall Horizons seuerall variations , yet in any one Horizon the néedle alwaies respecteth one only point without alteration , and that in his declining it kéepeth the like order and that certainlie in euery place . And although the néedle of the Compasse by reason of the waightines of the Flie cannot decline downward according to his own propertie , but onlie sheweth the point respectiue alwayes vpon the Horizon , which indéed , as they say , is most necessarie for nauigation , yet by such meanes and conclusions as are set down in the foresaid book , a diligent Pilot hauing with him a perfect stone may by exact obseruations find out the increasing or decreasing of the declnation of the néedle , which declination you shall find , as they thinke to bée more or lesse according as the point respectiue is more or lesse distant from the place whereas the triall is made , which being diligently obserued in sundrie places , with the certaine variation of the néedle from the Meridian , the true place of the point respectiue may be found out as they thinke .

Of the Mariners Card , and of the making thereof . Chapter . 27.

THe Mariners Carde , which some cal a nautical Planisphear , is none other thing , but a description made in plano vpon paper or parchment of the places that be in the sea , or on the land next adioyning to the sea , as points , Capes , Bayes , Portes , Floods , Ilandes , Rockes , Sandes , and such like , And such Cardes are either vniuersall or particular , the vniuersall Cards are those wherin are described the most parts of the world , such as is the vniuersall Carde or Map of Mercator , or of Plancius . The particular are those wherin some speciall partes of the sea and land are described . And both these kindes of Cardes are traced with certaine lines , whereof some are called Meridians , some Parallels , and some the Lines of the Mariners compasse , shewing the 32. winds before described in the beginning of this Treatise . The order of making which cardes in times past was woont to be thus .

First drawe with a paire of compasses a secrete circle that may be put out , so great as you shall thinke méet for your carde , which circle shall signifie the Horizon , then diuide that circle into foure equall quarters , by drawing two Diameters crossing one another , in the center of the foresaide circle with right angles , whereof the perpendicular line is the line of North and South , and the other crossing the same is the line of East and West , at the foure ends of which crosse Diameters you must set downe the foure principall windes , that is , East , West , North and South , marking the North part with a flower deluce in the toppe , and the East part with a crosse , as you may sée in the figure following . Then diuide euerie quarter of the saide circle with your compasses into two equall partes , setting downe prickes in the middest of euerie quarter , through which prickes , and also through the center of the circle , draw two other crosse lines , which must extende somewhat beyond the circumference of the Horizon , which two crosse lines together with the first two crosse lines shall diuide the circle into 8. partes , and thereby you shall haue the eight principall Windes . That done , diuide euerie eight parte of the saide Horizon into two equall partes by drawing other two crosse lines through the center , and extending somewhat beyond the circumference of the Horizon , as before ▪ wherby the whole circle shall be diuided into 16. partes , which shall suffice without making anie more diuisions , which woulde cause a confusion of lines , & at the end of euery one of these 16. lines you must drawe a little circle , whose center must stande vpon the circumference of the Horizon , euerie one whereof must bee also diuided into 16. parts by the helpe of 16. lines , diuersly drawne from the center of one little circle to another , in such order as the figure here placed more plainely sheweth to the eie , than I can expresse the same by mouth . And these little circles do signifie 16 litle Mariners compasses , the lines wherof signifying the winds , do shew how one place beareth from another , and by what winde the shippe hath to saile . But besides those little circles there is woont to be drawne also another circle somewhat greater than the rest vpon the verie center of the Horizon , which Circle by reason of the 16. lines that were first drawne passing through the same , is diuided into 16. parts , and the Mariners doe call this circle the mother compas .

mariners 'card
The shape and figure of the first liniaments of the Mariners Carde drawne after the olde maner , and how to set downe the places of the land or sea therein . Chapter . 28.

NOw the true setting downe of the places in the Mariners carde , as Points , Capes , Bayes , Floods , Ilands , and such like is to be done by knowing what latitude and longitude euery place hath , which is to be learned either by modern tables made of purpose : for the ancient Tables make no mention of any longitudes or latitudes of such places as are in the new found land , which land to them was neuer knowne , or els by such Mariners cardes already made as doe shew the true longitudes and latitudes of those places which you woulde describe in your Carde , of which longitudes and latitudes , and especially the longitudes of the places in the West Indies , few or none are as yet truely set down ,

Moreouer , to knowe the distance of places , that is , how many leagues or miles one place is distant from another there is woont to be set downe in the Mariners Carde a scale , otherwise called by the Mariners a Tronke , the making wherof is plainly taught by Martin Cortes in his arte of Nauigation , in the second chapter of his third booke , and also how to graduate the Cardes , to shew what latitude euerie place hath , and there also hée teacheth how to translate one Carde into another , and howe to reduce a greater Carde into a lesser , and contrariwise . To which booke I referre you , and the rather for that it is in English , translated many yeares since out of Spanish by M. Richard Eden .

But for so much as the sea and the earth doe make together one whole rounde body , the lines of the 32. rombes in the Carde being drawne right , and made to signifie great circles , can neuer shew the true course that the ship hath to hold , which Michaell Cogniet prooueth by a Figure demonstratiue , and thereof gathereth thrée conclusions . First , that a man may sayle right north and South round about the worlde , if the Sea in all that course be nauigable , and so returne againe to the Porte from whence he first departed . Secondlie , that making the Equinoctiall his Parallell hée may sayle East and West round about the worlde , and so returne to the porte from whence hee departed , Also if hée sayle East and West in any other Parallell that is distant from the Equinoctiall , hee may returne by the same Parallell to the porte from whence hée first departed , and yet not about the whole worlde , for that cannot be done , but onelie when the Equinoctiall is his Parallel , Thirdlie , that whosoeuer sayleth by any other rombe than by one of the foure principall , hée by often changing his Meridian and Horizon must needs saile by a line Spirall , which is neither perfectlie right , nor perfectlie rounde , and thereby hée may well approch the Pole , and also goe rounde about it , but yet with vnequall distance , so as he shall bee nigher beyonde it than on this side , by meanes whereof hee cannot returne to the place from whence hée came , as you may plainelie perceiue by this Figure demonstratiue here placed .

In which Figure the letter A. doeth signifie the North Pole , and the letters B. C. the Meridian passing through the Pole A. then suppose your ship to bée in Q. whereas the Pole is eleuated 30. degrées . and Q. to be your Zenith , and the right line G. Q. D to bee your right line of East and West cutting the foresaide Meridian with right angles , and let D. be the East point , and E. the West point .

Nowe you may sayle from Q. towardes the North with a South winde , and from A. you may saile againe Southwarde with a North winde vntill you come to the South Pole , and from thence you may sayle againe Northwarde with a South wind vntill you come againe to Q from whence you first departed , and so you shall haue gone rounde about the worlde vnder one selfe Meridian B. A. C. But in sayling East or West , you shall continuallie change your Meridian , and thereby change your East point , for in sayling from Q. towardes D. you come immediatelie to the Meridian A. F. whose right East point is G. and in sayling further Eastward , you come to the Meridian A. H. whose right East point is I. and so foorth from one Meridian to another , notwithstanding in kéeping still in one selfe parallell marked with K. L. M. you may saile rounde about the Pole A. which is the center , and so come againe to the Port Q. from whence you first departed , but not about the whole worlde : for that you cannot doe vnlesse the Equinoctiall were your Parallell , as hath bene saide before , but if you sayle by any other rombe than by one of the foure principall , that is , East , West , North , or South , your course shall neither be by right line , nor yet by true circle , but by a spirall line , which is partly right , and partlie round , so as you cannot with like course returne to the place from which you departed , by reason that you change so often both Meridian and Horizon , as by the Spirall line Q. N. O. drawne in this Figure you may easilie perceiue , which approcheth nigher to the Pole beyond it than on this side . Moreouer the sea men by making the Meridians and parallels in their cardes all of equall distance , they make some countries far greater than they should be by the one halfe . Also by that meanes he that sayleth East and West round about the Pole in the Parallell whose latitude is 60. degrées , should make as long a voyage as hee that sayleth East and West alongst the Equinoctiall , the voyage whereof is twise as long as the other , for the redresse and remedie of which faultes , Cogniet hopeth to finde out some more perfect rule of making cardes when opportunitie of time shall serue : in the meane time to reforme the saide faults , Mercator hath in his vniuersal carde or Mappe made the spaces of the Parallels of latitude to bée wider euerie one than other from the Equinoctiall towards either of the Poles , by what rule I knowe not , vnlesse it be by such a Table , as my friende M. Wright of Caius colledge in Cambridge at my request sent me ( I thanke him ) not long since for that purpose , which Table with his consent , I haue here plainlie set downe together with the vse thereof as followeth .

The Table followeth on the other side of the leafe .

A Table to drawe there by the parallelles in the Mariners Carde together with the vse thereof in trewer sort than they haue bene drawne heretofore , and the vse thereof . Chapter . 29.

Degrees of the Meridian , beginning from the Equinoctial circle . Equall partes of the Meridian in the Mariners Card ▪ of which partes euery degree of the Equinoctial contayneth 60 miles . 1 60 2 120 3 180 4 240 5 300 6 361 7 421 8 482 9 542 10 603 11 664 12 725 13 787 14 849 15 911 16 973 17 1035 18 1098 19 1162 20 1225 21 1289 22 1354 23 1419 24 1484 25 1551 26 1617 27 1684 28 1752 29 1820 30 1889 31 1959 32 2029 33 2100 34 2173 35 2245 36 2319 37 2394 38 2470 39 3546 40 2624 41 2703 42 2783 43 2865 44 2948 45 3032 46 3118 47 3205 48 3294 49 3385 50 3477 51 3572 52 3668 53 3767 54 3868 55 3972 56 4078 57 4187 58 4299 59 4414 60 4532 61 4655 62 4781 63 4911 64 5046 65 5189 66 5331 67 5482 68 5639 69 5804 70 5976 71 6156 72 6346 73 6547 74 6759 75 6985 76 6226 77 7484 78 7764 79 8067 80 8399

THe vse of this Table for making the sea Carde is thus . Ouerthwart the middest of the plaine superficies wherein you would drawe the lineamentes of the Carde , describe a right line marked with the letters A. B. C. whereof B. is the verie middest or center , and this line representeth the Equinoctiall , the ende whereof on the right hand , marked with the letter C. signifieth the East , and the other end on the left hand marked with the letter A. signifieth the West , which Equinoctiall line must bee diuided into 36. degrées , then crosse the same squirewise with perpendicular lines , passing through euerie tenth or fift degrée , as you sée in the example following . Then take with your compasses the length of halfe the Equinoctiall , that is , 180. degrées , and set one foote of your Compasses in the mutuall intersection of the Equinoctiall , and of that perpendicular or Meridian which passeth through the East ende of the Equinoctiall line , marked with the letter C. and with the other foote make a pricke in the same perpendicular or Meridian , and marke that pricke with the letter D. that done , diuide the space contayned betwixt this pricke and the Equinoctiall firste into thrée equall partes , and euerie one of those into other thrée equall partes , so haue you 9. partes . And againe , euerie one of those into thrée , so haue you twentie seuen partes , and diuide euerie one of those partes into foure partes , so shall you haue a hundred and eight partes , and if there bee space inough , diuide again euerie one of those into ten , so shal you haue a 1080. partes , and if it be possible , diuide againe euerie one of those partes into other tenne , so shall you haue in all 10/800. partes , but this can hardlie be done , vnlesse the Carde be verie large , wherein euerie degree of the Equinoctiall is neere an inch long , which happeneth verie seldome , and therefore 1080. partes shall suffice .

