Cosmographia, or, A view of the terrestrial and cœlestial globes in a brief explanation of the principles of plain and solid geometry applied to surveying and gauging of cask : the doctrine of primum mobile : with an account of the Juilan & Gregorian calendars, and the computation of the places of the sun, moon, and fixed stars ... : to which is added an introduction unto geography / by John Newton ... Newton, John, 1622-1678. This text is an enriched version of the TCP digital transcription A52257 of text R17177 in the English Short Title Catalog (Wing N1055). 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. 1185 KB of XML-encoded text transcribed from 286 1-bit group-IV TIFF page images. EarlyPrint Project Evanston,IL, Notre Dame, IN, St. Louis, MO 2018 A52257 Wing N1055 ESTC R17177 12546927 ocm 12546927 63077 2240972848

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Early English books online. (EEBO-TCP ; phase 1, no. A52257) Transcribed from: (Early English Books Online ; image set 63077) Images scanned from microfilm: (Early English books, 1641-1700 ; 326:7) Cosmographia, or, A view of the terrestrial and cœlestial globes in a brief explanation of the principles of plain and solid geometry applied to surveying and gauging of cask : the doctrine of primum mobile : with an account of the Juilan & Gregorian calendars, and the computation of the places of the sun, moon, and fixed stars ... : to which is added an introduction unto geography / by John Newton ... Newton, John, 1622-1678. [15], 510, [16] p., [12] leaves of folded plates : ill. Printed for Thomas Passinger ..., London : 1679. "Tables for the measuring of timber" and "Astronomy, the second part, or, An account of the civil year" have special title pages. Advertisement: p. [13]-[16] at end. Reproduction of original in Yale University Library. eng Geometry -- Early works to 1800. Calendar -- Early works to 1800. Geography -- Early works to 1800. Astronomy -- Early works to 1800. eebo Cosmographia, or A vievv of the terrestrial and cœlestial globes, in a brief explanation of the principles of plain and solid geometry, applied to sur Newton, John, 1679 519 122874 385 0 0 0 0 0 31.33 C The rate of 31.33 defects per 10,000 words puts this text in the C category of texts with between 10 and 35 defects per 10,000 words. 2004-04 Assigned for keying and markup 2004-05 Keyed and coded from ProQuest page images 2004-08 Sampled and proofread 2004-08 Text and markup reviewed and edited 2004-10 Batch review (QC) and XML conversion

Imprimatur ;

Guil. Iane R. P. D. Hen. Episc. Lond. à Sacris Dom.

COSMOGRAPHIA , OR A VIEW OF THE Terrestrial and Coelestial GLOBES , IN A Brief Explanation OF THE PRINCIPLES Of plain and solid GEOMETRY , Applied to Surveying and Gauging of CASK .

The Doctrine of the Primum Mobile .

With an Account of the Juilan & Gregorian Calendars , and the Computation of the Places of of the Sun , Moon , and Fixed Stars , from such Decimal Tables of their Middle Motion , as supposeth the whole Circle to be divided into an hundred Degrees or Parts .

To which is added an Introduction unto GEOGRAPHY .

By John Newton , D. D.

London , Printed for Thomas Passinger , at the Three Bibles on London-Bridge 1679.

TO THE Most Honourable HENRY SOMERSET , Lord Herbert , Baron of Chepstow , Raglan , and Gower , Earl and Marquess of Worcester , Lord President and Lord Lieutenant of Wales and the Marches , Lord Lieutenant of Gloucester , Hereford and Monmouth , and of the City and County of Bristol , Knight of the Most Noble Order of the Garter , and one of His Majestie 's Most Honourable Privy Council .

HE that adventures upon any thing contrary to the General received practice , what ever his own courage and resolutions are , had need to be supported , not only by the most Wise and Honourable , but also the most Powerful Persons that are in a Nation or Kingdom ; For let the Proposals be never so advantagious to the Publick , they shall not only be decried and neglected , but it is well , if the Promoter be not both abused and ruined : Yet I , notwithstanding all these discouragements , have not been silent , but in order to Childrens better Education , have long since published my thoughts , and have and do declare , that the multitude of Schools for the learning of the Latine and Greek Tongues , are destructive both to our youth and the Commonwealth ; and if the Opinion of Sir Francis Bacon in his Advice to King Iames concerning Sutton's Hospital , be not sufficient to warrant my Assertion , I could heartily wish that no such Evidence could have been produced , as the late unhappy Wars , in the Bowels of this Kingdom hath afforded us ; for what he saith there by way of Advice , we by woful Experience have found too true ; that by reason of the multitude of Grammar Schools , more Scholars are dayly brought up , than all the Preferments in this Nation can provide for , and so they become uncapable of other Professions , and unprofitable in their own , and at last become , materia rerum novarum ; whether this be an essential or an accidental Effect , I will not here dispute ; the truth of it , I am sure , cannot be denied : but that is not all ; by this means it comes to pass , that four of the seven Liberal Arts , are almost wholly neglected , as well in both Universities , as in all Inferiour Schools ; and setting aside the City of London , there are but few Places in this Nation , where a man can put his Son , to be well instructed in Arithmetick , Geometry , Musick and Astronomy ; and even that Famous City was without a Publick School for Mathematical Learning , till His present Majesty was pleased to lay the Foundation ; nay so averse are men in the general to these Arts ( which are the support of all Trade ) that without a high hand , it will be almost impossible , to make this People wise for their own good : I come therefore to your Honour , humbly to beg your Countenance and Assistance , that the Stream of Learning may be a little diverted , in those Schools that are already erected , and to be instrumental for the erecting more , when they shall be wanting ; that we may not be permitted still to begin at the wrong end ; but that according to the practice of the Ancient Philosophers , Children may be instructed in Arithmetick , Geometry , Musick and Astronomy ; before the Latine and Greek Grammars are thought on , these Arts in themselves , are much more easie to be learned , tend more to a general good , and will in a great measure facilitate the Learning of the Tongues , to as many as shall after this Foundation laid , be continued at School , and provided for in either Universities . Your Honour was instrumental to enlarge the Maintenance for God's Minister in the Place where I live , and perhaps it may please God to make you so , not only in making this Place in particular , but many other Places in this Land happy , by procuring Schools for these Sciences , and not only so , but by your Loyal and Prudent managing the several Trusts committed to you , you may do much for God's Glory , your Countries Good , and the continuance of your own Honour to all Future Generations , which is , and shall be the Prayer of ,

Your Honour 's Obliged and Devoted Servant , JOHN NEWTON .
TO THE READER .

MY Design in publishing these Introductions to Geometry and Astronomy , is so well known by all the Epistles , to my other Treatises of Grammar , Arithmetick , Rhetorick , and Logick , that I think it needless to tell thee here , that it is my Opinion , that all the Arts should be taught our Children in the English Tongue , before they begin to learn the Greek or Latin Grammar , by which means many thousands of Children would be fitted for all Trades , enabled to earn their own Livings , and made useful in the Commonwealth ; and that before they attain to twelve years of age ; and by consequence the swarming of Bees would be prevented , who being compelled to leave their Hives , for want of room , do spread themselves abroad , and instead of gathering of Honey , do● sting all that come in their way . We should not have such innumerable company of Gown-men to the loss and prejudice of themselves and the Common-wealth ; and those we had would probably be more learned , and better regarded .

His Majesty being pleased to begin this Work , by His Bounty towards a Mathematical School in Christ's Church London ; I am not now without hopes , to see the same effected in many other Places in this Kingdom ; and to this purpose I have to my Introductions to the other Arts , added these also to Geometry and Astronomy ; which I call by the name of Cosmographia ; and this I have divided into four Parts ; in the first I have briefly laid down the first Principles belonging to the three kinds of Magnitude or continued Quantity , Lines , Planes and Solids ; which ought in some measure to be known , before we enter upon Astronomy , and this part I call an Introduction unto Geometry :

The second and third Parts treat of Astronomy ; the first of which sheweth the Doctrine of the Primum Mobile , that is , the Declination , Right Ascension , and Oblique Ascensions of the Sun and Stars , and such other Problems , as do depend upon the Doctrine of Spherical Triangles .

The second Part of Astronomy , treateth of the motion of the Sun , Moon and Fixed Stars ; in order whereunto , I have first given thee a brief account of the Civil Year , with the cause of the difference between our Julian and Gregorian Calendar , and of both from the true ; for it must be acknowledged that both are erroneous , though ours be the worse of the two ; yet not so bad , but that our Dissenting Brethren have I hope some better Arguments to justifie their Non-conformity ▪ than what I see published in a little Book without any name to it , concerning two Easters in one Year ; by the General Table , saith this learned man , who owneth the Feast of Easter was to be observed Anno 1674. upon the 19 day of April , so the Almanacks for that Year , as well as the General Table set before the Book of Common Prayer ; but by the Rule in the said Book of Common Prayer given , the Feast of Easter should have been upon the twelfth of April , for Easter-Day must always be the first Sunday after the first Full Moon , which happeneth next after the one and twentieth day of March , and if the Full Moon happen upon a Sunday , Easter-Day is the Sunday after ; Now in the Year 1674. the 19 of April being Friday was Full Moon , therefore by this Rule , Easter-Day should be the twelfth , and by the Table and the Common Almanacks April the tenth ; but this learned man must know , that the mistake is in himself , and not in the Rule or Table set down in the Book of Common Prayer ; for if he please to look into the Calendar , he will find that the Golden Number Three , ( which was the Golden Number for that Year ) is placed against the last day of March , and therefore according to the supposed motion of the Moon , that Day was New Moon ; and then the Full Moon will fall upon the fourteenth day of April , and not upon the tenth , and so by consequence the Sunday following the first Full Moon after the 21 day of March was the nineteenth of April and not the twelfth . And thus the Rule and the Table in the Book of Common Prayer for finding the Feast of Easter are reconciled ; and when Authority shall think sit , the Calendar may be corrected and all the moveable Feasts be observed upon the days and times at first appointed ; but till that be , a greater difference than one Week will be found in the Feast of Easter between the Observation thereof according to the Moons true motion , and that upon which the Tables are grounded ; for by the Fathers of the Nicene Council it was appointed , that the Feast of Easter should be observed upon the Sunday following the first Full Moon after the Vernal Equinox , which then indeed was the 21 of March ; but now the tenth , and in the Year 1674. Wednesday the 11 of March was Full Moon , and therefore by this Rule , Easter-Day should have been upon March the fifteenth , whereas according to the Rules we go by , it was not till April the nineteenth .

The Tables of the Sun and Moons middle motions are neither made according to the usual Sexagenary Forms , nor according to the usual Degrees of a Circle and Decimal Parts , but according to a Circle divided into 100 Degrees and Parts , and this I thought good to do , to give the World a taste of the excellency of Decimal Numbers , which if a Canon of Sines and Tangents were fitted to it , would be found much better , as to the computing the Places of the Planets ; but as to the Primum Mobile , by reason of the general dividing a Circle into 360 Degrees , I should think such a Canon with the Decimal Parts most convenient , and in some cases the common Sexagenary Canon may be very useful , and indeed should wish and shall endeavour to have all printed together , one Table of Logarithms will serve them all , and two such Canons , one for the Study and another for the Pocket , would be sufficient for all Mathematical Books in that kind ; and then men may use them all or either of them as they shall have occasion , or as every one is perswaded in his own mind .

What I have done in this particular , as it was for mine own satisfaction , so I am apt to believe , that it will be pleasing to many others ; and although I shall leave every one to abound in his own sense , yet I cannot think that Custom should be such a Tyrant , as to force us always to use the Sexagenary form , if so , I wonder that men did not always use the natural Canon ; if no alteration may be admitted , what reason can be given for the use of Logarithms ; and if that be found more ready than the natural , in things of this kind , where none but particular Students are concerned , I should think it reasonable , to reduce all things hereafter , into that form , which shall be found most ready and exact ; now the Part Proportional in the Artificial Sines and Tangents in the three first Degrees cannot be well taken by the common difference , and the way of finding them otherwise will not be so easie in the Sexagenary Canon , as in either of the other , and this me thinks , should render that Canon which divides each Degree into 100 Parts more acceptable ; but thus to retain the use of Sines , Degrees , and Decimal Parts , doth not to me seem convenient , and to reckon up , a Planets middle motion , by whole Circles will sometimes cause a Division of Degrees by 60 , which hath some trouble in it also , but if a Circle be divided into 100 Degrees , this inconvenience is avoided , and were there no other reason to be given , this me thinks should make such a Canon to be desirable ; but till I can find an opportunity of publishing such an one , I shall forbear to shew any further uses of it , and for what is wanting here in this subject , I therefore refer thee to Mr. Street's Astronomia Carolina , and the several Books written in English by Mr. Wing .

The fourth Part of this Treatise is an Introduction unto Geography , in which I have given general Directions , for the understanding how the habitable part of the World is divided in respect of Longitude and Latitude in respect of Climes and Parallels with such other Particulars as will be found useful unto such as shall be willing to understand History ; in which three things are required ; The time when , and this depends upon Astronomy ; the place where , and this depends upon Geography ; and the Person by whom any memorable Act was done , and this must be had from the Historical narration thereof ; and he that reads History without some knowledge in Astronomy and Geography will find himself at a loss , and be able to give but a lame account of what he reads ; but after the learning of these Arts of Grammar , ( I mean so much thereof , as tends to the understanding of every ones Native Language ) Arithmetick , Geometry and Astronomy ; a Child may proceed profitably to Rhetorick and Logick , the reading of History , and the learning of the Tongues ; and sure there is no studious and ingenious man , but will stand in need of some Recreation , and therefore if Musick in the Worship and Service of God be not Argument enough to allow that a place among the Arts , let that poor end of Delight and Pleasure be her Advocate ; and although that all men have not Voyces , yet I can hardly believe , that he expects any Melodious Harmony in Heaven , that will not allow Instrumental Musick a place on Earth ; and as for those that have Voyces , surely the time of learning Vocal Musick , must be in Youth , and I am perswaded that the Arts and Sciences to some good degree may be learned by Children before they be full twelve years old , and would our Grammer Masters leave off their horrible severity , and apply themselves to such ways of teaching Youth , as the World is not now unacquainted with , I am perswaded that it is no difficult matter , in four years time more to fit Children in some good measure for the University .

The great Obstruction in this Work , is the general Ignorance of Teachers , who being unacquainted with this Learning , cannot teach others what they know not themselves . I could propound a remedy for this , Sed Cynthius aurem vellit ; Therefore I will forbear and leave what I have written , to be perused and censured as thou shall think fit .

John Newton .
Practical Geometry ; OR , THE ART of SURVEYING .
CHAP. I.

Of the Definition and Division of Geometry .

GEometry is a Science explaining the kinds and properties of continued quantity or magnitude .

2. There are three Kinds or Species of Magnitude or continued Quantity , Lines , Superficies and Solids .

3. A Line is a Magnitude consisting only of length without either breadth or thickness .

4. In a Line two things are to be considered , the Terms or Limits , and the several Kinds .

5. The term or limit of a Line is a Point .

6. A Point is an indivisible Sign in Magnitude which cannot be comprehended by sense , but must be conceived by the Mind .

7. The kinds of Lines are two , Right and Oblique .

8. A Right Line is that which lieth between his Points , without any going up or going down on either side . As the Line AB lieth streight and equally between the Points A and B. Fig. 1.

9. An Oblique Line is that which doth not lie equally between its Points , but goeth up and down sometimes on the one side and sometimes on the other . And this is either simple or various .

10. A simple Oblique Line , is that which is exactly Oblique , as the Arch of a Circle ; of Various Oblique Lines there is but little use in Geometry .

11. Thus are Lines to be considered in themselves , they may be also considered as compared to one another , and that either in respect of their distances , or in respect of their meetings .

12. In respect of their distances , they may be either equally distant , or unequally .

13. Lines equally distant are two or more , which by an equal space are distant from one another , and these are called Parallels ; and these though infinitely extended will never concur .

14. Lines unequally distant , are such as do more or less incline to one another , and these being extended will at last concur .

15. Concurring Lines are either perpendicular or not perpendicular .

16. A Perpendicular Line , is a Right Line falling directly upon another Right Line , not declining or inclining to one side more than another ; as the Line AB in Fig. 1.

17. A Perpendicular Line is twofold , to wit , either falling exactly in the middle of another Line , or upon some other Point which is not the middle .

18. A line exactly Perpendicular , may be drawn in the same manner , as any Right Line may be divided into two equal Parts ; the which may thus be done . If from the two Terms or Points of the Right Line given , there shall be described two Arches crossing one another above and below , a Line drawn through the Intersections of those Arches , shall be exactly Perpendicular , and also divide the Right Line given into her equal Parts . Fig. 1.

For Example ; Let CD be the Right Line given , and let it be required , to bisect this Line , and to erect a Perpendicular in the middle thereof . 1. Then setting one of your Compasses in the Points C , draw the Arches E and F. 2. Setting one Foot of your Compasses in D , draw the Arches G and H , and from the Intersections of these Arches draw the Right Line KL , so shall the Right Line KL be Perpendicular to the Right-Line CD , and the Right Line CD also divided into two equal Parts , in the Point A.

19. A Line Perpendicular to any other Point than the middle is twofold : for it is either drawn from some Point given in the Line ; or from some Point given without the Line .

20. From a Point given in the Line , at Perpendicular may thus be drawn . In Fig. 2. Let the given Line be CD , and let it be required to draw a Perpendicular Line to the Point C , your Compasses being opened to any reasonable distance , set one Foot in the Point C , and the other in any place on either side the Line CD , suppose at A , then describe the Arch ECF , this done draw the Line EA , and where that Line being extended shall cut the Arch ECF , a Right Line drawn from C to that Intersection shall be Perpendicular to the Point C in the Line CD , as was required .