And for the easier numbring of these partes , set to them Arithmeticall figures with blacke leade , which may afterwarde be put out when your worke is done , beginning at the Equinoctiall and so procéede from thence both Northward and Southwarde , then looke what number standeth right against euerie degrée in the former Table , which degrées doe extende from 1. to fourescore degrées , and omitting alwaies the first figure on the right hande of that number which you finde , for so you must alwaies doe when your diuision containeth no more but 1080. partes , count that number which remayneth vpon the line of diuision , and there make one pricke , and make another pricke of the same distance from the Equinoctiall vpon the outermost Meridian on the left hand , through which two prickes drawe a right line , and that shall be your first Parallell of latitude , and so procéede with all the rest first Northwarde , and then Southward , if you mind to make an vniuersall Carde . But the example following containeth no more but the one halfe of an vniuersall Carde procéeding from the Equinoctiall line Northwarde vnto the 80. degrées of latitude : for further Northwarde than 80. degrées is no land as yet discouered or knowne , nor yet all that , so farre as euer I could learne .

The draught of the Meridians and parallels of the Mariners Carde or nauticall Planispheare according to the former Table .

IN this last figure or Table is first drawn ( as you sée ) the Equinoctiall line marked with the letters A. B. C. and that line is deuided into 360. degrées , and therein also are drawne perpendicular lines , as well thorough the beginning and ending of the sayde Equinoctiall lyne , as also through euery tenth degrée thereof , which be the Meridians , and are euery where equidistant each one from other , then take halfe the length of the Equinoctiall which is A. B. or B. C. with your Compasses and setting one foote in the end of the Equinoctiall marked with C. make with the other foote a pricke at D. in the Meridian or perpendicular line marked with the letters C. D. E. then deuide the space contained betwixt C. & D. into 1080. parts in such sort as before hath béene shewed , and set the figures vnto them as here you sée , to the intent that you may the more readily number the parts . Then looke in the first Table what number answereth to euery 10. degrée of the Equinoctiall , & casting away the first figure of that number on the right hand , find out the parts answerable to the number remaining in the line C. D. and at those parts set prickes in both the outermost Meridians through which prickes you shall drawe the Parallels . As for example in the first Table you sée that the number right against 10. degrées is 60. ( the first figure 3. towardes the right hande being reiected ) therefore looke 60. in the line C. D. and by that part draw the first Parallel distant 10. degrées from the Equinoctiall . And after this maner all the rest of the Parallels are to be drawne .

Many doe vse to paint the vacant places in their Cardes with ouer many flags , and the compasses thereof with diuerse and superfluous colours which William Borne misliketh , wishing that in stead thereof they would shewe by letters or other Characters what moone doth make a full sea , in such places as are necessary to be knowne , and also to drawe the true shape and fashion of euery Cape or headlande that is néedefull about the coast , and at what point of the compasse the land riseth of this or that fashion , for being néere the land it wil séeme to be of one fashion , & being far off to be of an other fashion , and to mistake any place on the sea is very dangerous to the Mariner . But aboue al things let him that saileth by Card & Compasse be sure that the néedle of his compasse haue the like declination that his néedle had which made the Card. For Cogniet reporteth that certaine Mariners being in the west Indies , and séeing the North starre to be Northeast , maruailed thereat very much not knowing the cause of that error , which indéed was for that the néedle of their compasse was made to decline Northeast , whereas it should rather haue declined Northwest , for the West Indies do stand Westernly from the Azores . It is necessarie also to set downe in the Card such places as are daungerous , as sands , flats , or shoulds , rockes , and such like things as are not alwayes to be séene with the eye , to the intent that the Mariner being aduertised thereof may shunne the same . All which things before mentioned , Wagoner hath in all places contayned in his booke very well obserued . Thus hauing spoken sufficiently of making the Mariners Carde , Now I thinke it good to shewe the most necessary and chiefest vses thereof .

The chiefest vses of the Mariners Carde . Chap. 31.

THe chiefest vses of the sea Card are these foure here following :

The first is to know thereby how that place whereunto you would sayle , beareth from the place or Port from whence you set off , or depart . And that is to be knowne by the lines of the Mariners Compasses painted in the Card in this manner following : Take a paire of Compasses and hauing opened them , set the one foote thereof in the very place from whence you depart , and the other foote in the next line of that Compasse which is nearest vnto your place of departing , I meane such line as doth most rightly direct you to the place to which you would go , and your Compasse being opened at the fit widenesse to serue that line , draw them from the place of your departing vnto the place whereto you would go , suffering that foote of the Compasse which standeth vpon the line of the winde whilest you draw it forward , not to swerue one iot from that line , and that line will eyther rightly direct you to the place assigned or fall short thereof , or else ouerrech the same , if it fall short , then take another line nearer to the place from which you departed , and if it ouerreach , take some line that is further off from the place of your departing , and hauing found a line that pointeth directly to the place , consider what winde or rombe it is , for by that winde the place assigned beareth from you , and the rombe or winde opposite to that is the winde whereby you haue to sayle .

The second vse is to knowe by the Carde how farre the place whereto you go is distant from the place of your departing , which is done by helpe of the skale or trunke set downe in the Carde thus : Take the iust distance betwixt the two places with your compasses by setting the one foote in the one place , and the other foote in the other place , and apply that widenesse of the compasses to the skale or trunke , and the trunke wil shew how many leagues the one place is distant from the other , and if the distance betwixt the places be longer then the trunke , then take first the length of the trunke with your Compasses , and looke how many times that is contained in the space betwixt the two places , and if there doe remaine any odde measure , then hauing taken that odde measure with your compasses by setting them at such widenes as is answerable to that odde measure , apply that widenesse to the first part of the trunke , so shall you know the iust measure of the whole . And this rule serueth to take the true distance of any other two places whatsoeuer set downe in the Card.

The third vse is to know by the Card what Latitude or altitude of the Pole any place set downe in the Carde hath , which is done by helpe of the line of degrées of Latitude , otherwise called the Graduation of the Carde in this manner following . Set the one foote of your compasses in the very place whereof you would know the Latitude , and the other foote in the line of East & West , which is next vnto that place , and kéeping that foote still vpon that line , draw your compasses forward vntill you come to the line of degrées , and marke what degrée of the said line the foot of the compasse which was first set in the place doth crosse or touch , for that is the degrée of Latitude for that place , numbring from the lowest degrée of Graduation vpwarde , so shall you finde the Latitude of Lisbone in Portingale by Mercator his vniuersall Carde , and by the Carde set downe in Martin Cortes booke , and also by Medina his Carde drawne in his booke of Nauigation to be 38. degrées 30. minutes and somewhat more . But by the Tables of Ptolomie you shall finde it to be 40. degrées 24. minutes , and by Appian his Tables to be 39. degrées and 38. minutes .

The fourth vse chanceth when you are driuen out of your right course by stormes or tempest , which stormes how to foresée and to prognosticate , is plainely taught by Martin Cortes in the 19. Chap. of his second booke . Also you may be driuen by force of contrary windes , by surging of the sea , or by ouerthwart tides , currants , & such like impediments , so as you can not lay your course right to the place assigned , for remedy whereof you must séeke in what place you are , & to note the same in your Card which as the Mariners terme it , is to make a pricke in the Carde , which to doe truely in time of néede many things are to be known & well obserued and kept in memorie . First to know what latitude the place from whence you first departed hath , then to kéepe in minde what way your shippe did make good at euery shift of winde , that is to say , how many leagues , and in how long time you sayled by euery seuerall winde : and then not knowing well where you are , nor how farre you are distant from the place whereto you would go , learne to knowe by helpe of your Astrolabe or crosse staffe , in such sort as is before taught , the altitude of the Pole in that place where you are , which if you finde to be all one with the Latitude of the place of your departure , then you may assure your selfe that you haue sayled by the line of East and West without altering your Latitude , but if you finde the Latitude of the place where you are , to be more or lesse then the Latitude of the place from whence you departed , then resort to your Card , and take two payre of Compasses opened at such widenes as the one foote of the one Compasse may stand in the place from which you departed , and the other foote of the same Compasse to stand in the line of the rombe whereby you sayled : and let the one foote of the other Compasse stand in that degrée of Latitude which you last found , and the other foote of the same Compasse in the next line of East & West , and holding the Compasses so ordered in ech hande one paire : draw them both so as they may méete together , taking good héede in drawing them , that the foote of that compasse which was placed in the line of the winde , may at no time swarue from that line , nor the one foote of the other Compasse to swarue from the line of East and West , wherein it was first placed , and whereas the two féete of those Compasses doe méete , that is to say , that foote of the one compasse which was drawne from the place of your departing doe méete with that foote of the other compasse which came from the degrée of Latitude last found , where these two féete ( I say ) doe méete , there make a pricke or marke in your Carde , for that is the place where you shippe is at that instant : And from thence you must take your right course againe to the place whereunto you would go . But because it is necessarie aswell at this time as at all other times , to knowe what way your shippe hath made , and that the same is not in mine opinion , so plainely nor so commodiously taught by any one that I haue read , as by Michaell Cogniet , I minde therfore in the two chapters next following to set downe his way not only how to find out the way of your shippe when you saile South and North vnder one selfe Meridian , or in any other place where you are to change in your gate the latitude or altitude of the Pole , but also how to finde out the way of your shippe in sayling right East and West , without changing the altitude of the Pole , which way as he sayth , was neuer heretofore knowne to any Pylot but to him selfe first author and inuentor thereof .

How to know the way of your ship , and how many leagues are to be counted for one degree of Latitude in euerie Rombe whereby you sayle . Chap. 32.

Then suppose C. D. to be another Parallell equally distant from A. B. by one degrée of altitude . Now if you saile right East or West , then you shall alwayes remaine in the Parallel A. B. equally distant from the Pole . But if you saile from A. right North so farre as the altitude of the Pole is augmented one degrée , then your ship shal be in C. and if you sayle by the first rombe towardes the East so farre as the Pole doth altar in altitude one degrée , then your shippe shall be in E. and thereby your way must néedes , be longer , and so consequently the more rombes you decline from North to East , the longer is your way , and the more leagues must be accounted to one degrée of altitude , as the lines drawne from A. to E. F. G. H. I. K. and D. do shew . And what same euer is saide here of the quarter from North to East , the same is to be vnderstoode in all the other thrée quarters , that is , from North to West , from South to East , and from South to West . But if you will knowe how many leagues doe belong to euery degrée according to the rombe whereby you sayle , then consider well this Table here following .

ThefirstTable. Rombes . Leagues . 1 17 ⅚ 2 18 14/15 3 21 ½0 4 24 ¾ 5 31 ½ 6 45 ¾ 7 89 ⅔

For in sayling from North or South towardes East or West so farre as you come to change one degrée of altitude of the Pole , the saide degrée doth require for the first rombe , &c.

Now then to know how much way you haue made in sayling , you must first know aswell the Latitude of the place from which you departed as of that place whereunto you be arriued : then by the foresaid Table séeke to knowe how many leagues doe belong to a degrée of that rombe whereby you haue sayled , for in multiplying the number of the leagues by the degrées of the difference of the two Latitudes , the product thereof will shew you how many leagues you haue sayled , notwithstanding sith the way may be made longer or shorter by changing or shifting of the winde , it is néedefull that the Pylot haue consideration thereof , who by skilfull coniecture must sometime eyther adde to , or take fro according as néede shall require . Moreouer by the foresaid figure marked with letters , you may also easily vnderstande how much you change in Longitude , that is to say , how much you are distant from the Meridian of that place from whence you departed , be it eyther towardes the East or West , for he that sayleth from the point A. as is aforesaid right North or South , he remaineth alwayes vnder one selfe Meridian : but he that sayleth by the first rombe towards the East or West so farre as he changeth one degrée of Altitude , and arriueth to the point E. is now distant from his first Meridian so much as is the space betwixt C. and E. which we finde by computation to be thrée leagues and a halfe for one degrée of Latitude , which amounteth to twelue minutes of a degrée , and so of the rest of the rombes , as appeareth by this Table here following .

ThesecondTable. Rombes leagues Degrees and minutes or Longitude . first 3 ½ 0 12 second 7 ¼ 0 25 third 11 ⅔ 0 40 fourth 17 ½ 1 0 fift 26 ⅕ 1 30 sixt 42 ¼ 2 25 seuenth 880 5 2

For in sayling from North or Southe towardes East or West so farre as you change one degrée of altitude of the Pole you change also your Meridian , and thereby your Longitude , the quantitie whereof answerable to euery rombe , is set downe on the right side of this Table .

And to make this more plaine by example suppose that you sayle from Lisbone , which is a famous port in Portingale , by the winde Southwest and by west , which is the fift rombe from south to West , so farre as you finde the altitude of the Pole to be 18. degrées lesse then at Lisbone . Now if you would know how many leagues you haue sayled , and also how much the Meridian of that place is more westward then the Meridian of Lisbone , then doe thus . Looke in the first Table , and you shall finde that to one degrée of the fift Rombe doe belong 31. leagues and a halfe , which leagues being multiplyed by 18. doe make in all 582. leagues and ¾ . which you haue sayled . then looke in the second Table , and you shall finde for the fifth rombe one degrée and 30. minutes of Longitude , which being multiplyed by 18. and a halfe , do make 27. degrées and ¾ . of a degrée , for by so much is Lisbone more Eastward , then the place where you are . And whereas the first Table is made according to the proportion of right lines such as are commonly drawne in Mariners Cardes , Cogniet maketh another Table according to the proportion of circular lines , which for that it differeth very little or nothing from the first Table , I omit here to set it downe . But now because the first Table doth chiefely serue those that fayle either East or West in any Parallell betwixt the Equinoctiall and the 60. degrée of Latitude or Altitude of the Pole : And that from thence foorth by reason that they sayle by more oblique and spirall Circles doe make the longer voyage , Cogniet thought good to adde a thirde Table shewing how many leagues be answerable to one degrée of altitude to those that sayle eyther East or West in any Parallell that is betwixt the Pole and the 60. degrée of altitude , which Table differeth not much from the others in the foure first rombs , but in the thrée last , that is in the 5. 6. and seuenth rombe it differeth greatly , and most in the seuenth as you may easily perceiue by comparing this and the first Table together .

ThethirdTable. Deg M Leagues . The first rombe hath 1 1 Which according to the proportion of 17. leagues & a halfe , for one degree do make for euerye rombe so many leagues as this Table sheweth . 17 19/24 the second rombe 1 5 18 23/24 the third rombe 1 12 21 0 the fourth rombe 1 26 25 1/12 the fift rombe 1 50 32 3/12 the sixt rombe 2 42 47 ¾ the seuenth rombe 5 44 100 3/3

Howe to account the leagues in sayling directly East or West without changing the latitude or altitude of the pole . Chap. 33.

IN sayling ritht East or West , you continue still in one selfe Parallel without making any charge of Latitude of the Pole . Most men therefore thinke it impossible for you to make any true account of the leagues but onely by coniecture , for remedie whereof Cogniet hath inuented a rule most certaine , as he saith , the foundation whereof is thus : First you must suppose the Meridian of that place from whence you depart to be a firme and fixed point , and to be the very beginning of the Parallell wherein you sayle , then you must knowe what houre it is aswell at the place from whence you first departed , as also at the place whereunto you are arriued , and hauing the difference of the houres , you must knowe howe many leagues euery houre yéeldeth , according to the Parallell of that altitude of the Pole vnder which you sayle , so shall you easily knowe how manie leagues you haue sayled . This foundation being first laide , Cogniet setteth downe his generall rule in this manner following : When soeuer you haue to sayle ( sayth he ) right East or West , you must first prouide your selfe of these two thinges , the one is of an Astronomicall ring that may iustly shew the houre at anie time : In stéede whereof I thinke it better to haue an vniuersall Dyall , such a one as is described by William Borne in the one and twentieth Chapter of his Regiment : the other thing is to haue a true houre glasse that will run continually foure and twentie houres , and therefore had néede to be thrée times more long and large then common houre glasses be , whereof the glasse makers can quickly prouide you , and because the ship leaneth sometime on the one side , and sometime on the other , it shall be néedefull to hange the saide houre-glasse with rings of brasse or latten , to the intent that it may alwayes hange leuell like as the Mariners Compasses are wont to be hanged in their boxes . Now being thus prouided of these two instruments , you must at your departing prepare your houre glasse , that is to say , you must set it a running iust at noone when the sunne is right South , not forgetting to turne it in your voyage once a day in the very instant that it is readie to runne out . Then hauing sayled certaine daies , and being arriued at the place where you would knowe what way you haue made , looke to your houre-glasse , and tarry vntill it be full runne out , at which instant séeke to know what houre it is by your Astronomicall ringe , or rather by the vniuersall Dyall , for if you sayle East , you shall find it to be past noone , but if you sayle West , then it will want of noone , and kéepe those houres in minde that you may know how much it is more or lesse then noone , for those houres by shewing the difference of the houres that are betwixt the Meridian of the place from whence you departed , and the Meridian of the place whereto you are arriued , will certifie you with the helpe of this Table here following , what quantitie of way your shippe hath made .

A Table to help you to know what way your ship hath made in sayling right East or West without changing your Latitude , together with a briefe description and vse thereof .

De. of Latit . leagues . 0 262 ½ 1 262 5/12 2 262 ¼ 3 262 ⅛ 4 262 5 261 ½ 6 261 7 260 ½ 8 259 ⅞ 9 259 10 258 ½ 11 257 ½ 12 256 ⅔ 13 255 ¾ 14 254 ⅔ 15 253 ½ 16 252 ¼ 17 251 ⅛ 18 249 ⅔ 19 248 ¼ 20 246 ¾ 21 245 22 243 ¼ 23 241 ½ 24 239 ¾ 25 238 26 236 27 234 28 231 ⅞ 29 229 ½ 30 227 ¼ 31 225 32 222 ½ 33 220 ¼ 34 217 ½ 35 215 36 212 ⅓ 37 209 ¾ 38 206 ⅞ 39 203 ⅚ 40 201 41 198 42 195 43 192 44 188 ¾ 45 185 ½ 46 182 ¼ 47 179 48 175 ½ 49 172 50 168 ½ 51 165 52 161 ½ 53 157 54 154 ¼ 55 150 ½ 56 146 ⅔ 57 143 58 139 59 135 60 131 61 127 62 123 63 119 64 115 65 111 66 106 ¾ 67 102 ½ 68 98 ¼ 69 94 ⅓ 70 89 ⅚ 71 85 ½ 72 81 73 76 ⅔ 74 72 ⅓ 75 68 76 63 ½ 77 59 78 54 ½ 79 50 80 45 ½ 81 41 82 36 ½ 83 32 84 27 ½ 85 22 ¾ 86 18 ⅜ 87 13 ¾ 88 9 ¼ 89 4 ⅔ 90

This Table as you sée consisteth of foure collums , in euery one whereof are set downe on the left hande the degrées of altitude of the Pole , and next to that on the right hande the leagues answerable to euery degrée , the vse whereof is thus : First séeke out in the said Table the degrée of altitude belonging to that Parallell vnder which you sayled , and next to that on the right hande you shall finde the number of leagues incident to that degrée : which number of leagues , if you multiply by the number of houres before found , the product thereof will shew you how many leagues you haue sailed . And if there be any minutes annexed to the hours , then multiply also the number of the foresaid leagues found in the Table , by those minutes , and deuide the product therof by 60. and the quotient shall be leagues , which you must adde to the former leagues , and the summe thereof will shew how many leagues you haue in all sayled .

An example of counting the way of your shippe in sayling right west . Chap. 35.

SVppose that you haue to saile from the Cape Saint Vincent in Spaine right West , and therefore you prepare your houre glasse as it may begin to run iust at noone , and hauing sailed 8. or 9. dayes , ( not forgetting once euery day to turne your houre-glasse ) you doe arriue at one of the Iles of the Azores called S. Mary , and there hauing tarried vntill your houre-glasse be cleane run out , and séeking to know at that instant by your Astronomicall ring , or by some vniuersall Dyall what houre it is , you finde that it is a 11. of the clocke and 10. minutes past , which wanteth 50. minutes of iust noone , and that is the difference betwixt the two Meridians , that is to say , the Meridian of Cape S. Vincent , and the Meridian of the Ile S. Mary , both which places being in one selfe Parallell must néedes haue one selfe Latitude or Altitude , which is 37. degrées , which altitude being found in the Table in the second collum on the left hande , you finde there also hard by it on the right hand , 209. leagues and ¾ . of a league , for euery houre of that Parallell , which number of leagues , if you multiply by the foresaid 50. minutes , the product shal be 10/487. ½ which if you deuide by 60. you shall finde in the quotient 174. leagues and 19/●4 . parts of a league , which is the iust quantitie of your whole voyage .

Another example of counting the way of your shippe in sayling right East . Chap. 36.