21. From a Point given without the Line , a Perpendicular may be drawn in this manner . In Fig. 2. Let the given Line be CD , and let it be required to draw another Line Perpendicular thereunto , from the Point F without the Line . From the Point F draw a streight Line to some part of the Line CD at pleasure , as FE , which being bisected , the Point of Bisection will be A , if therefore at the distance of AF , you draw the Arch ECF , the Right Line CF shall be Perpendicular to the Line CD , as was required .

22. Hitherto concerning a Perpendicular Line . A Right Line not Perpendicular , is a Right Line falling indirectly upon another Right Line , inclining thereto on the one side more , and on the other less .

23. Lines unequally distant , and at last concurring , do by their meeting make an Angle .

24. An Angle therefore is nothing else , then the place , where two Lines do meet or touch one another , and the two Lines which constitute the Angle , are in Geometry called the sides of the Angle .

25. Every Angle is either Heterogeneous , or Homogeneous : that is called an Hetorogeneous Angle , which is made by the meeting of one Right Line , and another that is Oblique and Crooked ; and that is called an Homogeneous Angle , which is made by the meeting of two Lines of the same kind , that is , of two Right Lines , or of two curved or Circular Lines .

26. An Homogeneous Angle made of two curved or Circular Lines , is to be considered in Geometry as in Spherical Triangles , but the other which is made of Right Lines , is in all the Parts of Geometry of more frequent use .

27. Right lined Angles are either Right or Oblique .

28. A Right Angle is that whose legs or sides are Perpendicular to one another , making the comprehended space on both sides equal . Thus in Fig. 1. the Line AK is Perpendicular to the Line CD , and the Angles KAC and KAD , are right and equal to one another .

29. An Oblique Angle is that , whose sides are not Perpendicular to one another .

30. An Oblique Angle is either acute or obtuse .

31. An Acute Angle is that which is less than a Right .

32. An Obtuse Angle , is that which is greater than a Right . Thus in Fig. 1. The Angle BAC is an Acute Angle because less than the Right Angle CAK . And the Angle BAD is an Obtuse Angle being greater than the Right Angle DAK .

The Geometrical Propositions concerning Lines and Angles are very many , but these following we think sufficient for our present purpose .

Proposition I.

To divide a Right Line given into any Number of equal Parts .

Let it be required to divide the Right Line AB into five equal Parts . From the extream Points of the given Line A and B , let there be drawn two Parallel Lines , then from the Point A at any distance of the Compasses , set off as many equal Parts wanting one , as the given Line is to be divided into , which in our Example is four , and are noted thus , 1. 2. 3. 4. and from the Point B set off the like Parts in the Line BC , and let them be noted likewise thus , 1. 2. 3. 4. then shall the Parallel Lines , 14. 23. 32 , and 41. divide the Right Line AB into 5 equal Parts , as was required .

Proposition II.

Two Right Lines being given , to find a Mean propertional between them .

Let the two Right Lines given be DB and CB , which let be made into one Line as CD , which being besected the Point of bisection is A , from which as from a Centre describe the Arch CED , and from the Point B erect the Perpendicular BE , so shall BE , be the Mean proportional required ; for , BC. BE ∷ BE. BD.

Proposition III.

Three Right Lines being given , to find a fourth proportional .

Let the three given Lines be AB . BC. and AD. Fig. 5. to which a fourth proportional is required : draw AE at any Acute Angle , to the Line AD in the Point A ; and make DE parallel to BC , so shall AE be the fourth proportional required ; for , AB . BC ∷ AD. AE.

Proposition IV.

Vpon a Right Line given , to make a right-lined . Angle , equal to an Angle given .

Let it be required upon the Line CD in Fig. 6. to make an Angle , equal to the Angle DAE in Fig. 5. From the Point A as a Center , at any extent of the Compasses describe the Arch BG , between the sides of the Angle given , and with the same extent describe the Arch HL from the Point D , and then make HL equal to BG , then draw the Line DL , so shall the Angle CDL be equal to the Angle DAE given , as was required .

CHAP. II.

Of Figures in the general , more particularly of a Circle and the affections thereof .

HItherto we have spoken of the first kind of Magnitude , that is , of Lines , as they are considered of themselves , or amongst themselves .

2. The second kind of Magnitude is that which is made of Lines , that is , a Figure consisting of breadth as well as length , and this is otherwise called a Superficies .

3. And in a Superficies there are three things to be considered . 1. The Term or Limit . 2. The middle of the Term. 3. The Thing or Figure made by the Term or Limit .

4. The Term or Limit is that which comprehendeth and boundeth the Figure , it is commonly called the Perimeter or Circumference .

5. The Term of a Figure is either Simple or various .

6. A Simple Term is that which doth consist of a Simple Line , and is properly called a Circumference or Periphery : A Periphery therefore is the Term of a Circle or most Simple Figure .

7. A various Term is that which hath bending or crooked Lines , making Angles , and may therefore be called Angular .

8. The middle of Term is that which is the Center of the Figure ; for every Figure , whether Triangular , Quadrangular , or Multangular , hath a Center as well as the Circular , differing in in this , that the Lines in a Circle drawn from the Center to the Circumference are all equal , but in other Figures they are not equal .

9. The Thing or Figure made by the Term or Limit , is all that Area or space which is included by the Term or Terms . And here it is to be observed , that the Term of a Figure is one thing , and the Figure it self another ; for Example , A Periphery is the Term of a Circle , but the Circle it self is not properly the Periphery , but all that Area or space which is included by the Periphery , for a Periphery is nothing but a Line , but the Circle is that which is included by that Line .

10. As the Term of a Figure is either Simple or Various ; so the Figure it self is either Simple and Round , or Various and Angular .

11. A Simple Figure is that which is contained by a Simple or Round Line , and is either a Circle or an Ellipsis .

12. A Circle therefore is such a Figure which is made by a Line so drawn into it self , as that it is every where equally distant from the middle or Center .

13. An Ellipsis is an oblong Circle .

14. In a Circle we are to consider the affections which are as it were the Parts or Sections thereof , as they are made by the various applications of Right Lines .

15. And Right Lines may be applied unto a Circle , either by drawing them within , or without the Circle .

16. Right Lines inscribed within a Circle , are either such as do cut the Circle into two equal or unequal Parts , as the Diameter and lesser Chords , or such as do cut the Diameter and lesser Chords into two equal or unequal Parts , as the Right and versed Sines .

17. A Diameter is a Right Line drawn through the Center from one side of the Circumference to the other , and divideth the Circle into two equal Parts , As in Fig. 7. The Right Line GD drawn through the Center B is the Diameter of the Circle GEDL dividing the same into the two equal Parts GED , and GLD : and this is also called the greatest Chord or Subtense .

18. A Chord or Subtense is a Right Line inscribed in a Circle , dividing the same into two equal or unequal Parts ; if it divide the Circle into two equal Parts , it is the same with the Diameter , but if it divide the Circle into two unequal Parts it is less than the Diameter , and is the Chord or Subtense of an Arch less than a Semi-circle , and also of an Arch greater than a Semi-circle . As in the former Figure , the Right Line CAK divideth the Circle into two unequal Parts , and is the Chord or Subtense of the Arch CDK , less than a Semi-circle , and of the Arch CGK greater than a Semi-circle : and these are the Lines which divide the Circle into two equal or unequal Parts . And as they divide the Circle into two equal Parts , so do they also divide one another ; The lesser Chords when they are divided by the Diameter into two equal Parts , those Parts are called Right Sines , and the two Parts of the Diameter made by the intersection of the Chords are called versed Sines .

19. Sines are right or versed .

20. Right Sines are made by being besected , by the Diameter , and are twofold , Sinus totus , the whole Sine or Radius , and this is the one half of the Diameter , as the Lines BE or BD , and all Lines drawn from the Center to the Circumference .

21. Sinus simpliter , or the lesser Sines , are the one half of any Chord less than the Diameter , as in the former Figure CA or AK , which are the equal Parts of the Chord CAK , are the Sines of the Arches CD . and DK less than a Quadrant , and also the Sines of CEG and KLG greater than a Quadrant .

22. Versed Sines are the Segments of the Diameter , made by the Chords intersecting it , at Right Angles , as AD is the versed Sine of CD or DG and the other Segment AG is the versed Sine of the Arch CEG or KLG .

23. The Right Lines drawn without the Circle are two , the one touching the Circle , and is called a Tangent , and the other cutting the Circle , and is called a Secant .

24. A Tangent is a Right Line touching the Circle , and drawn perpendicular to the Diameter , and extended to the Secant .

25. A Secant is a Right Line drawn from the Center through the Circumference , and extended to the Tangent . As in the former Figure , the Right Line DF is the Tangent of the Arch CD , and the Right Line BF is the Secant of the same Arch CD .

Proposition I.

The Arch of a Circle being given to describe the whole Periphery .

Let ABC be an Arch given , and let the Circumference of that Circle be required . Let there be three Points taken in the given Arch at pleasure , as A , B , C ; open your Compasses to more than half the distance of A , B , and setting one Foot in A describe the Arch of a Circle , and the Compasses remaining at the same distance , setting one Foot in B , describe another Arch so as it may cut the former in two Points , suppose G , and H , and draw the Line HG towards that Part on which you suppose the Center of the Center of the Circle will fall .

In like manner , opening your Compasses to more than half your distance of B , C , describe two other Arches from the Points E and C , cutting each other in E and F , then draw the Line EF till it intersect the former Line HG , so shall the Point of Intersection be the Center of the Circumference or Circle required , as in Fig. may be seen .

Proposition II.

The Conjugate Diameters of an Ellipsis being given , to draw the Ellipsis .

Let the given Diameter in Fig. 24. be LB and ED , the greatest Diameter . LB being bisected in the Point of Bisection , erect the Perpendicular AD. which let be half of the lesser Diameter ED , then open your Compasses to the extent of AB , and setting one Foot in D , with the other make a mark at M and N in the Diameter BL , then cutting a thred to the length of BL , fasten the thred with your Compasses in the Points NM , and with your Pen in the inside of the thred describe the Arch BFKL , so shall you describe the one half of the Ellipsis required , and turning the Thred on the other side of the Compasses , you may with your Pen in the like manner describe the other half of the Ellipsis GBHL .

CHAP. III.

Of Triangles .

HItherto we have spoken of the most Simple Figure , a Circle . Come we now to those Figures that are Various or Angular .

2. And an Angular Figure is that which doth consist of three or more Angles .

3. An angular Figure consisting of three Angles , otherwise called a Triangle , is a Superficies or Figure comprehended by three Right Lines including three Angles .

4. A Triangle may be considered either in respect of its Sides , or of its Angles .

5. A Triangle in respect of its Sides , is either Isopleuron , Isosceles , or Scalenum .

6. An Isopleuron Triangle , is that which hath three equal sides . An Isoscecles hath two equal Sides . And a Scalenum hath all the three Sides unequal .

7. A Triangle in respect of its Angles is Right or Oblique .

8. A Right angled Triangle is that which hath one Right Angle and two Acute .

9. An Oblique angled Triangle , is either Acute or Obtuse .

10. An Oblique acute angled Triangle , is that which hath all the three Angles Acute .

11. An Oblique obtuse angled Triangle , is that which hath one Angle Obtuse , and the other two Acute .

Proposition I.

Vpon a Right Line given to make an Isopleuron or an Equilateral Triangle .

In Fig. 8. let it be required to make an Equilateral Triangle upon the Right Line AB . Open your Compasses to the extent of the Line given , and setting one Foot of your Compasses in A , make an Arch of a Circle above or beneath the Line given , then setting one Foot of your Compasses in B , they being full opened to the same extent , with the other foot draw another Arch of a Circle crossing the former , and from the Intersection of those Arches draw the Lines AC and AB , so shall the Triangle ACB be Equilateral as was desired .

Proposition II.

Vpon a Right Line given to make an Isosceles Triangle , or a Triangle having two Sides equal .

In Fig. 8. let AB be the Right Line given , from the Points A and B as from two Centers , but at a lesser extent of the Compasses than AB ; if you would have AB the greatest Side , at a greater extent ; if you would have it to be the least Side , describe two Arches cutting one another , as at F , and from the Intersection draw the Lines AF , and FB , so shall the Triangle AFB have two equal Sides , as was required .

Proposition 3.

To make a Scalenum Triangle , or a Triangle , whose three Sides are unequal .

In Fig. 9. let the three unequal Sides be EFG make AB equal to one of the given Lines , suppose G , and from A as a Center , at the extent of E describe the Arch of a Circle ; in like manner from B at the extent of F describe another Arch intersecting the former , then shall the Right Lines AC . CB and BA comprehend a Triangle , whose three sides shall be unequal , as was required .

CHAP. IV.

Of Quadrangular and Multangular Figures .

WE have spoken of Triangles or Figures consisting of three Angles , come we now to those that have more Angles than three , as the Quadrangle , Quinquangle , Sexangle , &c.

2. A Quadrangle is a Figure or Superficies , which is bounded with four Right Lines .

3. A Quadrangle is either a Parallelogram or a Trapezium .

4. A Parallelogram is a Quadrangle whose opposite Sides are parallel having equal distances from one another in all Places .

5. A Parallelogram is either Right angled or Oblique .

6. A Right angled Parallelogram , is a Quadrangle whose four Angles are all Right , and is either Square or Oblong .

7. A Square Parallelogram doth consist of four equal Lines . The Parts of a Square are , the Sides of which the Square is made , and the Diagonal or Line drawn from one opposite Angle to another through the middle of the Square .

8. An Oblong is a Right angled Parallelogram , having two longer and two shorter Sides .

9. An Oblique angled Parallelogram , is that whose Angles are all Oblique , and is either a Rhombus or a Rhomboides .

10. A Rhombus is an Oblique angled and equilateral Parallelogram .

11. A Rhomboides is an Oblique angled and inequilateral Parallelogram .

12. A Trapezium is a Quadrangular Figure whose Sides are not all parallel ; it is either Right angled or Oblique .

13. A Right angled Trapezium hath two opposite Sides parallel , but unequal , and the Side between them perpendicular .

14. An Oblique angled Trapezium is a Quadrangle , but not a Parallelogram , having at least two Angles Oblique , and none of the Sides parallel .

15. Thus much concerning Quadrangles or four sided Figures . Figures consisting of more than four Angles are almost infinite , but are reducible unto two sorts , Ordinate and Regular , or Inordinate and Irregular .

16. Ordinate and Regular Polygons are such , as are contained by equal Sides and Angles , as the Pentagon , Hexagon , and such like .

17. Inordinate or irregular Polygons , are such as are contained by unequal Sides and Angles . The construction of these Quadrangular and Multangular Figures is explained in the Propositions following .

Proposition . I.

Vpon a Right Line given to describe a Right angled Parallelogram , whether Square or Oblong .

In Fig. 10. let the given Line be AB , upon the Point A erect the Perpendic●lar AD equal to AB if you intend to make a Square , but longer or shorter , if you intend an oblong , and upon the Points D and B at the distance of AB and AD describe two Arches intersecting one another , and from the Intersection draw the Lines ED and EB , so shall the Right angled Figure AE be a Square , if AB and AD be equal , otherwise an Oblong , as was desired .

Proposition II.

To describe a Rhombus or Rhomboides .

In Fig. 11. To the Right Line AB draw the Line AD at any Acute Angle at pleasure , equal to AB if you intend a Rhombus , longer or shorter if you intend a Rhomboides , then upon your Compasses to the extent of AD and upon B as a Center describe an Arch ; in like manner , at the extent of AB upon D as a Center describe another Arch intersecting the former , then draw the Lines ED and EB , so shall AE be the Rhombus or Rhomboides , as was required .

Proposition III.

Vpon a Right Line given to make a Regular Pentagon , or five sided Figure .

In Fig. 12. Let the given Line be AB , upon A and B as two Centers describe the Circles EBGH and CAGK , then open your Compasses to the extent of BC , and making G the Center , describe the Arch HAFK , then draw the Lines KFE and HFC : so shall AE and BC be two sides of the Pentagon desired , and opening your Compasses to the extent of AB , upon E and C as two Centers describe two Arches intersecting one another , and from the Point of Intersection draw the Lines ED and DC , so shall the Figure AB and DE be the Pentagon required .

Proposition IV.

To make a Regular Pentagon and Decagon in a given Circle .

In Fig. 13. upon the Diameter CAB describe the Circle CDBL , from the Center AErect the Perpendicular AD , and let the Semidiameter AC be bisected , the Point of Bisection is E , set the distance ED from E to G , and draw the Line GD , which is the side of a Pentagon , and AG the side of a Decagon inscribed in the same Circle .

Proposition V.

In a Circle given to describe a Regular Hexagon .

The side of a Hexagon is equal to the Radius of a Circle , the Radius of a Circle therefore being six times applied to the Circumference , will give you six Points , to which Lines being drawn from Point to Point , will constitute a Regular Hexagon , as was desired .

Proposition VI.

In a Circle given to describe a Regular Heptagon or Figure consisting of seven equal sides .

The side of a Heptagon is equal to half the side of a Triangle inscribed in a Circle , having therefore drawn an Hexagon in a Circle , the Chord Line subtending two sides of the Hexagon lying together , is the side of a Triangle inscribed in that Circle , and half that Chord applied seven times to the Circumference , will give seven Points , to which Lines being drawn from that Point , will constitute a Regular Heptagon , as in Fig. 14. is plainly shewed .

CHAP. V.

Of Solid Bodies .

HAving spoken of the two first kinds of Magnitude , Lines and Superficies , come we now to the third , a Body or Solid .