SVppose then that you haue to sayle from the newe found lande right East in the Parallel of 50. degrées : and hauing caused your houre glasse to begin to run iust at noone , you set foorth and sayle the space of 15. dayes , not forgetting euery day once to turne your houre glasse . Now if at the 15. dayes end , you would know how many leagues you haue sayled , to finde in your Card in what place you are , you must first tarrie vntill your houre-glasse be cleane runne out , and at that instant séeke by your Astronomicall ring , or by some vniuersall Dyall , to know what houre it is , which because you haue sayled right East , you finde to be two houres and twelue minutes afternoone , then resorting to the Table aforesaide , you finde in the third colum the 50. degrées of altitude , together with the number 168. leagues and a halfe , annexed to the saide degrée for one houre of that Parallell , which summe being multiplyed by two houres , maketh 337. leagues , then multiply once againe 168. by the odde 12′· and deuide the product thereof by 60. so shal you find in the quotient 33. leagues and somewhat more , which being added to the former summe 337. leagues will make in all 370. leagues and a little more , and that is the true quantitie of your voyage . And Cogniet sayth that by practizing this way of counting , you may know euery day , yea euery houre , what way your shippe maketh in sayling right East or West .

To knowe how much you goe out of your way in sayling by one wrong rombe or by more . Chap. 37.

IF in sayling to any place you direct your course eyther too high or too lowe by one rombe you loose in euery 100. leagues 19. leagues and ⅗ . of a league , so as you go out of your way almost the fift part of your voyage . And if by two rombes , then in euery hundred leagues you loose 39. leagues , that is to say you go so far out of the right way : And if you fal thrée rombes out of your right course , then in euery hundred leagues you loose 58. leagues : and if you fall foure rombes , then you loose in euery hundred leagues 76. leagues , and the more rombes that you mistake in your direction , the more you wander out of your right course . Thus much touching the Mariners compasse and his Carde : nowe wee will speake of the North starre , and of his guardes , and then of the sunne and of the Moone , and so ende .

Of the North starre , otherwise called the loadstarre , and of his guardes , and how to know the same . Chap. 38.

THough euery Mariner knoweth the North star so soone as he séeth it in the firmament , because it is his chiefe guide to direct thereby his shippe in the night season in all the North partes of the world : yet euery man that is no sayler knoweth him not , and therefore minding here to treate thereof , I thinke it not amisse first to teach him how to knowe it , and in what part of the heauen it is placed , and how farre it is distant from the North Pole .

The Poets do faigne that of the 48. Images of the fixed stars that be in the heauen , which are otherwise called Constellations , there be two Beares hauing tayles , whereof the one is called the great Beare , and the other the little Beare , in euery of which are seuen principall stars : And that bright starre which is in the very tippe of the little Beares tayle , is commonly called of our English Mariners the Loadstar , of the Latines Stella polaris , of the Gréekes Cinosura , and of the Arabians Alrucuba . But to know this star , you must first find out the seuen stars of the great Beare by looking towards the North part of the firmament , of which stars foure being placed in his bodie , do make ( as it were ) a 4. square , and the other thrée stars being placed in his tayle , which is somewhat rising in the midst , doe represent the portion of a Circle , the little Beare hath also the like shape sauing that her féete & belly do turne vpwards , so as the tippe of her taile is answerable to the two hindermost starres of the great Beare , as you may sée in these figures following .

Some do call the great beare great Charles Waine or Wagon , and the little Beare little Charles wain , because in each figure the foure square stars do signifie the four whéeles , & the other thrée stars the 3. horses : Nowe to finde out the Load-starre , imagine a right line to passe thorough the two hindermost starres otherwise called the guardes or pointers of the great Beare , as you sée here in this figure , and that line will rightly direct your eye to the Loadstarre which standeth in the tippe of the little Beares taile , for there is no bright star that can be séene betwixt the two gards of the great Beare and the Loadstarre . Notwithstanding some do affirme that there is another starre nigher vnto the Pole then the foresaide Loadstarre , and that star is as they say no more but 50. minutes distant from the Pole : I haue heard that there is a Priest and Astrologer with the King of Demnarke , who is woont to finde out this starre by the helpe of an instrument whereby hee getteth first the sight of sixe other stars at one instant , which way I feare mee is more troublesome then profitable , and if there were any such starre indéede , and therewith so néedefull and méete to bee vsed as is the Loadstarre , I doubt not but that some of our learned Pilots would haue found it out , and also haue had some vse thereof long ere this time : But leauing this matter , I will returne to the description of our Mariners common Loadstarre , and shew of what bignesse it is : also what Longitude , Latitude , declination , and motion it hath , and finally the chiefest and most necessarie vses thereof .

This starre according to the prutenicall Tables is of the third bignesse , and hath in Longitude 53. degrées 30. minutes , counting from the first starre of the Rams horne , whose place in these our dayes is in the 27. degrée 30. minutes of Aries , and by this meanes the place or Longitude of the Loadstarre is in the 21. degrée of Gemini , & his Latitude , counting from the Ecliptique line towardes the North Pole of the Zodiaque is 66. degrées 0. minutes , and his declination counting from the Equinoctiall towardes the North pole of the worlde is 86. so as he is in these dayes distant from the North pole 4. degrées , some say 4. degrées and 9. minutes , but Cogniet sayth , that he is distant from the pole but 3. degrées and a halfe , who findeth by the Astronomicall Tables that this starre at the birth of Christ was 12. degrées 36. minutes distant from the Pole , and euer since hath gone decreasing , and shall decrease euery day more then other , vntill it come to be but 26. minutes distant from the Pole , which is as nigh as it can approch to the Pole , for then his distance shall beginne againe to encrease . This starre maketh his daily reuolution from East to West in 24. houres , as all other starres doe by vertue of the first mooueable , but his circuit is so small , and his gate so slow , as in the space of 24. houres he goeth not much more then 24. degrées of the Equinoctiall , and in his turning round about he maketh as it were a little Circle , in the middest whereof is the North pole it selfe , which is inuisible & can not be séene .

And though the image or constellation of the little Beare containeth many stars , yet the seuen before mentioned and set foorth in figure are most obserued , of which seuen starres in the little beare , two stars called the gards of the North star are to the Mariners most familiar , for diuers respects hereafter declared . And of those gards the one is Northerne & the other Southern , and both are said to be of the second bignesse , & yet the Southerne gard séemeth to the eye both lesser and darker then the other . Of which two stars I thinke it not amisse to set downe here the Longitude , Latitude , and declination as I haue done before of the Loadstarre .

The Longitude of the South garde is 100. degrées 30. minutes , whereby his place is in the ninth degrée of Leo. And his latitude is 72. degrées 40. minutes , and his declination is 73. degrées , so as he is distant from the North pole 17. degrées .

The longitude of the North guard is 109. degrées , 30. minutes , so as his place is in the 20. degrée of Leo , and his latitude is 74. degrées , 50 , minutes , and his declination is 76. degrées , so as his distance from the North Pole is 14. degrees . Thus hauing shewed the greatnesse , the longitude , the true place , the latitude , and the declination aswell of the Lodestarre , as of his two guardes , I thinke good now to set downe the chiefest vses that sea men haue of them .

The vses of the North starre , and of his guardes . Chapter . 39.

THe first vse is to knowe thereby the variation of the Mariners Compasse .

The second is to know by his guardes when the North starre is aboue or beneath the Pole .

The third is to know by the North starre and his guardes with the help of an instrument called a Nocturlabe the houre of the night .

The fourth is to know the eleuation of the Pole . And you haue to note that the North starre and his guardes are alwayes aboue our Horizon , and doe neuer goe downe in any place whereas the North Pole hath any eleuation , be it neuer so little . But besides these starres there be also in this our latitude diuers other images of starres that are alwayes aboue our Horizon , and doe neuer go downe , as the Dragon , the great Beare , the image of Cepheus , of Cassiopeia , the image of Auriga , hauing the Goat at his back which is a faire bright starre of the first bignesse and many others , which I leaue to name , because I haue heretofore described them all at large in my treatise of the Globes , by which starres by reason that with vs they neuer goe downe , the latitude of any place and also the houre of the night may bee knowne so well as by the Lodestarre .

But now as touching them vses of the foresaide starres before mentioned , the first whereof shewing how to find out by the Lodestarre the variation of the Compasse , is taught before in the 25. Chapter , according to William Boorne his rule set downe in his Regiment , and as for the other thrée vses they are plainly taught in the two chapters next following .

To knowe by helpe of a little Table made according to the Mariners rule touching the 8. principall rombes , shewing how and when the Lodestarre is either aboue or beneath the Pole , that you may knowe thereby the true altitude of the Pole in taking the height of the Lodestarre with your Astrolabe or crosse staffe . chap. 40. Chapter 40.

TO know this you must alwayes haue due consideration of the two guards of the little Beare , for when those guards are iust Southwest from the Lodestarre , then is the Lodestarre at his highest in the verie Meridian , and therefore it is right aboue the Pole , and when those guards be iust Northeast from the Lodestar then the Lodestarre is againe in the Meridian at his lowest , and thereby right vnder the Pole , and in both places his distance from the Pole is but thrée degrées and a halfe , as Cogniet saith , according to which account he setteth downe this Table following .

This Table is diuided into two collums , whereof that on the left hand contayneth the eight principall rombes or windes , and that on the right hand contayneth the degrées and minutes of distance of the Lodestarre from the Pole , being either aboue or beneath the Pole .

If the guards be The rombes or windes . The degrees and minutes of the declination of the Lodestarre from the Pole . West . Southwest South Southeast . Then the Lodestar is 1 ½ 3 ½ 3 0 1 0 Aboue the Pole . East Northeast North Northwest 1 ½ 3 ½ 3 0 1 0 Beneath the Pole .

This table differeth in one point from the Mariners common rule set downe as well by Medina , and by Martin Cortes , as also by William Boorne touching the guards , for they all appoint but halfe a degrée of declination of the North starre from the Pole , the guardes being either in the rombe Southeast , or Northwest , to both which rombes Cogniet appointeth one whole degreé of declination . The vse of which table in seeking to know , the eleuation of the Pole is thus . First , hauing taken with your Astrolabe the altitude of the Lodestarre aboue your Horizon , obserue immediatelie in which of the 8. former rombes his guardes be . For if they be in any of the 4. vpper rombes of the Table , then you must subtract from the height of the Lodestarre taken with your Astrolabe , so much as is set downe for that rombe in the collum on the right hand of the said table , and the remainder shall be the true altitude of the Pole , but if the guardes bee in any of the foure nether rombes , then you must adde so much to 〈◊〉 height of the Lodestar , and the summe thereof shall be the altitude of the Pole . But because this table serueth only for the 8. principall winds and for no more . Cogniet therefore setteth downe the making of a twofolde instrument , whereby you shall not onely know ( as he saith ) how much the Lodestarre is aboue or beneath the Pole in euery other rombe as wel as in the 8. principall rombes , but also you shall know therby the true houre of the night more exactlie than the Nocturlabe of Munster or Appian doth shew .

How to make an Instrument which wil shew at any houre of the night how much the Lodestar is either aboue or beneath the Pole in everie other rombe as well as in the 8. principall rombes , which are only contained in the former Table , and also the true houre of the night , the shape whereof followeth . Chapter 41.
The shape or figure of the Rectifier of the North star .

COgniet calleth this instrument Rectificatorium Stellae Polaris , that is to say , the Rectifier of the North starre , ioyning thereunto a Nocturlabe differing nothing at all from the Nocturlabe of Munster and of others , but onelie in placing the 21. of October in stead of the 28. of the same moneth vppon the line of North and South towards the handle of the instrument for such cause as is hereafter declared .

The making of the rectifier of the Lodestarre , is thus : Vpon a smooth peece of boord of firme wood , or vpon a péece of polished plate of brasse or Latten being sixe or seuen inches broade , hauing a handle , as you sée in the former figure , draw a circle diuided into 4. quarters by help of two crosse Diameters cutting one another in the center with right angles , the perpendicular whereof shall signifie the Meridian line , that is to say , the line of North and South , at whose vpper end set North , and at the nether end South , and the other ouerthwart Diameter shall bee the line of East and West , hauing East marked on the right hand and West on the left hand , and in euerie quarter of the foresaide circle you may if you will place the like rombes that are in the Flie of the Mariners Compasse , as you may sée done in the foresaid figure . Moreouer , you must diuide the vpper quarter on the left hand into thrée equall partes , and hauing taken two of those partes with your Compasses , measuring from the North point downe toward the West , and there make a pricke , marking the same with a little blacke crosse , and from that crosse drawe a right line that may passe through the Center vnto the circumference of the Circle , and there make another little blacke crosse , and this line shal diuide the circle into two equall partes , which line if you crosse with another right line passing through the center , and making thereby right angles , you shall diuide the circle into other foure quarters differing from the first foure quarters , though not in quantitie , yet in place , of which foure last quarters you must diuide that which is on the left hand into 90. equall degrées , beginning your account at the little blacke crosse on the left hand , and so procéede downward towards the South point of the circle .

Now to know how much the North star in euery rombe mounteth or descendeth and how to place the same vpon the instrument so as you may knowe how much to ad to the altitude of the Lodestarre , or to subtract from the same , you must vse this Table following , which Table consisteth The degrees and minutes of the de●lination of the lodestarre from the pole . The degrees and minutes of the Quadrant , diuided by divers portions of numbers into 90 degrees . Deg. Min. ¼ 4 6 ½ 8 13 ¾ 12 22 1 0 16 36 1 ¼ 20 55 1 ½ 25 23 1 ¾ 30 0 2 0 34 51 2 ¼ 40 0 2 ½ 45 35 2 ¾ 51 48 3 0 59 0 3 ¼ 68 13 3 ½ 90 0
of two collums , whereof that on the left hand containeth the degrées and minutes of the declination of the Lodestarre , procéeding from one quarter of a degrée to another , vntill you come to 3. deg. and a half , which is ( as Cogniet saith ) the greatest declination of the Lodestar , from the Pole : and the other collum on the right hande containeth the degrées and minutes of a quarter or quadrant , diuided by diuers and sundrie proportions of numbers into 90. degrées which proportional degrées and minutes are to bee reckoned in the first quadrant diuided into 90 equal degrées , & next descending from the first crosse on the left hande thus . First , beginning at the said crosse , and descending downwarde , tell out 4. degrées , and 6. minutes , & right against that set downe ¼ . then tell in the same quadrant from the saide crosse 8. degrées , and 13. minutes , and right against that set downe ½ and so procéede according as the Table directeth you , vntill you come to 90. against which you must set down 3. deg. & ½ And as this quarter is diuided , so you may deale with the other 3. quarters , and in each of the 2 quarters which are beneath the 2 crosses forget not to write this word Subtract , and in each of the 2. quarters aboue the 2. crosses write this word Ad , as you sée in the former figure , Then ther shal want nothing but only a ruler or index , which must be fastened to the center of the instrument , so as it may be turned round about a pinne hauing a round hole in it , through which you may sée the Lodestar , and also at the same instant by lifting vp and downe the index you may sée on the outside of the instrument the foreguard of the said star appearing euen with the right edge of the index called Linea fiduciae , or the fiducial line , drawn from the center of the instrument alongst the inward edge of the index , and so is the instrument for that purpose fully perfected . But if you wold ad therunto a nocturlabe , then you must draw vpon the center therof diuers circles next to the space containing the winds : and the spaces betwixt those circles must be some wider , & some narrower For the first vpper space must be narrow , containing the daies of euery moneth , and the next space somewhat wider , containing the number of those daies set down in Arithmeticall figures , and the third space wider than that , containing the names of the moneths then next to the lowest circle of the nethermost space place a little rundle fastened to the instrument with the foresaid hoslow pin , so as it may turne round about the sawe . And this rundle must be diuided into 24. hours , that is to say , 12 for the day , & 12. for the night , which rundle would be made with téeth wherof one must be longer than al his fellowes signifying alwaies the 12. houre of the night , which is alwaies to be laide vpon the day of the moneth wherein you séeke to know the houre of the night . And remember in distributing the dayes of the moneth , that you alwayes set the 21 day of October beneath towards the handle in the verie line of North and South passing right through the middest of the handle , so shal the instrument shew the houre of the night more trulie than when the 28. day of October standeth beneath vpon the line of South and North , as it doth in the common Nocturlabes , the makers whereof had respect to the Pole it selfe which is inuisible , and not to the North starre which is apparant to the eie , by meanes wherof the guards of the Lodestarre sometime doe shew sooner or latter by 7. degrées and 18. minutes , if you count from the Pole , which is almost halfe an houre difference , and for that cause also Cogniet maketh his account in his foresaid rectifier to begin at the little blacke crosse , seuen degrées more forwarde than it ought to doe , if he should count from the Pole , and not from the North starre .

How to know by the foresaid twofold instrument as wel the mounting and descending of the North star as the true houre of the night both at one instant and also the elevation of the Pole , Chapter 42.

FIrst then hauing layed and stayed the great tooth of the moouable rundle marked with 12. vppon the day of the moneth wherein you séeke , and holding the instrument by the handle with your one hand right before your face , leaue not to put that hand forward from you , or to bring it backwards towards you , vntill you may sée with the one eie , winking with the other , the North starre through the hole of the pin , which is in the center of the Instrument : and so soone as you sée the North star , lift with your other hand the index vp and downe vntill you sée also at that instant the North guard of the Lodestarre on the outside of the instrument appearing euen with the fiduciall line or inwarde edge of the said index . Then staying the index there , looke vpon what houre it falleth , for that shall be the houre of the night . And look also at that instant vpon what degrée of distance it falleth in the outermost border of the instrument wherein those degrées are set downe together with these wordes adde and subtract . For from the point marked with a little blacke crosse nigh vnto the Northwest descending through the nether moyetie or halfe deale of the instrument vntill you come to the other blacke crosse , placed nigh vnto the Southeast , the North starre is alwaies aboue the Pole so much as the index sheweth , which you must alwaies subtract from the height of the North starre to knowe the eleuation of the Pole , hauing first taken with your Astrolabe the height of the North starre . But if the index doe fall vppon any degrée or part of a degrée in the vpper moyetie or halfe deale of the instrument then to know the eleuation of the Pole , you must adde so much as the index sheweth vnto the altitude of the Lodestarre , as the words Adde and Subtract written in the outermost border of the instrument doe plainely shew .

But now if you would know at any houre of the day or night , in what rombe the foresaid guardes be without séeing them and also how much the Lodestarre is declined from the Pole , you néede doe no more but to lay the great tooth of the mooueable rundle vppon the day of the moneth , and then to bring the index vnto the houre which you require , and the said index will shewe in the border of the instrument in what rombe the guardes be , and how much the North starre is aboue or beneath the Pole .

I haue found by often triall that this instrument wil shew the true houre of the night , and also in what rombe the guardes bée , and thirdlie , how much the North starre is at any time either aboue or beneath the Pole , and by adding the degrée of distance found in the limbe of the instrument , or by subtracting the same according to the rule before giuen from that altitude of the north star , which I haue before taken with my Astrolabe , I haue found at all times the true eleuation of the Pole wheresoeuer I haue made triall thereof . But sith other Starres may perhaps appeare when the North Starre with his guardes shal be hidden , I woulde wish all carefull Mariners to acquaint themselues with as manie bright Starres as they can , and especiallie with those which do both rise and set , and also to learne by some Table the declination of euerie such Starre and whether it be Southernlie or Northernlie , for by taking the Meridian altitude of anie such starre and by adding to , or by taking from the altitude thereof , his declination according as the same declination is either Northernlie or Southernlie ( for if it be Northernlie , then you must subtract his declination , and if it be Southernlie , you must adde the same to the altitude of the Starre ) you shall finde thereby the altitude of the Equinoctiall , which being taken out of 90. the remainder will shew the eleuation of the Pole . The meetest starres for this purpose in these our North partes of the worlde are these , Hircus the Goate , Canis minor , the little Dogge , Canis maior , the great Dogge , Dexter humerus Orionis , the right shoulder of Orion . Cingulum Orionis , the girdle of Orion . Cor Leonis , the Lyons hart . Bubulcus , the Bearward . Spica Virginis , the wheat eare in the hand of Virgo . Aquila volans , the flying Eagle . Caput Andromedae , the head of Andromeda . Ras Algol , the head of Medusa . Oculus Tauri , the Buls eye , and diuers others ,

And in anie case consider whether the declination of the starre be greater or lesser than his Meridian altitude , for if his declination be greater then his meridian altitude , as of those stars which are nigh the Pole , then you must take his Meridian altitude with your Astrolabe at two seuerall times , that is to say , when he is at his highest in the meridian , and also when he is at the lowest point of the same Meridian called the depression , and hauing added those Meridian altitudes together take halfe thereof , and that half shall be the eleuation of the Pole , which way of finding out the eleuation of the Pole is nothing méet for a Mariner that is vnder saile . But now to procéede according to Cogniet his direction in this matter , hee saith , that if in sayling you approch so nigh vnto the Equinoctiall as the eleuation of the Pole is not aboue 17. degrées , then the guardes are not so easilie séene , wherefore it shall be néedfull to take som other starre whereby you may both know the houre of the night , and also how much the Lodestarre is eyther aboue or beneath the Pole , and Cogniet thinketh none of the starres more méet for that purpose than one of these two , that is , either the starre called Caput Medusae , that is the head of Medusa called of the Arabians , Ras Algol , or els the starre called Hircus , that is , the Goat , both which are faire bright starres and of the first bignesse : and the cause why he appointeth the head of Medusa , is for that this starre is directlie opposite to the former guard , in such sort as this starre is alwayes aboue the Horizon , when both the guardes are vnder the Horizon . Wherefore hauing prepared your instrument , that is to say , hauing laid the longest tooth of the mooueable rundle vpon the day of the moneth wherein you séeke , and hauing found out thereby in such order as is before taught , the north star and also the star called the head of Medusa , both at one instant , marke where the index falleth , and immediately turne the index from that point to the point opposite , abating twelue houres , and so the Index shall shew you thrée things at once . First , the rombe wherein the guardes are at that present . Secondlie , the houre of the night . And thirdly how much the North starre is aboue or beneath the Pole . But because this starre is not knowne perhaps to all Mariners , Cogniet would haue you to take the other bright Starre called Hircus , which ( as hée saith ) goeth 9. houres and ½ before the guardes , in such sort , as when the former guarde is East , , you shall finde this Starre , counting from the North starre , to be Northwest , and almost 45. degrées distant from the Pole , with which starre if you worke , as you did before with the heade of Medusa , sauing that you shall not néede to turne the index vnto the opposite point , but onelie to rebate from the point on which it falleth 9. houres and ½ you shall knowe all the thrée thinges last mentioned . And this rule ( as he saith ) is so generall as you may haue your desire by working in like manner with any other starre that is to you certainlie knowne , & is at that time aboue the Horizon .