2. A Body or Solid is a Magnitude consisting of length , breadth and thickness .

3. A Solid is either regular or irregular .

4. That is called a regular Solid , whose Bases , Sides and Angles are equal and like .

5. And this either Simple or Compound .

6. A simple regular Solid , is that whith doth consist of one only kind of Superficies .

7. And this is either a Sphere or Globe , or a plain Body .

8. A Globe is a Solid included by one round and convex Superficies , in the middle whereof there is a Point , from whence all Lines drawn to the Circumference are equal .

9. A simple plain Solid , is that which doth consist of plain Superficies .

10. A plain Solid is either a Pyramid , a Prism , or a mixt Solid .

11. A Pyramid is a Solid , Figure or Body , contained by several Plains set upon one right lin'd Base , and meeting in one Point .

12. Of all the several sorts of Pyramids , there is but one that is Regular , to wit a Tetrahedron , or a Pyramid consisting of four regular or equilateral Triangles ; the form whereof ( as it may be cut in Pastboard ) may be conceived by Figure 15.

13. A Prism is a Solid contained by several Plains , of which those two which are opposite , are equal , like and parallel , and all others are Paralellogram .

14. A Prism is either a Pentahedron , a Hexahedron , or a Polyhedron .

15. A Pentahedron Prism , is a Solid comprehended of five Sides , and the Base a Triangle , as Fig. 16.

16. An Hexahedron Prism , is a Solid comprehended of six Sides , and the Base a Quadrangle , as Fig. 17.

17. An Hexahedron Prism , is distinguished into a Parallelipipedon and a Trapezium .

18. An Hexahedron Prism called a Trapezium is a Solid , whose opposites Plains or Sides , are neither opposite nor equal .

19. A Parallelipipedon is either right angled or oblique .

20. A right angled Parallelipipedon is an Hexahedron Prism , comprehended of right angled Plains or Sides ; and it is either a Cube or an Oblong .

21. A Cube is a right angled Parallelipipedon comprehended of six equal Plains or Sides .

22. An Oblong Parallelipipedon , is an Hexahedron Prism , comprehended by unequal Plains or Sides .

23. An Oblique angled Parallelipipedon , is an Hexahedron Prism , comprehended of Oblique Sides .

24. A Polyhedron Prism , is a Solid comprehended by more than six Sides , and hath a multangled Base , as a Quincangle , Sexangle , &c.

25. A regular compound or mixt Solid , is such a Solid as hath its Vertex in the Center , and the several Sides exposed to view , and of this sort there are only three ; the Octohedron , the Icosahedron , of both which the Base is a Triangle ; and the Dodecahedron , whose Base is a Quincangle .

26. An Octohedron is a Solid Figure which is contained by eight equal and equilateral Triangles , as in Fig. 18.

27. An Icosahedron is a Solid , which is contained by twenty equal and equilateral Triangles , as Fig. 19.

28. A Dodecahedron is a Solid , which is contained by twelve equal Pentagons , equilateral and equiangled , as in Fig. 20.

29. A regular compound Solid , is such a Solid as is Comprehended both by plain and circular Superficies , and this is either a Cone or a Cylinder .

30. A Cone is a Pyramidical Body , whose Base is a Circle , or it may be called a round Pyramis , as Fig. 21.

31. A Cylinder is a round Column every where comprehended by equal Circles , as Fig. 22.

32. Irregular Solids are such , which come not within these defined varieties , as Ovals , Frustums of Cones , Pyramids , and such like .

And thus much concerning the description of the several sorts of continued Quantity , Lines , Plains and Solids ; we will in the next place consider the wayes and means by which the Dimentions of them may be taken and determined , and first we will shew the measuring of Lines .

CHAP. VI.

Of the Measuring of Lines both Right and Circular .

EVery Magnitude must be measured by some known kind of Measure ; as Lines by Lines , Superficies by Superficies , and Solids by Solids , as if I were to measure the breadth of a River , or height of a Turret , this must be done by a Right Line , which being applied to the breadth or height desired to be measured , shall shew the Perches , Feet or Inches , or by some other known measure the breadth or height desired : but if the quantity of some Field or Meadow , or any other Plain be desired , the number of square Perches must be enquired ; and lastly , in measuring of Solids , we must use the Cube of the measure used , that we discover the number of those Cubes that are contained in the Body or Solid to be measured . First , therefore we will speak of the several kinds of measure , and the making of such Instruments , by which the quantity of any Magnitude may be known .

2. Now for the measuring of Lines and Superficies , the Measures in use with us , are Inches , Feet , Yards , Ells and Perches .

3. An Inch is three Barley Corns in length , and is either divided into halves and quarters , which is amongst Artificers most usual , or into ten equal Parts , which is in measuring the most useful way of Division .

4. A Foot containeth twelve Inches in length , and is commonly so divided ; but as for such things as are to be measured by the Foot , it is far better for use , when divided into ten equal Parts , and each tenth into ten more .

5. A Yard containeth three Foot , and is commonly divided into halves and quarters , the which for the measuring of such things as are usually sold in Shops doth well enough , but in the measuring of any Superficies , it were much better to be divided into 10 or 100 equal Parts .

6. An Ell containeth three Foot nine Inches , aud is usually divided into halves and quarters , and needs not be otherwise divided , because we have no use for this Measure , but in Shop Commodities .

7. A Pole or Perch cotaineth five Yards and an half , and hath been commonly divided into Feet and half Feet . Forty Poles in length do make one Furlong , and eight Furlongs in length do make an English Mile , and for these kinds of of lengths , a Chain containing four Pole , divided by Links of a Foot long , or a Chain of fifty Foot , or what other length you please , is well enough , but in the measuring of Land , in which the number of square Perches is required ; the Chain called Mr. Gunters , being four Pole in length divided into 100 Links , is not without just reason reputed the most useful .

8. The making of these several Measures is not difficult , a Foot may be made , by repeating an Inch upon a Ruler twelve times , a Yard is eight Foot , and so of the rest ; the Subdivision of a Foot or Inch into halves and quarters , may be performed by the seventeenth of the first , and into ten or any other Parts by the first Proposition of the first Chapter , and all Scales of equal Parts , of what scantling you do desire . And this I think is as much as needs to be said concerning the dividing of such Instruments as are useful in the measuring Right Lines .

9. The next thing to be considered is the measuring of Circular Lines , or Perfect Circles .

10. And every Circle is supposed to be divided into 360 Parts called Degrees , every Degree into 60 Minutes , every Minute into 60 Seconds , and so forward this division of the Circle into 360 Parts is generally retained , but the Subdivision of those Parts , some would have be thus and 100 , but as to our present purpose either may be used , most Instruments not exceeding the fourth part of a Degree .

11. Now then a Circle may be divided into 360 Parts in this manner , Having drawn a Diameter through the Center of the Circle dividing the Circle into two equal Parts , cross that Diameter with another at Right Angles through the Center of the Circle also , so shall the Circle be divided into four equal Parts or Quadrants , each Quadrant containing 90 Degrees , as in Fig. 7. GE. ED. DL and LG , are each of them 90 Degrees ; and the Radius of a Circle being equal to the Chord of the sixth Part thereof , that is to the Chord of 60 Degrees , as in Fig. 14. if you set the Radius GB from L towards G , and also from G towards L , the Quadrant GL will be subdivided into three equal Parts , each Part containing 30 Degrees , GM . 30. MH 30 and HL 30 , the like may be done in the other Quadrants also ; so will the whole Circle be divided into twelve Parts , each Part containing 30 Degrees .

And because the side of a Pentagon inscribed in a Circle is equal to the Chord of 72 Degrees , or the first Part of 360 , as in Fig. 13. therefore if you set the Chord of the first Part of the Circle given from G to L or L to G , in Fig. 7. you will have the Chord of 72 Degrees , and the difference between GP 72 and GH 60 is HP 12 , which being bisected , will give the Arch of 6 Degrees , and the half of six will give three , and so the Circle will be divided into 120 Parts , each Part containing three Degrees , to which the Chord Line being divided into three Parts , the Arch by those equal Divisions may be also divided , and so the whole Circle will be divided into 360 , as was desired .

12. A Circle being thus divided into 360 Parts , the Lines of Chords , Sines , Tangents and Secants , are so easily made ( if what hath been said of them in the Second Chapter be but considered ) that I think it needless to say any more concerning their Construction , but shall rather proceed unto their Use.

13. And the use of these Lines and other Lines of equal Parts we will now shew in circular and right lined Figures ; and first in the measuring of a Circle and Circular Figures .

CHAP. VII .

Of the Measuring of a Circle .

THe squaring of a Circle , or the finding of a Square exactly equal to a Circle given , is that which many have endeavoured , but none as yet have attained : Yet Archimedes that Famous Mathematician hath sufficiently proved , That the Area of a Circle is equal to a Rectangle made of the Rodius and half the Circumference : Or thus , The Area of a Circle is equal to a Rectangle made of the Diameter and the fourth part of the Circumference . For Example , let the Diameter of a Circle be 14 and the Circumference 44 ; if you multiply half the Circumference 22 by 7 half the Diameter , the Product is 154 ; or if you multiply 11 the fourth part of the Circumference , by 14 the whole Diameter , the Product will still be 154. And hence the Superficies of any Circle may be found though not exactly , yet near enough for any use .

2. But Ludolphus Van Culen finds the Circumference of a Circle whose Diameter is 1.00 to be 3.14159 the half whereof 1.57095 being multiplied by half the Diameter 50 , &c. the Product is 7.85395 which is the Area of that Circle , and from these given Numbers , the Area , Circumference and Diameter of any other Circle may be found by the Proportions in the Propositions following .

Proposition I.

The Diameter of a Circle being given to find the Circumference .

As 1. to 3.14159 : so is the Diameter to the Circumference . Example . In Fig. 13. Let the Diameter IB be 13. 25. I say as 1. to 3. 14159. so IB . 13.25 to 41.626 the Circumference of that Circle .

Proposition II.

The Diameter of a Circle being given to find the Superficial Content .

As 1. to 78539 ; so is the Square of the Diameter given , to the Superficial Content required . Example , Let the Diameter given be as before IB 13.25 the Square thereof is 175.5625 therefore .

As 1. to 78539 : so 175.5625 to 137.88 the Superficial Content of that Circle .

Proposition III.

The Circumference of a Circle being given , to find the Diameter .

This is but the Converse of the first Proposition : Therefore as 3.14159 is to 1 : so is the Circumference to the Diameter ; and making the Circumference an Unite , it is . 3. 14159. 1 ∷ 1. 318308 , and so an Unite may be brought into the first place . Example , Let the given Circumference be 41. 626. I say ,

As 1. to 318308 : so 41.626 to 13. 25. the Diameter required .

Proposition IV.

The Circumference of a Circle being given to find the Superficial Content .

As the Square of the Circumference of a Circle given is to the Superficial Content of that Circle : so is the Square of the Circumference of another Circle given to the Superficial Content required . Example , As the Square of 3.14159 is to 7853938 : so is 1. the Square of another Circle to 079578 the Superficial Content required , and so an Unite for the most easie working may be brought into the first place : Thus the given Circumference being 41. 626. I say ,

As 1. to 0.79578 : so is the Square of 41.626 to 137.88 the Superficial Content required .

Proposition V.

The Superficial Content of a Circle being given , to find the Diameter .

This is the Converse of the second Proposition , therefore as 78539 is to 1. so is the Superficial Content given , to the Square of the Diameter required . And to bring an Unite in the first place : I say .

As 7853978. 1 ∷ 1. 1. 27324 , and therefore if the Superficial Content given be 137.88 , to find the Diameter : I say ,

As 1. to 1.27324 : so 137.88 to 175.5625 whose Square Root is 13.25 , the Diameter sought .

Proposition VI.

The Superficial Content of a Circle being given , to find the Circumference .

This is the Converse of the Fourth Proposition , and therefore as 079578 is to 1 : so is the Superficial Content given , to the Square of the Circumference required , and to bring an Unite in the first place : I say ,

As 079578. 1 : : 1. 12.5664 , and therefore if the Superficial Content given be 137.88 , to find that Circumference : I say ,

As 1. to 12.5664 : so is the 137.88 to 1732.7 whose Square Root is 626 the Circumference .

Proposition VII .

The Diameter of a Circle being given to find the Side of the Square , which may be inscribed within the same Circle .

The Chord or Subtense of the Fourth Part of a Circle , whose Diameter is an Unite , is 7071067 , and therefore , as 1. to 7071067 : so is the Diameter of another Circle , to the Side required . Example , let the Diameter given be 13.25 to find the side of a Square which may be inscribed in that Circle : I say ,

As 1. to 7071067 : so is 13.25 to 9.3691 the side required .

Proposition VIII .

The Circumference of a Circle being given , to find the Side of the Square which may be inscribed in the same Circle .

As the Circumference of a Circle whose Diameter is an Unite , is to the side inscribed in that Circle ; so is the Circumference of any other Circle , to the side of the Square that may be inscribed therein . Therefore an Unite being made the Circumference of a Circle .

As 3.14159 to 7071067 : so 1. to 225078.

And therefore the Circumference of a Circle being as before 41.626 , to find the side of the Square that may be inscribed : I say ,

As 1. to 225078. so is 41.626 to 9.3691 the side inquired .

Proposition IX .

The Axis of a Sphere or Globe being given , to find the Superficial Content .

As the Square of the Diameter of a Circle , which is Unity , is to 3.14159 the Superficial Content , so is the Square of any other Axis given , to the Superficial Content required . Example , Let 13.25 be the Diameter given , to find the Content of such a Globe : I say ,

As 1. to 3.14159 : so is the Square of 13.25 to 551.54 the Superficial Content required .

Proposition X.

To find the Area of an Ellipsis .

As the Square of the Diameter of a Circle , is to the Superficial Content of that Circle ; so is the Rectangle made of the Conjugate Diameters in an Ellipsis , to the Area of that Ellipsis ; And the Diameter of a Circle being one , the Area is 7853975 , therefore in Fig. 26. the Diameters AC8 and BD5 being given , the Area of the Ellipsis ABCD may thus be found .

As 1. to 7853975 : so is the Rectangle AC in BD40 to 3.1415900 , the Area of the Ellipsis required .

CHAP. VIII .

Of the Measuring of Plain Triangles .

HAving shewed the measuring of a Circle , and Ellipsis , we come now to Right lined Figures , as the Triangle , Quadrangle , and Multangled Figures , and first of the measuring of the plain Triangles .

2. And the measuring of Plain Triangles is either in the measuring of the Sides and Angles , or of their Area and Superficial Content .

3. Plain Triangles in respect of their Sides and Angles are to be measured by two sorts of Lines , the one is a Line of equal Parts , and by that the Sides must be measured , the other is a Line of Chords , the Construction whereof hath been shewed in the sixth Chapter , and by that the Angles must be measured , the Angles may indeed be measured by the Lines of Sines , Tangents or Secants , but the Line of Chords being not only sufficient , but most ready , it shall suffice to shew how any Angle may be protracted by a Line of Chords , or the Quantity of any Angle found , which is protracted .

4. And first to protract or lay down an Angle to the Quantity or Number of Degrees proposed , do thus , draw a Line at pleasure as AD in Figure 5 , then open your Compasses to the Number of 60 Degrees in the Line of Chords , and setting one Foot in A , with the other describe the Arch BG , and from the Point A let it be required to make an Angle of 36 Degrees : open your Compasses to that extent in the Line of Chords , and setting one Foot in B , with the other make a mark at G , and draw the Line AG , so shall the Angle BAG contain 36 Degrees , as was required .

5. If the Quantity of an Angle were required , as suppose the Angle BAG , open your Compasses in the Line of Chords to the extent of 60 Degrees , and setting one Foot in A , with the other draw the Arch BG , then take in your Compasses the distance of BG , and apply that extent to the Line of Chords , and it will shew the Number of Degrees contained in that Angle , which in our Example is 36 Degrees .

6. In every Plain Triangle , the three Angles are equal to two right or 180 Degrees , therefore one Angle being given , the sum of the other two is also given , and two Angles being given , the third is given also .

7. Plain Triangles are either Right Angled or Oblique .

8. In a Right Angled Plain Triangle , one of the Acute Angles is the Complement of the other to a Quadrant or 90 Degrees .

9. In Right Angled Plain Triangles , the Side subtending the Right Angle we call the Hypotenuse , and the other two Sides the Legs , thus in Fig. 5 ▪ AE is the Hypotenuse , and AD and ED are the Legs ; these things premised , the several cases in Right Angled and Oblique Angled Plain Triangles may be resolved , by the Propositions following .

Proposition I.

In a Right Angled Plain Triangle , the Angles of one Leg being given , to find the Hypotenuse and the other Leg.

In the Right Angled Plain Triangle ADE in Fig. 5. Let the given Angles be DAE 36 , and DEA 54 , and let the given Leg be AD 476 ; to find the Hypotenuse AE , and the other Leg ED.

Draw a Line at pleasure , as AD , and by your Scale of equal Parts set from A to D 476 the Quantity of the Leg given , then erect a Perpendicular upon the Point D , and upon the Point A lay down your given Angle DAE 36 by the fourth hereof , and draw the Line AE till it cut the Perpendicular DE , then measure the Lines AE and DE upon your Scale of Equal Parts , so shall AE 588.3 be the Hypotenuse , and DE 345.8 the other Leg.

Proposition II.

The Hypotenuse and Oblique Angles given , to find the Legs .