What starres are observed by those that sayle beyond the Equinoctiall vnder the South Pole . Chapter . 43.

THe ancient Astronomers , as Ptolomey , Timochares , Hypparchus and others did neuer describe any star to be more nigh vnto the South pole , than that which is called Canopus , which is a faire bright starre of the first bignesse , and according to the Tables of Copernicus , is distant from the South Pole 38. degrées , and ¼ . But those that haue sayled in the South seas of later dayes , haue founde out other starres vnknowne to the ancient Astronomers , which are much néerer vnto the said Pole . For Albericus Vesputius writeth of thrée Starres , making together a Triangle Orthogonall , that is to say , hauing one right angle , now called the southern Triangle , the middle star wherof is distant from the south Pole 9. degrées , ⅖ .

There be also latelie found out diuers images of other starres nigh vnto the South pole , as that which is called Noah his Doue , or Pigeon , and another called Polophilax , made in the shape of a man , whose longitude and latitude hath not as yet bene rightlie set downe by any that I haue read .

And he saith that these starres are neither anie of those starres that are appointed to the twelue Signes in the Zodiaque , nor yet any of the 36. Images or constellations that bee in heauen . Moreouer he saith , that in taking the altitude of the great Starre called the foote béeing in his right place , that is to say , when hée is directlie opposite to the head , and that you finde his altitude to be 30. degrées , then you may assure your selfe that you are right vpon the Equinoctiall . And if you finde his altitude to bee more than 30. degrées , then you are past the Equinoctiall towardes the South Pole . But if you find it to bee lesse than 30. degrées , then you are still on the North side of the Equinoctiall .

Besides the starres aboue mentioned , our Mariners in these North parts of the world are woont to obserue diuers other stars , to the number of 32. whose longitude and declination together with their bignesse and also when they rise and set , and when they are mounted to the Meridian , that is to say , are iust South , is plainlie set forth by Tables collected of purpose out of the Astronomicall Tables by William Boorne , which Tables you shall finde in the 20. chapter of his book called the regiment of the sea , And Robert Norman doeth also set downe the like tables in his booke called the Attractiue , and therefore I thinke it superfluous to repeat the same againe here , and speciallie sith I haue described vnto you all the starres that bee in the firmament that were knowne to the ancient Astronomers , and haue shewed you how to find out by the Globe their longitudes , their latitudes , their declinations their greatnes and all other accidentes belonging to the Starres in my treatise of the Celestiall Globe , which I wrote of purpose to further yong sea men .

The knowledge of the Stars serueth sea men chiefly to knowe thereby the latitude of any place , and also to knowe the houre of the night : And thirdlie , to conicecture by their manner of rising and setting , and other their aspectes what weather is like to followe , either foule or faire : the rules whereof to teach trulie bebelongeth to Astronomers , yet many sea men by diligent obseruation doe attaine to right good iudgment therein . Wherefore leauing to speake any further of the Starres , I will now briefly speake of the Sunne and of his motions , of his rising and setting in euerie latitude , and of his declination from the Equinoctiall , and of other his like apparances . I say here brieflie , because I haue alreadie spoken of him at large in the 1. part of my spheare wheras I treat of the Zodiaque .

Of the Sunne , and of his motion , and of the chiefest apparances belonging to him . Chapter 44.

THe Sunne according to the moouing of the first mooueable which is from East to West maketh his dayly reuolution in 24 ▪ houres , as all other Starres doe , but according to his owne motion , which is from West to East in going through the twelue Signes of the Zodiaque he spendeth a whole yeare , for his daylie mouing vpon his own center , called the excentrick , because it is out of the center of the world , is little more then 59. minutes , 8. secondes , making thereby the whole yeare to consist of 365. dayes , fiue houres 49. minutes , 8. seconds , 19. thirdes , 37. fourths , 24. fiftes , and this is called of Copernicus , the equall tropicall yeare , which taketh his beginning from the first point of Aries , otherwise called the Vernall Equinoxe , into which point the Sunne entreth not euery yeare at one selfe day of the moneth , for sometime hee entreth into that point the tenth day , and sometime the eleuenth day of March , which day of his entring is alwayes trulie set downe in euerie Ephemerides , and because the beginning of the tropicall yeare is so vncertaine , the Astronomers doe make their yeare called the syderal yeare , to begin at the former star of the Rams horne , and thereby doe make the yeare to consist of 365. dayes . 6 houres 9. minutes and 39. secondes , according to which yeare they alwaies rectifie or bring to equalitie , aswell the equall as the vnequal tropicall yeare , of both which I haue spoken in the first part of my Spheare , the 38. and 39. chapters .

Moreouer , the Sunne hath thrée motions , that is , slowe , swift and meane . His slowe motion is when he is in the point called Auge or Apogeon , which is a point imagined to bee nigh vnto the outermost edge of the circle which carrieth the bodie of the Sunne called Deferens Solis , and is furthest distant from the center of the world , which point in these our daies is in the 9. degrée of Cancer , or there abouts . And béeing in that point , he goeth little more than 57. minutes in 24. houres . Againe , his swift motion is when he is in the opposite point to the Auge , called Perigeon , which point in these dayes is in the 9. degree of Capricorne , and being in this point , he goeth one whole degree and almost two minutes in 24 houres , which is almost fiue minutes more than he maketh in his slow motion . His meane motion is when he is in the midst betwixt the two foresaid points , wheras in 24. houres he goeth one whole degree & somewhat more than two minutes . And these thrée sundrie motions doe cause the Equinoctiall pointes not to be of equall distance . For the Sunne spendeth seuen dayes and somewhat more in going from the Equinoxe of March , to the Equinoxe of September , than he doth in going from the Equinoxe of September to the Equinoxe of March . For if in this present yeare 1592. which is leape yeare , you count the dayes from the Vexnall Equinoxe , which is the eleuenth of March , vnto the Autumne Equinoxe , which is the 13. of September , you shall find the number of the dayes to be 186. and the other number from the 13. of September to the eleuenth of March , to be but 180. which is lesse than the first number by 6 dayes , and if it were not leape yeare the difference would be seuen dayes , because that Februarie in the leape yeare hath 29. dayes .

It is necessarie that sea men haue some vnderstanding of the thrée foresaide motions to the intent that they may the better know the true place of the Sunne and thereby his true declination . And note that no calculation of his declination can continue without errour aboue twentie foure yeares . For as often as the leape yeare commeth about , which is euerie foure yeares , the Sunne is vppon the Equinoctiall sooner by halfe an houre . But as for the true place of the Sunne , and especially euery day at noontide , the Ephemerides doth most trulie shewe , and hauing his place , you shall easilie finde his declination by this Table following , which will serue for these twentie yeares and more , the like whereof , together with the vse of the same is set downe in the first part of my Spheare the 13. chapter .

A Table shewing the declination of the Sunne euerie day throughout the yeare , and the vse thereof . Chapter 45.

Degrees of ●he Signes . ♈ ♎ ♉ ♏ ♊ ♐ Degrees of the Signes D M S D M S D M S 1 0 23 53 11 50 6● 20 22 57 29 2 0 47 46 12 10 56 20 35 7 28 3 1 11 39 12 31 34 20 46 55 27 4 1 35 30 12 51 59 20 58 20 26 5 1 59 20 13 12 12 21 9 21 25 6 2 23 8 13 32 12 21 19 59 24 7 2 46 54 13 51 58 21 30 13 23 8 3 1● 37 14 11 30 21 40 3 22 9 3 34 18 14 30 48 21 49 29 21 10 3 57 54 14 49 51 21 58 29 20 11 4 21 28 15 8 40 22 7 6 19 12 4 44 57 15 27 13 22 15 17 18 13 5 8 22 15 45 30 22 23 3 17 14 5 31 42 16 3 32 22 30 24 16 15 5 54 57 16 21 17 22 37 19 15 16 6 18 6 16 38 44 22 43 48 14 17 6 41 9 16 55 55 22 49 50 13 18 7 4 6 17 12 48 22 55 27 12 19 7 26 57 17 29 23 23 0 38 11 20 7 49 40 17 45 40 23 5 22 10 21 8 12 16 18 1 39 23 9 39 9 22 8 34 45 18 17 18 23 13 29 8 23 8 57 5 18 32 37 23 16 53 7 24 9 16 16 18 47 38 23 19 50 6 25 9 41 19 19 2 18 33 22 19 5 26 10 3 12 19 16 37 23 24 22 4 27 10 24 56 19 30 36 23 25 57 3 28 10 46 30 19 44 14 23 27 5 2 29 11 7 53 19 57 30 23 27 46 1 30 11 29 5 20 10 25 23 28 9 0 ♍ ♓ ♌ ♒ ♋ ♑

THe vse of this Table is thus : If the sunne be in any of the signes set down in the front of the Table , then séeke his degrée ( first found by the Ephemerides ) in the left collum , the degrées whereof do descend from one to 30. and the square Angle answerable to the signe and degrée will shew his declination . But if the sunne be in any of the signes that are in the foote of the Table , then séeke his degrée in the right collum , the degrées whereof do ascend , and the square Angle answerable to that signe and degrée will shew his declination .

There be other things also méete for Sea-men to knowe , touching the sunne as these : First to knowe the foure seasons of the yeare , secondly to know by his declination , the length of both the day and the night in euery latitude , and how it doth encrease and decrease : Item to know in what rombe or winde , and at what houre he riseth and setteth , and also his Meridian altitude , that is , when he is right South euery day , to finde out by the helpe of that , and by knowing his declination , the true Latitude of any place , & by his shadow to know the houre of the day , which are two chiefe pointes that the Mariner hath most néede to knowe . Of all which thinges I minde here to treate both plainely and briefely .

Of the foure seasons of the yeare , that is , Spring time , Summer , fall of the leafe , called otherwise Autumne , and winter . Chap. 46.