Let the given Hypotenuse be 588 , and one of the Angles 36 degrees , the other will then be 54 degrees , Draw a Line at pleasure , as AD , and upon the Point A by the fourth ▪ hereof lay down one of the given Angles suppose the less , and draw the Line AC , and from your Scale of equal Parts , set off your Hypotenuse 588 from A to E , and from the Point E to the Line AD let fall the Perpendicular ED , then shall AD being measured upon the Scale be 476 for one Leg , and ED 345.8 the other .

Proposition III.

The Hypotenuse and one Leg given to find the Angles and the other Leg.

Let the given Hypotenuse be 588. and the given Leg 476. Draw a Line at pleasure as AD , upon which set the given Leg from A to D. 476 , and upon the Point D , erect the Perpendicular DE , then open your Compasses in the Scale of Equal Parts to the Extent of your given Hypotenuse 588 , and setting one Foot of that Extent in A , move the other till it touch the Perpendicular DE , then and there draw AE , so shall ED be 345.8 the Leg inquired , and the Angle DAE , will be found by the Line of Chords to be 36 ▪ whose Complement is the Angle DEA . 54.

Proposition IV.

The Legs given to find the Hypotenuse , and the Oblique Angles .

Let one of the given Legs be 476 , and the other 345.8 , Draw the Line AD to the extent of 476 , and upon the Point D , erect the Perpendicular DE to the extent of 345.8 , and draw the Line AE , so shall AE be the Hypotenuse 588 , and the Angle DAE will by the Line of Chords be found to be 36 Degrees , and the Angle DEA 54 , as before .

Hitherto we have spoken of Right angled plain Triangles : the Propositions following concern such as are Oblique .

Proposition V.

Two Angles in an Oblique angled plain Triangle , being given , with any one of the three Sides , to find the other two Sides .

In any Oblique angled plain Triangle , let one of the given Angles be 26.50 and the other 38. and let the given Side be 632 , the Sum of the two given Angles being deducted from a Semi-circle , leaveth for the third Angle 115.50 Degrees , then draw the Line BC 632. and upon the Points B and C protract the given Angles , and draw the Lines BD and CD , which being measured upon your Scale of equal Parts BD will be fou●d to be 312.43 , and BD 431.09 ,

Proposition VI.

Two Sides in an Oblique Angled Triangle being given , with an Angle opposite to one of them , to find the other Angles and the third Side , if it be known whether the Angle Opposite to the other Side given be Acute or Obtuse .

In an Oblique Angled Plain Triangle , let the given Angle be 38 Degrees , and let the Side adjacent to that Angle be 632 , and the Side opposite 431. 1. upon the Line BC in Fig. 25. protract the given Angle 38 Degrees upon the Point C , and draw the Line DC , then open your Compasses to the Extent of the other Side given 431. 1. and setting one Foot in B , turn the other about till it touch the Line DC , which will be in two places , in the Points D and E ; if therefore the Angle at B be Acute the third Side of the Triangle will he CE , according therefore to the Species of that Angle you must draw the Line BD or BE to compleat the Triangle , and then you may measure the other Angles , and the third Side as hath been shewed .

Proposition VII .

Two Sides of an Oblique Angled Plain Triangle being given , with the Angle comprehended by them to find the other Angles and the third Side .

Let one of the given Sides be 632 , and the other 431.1 , and let the Angle comprehended by them be Deg. 26.50 , draw a Line at pleasure , as BC , and by help of your Scale of Equal Parts , set off one of your given Sides from B to C 632. then upon the Point B protract the given Angle 26. 50. and draw the Line BD , and from B to D , set off your other given Side 431. 1. and draw the Line DC , so have you constituted the Triangle BDC , in which you may measure the Angles and the third Side , as hath been shewed .

Proposition VIII .

The three Sides of an Oblique Angled Triangle being given , to find the Angles .

Let the length of one of the given Sides be 632 , the length of another 431.1 , and the length of the third 312.4 , and Draw a Line at pleasure , as BC in Fig. 25 , and by help of your Scale of Equal Parts , set off the greatest Side given 632 from B to C. then open your Compasses in the same Scale to the extent of either of the other Sides , and setting one Foot of your Compasses in B , with the other describe an occult Arch , then extend your Compasses in the same Scale to the length of the third Side , and setting one Foot in C with the other describe another Arch cutting the former , and from the Point of Intersection draw the Lines BD and DC . to constitute the Triangle BDC , whose Angles may be measured , as hath been shewed .

And thus may all the Cases of Plain Triangles be resolved by Scale and Compass , he that desires to resolve them Arithmetically , by my Trigometria Britannica , or my little Geometrical , Trigonometry ; only one Case of Right Angled Plain Triangles which I shall have occasion to use , in the finding of the Area of the Segment of a Circle I will here shew how , to resolve by Numbers .

Proposition IX .

In a Right Angled Plain Angle the Hypotenuse and one Leg being given to find the other Leg.

Take the Sums and difference of the Hypotenuse and Leg given , then multiply the Sum by the Difference , and of the Product extract the Square Root , which Square Root shall be the Leg inquired .

Example . In Fig. 5. Let the given Hypotenuse be AE 588.3 , and the given Leg AD 476 , and let DE be the Leg inquired . The Sum of AE and AD is 1064.3 , and their Difference is 112.3 , now then if you multiply 1064.3 by 112.3 , the Product will be 119520.89 , whose Square Root is the Leg DE. 345. 8.

Proposition X.

The Legs of a Right Angled Plain Triangle being gived , to find the Area or Superficial Content thereof .

Multiply one Leg by the other , half the Product shall be the Content . Example , In the Right angled plain Triangle ADE , let the given Legs be AD 476 , and DE 345 , and let the Area of that Triangle be required , if you multiply 476 by 345 the Product will be 164220 , and the half thereof 82110 is the Area or Superficial Content required .

Proposition XI .

The Sides of an Oblique angled plain Triangle being given to find the Area or Superficial Content thereof .

Add the three Sides together , and from the half Sum subtract each Side , and note their Difference ; then multiply the half Sum by the said Differences continually , the Square Root of the last Product , shall be the Content required .

Example . In Fig. 9. Let the Sides of the Triangle ABC be AB 20. AC 13 , and BC 11 the Sum of these three Sides is 44 , the half Sum is 22 , from whence subtracting AB 20 , the Difference is 2 , from whence also if you substract AC 13 , the Difference is 9 , and lastly , if you subtract BC 11 from the half Sum 22 , the Difference will be 11. And the half Sum 22 being multiplied by the first Difference 2 , the Product is 44 , and 44 being multiplied by the Second Difference 9 , the Product is 396 , and 396 being multiplied by the third Difference 11 , the Product is 4356 , whose Square Root 66 , is the Content required .

Or thus , from the Angle C let fall the Perpendicular DC , so is the Oblique angled Triangle ABC , turned into two right , now then if you measure DC upon your Scale of Equal Parts , the length thereof will be found to be 6.6 , by which if you multiply the Base AB 20 , the Product will be 132.0 , whose half 66 , is the Area of the Triangle , as before .

Proposition XII .

The Sides of any Oblique angled Quadrangle being given , to find the Area or Superficial Content thereof .

Let the Sides of the Oblique angled Quadrangle ABED in Fig. 11. be given , draw the Diagonal AE , and also the Perpendiculars DC and BF , then measuring AE upon the same Scale by which the Quadrangular Figure was protracted , suppose you find the length to be 632 , the length of DC 112 , and the length of BF 136 , if you multiply AE 632 by the Half of DC 56 , the Product will be 35392 the Area of ACED . In like manner if you multiply AE 632 , by the half of BF 68 , the Product will be 42976 the Area of ACEB , and the Sum of these two Products is the Area of ABED as was required .

Or thus , take the Sum of DC 112 , and BF 136 ; the which is 248 , and multiply AE 632 by half that Sum , that is by 124 , the Product will be 78368 the Area of the Quadrangular Figure ABED , as before .

Proposition XIII .

The Sides of a plain irregular multangled Figure being given , to find the Content .

In Fig. 26. Let the Sides of the multangled Figure . A. B. C. D. E. F. G. H. be given , and let the Area thereof be required , by Diagonals drawn from the opposite Angles reduce the Figure given , into Oblique angled plain Triangles , and those Oblique angled Triangles , into right by letting fall of Perpendiculars , then measure the Diagonals and Perpendiculars by the same Scale , by which the Figure it self was protracted , the Content of those Triangles being computed , as hath been shewed , shall be AF the Content required : thus by the Diagonals AG. BE and EC the multangled Figure propounded is converted into three Oblique angled quadrangular Figures , AFGH . AFEB and BEDC , and each of these are divided into four Right angled Triangles , whose several Contents may be thus computed . Let GA 94 be multiplied by half HL 27 more Half of KF 29 , that is by 23 , the Product will be 21 , be the Area of AHGF. Secondly , OB is 11 , and FN 13 , their half Sum 12 , by which if you multiply AE 132 , the Product will be 1584 the Area of AFEB . Thirdly , let Bp be 18 m D 32 , the half Sum is 25 , by which if you multiply AEC 125 the Product will be 3125 the Area of BEDC , and the Sum of these Products is 6871 the Area of the whole irregular Figure . ABCDEFGH , as was required .

Proposition XIV .

The Number of Degrees in the Sector of a Circle being given , to find the Area thereof .

In Fig. 27. ADEG is the Sector of a Circle , in which the Arch DEG , is Degrees . 23.50 , and by 1. Prop. of Archimed . de Dimensione Circuli , the length of half the Arch is equal to the Area of the Sector of the double Arch , there the length of DE or EG is equal to the Area of the Sector ADEG : and the length or circumference of the whole Circle whose Diameter is 1 according to Van Culen , is 3.14159265358979 , therefore the length of one Centesme of a Degree , is . 0. 01745329259. Now then to find the length of any Number of Degrees and Decimal Parts , you must multiply the aforesaid length of one Centesme by the Degrees and Parts given , and the Product shall be the length of those Degrees and Parts required , and the Area of a Sector containing twice those Degrees and Parts . Example , the half of DEG 23.50 is DE or EG 11.75 , by which if you multiply 0.01745329259 , the Product will be 2050761879325 , the length of the Arch DE , and the Area of the Sector ADEG .

Proposition XV.

The Number of Degrees in the Segment of a Circle being given , to find the Area of the Segment .

In Fig. 27. Let the Area of the Segment DEGK be required , in which let the Arch DEG be Degrees 23.50 , then is the Area of the Sector ADEG 2050761879325 by the last aforegoing , from which if you deduct the Area of the Triangle ADG , the remainer will be the Area of the Segment DEGK . And the Area of the Triangle ADG may thus be found . DK is the Sine of DE 11.75 , which being sought in Gellibrand's Decimal Canon is . 2036417511 , and AK is the Sine of DH 78.25 , or the Cosine of DE. 9790454724 , which being multiplied by the Sine of DE , the Product will be 1993745344 , or if you multiply AG

the Radius by half DF the Sine of the double Arch DEG , the Product will be 19937453445 as before , and this Product being deducted from the Area of the Sector ADEG 2050761879325 , the remainer will be 57016434875 the Area of the Segment DEGL , as was desired .

Proposition XVI .

The Diameter of a Circle being cut into any Number of Equal Parts , to find the Area of any Segment made by the Chord Line drawn at Right Angles through any of those equal Parts of the Diameter .

In Fig. 28. The Radius AD is cut into five Equal Parts , and the Segment EDFL is made by the Chord Line ELF at Right Angles to AD in the fourth Equal Part , or at eight tenths thereof : now then to find the Area of this Segment we have given AE Radius , and AL 8 , and therefore by the ninth hereof EL will be found to be 606000 , the Sine of ED 36.87 , by which if you multiply 0.0174532 , the Product is the Area of the Sector AEDF 64350286 , and the Area of the Triangle AEF is 48 , which being deducted from the Area of the Sector , the Remainer 16350286 is the Area of the Sector EDFL , as was required . And in this manner was that Table of Segments made by the Chord Lines cutting the Radius into 100 Equal Parts .

Another way .

In Fig. 28. Let the Radius AD be cut into 10.100 or 1000 Equal Parts , and let the Area of the Segments made by the Chord Lines drawn at Right Angles through all those Parts be required : first find the Ordinates GK and M. PN . EL , the double of each Ordinate , will be the Chords of the several Arches , and the Sum of these Chords beginning with the least Ordinate , will orderly give you the Area of the several Segments made by those Chord Lines , but the Diameter must be be divided into 100000 Equal Parts , because of the unequal differences at the beginning of the Diameter : but taking the Area of the Circle to be 3. 1415926535 , &c. as before , the Area of the Semicircle will be 1.57079632 , from which if you deduct the Chord GH1999999 , the Chord answering to 999 Parts of the Radius , the remainer is . 1.56879632 the Area of the Segment GDH . And in this manner by a continual deduction of the Chord Lines from the Area of the Segment of the Circle given , was made that Table shewing the Area of the Segments of a Circle to the thousandth part of the Radius .

And because a Table shewing the Area of the Segments of a Circle to the thousandth part of the Radius , whose whole Area is Unity , is yet more useful in Common Practice , therefore from this Table , was that Table also made by this Proportion .

As the Area of the Circle whose Diameter is . Unity , to wit 3.14149 is to the Area of any part of that Diameter , so is Unity the supposed Area of another Circle , to the like part of that Diameter . Example , the Area answering to 665 parts of the Radius of a Circlewhose Area is 3.14159 is 0.91354794 therefore ,

CHAP. IX .

Of the Measuring of Heights and Distances .

HAving shewed in the former Chapter , how all plain Triangles may be measured , not only in respect of their Sides and Angles , but in respect of their Area , and the finding of the Area of all other plain Figures also , that which is next to be considered , is the practical use of those Instructions , in the measuring of Board , Glass , Wainscot , Pavement , and such like , as also the measuring or surveying of Land ; and first we will shew the measuring of Heights and Distances .

2. And in the measuring of Heights and Distances , besides a Chain of 50 or 100 Links , each Link being a Foot , it is necessary to have a Quadrant of four or five Inches Radius , and the larger the Quadrant is , the more exactly may the Angles : be taken , though for ordinary Practice , four or five Inches Radius will be sufficient .

Let such a Quadrant therefore be divided in the Limb into 90 Equal Parts or Degrees , and numbred from the left hand to the right , at every tenth Degree , in this manner 10. 20. 30. 40. 50. 60. 70. 80. 90. and within the Limb of the Quadrant draw another Arch , which being divided by help of the Limb into two Equal Parts , in the Point of Interfection set the Figure 1. representing the Radius or Tangent of 45 Degrees , and from thence both ways the Tangents of 63.44 Deg. 71.57 Deg. 75.97 Deg. 78.70 Deg. 80.54 Deg. that is , 2. 3. 4. 5 and 6 being set also , your Quadrant will be fitted for the taking of Heights several ways , as shall be explained in the Propositions following .

Proposition I.

To find the Height of a Tower , Tree , or other Object at one Station .

At any convenient distance from the Foot of the Object to be measured , as suppose at C in Fig. 30. and there looking through the Sights of your Quadrant till you espie the top of the Object at A , observe what Degrees in the Limb are cut by the Thread , those Degrees from the left Side or Edge of the Quadrant to the Right , is the Quantity of the Angle ACB , which suppose 35 Degrees ; then is the Angle BAC 55 Degrees , being the Complement of the former to 90 Degrees . This done with your Chain or otherwise measure the distance from B the Foot of the Object , to your Station at C , which suppose to be 125 Foot . Then as hath been shewed in the 1. Prop. Chap. 8. draw a Line at pleasure as BC , and by your Scale of Equal Parts , set off the distance measured from B to C 125 Foot , and upon the Point C lay down your Angle taken by observation 35 Degrees , then erect a Perpendicular upon the Point B , and let it be extended till it cut the Hypothenusal Line AC , so shall AB measured on your Scale of Equal Parts , be 87.5 Foot for the Height of the Object above the Eye ; to which the Height of the Eye from the Ground being added , their Sum is the Height required .

Another way .

Let AB represent a Tower whose Altitude you would take , go so far back from it , that looking through the Sights of your Quadrant , to the top of the Tower at A the Thread may cut just 45 Degrees in the Limb , then shall the distance from the Foot of the Tower , to your Station , be the Height of the Tower above the Eye .

Or if you remove your Station nearer and nearer to the Object , till your Thread hang over the Figures 2. 3. 4 or 5 in the Quadrant , the Height of the Tower at 2. will be twice as much as the distance from the Tower to the Station , at 3. it will be thrice as much , &c. As if removing my Station from C to D , the Thread should hang over 2 in the Quadrant , and the distance BD 62 Foot , then will 124 Foot be the Height of the Tower , above the Eye .

In like manner if you remove your Station backward till your Thread fall upon one of those Figures in the Quadrant ; between 45 and 90 Degrees , the distance between the Foot of the Tower , and your Station will at 2. be twice as much as the Height , at 3. thrice as much , at 4. four times so much , and so of the rest .

A Third way by a Station at Random .

Take any Station at pleasure suppose at C , and looking through the Sights of your Quadrant , observe what Parts of the Quadrant the Thread falls upon , and then measure the distance between the Station , and the Foot of the Object , that distance being multiplied by the parts cut in the Quadrant , cutting off two Figures from the Product shall be the Height of the Object above the Eye ?

Example , Suppose I standing at C , that the Thread hangs upon 36 Degrees , as also upon 72 in the Quadrant which is the Tangent of the said Arch , and let the measured distance be CB 125 Foot , which being multiplied by 72 , the Product is 9000 , from which cutting off his Figures because the Radius is supposed to be 100 , the Height inquired will be 90 Foot , he that desires to perform this work with more exactness , must make use of the Table of Sines and Tangents Natural or Artificial , this we think sufficient for our present purpose .

Proposition II.

To find an inaccessible Height at two Stations .

Take any Station at pleasure as at D , and there looking through the Sights of your Quadrant to the top of the Object , observe what Degrees are cut by the Thread in the Limb , which admit to be 68 Degrees , then remove backward , till the Angle taken by the Quadrant , be but half so much as the former , that is 34 Degrees , then is the distance between your two Stations equal to the Hypothenusal Line at your first Station , viz. AD. if the distance between your two Stations were 326 foot , then draw a Line at pleasure as BD , upon the Point D protract , the Angle ADB 68 Degrees , according to your first Observation , and from your Line of equal parts set off the Hypothenusal 326 Foot from D to A , and from the Point A let fall the Perpendicular AB which being measured in your Scale of Equal Parts , shall be the Altitude of the Object inquired .

Or working by the Table of Sines and Tangents , the Proportion is .

As the Radius , is to the measured distance or Hypothenusal Line AD ; so is the Sine of the Angle ADE , to the height AB inquired .

Another more General way , by any two Stations taken at pleasure .

Admit the first Station to be as before at D , and the Angle by observation to be 68 Degrees , and from thence at pleasure I remove to C , where observing aim I find the Angle at C to be 32 Degrees , and the distance between the Stations 150 Foot . Draw a Line at pleasure as BC , and upon Clay down your last observed Angle 32 Degrees , and by help of your Scale of Equal Parrs , set off your measured distance from C to D 150 Foot , then upon D lay down your Angle of 68 Degrees , according to your first Observation , and where the Lines AD and AC meet , let fall the Perpendicular AB , which being measured in your Scale of Equal Parts , shall be the height of the Object as before .

Or working by the Tables of Sines and Tangents , the Proportions .

1. As the Sine of DAC to the Distance DC . So the Sine of ACD , to the Side AD.

2. As the Radius , to the Side AD ; so the Sine ADB , to the Perpendicular height AB inquired .

The taking of Distances is much after the same manner , but because there is required either some alteration in the sights of your Quadrant or some other kind of Instrument for the taking of Angles , we will particularly shew , how that may be also done several ways , in the next Chapter .

CHAP. X.

Of the taking of Distances .

FOr the taking of Distances some make use of a Semicircle , others of a whole Circle , with Ruler and Sights rather than a Quadrant , and although the matter is not much by which of these Instruments the Angles be taken , yet in all Cases the whole Circle is somewhat more ready , than either a Semicircle or Quadrant , the which with its Furniture is called the Theodolite .

2. A piece of Board or Brass then about twelve or fourteen Inches Diameter , being made Circular like a round Trencher , must be divided into four Quadrants , and each Quadrant divided into 90 Degrees , or the whole Circle into 360 , and each Degree into as many other Equal Parts , as the largeness of the Degrees will well permit : let your Circle be numbred both ways to 360 , that is from the right hand to the left , and from the left to the right .

3. Upon the backside of the Circle there must be a Socket made fast , that it may be set upon a three legged Staff , to bear it up in the Field .

4. You must also have a Ruler with Sights fixed at each end , for making of Observation , either fixed upon the Center of your Circle , or loose , as you shall think best ; your Instrument being thus made , any distance whether accessible or inaccessible may thus be taken .

5. When you are in the Field , and see any Church , Tower , or other Object , whose Distance from you , you desire to know , choose out some other Station in the same Field , from whence you may also see the Object , and measure the distance between your Stations ; then setting your Ruler upon the Diameter of your Circle , set your Instrument so , as that by the Sights on your Ruler , you may look to the other Station , this done turn your Ruler to that Object whose distance you desire to know , and observe how many Degrees of the Circle are cut by the Ruler , as suppose 36 Degrees , as the Angle ACD in Fig. 30. Then removing your Instrument to D , lay the Ruler on the Diameter thereof , and then turn the whole Instrument about till through your Sights you can espy the mark set up at your first Station at C , and there fix your Instrument , and then upon the Centre of your Circle turn your Ruler till through the Sights you can espy the Object whose distance is inquired , suppose at A ; and observe the Degrees in the Circle cut by the Ruler , which let be 112 , which is the Angle ADC , and let the distance between your two Stations be DC 326 Foot ; so have you two Angles and the side between them , in a plain Triangle given , by which to find the other sides , the which by protraction may be done as hath been shewed , in the fifth Proposition of Chapter 8. but by the Table of Sines and Tangents , the Proportion is .

As the Sine of DAC , is to DC ; so is the Sine of ACD to the Side AD.

Or , as the Sine of DAC , is to the given Side DC .

So is the Sine of ADC to the Side AC .

6. There is another Instrument called the plain Table , which is nothing else , but a piece of Board , in the fashion and bigness of an ordinary sheet of paper , with a little frame , to fasten a sheet of paper upon it , which being also set upon a Staff , you may by help of your Ruler , take a distance therewith in this manner .

Having measured the distance between your two Stations at D and C , draw upon your paper a Line , on which having set off your distance place your Instrument at your first Station C , and laying your Ruler upon the Line so drawn thereon , turn your Instrument till through the Sights you can espy the Station at D , then laying your Ruler upon the Point C , turn the same about till through the Sights you can espy the Object at A , and there draw a Line by the side of your Ruler , and remove your Instrument to D , and laying your Ruler upon the Line DC , turn the Instrument about , till through the Sight you can espy the Mark at C , and then laying your Ruler upon the Point D , turn the same , till through the Sights you can espy the Object at A , and by the side of your Ruler draw a Line , which must be extended till it meet with the Line AC , so shall the Line AD being measured upon your Scale of Equal Parts , be the distance of the Object from D , and the Line AC shall be the distance thereof from C.

7. And in this manner may the distance of two , three or more Objects be taken , from any two Stations from whence the several Objects may be seen , and that either by the plain Table , or Theodolite .

CHAP. XI .

How to take the Plot of a Field at one Station , from whence the several Angles may be seen .

ALthough there are several Instruments by which the Plat of a Field may be taken , yet do I think it sufficient to shew the use of these two , the plain Table and Theodolite .

2. In the use of either of which the same chain which is used in taking of heights and distances , is not so proper . I rather commend that which is known by the Name of Gunter's Chain , which is four Pole divided into 100 Links ; being as I conceive much better for the casting up the Content of a Piece of Ground , than any other Chain that I have yet heard of , whose easie use shall be explained in its proper place .

3. When you are therefore entered the Field with your Instrument , whether plain Table , or Theodolite , having chosen out your Station , let visible Marks be set up in all the Corners thereof , and then if you use the plain Table , make a mark upon your paper , representing your Station , and laying your Ruler to this Point , direct your Sights to the several Corners of the Field , where you have caused Marks to be set up , and draw Lines by the side of the Ruler upon the paper to the point representing your station , then measure the distance of every of these Marks from your Instrument , and by your Scale set those distances upon the Lines drawn upon the paper , making small marks at the end of every such distance , Lines drawn from Point to Point , shall give you upon your paper , the Plot of the Field , by which Plot so taken the content of the Field may easily be computed .

Example . Let Fig. 31. represent a Field whose Plot is required ; your Table being placed with a sheet of paper thereupon , make a Mark about the middle of your Table , as at A. apply your Ruler from this Mark to B and draw the Line AB , then with your Chain measure the distance thereof which suppose to be 11 Chains 36 Links , then take 11 Chains 36 Links from your Scale , and set that distance from A to B , and at B make a mark .

Then directing the Sights to C , draw a Line by the side of your Ruler as before , and measure the distance AC , which suppose to be 7 Chains and 44 Links , this distance must be taken from your Scale , and set from A to C upon your paper .

And in this manner you must direct your Sights from Mark to Mark , until you have drawn the Lines and set down the distances , between all the Angles in the Field and your station , which being done , you must draw the Lines from one Point to another , till you conclude where you first began , so will those Lines BC. CD . DE. FG. and GB , give you the exact Figure of the Field .

4. To do this by the Theodolite , in stead of drawing Lines upon your paper in the Field , you must have a little Book , in which the Pages must be divided into five Columns , in the first Column whereof you must set several Letters to signifie the several Angles in the Field , from which Lines are to be drawn to your place of standing , in the second and third Columns the degrees and parts taken by your Instrument , and the fourth and fifth , to set down your distances Chains and Links , this being in readiness , and have placed your Instrument direct your Sights to the first mark at B , and observe how many Degrees are comprehended between the Diameter of your Instrument , and the Ruler , and set them in the second and third Columns of your Book against the Letter B , which stands for your first Mark , then measure the distance AB as before , and set that down , in the fourth and fifth Columns , and so proceed from Mark to Mark , until you have taken all the Angles and Distances in the Field , which suppose to be , as they are expressed in the following Table .

Degr. Part Chains Links B 39 75 11 56 C 40 75 7 44 D 96 00 7 48 E 43 25 8 92 F 80 00 6 08 G 59 25 9 73

5. Having thus taken the Angles and Distances in the Field , to protract the same on Paper or Parchment , cannot be difficult ; for if you draw a Line at pleasure as EB representing the Diameter of your Instrument about the middle thereof , as at A , mark a Mark , and opening your Compasses to 60 Degrees in your Line of Chords , upon A as a Center describe a Circle , then lay your Field book before you seeing that your first Observattion cut no Degrees , there are no Degrees to be marked out in the Circle , but the Degrees at C are 40.75 which being taken from your Line of Chords , you must set them from H to I , and draw the Line AI. the Degrees at D are 96 which must in like manner be set from I to K , and so the rest in order .

This done observe by your Field-book the length of every Line , as the Line AB at your first Observation was 11 Chains and 36 Links , which being by your Scale set from A will give the Point B in the Paper , the second distance being set upon AI will give the Point C , and so proceeding with the rest , you will have the Points BCDEF and G , by which draw the Lines BC. CD . DE ; EF. FG and GB , and so at last you have the Figure of the Field upon your Paper , as was required .

And what is here done at one station , may be done at two or more , by measuring one or two distances from your first station , taking at every station , the Degrees and distances to as many Angles , as are visible at each station .

And as for taking the the Plot of a-Field by Intersection of Lines , he that doth but consider how the distances of several Objects may be taken at two stations , will be able to do the other also , and therefore I think it needless , to make any illustration by example .

CHAP. XII .

How to take the Plot of a Wood , Park or other Champion Plain , by going round the same , and making Observation at every Angle .

BY these Directions which have been already given , may the Plot of any Field or Fields be taken , when the Angles may be seen alone or more stations within the Field , which though it is the case of some Grounds , it is not the case of all ; now where observation of the Angles cannot be observed within , they must be observed without , and although this may be done by the plain Table , yet as I judge it may be more conveniently done by the Theodolite , in these cases thereof I chiefly commend that Instrument , I know some use a Mariners Compass , but the working with a Needle is not only troublesom , but many times uncertain , yet if a Needle be joyned with the Theodolite the joynt Observations of the Angles may serve to confirm one another .

2. Suppose the Fig. 32. to be a large Wood whose Plot you desire to take ; Having placed your Instrument at the Angle A , lay your Ruler on the Diameter thereof , turning the whole Instrument till through the Sights you espy the Angle at K , then fasten it there , and turn your Ruler upon the Center , till through the Sights you espy your second Mark at B , the Degrees cut by the Ruler do give the quantity of that Angle BAK , suppose 125 Degrees , and the Line AB 6 Chains , 45 Links , which you must note in your Field-book , as was shewed before .

3. Then remove your Instrument to B , and laying your Ruler upon the Diameter thereof , turn it about , till through the Sights you can espy your third mark at C , and there fasten your Instrument , then turn the Ruler backward till through the Sights you see the Angle at A , the Degrees cut by the Ruler being 106.25 the quantity of the Angle ABC , and the Line BC containing 8 Chains and 30 Links , which note in your Field-book , as before .

4. Remove your Instrument unto C , and laying the Ruler on the Diameter thereof , turn the Instrument about till through the Sights you see the Angle at D , and fixing of it there , turn the Ruler upon the Center till you see your last station at B , and observe the Degrees cut thereby , which suppose to be 134 Degrees , and the Line CD 6 Chains 65 Links , which must be entered into your Field-book also , and because the Angle BCD is an inward Angle , note it with the Mark for your better remembrance .

5. Remove your Instrument unto D , and Iaying the Ruler on the Diameter , turn the Instrument about , till through the Sights , you see the Angle at E , and there fixing your Instrument , turn your Ruler backward till you espy the Mark at C , where the Degrees cut are , suppose 68.0 and the Line DE 8 Chains and 23 Links .

6. Remove your Instrument unto E , and laying the Ruler on the Diameter , turn the Instrument about , till through the Sights you see the Angle at F , and there fix it , then turn the Ruler backward till you see the Angle at D , where the Degrees cut by the Ruler suppose to be 125 and the Line EF 7 Chains and 45 Links .

7. Remove your Instrument unto F , and laying your Ruler upon the Diameter , turn the Instrument about , till through the Sights , you see the Angle at G , where fix the same , and turn the Ruler backward till you see the Angle at E , where the Degrees cut by the Ruler are 70 , and the Line FG 4 Chains 15 Links , which must be set down with this or the like Mark at the Angle .

8. Remove your Instrument unto G , and laying your Ruler upon the Diameter , turn the Instrument about , till through the Sights you see the Angle at H , where fix the same , and turn the Ruler backward till you see the Angle at F , where the Degrees cut by the Ruler are 65.25 , and the Line GH 5 Chains 50 Links .

9. Remove your Instrument in like manner to H and K , and take thereby the Angles and Distances as before , and having thus made observation at every Angle in the Field , set them down in your Field-book , as was before directed , the which in our present Example will be as followeth .

A 151.00 6.45 B 106.25 8.30 C 134.00 6.65 D 68.00 8.23 E 125.00 7.45 F 70.25 4.15 G 65.25 5.50 H 130.00 6.50 K 140.00 11.00

The taking of the inward Angles BCD and EFG was more for Conformity sake than any necessity , you might have removed your Instrument from B to D , from E to G , the Length of the Lines BC. CD . EF and G , would have given by protraction the Plot of the Field without taking these Angles by observation ; many other compendious ways of working there are , which I shall leave to the discretion of the Ingenious Practitioner .

10. The Angles and Sides of the Field being thus taken , to lay down the same upon Paper , Parchment , another Instrument called a Protractor is convenient , the which is so well known to Instrument-makers , that I shall not need here to describe it , the chief use is to lay down Angles , and is much more ready for that purpose than a Line of Chords , though in effect it be the same .

11. Having then this Instrument in a readiness draw upon your Paper or Parchment upon which you mean to lay down the Plot of that Field , a Line at pleasure as AB . Then place the Center of your Protractor upon the Point A , and because the Angle of your first observation at A was 115 Degrees 00 Parts , turn your Protractor about till the Line AK lie directly under the 115 Degree ; and then at the beginning of your Protractor make a Mark , ând draw the Line AB , setting off 6 Chains 45 Links from A to B.

12. Then lay the Center of your Protractor upon the Point B , and here turn your Protractor about , till the line AB lie under 106 Degrees 25 Parts , and draw the Line BC , setting off the Distance 8 Chains , 30 Links from B to C.

13. Then lay the Center of your Protractor upon the Point C , and turn the same about till the Line BC lie under 134 Degrees , but remember to make it an inward Angle , as it is marked in your Field-Book , and there make a Mark , and draw the Line CD , setting off 6 Chains , 65 Links from C to D.

And thus must you do with the rest of the Sides and Angles , till you come to protract your last Angle at H , which being laid down according to the former Directions the Line HK will cut the Line AK making AK 11 Chains and HK 6 Chains , 50 Links . This work may be also performed by protracting your last observation first ; for having drawn the Line AK , you may lay the Center of your Protractor upon the Point K , and the Diameter upon the Line AK ; and because your Angle at K by observation was 140 Degrees , you must make a Mark by the Side of your Protractor at 140 Degrees ; and draw the Line KH , setting off 6 Chains , 50 Links from K to H. And thus proceeding with the rest of the Lines and Angles , you shall find the Plot of your Field at last to close at A , as before it did at K.

CHAP. XIII .

The Plot of the Field being taken by any Instrument , how to compute the Content thereof in Acres , Roods , and Perches .

THe measuring of many sided plain Figures hath been already shewed in the 13 Proposition of the 8 Chapter , which being but well considered , to compute the Content of a Field cannot be difficult ; It must be remembred indeed that 40 square Pearches do make an Acre .

2. Now then if the Plot be taken by a four Pole Chain divided into 100 Links , as 16 square Poles are the tenth part of an Acre ; so 10.000 square Links of such a Chain are equal to 16 square Pole , or Perches ; and by consequence 100.000 square Links are equal to an Acre , or the square Pearches .

3. Having then converted your Plot into Triangles , you must cast up the Content of each Triangle as hath been shewed , and then add the several Contents into one Sum , and from the aggregate cut off five Figures towards the right hand ; the remainer of the Figures towards the left hand are Acres , and the five Figures so cut off towards the right hand are parts of an Acre , which being multiplied by four , if you cut off five Figures from the Product , the Figures remaining towards the left hand are Roods , and the five Figures cut off are the parts of a Rood , which being multiplied by forty , if you cut off five Figures from the Product , the Figures remaining towards the left hand are Perches , and the Figures cut off are the Parts of a Pearch .

Example . Let 258.94726 be the Sum of several Triangles , or the Content of a Field ready cast up , the three Figures towards the left hand 258 are the Acres , and the other Figures towards the right hand 94726 are the Decimal Parts of an Acre , which being multiplied by 4 , the Product is 3.78904 , that is three Roods and 78904 Decimal Parts of a Rood , which being multiplied by 40 , the product is 31.56160 , that is 31 Perches and 56160 Decimal Parts of a Perch ; and therefore in such a Field there are Acres 258 , Roods 3 , Pearches 31 , and 56160 Decimal Parts of a Perch .