IN this our Clime the spring is said to begin when the Sun entreth into the first point of Aries , which is about the xi . of March , and continueth vnto the last point of Gemini , which time is saide to be hot and moyst , and therefore is likened to childhoode : And sommer beginneth when the Sun entereth into the first point of Cancer , which is about the 12. or 13. of Iune , and endeth when he is in the last degrée of Virgo : and this time is said to be hotte and dry , and therefore is likened to Adolescencie . Then Autumne or fall of the leafe beginneth when the Sunne entreth into the first point of Libra , which is about the 13. or 14. of September , and endeth when the Sun is in the last degrée of Sagittarius : and this time is said to be colde and dry , and therefore is likened to manhood : Finally winter beginneth when the sunne entreth into the first point of Capricorne , and endeth when he is in the last degrée of Pisces : and this time is said to be colde and moyst , and therefore is likened to olde age : notwithstanding Galen in his first booke de Elementis , sayth that the spring is temperately hot and moyst , and therefore a most wholsome time : And sommer is more hotte then colde , and more dry then moyst , and therefore is sayd to be hotte and dry : And Autumne is also saide to be dry because it is more dry then moyst , and yet neither hotte nor colde , but vnequally mixt , and thereby infectiue and causing sickenesse . And winter is sayd to be colde and moyst , not because it is colder or moyster then any other season , but because that in winter moysture excéedeth drinesse , and coldnesse excéedeth heate . But you haue to vnderstand that these foure seasons haue not like qualities in all the 5. Zones : For in the burnt Zone , and specially to those that dwell right vnder the Equinoctiall the sunne being in Aries or Libra causeth greatest heat , & thereby two sommers because he is then right ouer their heads , & being in either of the Solstices , that is in the beginning of Cancer or Capricorne he causeth two winters because he is then furthest from them as I haue declared vnto you in the second part of my treatise of the Spheare the 20. Chapter , whereas I treate of the seasons and shadowes incident to diuerse Climes and Parallels whereunto I referre you , and so I ende with this matter .

How to knowe when the Sunne riseth and setteth in everie latitude , and thereby the length of the day and night , and also in what rombe or wind he riseth and setteth , and how much he declineth everie day from the Equinoctiall either Northward or Southward . Also howe to know the elevation of the Pole , otherwise called the latitude of any place , by knowing the Meridian altitude of the Sunne , and his declination . Chap. 47.

THe most part of all these thinges haue béene taught before in my Spheare , in my Treatise of the Globes , and also in my treatise of the Astrolabes . And in the first two Treatises I shew also how to finde out the Longitude of any place , and therefore néedeth not here to be rehearsed : But if you would know how to handle the declination of the sunne being vpon the sea , then reade 7. 8. 9. and 10. chapters of William Borne his booke called the Regiment of the Sea , and you shal be fully instructed therein : The whole effect of all which Chapters Robert Norman setteth downe in few words in his new Attractiue in this manner as followeth .

First learne whether the sun haue South declination or North declination , which you shall know by his being in any of the Northerne or Southern signes : Then marke what shadow he casteth , and whether it striketh towardes the Pole whereunto he is néerest , or to the contrary . For if the sunne casteth his shadowe the same way that he is from the Equinoctiall , he shal be betwixt you and the Equinoctiall , & then hauing taken his Meridian altitude subtract the same from 90. & adde vnto the remainder the suns declination for that day , and the summe thereof shal be the eleuation of the Pole or the distance of your Zenith from the Equinoctiall otherwise called the Latitude which is alwaies equal to the eleuation of the Pole : But if the sun casteth his shadow to the contrary side of the Equinoctiall , that is to say , being in his North declination casteth his shadow Southward , or being in his south declination casteth his shadow Northward , then either the Equinoctiall shal be betwixt you & the sunne , or you in the Equinoctiall , or else you shal be betwixt the Equinoctial and the sunne , which you shall know thus : Adde the declination of the sunne for that day wherin you séeke vnto his Meridian altitude , and if the summe of the addition be lesser then 90. degrées , then so much as it wanteth of 90. degrées shall you be distant from the Equinoctiall on that side on which the shadow stréeketh : but if it amounteth iust to 90. degrées , then you shal be right vnder the Equinoctial . Againe if it be more then 90. degrées , then so much as is the ouerplus , so much shall you be from the Equinoctiall towards the sunne , at which time you shall be also betwixt the Equinoctiall and the sunne .

And if you finde the Meridian altitude of the sunne to be eauen with your Zenith , then looke what declination the sun hath at that instant , and so much shall you be from the Equinoctiall on that side wherein the sunne is : But if the sunne haue no declination , then shall you be right vnder the Equinoctiall line .

Of the shaddow of the Sunne , and how to know therby the houre of the day in any latitude by help of an vniversall diall . Chap. 48.

MY former order now requireth that I should speake somewhat of the shadow of the Sun , and of the diuersitie therof , according to the Clime or Parallel vnder which you sayle . But for so much as I haue shewed you in the second part of my Sphere from the 20. to the 27. chapter of the same , what diuerse shadowes the Sunne yéeldeth in diuerse Climes and Parallels , and also haue shewed what Vmbra recta and Vmbra versa is in the 40. Chapter of my Treatise of the Astrolabes : I minde not here therefore to make a new recitall therof , but only to shew how you shall find by his shadow the true houre of the day in euery Latitude by a generall Dyall , made to serue in all Latitudes , of which Dyals , though I haue séene diuerse and of diuerse shapes , yet none liketh me better then that which William Borne setteth downe in his Regiment the 21. Chapter , and calleth it the Equinoctiall Diall , which serueth not onely to know the houre of the day by the shadow of the sunne , but also the houre of the night by the shadowe of the Moone when she shineth cleare , which Diall being of small charge , I would wish all Mariners to haue : The making and vse whereof is so plainely set downe by himselfe in the foresaid Chapter as I thinke it superfluous to set it downe againe here . And thus ending with the the sunne , I will now turne my pen to the Moone .

Of the Moone and of all her diuers motions . Chap. 49.

THe Moone is a round thicke and darke bodie , hauing no light of her selfe , but onely such as she receiueth from the sunne , and shee maketh her dayly motion from East to West as all other starres doe in 24. houres , according to the moouing of the Primum mobile . But according to her owne motion , which is from West to East , the goeth but 13. degrées 12. minutes in the space of 24. houres , and that is according to her meane motion . And she passeth thorough the twelue signes of the Zodiaque in 27. dayes and eight houres : during which time the sunne by his natural motion which is also from West to East , is remooued from the place of coniunction almost 27. degrées , so as the Moone not finding the Sun there spendeth two dayes foure houres , and 44. minutes more in ouertaki●g him , which being added to 27. dayes and 8. houres doe make in all 29. dayes 12. houres and 44. minutes . Notwithstanding by reason that the Moone hath aswel as the Sunne thrée motions , that is , swift , meane , and flowe , she may change sometime sooner and sometimes later then in 29. dayes 12. houres 44. minutes , and yet one change counted with an other shall make vpp the selfe same summe . And note that this her thréefold motion dependeth vppon two pointes , the one called the Auge , and the other the point opposite to the Auge .

The point Auge of the Moone is when she is furthest distant from the earth , and the point opposite to that is when shée is nighest to the earth : For wheu she is in the point Auge , she goeth little more then twelue degrées in 24. houres : but when she is in the opposite point she goeth almost 15. degrées in 24. houres . And in her meane motion which is in the middest betwixt the two foresaide points , shee goeth 13. degrées and 12. minutes in 24. houres . Now because the Mariners doe account the moouing of the Moone by the pointes of their Compasses , they may thereby vnderstand that she goeth not alwaies in 24. houres one point and thrée minutes as they reckon , but sometime more and sometime lesse : For when she is in her slow motion she 〈…〉 more then 12. degrées in 24. houres , in which time the 〈…〉 one degrée , so as the Moone is distant from the 〈…〉 degrées , which is but 44. minutes of an houre , which wanteth 4. minutes of a whole point , whereto is attributed 48′· as hath beene said before : and in her swift motion she goeth 15. degrees , in which the sunne goeth one degree , so as she is distant from the sunne 14. degrees , which is more then a point and 3. minutes of the compasse . And you haue to note that the point Auge of the Moone is mooueable and passeth through the Zodiaque in the space of 19. yeares , and thereby sometime causeth the full of the Moone to happen sooner or later .

How to knowe in what signe the point Auge of the Moone is in any yeare . Chap. 50.

IF you would know in what signe the Auge of the Moone is in any yeare , then you must consider the prime or golden number of that yeare for when the prime or golden number is one , then her Auge is in Aries . And if the Moone be also then in Aries , she is in her slowe motion : and being in the point opposite which is ♎ , she is in her swift motion . And sith this Auge of the Moone goeth through the 12. signes in 19. yeares as hath béene said , it must néedes fall out that in 9. yeares and a halfe her Auge commeth to be in Libra : And then the Moone being there she is in her slowe motion : And being in the point opposite which is Aries , she is in her swift motion : Again when the prime is 5. then her point Auge is in Cancer : and the Moone being there she is in her slow motion , and when she commeth to be in the point opposite which is Capricorne , she is in her swift motion : And when the prime is 14. or 15. then her point Auge is in Capricorne , where if the Moone be also , then shee is in her slowe motion : and being in the point opposite which is Cancer , she is in her swift motion . And note that when the Moone is in her swift motion she maketh her change , or full , or any other aspect the sooner : And contrariwise when she is in her slow motion she maketh her change , or ful , or any other aspect the later . Moreouer you shal sée her at the time of her change eyther sooner or later , according to the time of the yeare , for from Ianuary to Iune you shal sée her within 24. houres after her change , because she hath during those monethes North declination from the sunne , & maketh a greater arch then the sunne doth . But from Iuly to December you shall not sée the Moone scant thrée dayes after the change . But you may sée her within 24. houres before her change , because that during those monethes she hath South declination from the sunne : And note that when the Moone is thrée dayes and 18. houres , which is the halfe quarter of the Moone , the sea men doe call that time the prime day , because the Moone is then 4. points to the Eastward of the Sunne , which is thrée houres , for to euery point is attributed thrée quarters of an houre as hath béene saide before . Moreouer it is necessary for sea men to knowe when the Moone riseth & setteth , and in what part of the Horizon in euery Latitude : and how long she shineth and when she is full South , and also what Latitude she hath , and whether it be South or North euery day and houre thoroughout the yeare . All which thinges are most easie to be found by the Globe , and by helpe of the Ephemerides in such sort as is before taught in my Treatise of the Globes , and also by helpe of M. Blagraue his Astrolabe , her true place in the Zodiaque being first knowne by the Ephemerides . It is méete also to know when she is in coniunction with the sunne , or at the full , and the rest of her quarters , which is easily knowne by the Ephemerides .

When the Moone is saide to be in Coniunction with the Sunne , or to be at the full , and what her greatest latitude is aswell from the Ecliptique line , as from the Equinoctiall . Chap. 51.