CHAP. XIV .

How to take the Plot of Mountainous and uneven Grounds , and how to find the Content .

WHen you are to take the Plot of any Mountainous or uneven piece of Ground , such as is that in Figure 33 , you must first place your Instrument at A , and direct your Sights to B , measuring the Line AB , observing the Angle GAB , as was shewed before , and so proceed from B to C , and because there is an ascent from C to D , you must measure the true length thereof with your Chain , and set that down in your Book , but your Plot must he drawn according to the length of the Horizontal Line , which must be taken by computing the Base of a right angled Plain Triangle , as hath been shewed before , and so proceed from Angle to Angle until you have gone round the Field , and having drawn the Figure thereof upon your Paper , reduce into Triangles and Trapezias , as ABC . CDE . ACEF and AFG . then from the Angles B. C. D. F and G ; let fall the Perpendiculars , BK . CN . DL . FM . and GH . This done you must measure the Field again from Angle to Angle , setting down the Distance taken in a straight Line over Hill and Dale , and so likewise the several Perpendiculars , which will be much longer than the streight Lines measured on your Scale , and by these Lines thus measured with your Chain cast up the Content ; which will be much more than the Horizontal Content of that Field according to the Plot , but if it should be otherwise plotted than by the Horizontal Lines , the Figure thereof could not be contained within its proper limits , but being laid down among other Grounds , would force some of them out of their places , and therefore such Fields as these must be shadowed off with Hills , if it be but to shew that the Content thereof is computed according to the true length of the Lines from Corner to Corner , and not according to their Distance measured by Scale in the Plot.

CHAP. XV.

How to reduce Statute Measure into Customary , and the contrary .

WHereas an Acre of Ground by Statute Measure is to contain 160 square Perches , measured by the Pole or Perch of sixteen foot and a half : In many places of this Nation , the Pole or Perch doth by custom contain 18 foot , in some 20. 24. 28 Foot ; it will be therefore required to give the Content of a Field according to such several quantities of the Pole or Perch .

2. To do this you must consider how many square Feet there is in a Pole according to these several Quantities .

In 16.5 to the Pole , there are 272.25 sq. feet . In 18 to the Pole there are 324 square feet . In 20 to the Pole there are 400 square feet . In 24 to the Pole there are 576 square feet . In 28 to the Pole there are 784 square feet .

Now then if it were desired to reduce 7 Acres , 3 Roods , 27 Perches , according to Statute Measure , into Perches of 18 Foot to the Perch ; first reduce your given quantity , 7 Acres . 3 Rods , 27 Poles into Perches , and they make 1267 Perches .

Then say , as 324. to 272. 25. so is 1267 to 1065. 6. that is 1065 Perches , and 6 tenths of a Perch . But to reduce customary Measure into statute measure , say as 272. 25. is to 324 so is 1267 Perches in customary measure , to 1507. 8 that is 1507 Perches and 8 tenths of a Perch in statute measure , the like may be done , with the customary measures of 20.24 and 28 or any other measure that shall be propounded .

CHAP. XVI .

Of the Measuring of solid Bodies .

HAving shewed how the content of all plains may be computed , we are now come to the measuring of solid Bodies , as Prisms , Pyramids and Spheres , the which shall be explained in the Propositions following .

Proposition . I.

The base of a Prism or Cylinder being given , to find the solid content .

The base of a Prism is either Triangular , as the Pentahedron ; Quadrangular , as the Hexahedron , or Multangular , or the Polyhedron Prism , all which must be computed as hath been shewed , which done if you multiply the base given by the altitude , the product shall be the solid content required .

Example . In an Hexahedron Prism , whose base is quadrangular , one side of the Base being 65 foot and the other 43 , the Superficies or Base will be 27. 95. Which being multiplyed by the Altitude , suppose 12. 5. the product . 359. 375. is the solid content required .

In like manner the Base of a Cylinder being 45. 6. and the altitude 15. 4. the content will be 702. 24.

And in this manner may Timber be measured whether round or squared , be the sides of the squared Timber equal or unequal .

Example . Let the Diameter of a round piece of Timber be 2. 75 foot . Then , As 1 it to 785397. so is the square of the Diameter 2. 75. to 5.9395 the Superficial content of that Circle .

Or if the circumference had been given 8. 64. then , As 1 is to 079578 , so is the square of 8. 64. to 5.9404 the superficial content .

Now then if you multiply this Base 5. 94. by the length , suppose 21 foot , the content will be 124. 74.

If the side of a piece of Timber perfectly square be 1.15 this side being multiplyed by it self , the product will be 1.3225 the superficial content , or content of the Base , which being multiplyed by 21 the length , the content will be 27. 7745.

Or if a piece of Timber were in breadth 1. 15. in depth 1.5 the content of the Base would be 1.725 which being multiplied by 21 the length , the content will be . 36. 225.

Proposition . II.

The Base and Altitude of a Pyramid or Cone being given , to find the solid content .

Multiply the Altitude by a third part of the Base , or the whole Base by a third part of the Altitude , the Product shall be the solid content required .

Example . In a Pyramid having a Quadrangular Base as in Fig. 22. The side CF 17. CD 9. 5. the Product is the Base CDEF . 161. 5 , which being multiplyed by 10.5 the third of the Altitude AB 31.5 the Product is 1695.75 the content . Or the third of the Base . viz. 53. & 3 being multiplied by the whole Altitude AB 31.5 the Product will be the content as before .

2. Example . In Fig. 21. Let there be given the Diameter of the Cone AB 3. 5. The Base will be 96. 25. whose Altitude let be CD 16.92 the third part thereof is 5.64 & 96.25 being multiplied by 5.64 , the Product 542.85 is the solid content required .

Proposition . III.

The Axis of a Sphere being given , to find the solid content .

If you multiply the Cube of the Axis given by 523598 the solid content of a Sphere whose Axis is an unite , the Product shall be the solid content required .

Example . Let the Axis given be ● , the Cube thereof is 27 , by which if you multiply . 523598 , the Product 14.137166 is the solid content required .

Proposition . IV.

The Basis and Altitude of the Frustum of a Pyramid or Cone being given , to find the content .

If the aggregate of both the Bases of the Frustan and the mean proportional betwe●n them , shall be multiplied by the third part of the Altitude , the Product shall be the solid content of the Frustum .

Example . In Fig. 22. Let CDEF represent the greater Base of a Pyramid , whose superficial content let be 1. 92 , and let the lesser Base be HGLKO . 85 the mean proportional between them is . 1. 2775 and the aggregate of these three numbers is . 4. 0475. Let the given Altitude be 15. the third part thereof is . 5 by which if you multiply 4.0475 the Product 20. 2375 is the content of the Frustum Pyramid .

And to find the content of the Frustum Cone . I say .

As. 1. ro 78539. so 20.23 to 15. 884397 , the content of the Cone required .

But if the Bases of the Frustum Pyramid shall be square , you may find the content in this manner .

Multiply each Diameter by it self and by one another , and the aggregate of these Products , by the third part of the altitude , the last Product shall be the content of the Frustum Pyramid .

Example . Let the Diameter of the greater Base be 144 , the Diameter of the lesser Base 108 , and the altitude 60.

20736 11664 15552 47952

Which being multiplyed by 20 the third part of the Altitude , the Product 959040 is the content of the Frustum Pyramid .

And this content being multiplied by .785 39 the content of the Frustum Cone will be .753 .228 .

Another way .

Find the content of the whole Pyramid of the greater and lesser Diameter , the lesser content deducted from the greater , the remain shall be the content of the Frustum . To find the content of the whole Pyramid , you must first find their several Altitudes in this manner .

As the difference between the Diameters ,

Is to the lesser Diameter .

So is the Altitude given , to the Altitude cut off .

Example . The difference between the former Diameter . 144. and 108 is 36 , the Altitude 60. now then As 36. 108 ∷ 60. 108. the altitude cut off .

Now then if you mnltiply the lesser Base 1 1664 by 60 the third part of 180 the Product 699840 is the content of that Pyramid .

And adding 60 to 180 the Altitude of the greater Pyramid is 240 , the third part whereof is 80 , by which if you multiply the greater Base before found , 70736 , the Product is the content of the greater Pyramid . 1658880 , from which if you deduct the lesser 699840 the remainer 959040 is the content of the Frustum Pyramid as before .

And upon these grounds may the content of Taper Timber , whether round or square , and of Brewers Tuns , whether Circular or Elliptical , be computed , as by the following Propositions shall be explained .

Proposition . V.

The breadth and depth of a Taper piece of Squared Timber , both ends being given together with the length , to find the content .

Let the given Dimensions . At the Bottom be A. 5.75 and B 2.34 At the Top. C. 2.16 and D. 1.83 . And let the given length be 24 Foot .

According to the last Proposition , find the Area or Superficial content of the Tree at both ends thus .

Multiply the breadth 3.75 0.574031 By the depth 2.34 0.369215 The Product 8.7750 0.943246 2. Multiply the breadth 2.16 0.334453 By the depth 1.82 0.262451 The Product is 3.9528 0.596904 3. Multiply the 1. Content . 87750 0.943246 by the second content . 3.9528 0.596904 And find the square root 5.8986 1.540150 0.770075

The Sum of these 18.6264 being multiplyed by 8 one third of the length , the content will be found to be 149. 0112. Thus by the Table of Logarithms the mean proportional between the two Bases is easily found , and without extracting the square Root , may by natural Arithmetick be found thus .

A 4 2/2 CX A half C multiplyed by B : And C more half A multiplyed by D being added together and multiplyed by 30 , the length shall give the content . Example .

A. 3.75 C 2.16 1/2 C. 1.08 1/2 A 1.875 Sum 4.83 Sum. 4.035 B - 2.34 D. 1.83 1932 12105 1449 32280 966 4035 11.3022 7.38405 11.30220 The sum of the Products 18.68625

Being multiplyed by 8 the third of the length , the content will be . 149. 49000. The like may be done for any other .

Proposition VI.

The Diameters of a piece of Timber being given at the Top and and Bottom , together with the length , to find the content .

The Proposition may be resolved either by the Squares of the Diameters , or by the Areas of the Circles answering to the Diameters given , for which purpose I have here annexed not only a Table of the Squares of all numbers under a thousand , but a Table sharing the third part of the Areas of Circles in full measure , to any Diameter given under 3 foot .

And therefore putting S = The Sum of the Tabular numbers answering to the Diameters at each end .

X = The difference between these Diameters .

L = the length of the Timber , C = The content .

Then 1 ½ S = ½ - XX. + L. = C.

If you work by the Table of the squares of Numbers . you must multiply the less side of the Equation , by 0.26179 the third part of 0.78539 the Product being multiplyed by the length , will give the content .

But if you work by the Table of the third parts of the Areas of Circles in full measure , the tabular Numbers being multiplyed by the length will give the content . Only instead of the square of the difference of the Diameter , you must take half the Tabular number answering to that Difference , and you shall have the content as before . Example .

Their difference is 0.83 7.5625 3.7249 . 11.2874 5.6437 16.9311 0.3362 16.5949 26179 1493541 1161643 165949 995694 331898 4.344378871 Or by the Table of Areas . 1.979857 0.975176 2.955033 1.477516 4.432549 0.088016 4.344533 24 17378132 8689066 104268792

But because that in measuring of round Timber the circumference is usually given and not the Diameter , I have added another Table by which the circumference being given , the Diameter may be found .

Example . Let the circumference of a piece of Timber be 8325220 looking this Number in the second column of that Table , I find the next less to be 8.168140 and thence proceeding in a streight Line , I find that in the seventh Column the Number given , and the Diameter answering thereunto to be 2. 65. and thus may any other Diameter be found not exceeding the three foot . The Proportion by which the Table was made , is thus . As 1. to 3.14159 so is the Diameter given , to the circumference required .

Or the Circumference being given , to find the Diameter , say : As. 1. to 0.3183 , so is the Circumference given to the Diameter required .

And although by these two Tables all round Timber may be easily measured , yet it being more usual to take the Circumference of a Tree , then the Diameter , I have here added a third Table , shewing the third part of the Areas of Circles answering to any circumference under 10 foot , and that in Natural and Artificial numbers , the use of which Table shall be explained in the Proposition following .

Proposition . VII .

The Circumference of a piece of round Timber at both ends , with the length being given , to find the content .

The Circumference of a Circle being given , the Area thereof may be found as hath been shewed , in the 7 Chapter , Proposition 4. and by the first Proposition of this ; and to find the third part of the Area , which is more convenient for our purpose I took a third part of the number given by which to find the whole , that is a third part of 07957747 that is 0.02652582 and having by the multiplying this number by the square of the Circumference computed three or four of the first numbers , the rest were found by the first and second differences .

The Artificial numbers were computed by adding the Logarithms of the Squares of the circumference , to 8.42966891 the Logarithm of 0. 02652582.

And by these Natural and Artificial numbers the content of round Timber may be found two ways

By the Natural numbers in the same manner as the content was computed , the Diameters being given , and by the Natural and Artificial numbers both , by finding a mean proportional between the two Areas at the top and bottom of the Tree , as by Example shall be explained .

Let the given Diamensions , or Circumferences be At the Bottom 9.95 Their difference is 6.20 At the Top 3.75

ThetabularNumbers. Natural Artificial . Answering to 9.95 2.626162 0.418931 And to 3.75 0.373019 9.571731

9.990662 9.995331 3.988481 2 . 9●9181 ● . 499190 4.498771 0.509826 3.988945

Which being multiplyed by the length 24 , the content will be 95. 73468.

Mr. Darling in his Carpenters Rule made easie , doth propound a shorter way , but not so exact , which is by the Circumference given in the middle of the piece to find the side of the Square , namely by multiplying the Circumference given by 28209 , or 2821. which side of the Square being computed in Inches , and lookt in his Table of Timber measure , doth give the content of the Tree not exceeding 31 foot in length , the which way of measuring may be as easily performed by this Table . Example .

The circumference at the top and bottom of the Tree being given 9.95 and 3.75 the Sum is 13.70 The half thereof is the mean circumfer . 6.85 Which sought in the Table , the Numbers are .

The Natural number is 1.244657 , which being multiplyed by 3 the Product is 3.733971 , which multiplyed by the length 24 , the content is 89. 615304.

0.095049 1.380211 1.475260 89613
Proposition . VIII .

The Diameters of a Brewers Tun at top and bottom being given with the height thereof , to find the content .

In Fig. 29. Let the given Diameter .

At the top be AC 136 BD 128

At the bottom . KG 152 HF 144 Altit . 51 Inches .

The which by the 5 Proposition of this Chap. may thus be computed . AC 139 + ½ KG 76 = 212 × BD 128 the Product is 27136.

And KG 1524 ½ AC 68 = 220 × HF 144 the Product is 31680. the Sum of these 2 Products is 58816 which being multiplyed by onethird of 51 , that is by 17 , and that Product multiplyed by 26179 the third of 78539 will give the content .

54.76949 1.230449 1.999944 9.417968 5.417912

Thus the content of a Tun may be found in Inches , which being divided 282 the number of Inches in an Ale Gallon , the quotient will be the content in Gallons .

Or thus ; divide the former . 26179 by 282 the quotient will be 00092836. by which the content may be found in Ale Gallons in this manner .

5.999944 6.967719 2.967663
Proposition . IX .

The Diameters of a close Cask , at head and bung with the length given , to find the content .

In the resolving of this Proposition , we are to consider the several forms of Casks , as will as the kind of the Liquor , with which it is filled , for one and the same Rule will not find the content in all Cask .

And a Coopers Cask is commonly taken , either for the middle Frustum of a Spheroid , the middle Frustum of a Parabolical Spindle , the middle Frustum of two Parabolick Conoids , or for the middle Frustum of two Cones abutting upon one common Base .

And the content of these several Casks may be found either by equating the Diameters , or by equating the Circles . for the one , a Table of Squares is necessary , and a Table shewing the third part of the Areas of a Circle to all Diameters . The making of the Table of Squares , every one knows , to be nothing else but the Product of a Number multiplyed , by it self , thus the Square of 3 is 9. the Square of 8 is 64 and so of the rest .

And the Area of a Circle to any given Diameter may be found , as hath been shewed , in Chap. 7 Proposition 2. But here the Area of a Circle in Inches , will not suffice , it will be more fit for use , if the third part of the Area be found in Ale and Wine Gallons both , the which may indeed be done by dividing the whole Area in Inches by 3 and the quotient by 282 to make the Table for Ale-measure , and by 231 to make the Table for Wine-measure ; but yet these Tables ( as I think ) may be more readily made in this manner .

The Square of any Diameter in Inches , being divided by 3.81972 will give the Area of the Circle in Inches : And this Division being multiplyed by 282 will give you 1077.161 for a common Division , by which to find the Area in Ale-Gallons , or being multiplyed by 231 the Product , 882.355 will be a commou Division by which to find the Area in Wine-Gallons .

But because it is easier to multiply then divide : If you multiply the several Squares by 26178 the third part of 78539 the Product will give the Area in Inches , or if you divide . 26179 by 282 the quotient will be . 00092886 for a common Multiplicator , by which to find the Area in Ale-Gallons , or being divided by 231 the quotient will be 0011333 a common Multiplicator , by which to find the content in Wine-Gallons . An Example or two will be sufficient for illustration . Let the Diameter given be 32 Inches , the Square thereof 1024 being divided by 3.81970 the quotient is 268.083 , and the same Square 1024 being multiplyed by 261799 , the Product will be 268. 082.

Again if you divide 1024 by 1077.161 the quotient will be 9508 , or being multiplied by 00092836 , the Product will be 9508.

Lastly if you divide 1024 by 882.755 , the quotient will be 1.1605 , or being multiplied by 00113333 the Product is 1.1605 ,

And in this manner may the Tables be made for Wine and Beer-measure , but the second differences in these Numbers being equal , three or four Numbers in each Table being thus computed , the rest may be found by Addition only .

Thus the Squares of 1. 2. 3. and 4 Inches are . 1. 4. 9 and 16 by which if you multiply 00113333 , the several Products will be third part of the Area , of the Circles answering to those Diameters in Wine-Gallons . Or 00092836 being multiplied by those Squares , the several Products , will be the third part of the Areas of the Circles answering to those Diameters in Ale-Gallons ; the which with their first and second differences are as followeth .

TheProductsorAreasinWine-Gallons: 1. 00113333 2. 00453332 33999 226666 3. 01019997 566665 226666 4. 01813328 796331

TheProductsinAle-Gallons. 1. 00092836 2. 00371344 278508 185672 3. 00835524 464180 185672 4. 01485376 649852

And by the continual addition of the second differences to the first , and the first differences to the products before found , the Table may be continued as far as you please .

The construction of the Tables being thus shewed : We will now shew their use in finding the content of any Cask .

Let S = the Sum of the Tabular Numbers answering to the Diameters at the Head and Bung. D = their difference X = the difference of the Diameters themselves . L = the length of the Vessel , and C = the content thereof .

1. If a Cask be taken for the middle Frustum of a Spheroid , intercepted between two Planes parallel , cutting the Axis at right Angles : Then 1 ½ S + ½ D × L = C.

2. If a Cask be taken for the middle Frustum of a parabolical Spindle , intercepted between two planes parallel cutting the Axis at right Angles . Then 1 ½ S + ½ D × L = C.

3. If a Cask be taken for the middle Frustum of two Parabolick Conoids , abutting upon one common Base , intercepted between two Planes parallel , cutting the Axis at right Angle : Then 1 ½ S : × L = C.

4. If a Cask be taken for the middle Frustum of two Cones , abutting upon one common bafe , intercepted between two Planes parallel cutting the Axis at Right Angles . Then 1 ½ S — ⅓ XX. × L = C.

In all these four Equations , if you work by the Table of Squares of numbers , you must multiply the less side of the Equation by 262 , if you would have the content in Cubical Inches ; by 001133 if you would have the content in Wine-Gallons ; and by 000928 , if you would have the content in Ale-Gallons .

But if you work by the Tables of the third parts of the Areas Circle , the Tabular Numbers being multiplyed by the length only will give the content required , only in the fourth Equation instead of half the Square of the Difference of the Diameters , take half the Tabular Number answering to that difference , and you shall have the content required ; as by the following Examples will better appear , then by many words .

Examples in Wine-measure by the Table of the Squares of Numbers .

The Diameter of a Vessel

At the Bung being 32 Inches .

At the Head 22 Inches .

The difference of the Diameters 10 Inches .

And the length of the Vessel 44 Inches .

Spheroid . Parabolick Spindle . 1024 1024 484 484 1508 1508 754 754 270 540 2532 23160 2532 23160 7596 69480 7596 69480 7596 69480 28695156 262472280 44 44 114780624 104988912 114780624 104988912 126.2586864 115.4878032

Parabolick Conoid Cone . 1024 1024 484 484 1508 1508 754 754 50 2262 2212 2262 2212 6786 6636 6786 6636 6786 6636 25635246 25068596 44 44 102540984 200274384 102540984 100274384 112.79508241 110.30182224

This which hath been done by the Table of Squares may be more easily performed , by the Table of the third part of the Areas of Circles , ready reduced to Wine-Gallons .

Spheroid Parabolick Spindle . 1.16053 1.16053 0.54853 0.54853 1.70906 1.70906 85453 85453 30600 61200 2.86959 2.624790 44 44 1147836 1049916 1147836 10499160 126.26196 115.490760

Parabolick Conoid Cone . 1.16053 1.16053 0.54853 0.54853 1.70906 1.70906 85453 85453 56666 2.56359 2.506924 44 44 1025436 10027696 1025436 10027696 112.79796 110.304656

Examples in Ale-measure by the Table of the Squares of Numbers .

Spheroid . Parabolick Spindle . 1024 1024 484 484 1508 1508 754 754 270 540 2532 2316.0 00092836 00092836 22758 20844 5064 4632 20256 18528 7596 6948 15192 138960 235660752 2.150081760 44 44 948623008 860032704 940643008 860032704 103.22673088 94.60359744

Parabolick Conoid Cone . 1024 1024 484 484 1508 1508 754 754 50 2262 2212 20358 19909 4524 4424 18096 17696 6786 6636 13527 13272 2.09995032 2.05423232 44 44 8.39980128 821692928 839980128 821692928 92.39781408 90.38622208

BytheAreasofCircles. Spheroid . Parabolick Spindle . 0.95052 0.95052 0.44930 0.44930 1.39982 1.39982 .69991 69991 .25061 050122 2.35034 2.149852 34 44 940136 8599408 940136 8599408 103.41496 94.593488

Parabolick Conoid . Cone . 0.95052 0.95052 0.44930 0.44930 1.39982 1.39982 69991 .69991 46425 209973 2.053305 44 44 839892 8213220 839892 8213220 90.345420 90.345420

And here for the Singularity of the Example , I will set the Dimensions of a Cask lately made in Herefordshire , for that excellent Liquor of Red streak Cyder , the like whereof either for the largeness of the Cask , or incomparable goodness of that kind of Drink , is not to be found in all England , nay and perhaps not in the World .

The length of the Cask is 104 Inches .

The Diameter at the Bung 92 Inches .

And the Diameter at the Head 74 Inches .

The Numbers in the Table of Ale-Gallons answering to these Dimensions are .

Spheroid Parabolick Spindle . Bung. 92 7.859639 7.859639 Head . 74 5.083699 5.083699 12.941338 12.941338 6.470669 6.470669 1.386770 .277394 20.798777 19.689401 104 104 83195108 78.757604 20798777 19689401 Con. 2163.072808 2047.697704

Parabolick Conoid . Cone . 7.857639 7.857639 5.083699 5.083699 12.941338 12.941338 6.470669 6.470669 0.150394 19.412007 19.261613 104 104 77648028 77046452 19412007 19261613 201. 8. 848728 2003.207752

And thus you have the content of this Cask by four several Ways of Gauging , but that which doth best agree with the true content , found by these that filled the same is the second way or that which takes a Cask to be the middle Frustum of a Parabolick Spindle , according to which the content is 2047 Gallons . That is allowing 64 Gallons to the Hogshead . 32 Hogsheads very near .

Proposition . X.

If a Cask be not full , to find the quantity of Liquor contained in it , the Axis being posited parallel to the Horizon .

To resolve this Proposition , there must be given the whole content of the Cask , the Diameter at the Bung , and the wet Portion thereof , then by help of the Table of Segments , whose Area is unity , and the Diameter divided into 10.000 equal parts , the content may thus be found .

As the whole Diameter , is to its wet Portion .

So is the Diameter in the Table . 10.000 to its like Portion , which being sought in the Table of Segments , gives you a Segment , by which if you multiply the whole content of the Cask , the Product is the content of the Liquor remaining in the Cask .

But in the Table of Segments in this Book , you have the Area , to the equal parts of one half of the Diameter only , when the Cask therefore is more then half full , you must make use of the dry part of the Diameter instead of the wet , so shall you find what quantity of Liquor is wanting to fill up the Cask , which being deducted from the whole content of the Cask ; the remainer is the quantity of Liquor yet remaining , an Example in each will be sufficient , to explane the use of this Table .

1. Example , In a Wine Cask not half full , let the great Diameter be as before 32 Inches , the content 126.25 Gallons , and let the wet part of the Diameter be 12 Inches , First I say .

As the whole Diameter 32. is to the wet part 12. so is 10.000 to 3750 , which being sought in the Table , I find , the Area of that Segment to be . 342518 which being multiplyed by the whole content of the Cask 126.25 , the Product is 43.24289750 and therefore there is remaining in the Cask 43 & 1/4 ferè .

2. Example . In the same Cask let the wet part of the Diameter be 18 Inches . I say .

As 32.18 : : 10000.5625 whose Complement to 10000 is 4375 which being sought in the Table , I find the Area answering thereto to be 420630 ; now then I say .

As the whole Area of the Circle 1000000 is to the whole content of the Cask 126. 25.

So is the Area of the Segment sought . 420630 , to the content 53.1044375 which is in this case the content of the Liquor that is wanting , this therefore being deducted from the content of the whole Cask , 136. 25. the part remaining in the Vessel is . 73. 1455625.

Thus may Casks be gauged in whole or in part , in which a Table of Squares is sometimes necessary , as being the Foundation , from whom the other Tables are deduced ; such a Table therefore is here exhibited , for all Numbers under 1000 , by help whereof the Square of any Number under 10.000 may easily be found in this manner .

The Rectangle made of the Sum and Difference of any two Numbers , is equal to the Difference of the Squares of these Numbers .

Example , Let the given Numbers be 36 and 85 their Sum is 121 , their difference 49 , by which if you multiply 121 , the Product will be 5929. The Square of 36 is 1296 , and the Square of 85 is 7225 , the difference between which Squares is 5929 as before .

And hence the Square of any Number under 10.000 may thus be found , the Squares of all Numbers under 1000 being given .

Example . Let the Square of 5715 be required . The Square of 571 by the Table is 326041 , therefore the Square of 5710 is 32604100 : the Sum of 5710 and 5715 is 11425 , and the difference 5 , by which if you multiple 11425 , the Product is 52125 which being added unto 32604100 the Sum 32656325 is the Square of 5715. The like may be done for any other .

TABLES FOR THE Measuring OF TIMBER , AND THE GAUGING OF CASKS AND Brewers Tuns .

LONDON , Printed for Thomas Passinger at the three Bibles on London-Bridge . 1679.