SHe is said to be in coniunction with the sunne when the sunne and she be both in one selfe signe and like degrée : But when she is at the full , then she is opposite to the sunne , and distant from him 6. signes , which is the one halfe of the Zodiaque , containing 180. degrées : And in euery quarter she is distant from the sunne thrée signes which is 90. degrées : Moreouer the Moone is said to haue Latitude both Northerne and Southerne from the Ecliptique line , which line the deferent of the Moone crosseth in 2. points , and thereby maketh two intersections , whereof the one tending towardes the North is called the head of the Dragon , and the other intersection towardes the South is called the tayle of the Dragon , so as when the Moone is passed 90. degrées from the Dragons head towardes the North , then her Latitude is 5. degrées Northward : And when she is distant 90. degrées from the tayle of the Dragon towardes the South , then her Latitude is also 5. degrées Southward , which is the greatest Latitude that she hath on eyther side of the Ecliptique line , whereof I haue written more at large in the first part of my Spheare the 25. chap. in which you shall finde a figure representing the saide Dragon both head and tayle : But her Latitude is to be considered in two respects , that is not onely from the Ecliptique line , but also from the Equinoctiall , for from the Ecliptique line her greatest Latitude is but 5. degrées on eyther side of the Ecliptique , as hath bin said before . But from the Equinoctiall her greatest Latitude is 28. degrées and a halfe on either side of the Equinoctiall , which in mine opinion might be more rightly called her greatest declination , which excéedeth by 5. degrees the declination of the sunne , for that is but 23. degrees and a halfe , or rather 28′· on either side of the Equinoctiall . But this great Latitude of the Moone is onely to be vnderstood when the prime is one , and that her Auge is in Aries : For when the prime is betwixt 9. or 10. yeares or more , the Moone declineth not from the Equinoctial on either side aboue 18. degrees and a halfe at the most .

How to know in what part of the Zodiaque the head of the Dragon is euerie yeare . Chap. 52.

IF you would know in what part of the Zodiaque the head of the Dragon is , then you must consider the prime , for when the prime is one then the Dragons head is in the first point of Aries , euen as the point Auge is . And in 19. yeares it passeth thorough the twelue signes aswell as the point Auge of the Moone , but with contrary course for the point Auge of the Moone mooueth according to the succession of the signes , that is , from Aries to Taurus , Gemini and so foorth : But the head of the Dragon hath a contrary motion , that is , from Aries to Pisces , and so into Aquarius and so foorth , so as in 9. yeares and a halfe it meeteth iust with the point Auge of the Moone in the signe Libra . Thus you see that by knowing the prime , you learne also to knowe in what signe the Auge of the Moone , and also the Dragons head is , and what Latitude the Moone hath , aswell from the Ecliptique line as from the Equinoctiall . But one of the chiefest points to be had by the Moone , is to know thereby the tides , that is when the sea floweth and ebbeth in any place : whereof we come now to speake .

How to know the tydes in any place by the Moone . Chap. 53.

BEfore that you enter into any Hauen or Riuer , it is necessary to knowe the true tydes of that place , which tydes are subiect to the motion of the Moone , for she causeth at one place or other alwaies in one certaine rombe full Sea . As for example it is alwayes full sea at Antwerp when the Moone is either right East or West : and it is ebbe or lowe water there when she is North or South , and because the Moone doth passe through all the rombes of the Mariners Compasse in 24. houres , they allow to euery rombe ¾ . of an houre , which is 45. minutes , and that being multiplyed by 32. do make iust 24. houres , wherefore if the two rombes , North and South do yéeld ech of them 12. houres , then the first rombe must néedes yéelde ¾ . of an houre , the second one houre and ½ . and the third two houres and ¼ . and so foorth of the rest : For by adding to euery rombe ¾ . you shall find that the East and West doe yéelde alwayes 6. houres : but then you must note that according to the age of the Moone the tydes do fall euery day later and later , wherfore to know the true time of the tyde in any place , you must first learne by some Rutter or by the relation of others that can tell in what rombe the Moone causeth full sea in that place , & then at what houre it is full sea , the Moone being eyther in the change or at the full , which you shall know by allowing to euery rombe ¾ . of an houre in such manner as is before set downe . But if you would knowe at what houre it is full sea in that place euery day , then you must first vnderstand that the Moone in 30. daies slacketh 24. houres , which amounteth to ⅘ . of an houre which is 48. minutes for euery day , for so much she declineth euery day from the sunne , then looke how many dayes the Moone is old , and hauing multiplyed the same by ⅘ . that is to say by 48. minutes , adde the product thereof to those houres at which it is full sea , and you shall haue the true time of full sea euery day . As for example , suppose th at it was full sea at the last new Moone at some place here in England at thrée of the clocke in the afternoone , and nowe I would know at what a clocke it shall be full sea 5. dayes after the new Moone . Now if you multiply 5. being the age of the Moone by 48. minutes , and adde to the product therof thrée houres , which was the time of the last change , the full sea shall be at that place at seuen of the clocke . But if the summe of such addition be aboue 12. houres , then you must cast away the 12. and the remainder shall shew you the true houre of full sea .

How to know by helpe of an instrument the tydes at any place . Chap. 54.

THere is also an easier way to knowe the tydes euery day in any place by the helpe of an instrument set downe by Cogniet , wherof both the making and vse here followeth . First vpon some bord well playned and made smooth , draw a Circle , and deuide the same into 30. parts signifying the dayes or age of the Moone , setting the number of 30. aboue in the toppe of the instrument , and place all the other numbers as 1. 2. 3. and so foorth towards your right hande , that done , make a mooueable roundle which may turne about within the verdge of the first Circle , and deuide that into 24. houres , and also into 32. rombes , setting the North point marked with the Flower deluce at the twelfth houre aboue , & the South point at the twelfth houre beneath , and the East point at the sixth houre on your right hand , and the West point at the sixth houre on your left hande , and so shall your instrument be perfect , the vse whereof is thus :

First you must knowe what rombe of the Moone in that place which you séeke maketh a full sea , and also the age of the Moone by some Almanacke or some other rule before taught , then hauing these two thinges turne the inward rondle of the houres & rombes vntill the foresaid knowne rombe doth iustly answere to the 30. day of the great rondle , and there staying it firme with your finger , séeke in the outermost border of the greater rondle the age of the Moone , and that will shew you in the rondle of houres the very houre of full sea that day in that place .

But one chiefe thing touching the knowing of the tydes is to be noted as Borne sayth , which is that the sea will flow more by one point of the Compasse in the spring tydes , then in any of the quarters of the Moone called Nep tydes in euery riuer that hath any indraft , and is of some reasonable distance from the sea : As for example , it floweth at Graues end at the change or full of the Moone when she is South Southwest : but in any of her quarters it skant floweth when she is South and by West , and this rule as he sayth is generall for euer .

TheSexagenarieTablewherebytheproductsinMultiplication,thequotientsinDiuision,andthesquareRootesarefoundAstronomicallFractions. 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 58 1 57 2 56 3 55 4 54 5 53 6 52 7 51 8 50 9 49 10 48 11 47 12 46 13 45 14 44 5 43 16 42 17 41 18 40 19 39 20 38 21 37 22 36 23 35 24 34 25 33 26 32 27 31 28 30 29 59 54 4 55 6 54 8 53 10 52 12 51 14 50 16 49 18 48 20 47 22 46 24 45 26 44 28 43 30 42 32 41 34 40 36 39 38 38 40 37 42 36 44 35 46 34 48 33 50 32 52 31 54 30 56 29 58 58 54 9 53 12 52 15 51 18 50 21 49 24 48 27 47 30 46 33 45 36 44 39 43 42 42 45 41 48 40 51 39 54 38 57 38 0 37 3 36 6 35 9 34 12 33 15 32 18 31 21 30 24 29 27 57 1 0 1 52 16 51 20 50 24 49 28 48 32 47 36 46 40 45 44 44 48 43 52 42 56 42 0 41 4 40 8 39 12 38 16 37 20 36 24 35 28 34 32 33 36 32 40 31 44 30 48 29 52 28 56 56 2 0 2 0 4 50 25 49 30 48 35 47 40 46 45 45 50 44 55 44 0 43 5 42 10 41 15 40 20 39 25 38 30 37 35 36 40 35 45 34 50 33 55 33 0 32 5 31 10 30 15 29 20 28 25 55 3 0 3 0 6 0 9 48 36 47 42 46 48 45 54 45 0 44 6 43 12 42 18 41 24 40 30 39 36 38 42 37 48 36 54 36 0 35 6 34 12 33 18 32 24 31 30 30 36 29 42 28 48 27 54 54 4 0 4 0 8 0 12 0 16 46 49 45 56 45 3 44 10 43 17 42 24 41 31 40 38 39 45 38 52 37 59 37 6 36 13 35 20 34 27 33 34 32 41 31 48 30 55 30 2 29 9 28 16 27 23 53 5 0 5 0 10 0 15 0 20 0 25 45 4 44 12 43 20 42 28 41 36 40 44 39 52 39 0 38 8 37 16 36 24 35 32 34 40 33 48 32 56 32 4 31 12 30 20 29 28 28 36 27 44 26 52 52 6 0 6 0 12 0 18 0 24 0 30 0 36 43 21 42 30 41 39 40 48 39 57 39 6 38 15 37 24 36 33 35 42 34 51 34 0 33 9 32 18 31 27 30 36 29 45 28 54 28 3 27 12 26 21 51 7 0 7 0 14 0 21 0 28 0 35 0 42 0 49 41 40 40 50 40 0 39 10 38 20 37 30 36 40 35 50 35 0 34 10 33 20 32 30 31 40 30 50 30 0 29 10 28 20 27 30 26 40 25 50 50 8 0 8 0 16 0 24 0 32 0 40 0 48 0 56 1 4 40 1 39 12 38 23 37 34 36 45 35 56 35 7 34 18 33 29 32 40 31 51 31 2 30 13 29 24 28 35 27 46 26 57 126 8 25 19 49 9 0 9 0 18 0 27 0 36 0 45 0 54 1 3 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24 35 25 34 26 33 27 32 28 31 29 30 59 60 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

How a generall Rutter shewing the tydes in al places should be made . Chap. 55.

WIlliam Borne doth set downe what Moone doth make a full sea , aswell here in the most parts of our English coasts as in some other parts of France & Spain , and so doe many others whose Tables touching the tydes are called Rutters , whereof some are truely set downe and some are false : But sith such Rutters do serue but for a few particular places , I would wish that some learned Pilot that hath sayled many and sundry long voyages , to make a generall Rutter that might serue for al places if it were possible , or at the least for so many places as are knowne in these daies : And I would wish such generall Rutter to be made in manner of an Alphabet : and that euery place might haue his true Longitude and Latitude added thereunto , to the intent that euery place might be the more easily found out in any Map or Card that is graduated with degrées of Longitude and Latitude : And then to shewe what Moone doth make a full sea in euery such place , which thing who soeuer would performe , he should in mine opinion deserue great commendation . And thus I end this Treatise , praying all the learned Seamen not to be offended or gréeued with mee for that I doe make young Gentlemen our owne Countrimen partakers of their most worthy knowledge , whereof the ignorant are not able to iudge , nor to yéelde them that prayse which they deserue : yea rather I hope that they will help to perfect what so euer I haue herein left vnperfected , to make the young Gentlemen the more skilfull , and thereby the more seruiceable to their Countrie , and in so doing they shall procure to them selues great good will and infinite thankes .

FINIS .
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