ATableofSquares. 1 1 3 2 4 3 09 5 4 16 7 5 25 9 6 36 11 7 49 13 8 64 15 9 81 17 10 100 19 11 121 21 12 144 23 13 160 25 14 196 27 15 225 29 16 256 31 17 287 33 18 324 35 19 361 37 20 400 39 21 441 41 22 484 43 23 529 45 24 576 47 25 625 49 26 676 51 27 729 53 28 784 55 29 841 57 30 900 59 31 961 61 32 1024 63 33 1089 65 34 1156 67 34 1156 69 35 1225 71 36 1206 73 37 1369 75 38 1444 77 39 1521 79 40 1600 81 41 1681 83 42 1764 85 43 1841 87 44 1936 89 45 2025 91 46 2116 93 47 2209 95 48 2304 97 49 2401 99 50 2500 101 51 2601 103 52 2704 105 53 2809 107 54 2916 109 55 3025 111 56 3136 113 57 3249 115 58 3364 117 59 3481 119 60 3600 121 61 3721 123 62 3844 125 63 3969 127 64 4096 129 65 4225 131 66 4356 133 67 4489 135 67 4489 135 68 4624 137 60 4761 139 70 4900 141 71 5041 143 72 5184 145 73 5329 147 74 5476 149 75 5625 151 76 5776 153 77 5929 155 78 6084 157 79 6241 159 80 6400 161 81 6561 163 82 6724 165 83 6889 167 84 7056 169 05 7225 171 06 7396 173 87 7559 175 88 7744 177 89 7921 179 90 8100 181 91 8281 183 92 8464 185 93 8649 187 94 8836 189 95 9025 191 96 9216 193 97 9409 195 98 9604 197 99 9801 199 100 10000 201 101 10201 203 102 10404 205 103 10609 207 104 10816 209 105 11025 211 106 11236 213 107 11449 215 108 11664 217 109 11881 219 110 12100 221 111 12321 223 112 12544 225 113 12769 227 114 12996 229 115 13225 231 116 13456 233 117 13689 235 118 13924 237 119 14161 239 120 14400 241 121 14641 243 122 14884 245 123 15129 247 124 15376 249 125 15625 251 126 15876 253 127 16129 255 128 16384 257 129 16641 259 130 16900 261 131 17161 263 132 17424 265 133 17689 267 134 17956 269 134 17956 269 135 18225 271 136 18496 273 137 18769 275 138 19044 277 139 19321 279 140 19600 281 141 19881 283 142 20164 285 143 20449 287 144 20736 289 145 21025 291 146 21316 293 147 21609 295 148 21904 297 149 22201 299 150 22500 301 151 22801 303 152 23104 305 153 23409 307 154 23716 309 155 24025 311 156 24336 313 157 24649 315 158 24964 317 159 25281 319 160 25600 321 161 25921 323 162 26244 325 163 26569 327 164 26896 329 165 27225 331 166 27556 333 167 27889 335 167 27889 335 168 28224 337 169 28561 339 170 28900 341 171 29241 343 172 29584 345 173 29929 347 174 30276 349 175 30625 351 176 30976 353 177 31329 355 178 31684 357 179 32041 359 180 32400 361 181 32761 363 182 33124 365 183 33489 367 184 33856 369 185 34225 371 186 34596 373 187 34969 375 188 35344 377 189 35721 379 190 36100 381 191 36481 383 192 36864 385 193 37249 387 194 37636 389 195 38025 391 196 38416 393 197 38809 395 198 39204 397 199 39601 399 200 40000 401 201 40401 403 202 40804 405 203 41209 407 204 41616 409 205 42025 411 206 42436 413 207 42849 415 208 43264 417 209 43681 419 210 44100 421 211 44521 423 212 44944 425 213 45369 427 214 45796 429 215 46255 431 216 46656 433 217 47089 435 218 47524 437 219 47961 439 220 48400 441 221 48841 443 222 49284 445 223 49729 447 224 50176 449 225 50625 451 226 51076 453 227 51529 455 228 51984 457 229 52441 459 230 52900 461 231 53361 463 232 53824 465 233 54289 467 234 54756 469 234 54756 469 235 55225 471 236 55696 473 237 56169 475 238 56644 477 239 57121 479 240 57600 481 241 58081 483 242 58564 485 143 59049 487 244 59536 489 245 60025 491 246 60516 493 247 61009 495 248 61504 497 249 62001 499 250 62500 501 251 63001 503 252 63504 505 253 64009 507 254 64516 509 255 65025 511 256 65536 513 257 66049 515 258 66564 517 259 67071 519 260 67600 621 261 68121 523 262 68644 525 263 69169 527 264 69696 529 265 70225 531 266 70756 533 277 71289 535 267 71289 535 268 71824 537 269 72361 539 270 72900 541 271 73441 543 272 73984 545 273 74529 547 274 75076 549 275 75625 551 276 76176 553 277 76729 555 278 77284 557 279 77841 559 280 78400 561 281 78961 563 282 79524 565 283 80089 567 284 80616 569 285 81225 571 286 81796 573 287 82369 575 288 82944 577 289 83521 579 290 84100 581 291 84681 583 292 85264 585 293 85849 587 294 86436 589 295 87025 591 296 87616 593 297 88200 595 298 88804 597 299 89401 599 300 90000 601 301 090601 603 302 091204 605 303 091809 607 304 092416 609 305 093025 611 306 093636 613 307 094249 615 308 094864 617 309 095481 619 310 096109 621 311 096721 623 312 97344 625 313 97969 627 314 98596 629 315 99325 631 316 99856 633 317 100487 645 318 101124 637 319 101761 639 320 102400 641 321 103041 643 322 103684 645 323 104329 647 324 104976 649 325 105625 651 326 106276 653 327 106929 655 328 107584 657 329 108241 659 330 108900 661 331 109561 663 332 110224 665 333 110889 667 334 111556 669 334 111556 669 335 112225 671 336 112896 673 337 113569 675 338 114244 677 339 114921 679 340 115600 681 341 116281 683 342 116964 685 343 117649 687 344 118336 689 345 119025 691 346 119716 693 347 120409 695 348 121104 697 349 121801 699 350 122500 701 351 123201 703 352 123904 705 353 124609 707 354 125316 709 355 126025 711 356 126736 713 357 127449 715 358 128164 717 359 128881 719 360 129600 721 361 138321 723 362 131044 725 363 131769 727 364 132496 729 365 133225 731 366 133956 733 367 134689 735 367 134689 735 368 135424 737 369 136161 739 370 136900 741 371 137641 743 372 138384 745 373 139129 747 374 139876 749 375 140625 751 376 141376 753 377 142129 755 378 142884 757 379 143641 759 380 144400 761 381 145161 763 382 145924 765 383 146689 767 384 147456 769 385 148225 771 386 148996 773 387 149769 775 388 150544 777 389 151321 779 390 152100 781 391 152881 783 392 153664 785 393 154449 787 394 155236 789 395 156025 791 396 156816 793 397 157609 795 398 158404 797 399 159201 799 400 160000 801 401 160801 803 402 161604 805 403 162409 807 404 163216 809 405 164025 811 406 164836 813 407 165649 815 408 166464 817 409 167281 819 410 168100 821 411 168921 823 412 169744 825 413 170569 827 414 171396 829 415 172225 831 416 173056 833 417 173889 835 418 174724 837 419 175561 839 420 176400 841 421 177241 843 422 178084 845 423 178929 847 424 179776 849 425 180625 851 426 181476 853 427 182329 855 428 183184 857 429 184041 859 430 184900 861 431 185761 863 432 186624 865 433 187489 867 434 188356 869 434 188356 869 435 189225 871 436 190096 873 437 190969 875 438 191844 877 439 192721 879 440 193600 881 441 194481 883 442 195364 885 443 196249 887 444 197136 889 445 198025 891 446 198916 893 447 199809 895 448 200704 897 449 201601 899 450 202500 901 451 203401 903 452 204304 905 453 205209 907 454 206116 909 455 207025 911 456 207936 913 457 208849 915 458 209764 917 459 210681 919 460 211600 921 461 212521 923 462 213444 925 463 214369 927 464 215296 929 465 216225 931 466 217156 933 467 218089 935 467 218089 935 468 219024 937 469 219961 939 470 220900 941 471 221841 943 472 222784 945 473 223729 947 474 224676 949 475 225625 951 476 226576 953 477 227529 955 478 228484 957 479 229441 959 480 230400 961 481 231361 963 482 232324 965 483 233289 967 484 234256 969 485 235225 971 486 236196 973 487 237169 975 488 238144 977 489 239121 979 490 240100 981 491 241081 983 492 242064 985 493 243049 987 494 244036 989 495 245025 991 496 246016 993 497 247009 995 498 248004 997 499 249001 999 500 250000 1001 501 251001 1003 502 252004 1005 503 253009 1007 504 254016 1009 505 255025 1011 506 256036 1013 507 257049 1015 508 258064 1017 509 259081 1019 510 260100 1021 511 261121 1023 512 262144 1025 513 263169 1027 514 264196 1029 515 265225 1031 516 266256 1033 517 267289 1035 518 268324 1037 519 269361 1039 520 270400 1041 521 271441 1043 522 272484 1045 523 273529 1047 524 274576 1049 525 275625 1051 526 276676 1053 527 277729 1055 528 278784 1057 529 279841 1050 530 288900 1061 531 281961 1063 532 283024 1065 533 284089 1067 534 285156 1069 534 285156 1069 535 286225 1071 536 287296 1073 537 288369 1075 538 289444 1077 539 290521 1079 540 291600 1081 541 292681 1083 542 293764 1085 543 294849 1087 544 295936 1089 545 297025 1091 546 298116 1093 547 299209 1095 548 300324 1097 549 301401 1099 550 302500 1101 551 303601 1103 552 304704 1105 553 305809 1107 554 306916 1109 555 308025 1111 556 309136 1113 557 310249 1115 558 311364 1117 559 312481 1119 560 313600 1121 561 314721 1123 562 315844 1125 563 316969 1127 564 318096 1129 565 319225 1131 566 320356 1133 567 321489 1135 567 321489 1135 568 322624 1137 569 323761 1139 570 324900 1141 571 326041 1143 572 327184 1145 573 328329 1147 574 329476 1149 575 330625 1151 576 331776 1153 577 332929 1155 578 334084 1157 579 335241 1159 580 336400 1161 581 337561 1163 582 338724 1165 583 339889 1167 584 341056 1169 585 342225 1171 586 343396 1173 587 344569 1175 588 345744 1177 589 346921 1179 590 348100 1181 591 349281 1183 592 350464 1185 593 351649 1187 594 352836 1189 595 354025 1191 596 355216 1193 597 356409 1195 598 357604 1197 599 358801 1199 600 369000 1201 601 361201 1203 602 362404 1205 603 963609 1207 604 364816 1209 605 366025 1211 606 367236 1213 607 368449 1215 608 369664 1217 609 370881 1219 610 372100 1221 611 373321 1223 612 374544 1225 613 375769 1227 614 376996 1229 615 378225 1231 616 379456 1233 617 380689 1235 618 381924 1237 619 383161 1239 620 384400 1241 621 385641 1243 622 386834 1245 623 388129 1247 624 389376 1249 625 390625 1251 626 391876 1253 627 393129 1255 628 394385 1257 629 395641 1259 630 396900 1261 631 398161 1263 632 399424 1265 633 400689 1267 634 401956 1269 634 401956 1269 635 403225 1271 636 404496 1273 637 405769 1275 638 407044 1277 639 408321 1279 640 409600 1281 641 410881 1283 642 412164 1285 643 413449 1287 644 414736 1289 645 416025 1291 646 417316 1293 647 418609 1295 648 419904 1297 649 421201 1299 650 422500 1301 651 423801 1303 652 425104 1305 653 426409 1307 654 427716 1309 655 429025 1311 656 430336 1313 657 431649 1315 658 432964 1317 659 434281 1319 660 435600 1321 661 436921 1323 662 438244 1325 663 439569 1327 664 440896 1329 665 442225 1331 666 443556 1333 667 444889 1335 667 444889 1335 668 446224 1337 669 447561 1339 670 448900 1341 671 450241 1343 672 451584 1345 673 452929 1347 674 454276 1349 675 455625 1351 676 456976 1353 677 458329 1355 678 459684 1357 679 461041 1359 680 462400 1361 681 463761 1363 682 465124 1365 683 466489 1367 684 467856 1369 685 469225 1371 686 470596 1373 687 471969 1375 688 473344 1377 689 474721 1379 690 476100 1381 691 477481 1383 692 478864 1385 693 480249 1387 694 481636 1389 695 483025 1391 696 484416 1393 697 485809 1395 698 487204 1397 699 488601 1399 700 490000 1401 701 491401 1403 702 492804 1405 703 494209 1407 704 495616 1409 705 497025 1411 706 498436 1413 707 499849 1415 708 501264 1417 709 502681 1419 710 504100 1421 711 505521 1423 712 506944 1425 713 508369 1427 714 509796 1429 715 511225 1431 716 512656 1433 717 514089 1435 718 515524 1437 719 516961 1439 720 518400 1441 721 519841 1443 722 521284 1445 723 522729 1447 724 524176 1449 725 525625 1451 726 527076 1453 727 528529 1455 728 529984 1457 729 531441 1459 730 532900 1461 731 534361 1463 732 535824 1465 733 537289 1467 734 538756 1469 734 538756 1469 735 540225 1471 736 541696 1473 737 543169 1475 738 544644 1477 739 546121 1479 740 547600 1481 741 549081 1483 742 550564 1485 743 552049 1487 744 553536 1489 745 555025 1491 746 556516 1493 747 558009 1495 748 559504 1497 749 561001 1499 750 562500 1501 751 564001 1503 752 565504 1505 753 567009 1507 754 568516 1509 755 570025 1511 756 571536 1513 757 573049 1515 758 574564 1517 759 576081 1519 760 577600 1521 761 579121 1523 762 580644 1525 763 582169 1527 764 583696 1529 765 585225 1531 766 586756 1533 767 588289 1535 767 588289 1535 768 589824 1537 769 591361 1539 770 592900 1541 771 594441 1543 772 595984 1545 773 597529 1547 774 599076 1549 775 600625 1551 776 602176 1553 777 603726 1555 778 605284 1557 779 606841 1559 780 608400 1561 781 609961 1563 782 611524 1565 783 613089 1567 784 614656 1569 785 616225 1571 786 617796 1573 787 619369 1575 788 620944 1577 789 622521 1579 790 624100 1581 791 625681 1583 792 627264 1585 793 628849 1587 794 630436 1589 795 632025 1591 796 633616 1593 797 635209 1595 798 636804 1597 799 638401 1599 800 640000 1601 801 641601 1603 802 643204 1605 893 644809 1607 804 646416 1609 805 648025 1611 806 649636 1613 807 651249 1615 808 652864 1617 809 654481 1619 010 656100 1621 811 657721 1623 812 659344 2625 813 560969 1627 814 562596 1629 815 564225 1631 816 565856 1633 817 567489 1635 818 569124 1637 819 570761 1639 820 672400 1641 821 674041 1643 822 675684 1645 823 677329 1647 824 678976 1649 825 680625 1651 826 682276 1653 827 683929 1655 828 685584 1657 829 687241 1659 830 688900 1661 831 690561 1663 832 692224 1665 833 693889 1667 834 695556 1669 834 695556 1669 835 697225 1671 836 668869 1673 837 700569 1675 838 702244 1677 839 703921 1679 840 705600 1681 841 707281 1683 842 708964 1685 853 710649 1687 844 712336 1689 845 714025 1691 846 715716 1693 847 717409 1695 848 719104 1697 849 729801 1699 850 722500 1701 851 724201 1703 852 725904 1705 853 727609 1707 854 729316 1709 855 731025 1711 856 732736 1713 857 734449 1715 858 736164 1717 859 737881 1719 860 739600 1721 861 741321 1723 862 743044 1725 863 744769 1727 864 746596 1729 865 748225 1731 866 749956 1733 867 751689 1735 867 751689 1735 868 753424 1737 869 755161 1739 870 756900 1741 871 658641 1743 872 760384 1745 873 762129 1747 874 763876 1749 875 765625 1751 876 767376 1753 877 769529 1755 878 770884 1757 879 772641 1759 880 774400 1761 881 776161 1763 882 777924 1765 883 779689 1767 884 781456 1769 885 783225 1771 886 754996 1773 887 786709 1775 888 786544 1777 889 790321 1779 890 792100 1781 891 793881 1783 892 795664 1785 893 797449 1787 894 799236 1789 895 801025 1791 896 802816 1793 897 894609 1795 808 806404 1797 899 808281 1799 900 810000 1801 901 811801 1803 902 813604 1805 903 815409 1807 904 817216 1809 905 819025 1811 906 820836 1813 907 822649 1815 908 824464 1817 909 826281 1819 910 828100 1821 911 829921 1823 912 831744 1825 913 833569 1827 914 835396 1829 915 837225 1831 916 839056 1833 917 840889 1835 918 842724 1837 919 844561 1839 920 846400 1841 921 848241 1843 922 850084 1845 923 851929 1847 924 853776 1849 925 855625 1851 926 857476 1853 927 859329 1855 928 861184 1857 929 863041 1859 930 864900 1861 931 866761 1863 932 868624 1865 933 870489 1867 934 872356 1869 934 872356 1869 935 874225 1871 936 876096 1873 937 877969 1875 938 879844 1877 939 881721 1879 940 883600 1881 941 885481 1883 942 887364 1885 943 889249 1887 944 891136 1889 945 893025 1891 946 894916 1893 947 896809 1895 948 898704 1897 949 900601 1899 950 902500 1901 951 904401 1903 952 906304 1905 953 908209 1907 954 910116 1909 955 912025 1911 956 913936 1913 957 915849 1915 958 917764 1917 959 919681 1919 960 921600 1921 961 923521 1923 962 925444 1925 963 927369 1927 964 929296 1929 965 931225 1931 966 933156 1933 967 935089 1935 967 935089 1935 968 937024 1937 969 938961 1939 970 940900 1941 971 942841 1943 972 944784 1945 973 946729 1947 974 948676 1949 975 950625 1951 976 952576 1953 977 954529 1955 978 956484 1957 979 958441 1959 980 960400 1961 981 962361 1963 982 964324 1965 983 966289 1967 984 968256 1969 985 970225 1971 986 972196 1973 987 974169 1975 988 976144 1977 989 978121 1979 990 980100 1981 991 982081 1983 992 984064 1985 993 986049 1987 994 988036 1989 995 990025 1991 996 992016 1993 997 994009 1995 998 996004 1997 999 998001 1999 1000 1000000 2001

0 1 2 3 4 5 6 7 8 9 1 0.001133 001371 001631 001915 002221 002549 002901 003275 003671 004091 2 0.004533 004997 005485 005995 006527 007083 007661 008261 008885 009531 3 0.010199 010891 011605 012341 013101 013883 014687 015515 016365 017237 4 0.018133 019051 019991 020955 021941 022949 023981 025035 026111 027211 5 0.028333 029477 030645 031835 033047 034283 035541 036825 038125 039451 6 0.040799 042171 043565 044981 046421 047883 049367 050875 052405 053957 7 0.051533 057131 058751 060395 062061 063749 065461 067195 068951 070731 8 0.072533 074357 076205 078075 079967 681883 083821 085781 087765 089771 9 0.091799 093851 095925 098021 100141 102283 104447 106635 108845 111077 10 0.113333 115610 117911 120234 122580 124949 127340 129754 132191 134650 11 0.137132 139637 142164 144714 147287 149882 152500 155141 157804 160490 12 0.163199 165930 168684 171461 174266 177082 179927 182794 185684 188597 13 0.191532 194490 197471 200474 203500 206549 229620 212714 215831 218970 14 0.222132 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26.202032 233234 264455 295694 326952 358229 389524 420837 452169 483520 169 26.514689 546277 577684 609109 640552 672014 703495 734994 766512 798049 170 26.829604 861177 892769 924380 956009 987657 019323 051008 082712 114432 171 27.146174 177933 209711 241508 273322 305156 337008 368879 400768 432675 172 27.464602 496547 528510 560492 592493 624512 656549 688606 720681 752774 173 27.784886 817016 849166 881333 913520 945724 977948 010190 042450 074729 174 28.107027 139343 171678 204031 236403 268794 301213 333630 366076 398541 175 28.431025 463526 496047 528586 561142 593720 626314 658928 691559 724210 176 28.756879 789566 822273 854997 881740 920502 953283 986082 018899 051735 177 29.084590 117463 150355 183265 216194 249142 282108 315092 348096 381117 178 29.414158 447217 480294 513390 546505 579638 612790 645960 679149 712356 179 29.745582 778827 812090 845372 878672 919991 945328 978684 012059 045452 180 30.078864 112294 145743 179210 212696 246200 279723 313265 346825 380404 181 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0 1 2 3 4 5 6 7 8 9 01 0.002617 003167 003769 004424 005131 005890 006702 007566 008482 009450 02 0.010471 011545 012671 013849 015079 016362 017697 019085 020525 022017 03 0.023561 025158 026808 028509 030264 032070 033929 035840 037803 039819 04 0.041887 044008 046181 048406 050684 053014 055396 057831 060318 062858 05 0.065449 068093 070790 073539 076340 079194 082100 085058 088069 091132 06 0.094247 097415 100635 103908 107232 110610 114039 117521 121055 124642 07 0.128281 131973 135716 139512 143361 147262 151215 155220 159278 163388 08 0.167551 171766 176033 180353 184725 189149 193626 198155 202737 207371 09 0.212057 216796 221586 226430 231325 236273 241274 246326 251432 256589 1.0 0.261799 267061 272376 277742 283162 288633 294157 299734 305362 311043 1.1 0.316777 322563 328401 334291 340234 346229 352277 358377 364529 370734 1.2 0.376991 383300 389662 396076 402542 409061 415632 422256 482932 435660 1.3 0.442440 449273 456159 463096 470086 477129 484224 491371 498570 505822 1.4 0.513126 520483 527892 535353 542867 550433 558051 565722 573445 581220 1.5 0.589048 596928 604861 612846 620883 628973 637114 645309 653555 661855 1.6 0.670206 678610 687066 695574 704135 712748 721414 730132 738902 747725 1.7 0.756600 765527 774507 783539 792623 801760 810949 820191 829485 838831 1.8 0.848230 857680 867184 876739 886348 896008 905721 915486 925303 935173 1.9 0.945097 955070 965097 975176 985308 995492 005728 016017 026358 036751 2.0 1.047197 057695 068246 078849 089504 100211 110971 121784 132648 143565 2.1 1.154535 165557 176631 187757 198936 210167 221451 232787 244175 255616 2.2 1.267109 278654 290252 301902 313604 325359 337166 349026 360937 372902 2.3 1.384918 396987 409109 421282 433508 445787 458117 470500 482936 495424 2.4 1.507964 520557 533201 545899 558648 571450 584305 597211 610170 623182 2.5 1.636246 649362 662530 675751 689024 702350 715728 729158 742641 756176 2.6 1.769763 783403 797095 810840 824637 838486 852387 866341 880347 894406 2.7 1.908517 922680 036896 951164 965485 979857 994283 008760 023290 037872 2.8 2.052507 067194 081933 096725 111569 126465 141414 156415 171468 186574 2.9 2.201732 216943 232206 247521 262889 278309 293781 309306 324883 340512 3.0 2.356194

0 1 2 3 4 5 6 7 8 9 10 0.314159 345575 376991 408407 439822 471238 502654 534070 565486 596902 20 0.628318 659734 691150 722566 753982 785398 816814 848230 879645 911061 30 0.942477 974893 006309 037725 069148 100557 131973 162389 193805 225221 40 1.256637 288052 319468 350884 382300 413716 445132 476548 507964 539380 50 1.570796 602212 633628 665044 696460 727875 759291 790707 822123 853539 60 1.884955 916371 947787 979203 010619 042035 073451 104867 136283 167698 70 2.199114 230530 261946 293362 324778 356194 387610 419026 450442 481858 80 2.513274 544690 576105 607521 638937 670353 701769 733185 764601 796017 90 2.827433 858849 890265 921681 953097 984513 015928 047344 078760 110176 1.00 3.141592 173008 204424 235840 267256 298672 330088 361504 392920 424335 10 3.455751 487167 518583 549999 581415 613831 644247 675663 707079 738495 20 3.769911 801327 832743 864158 895574 926990 958406 989822 021238 052654 30 4.084070 115486 146902 178318 209734 241150 272566 303981 335397 366813 40 4.398229 429645 461061 492477 523893 555309 586725 618141 649557 680973 50 4.712388 743804 775220 806636 838052 869468 900884 932300 963716 995132 60 5.026548 057964 089380 120796 152211 183627 215043 246459 277875 309291 70 5.340707 372123 403539 434945 466371 497787 529203 560618 592034 623450 80 5.654866 686282 717698 749144 780530 811946 843362 874778 906194 937610 90 5.969026 000441 031857 063273 094689 126105 157521 188937 220353 251769 2.00 6.283105 314601 346017 377433 408849 440264 471689 503096 534512 565928 10 6.597344 628760 660176 691592 723008 754424 785840 817256 848671 880087 20 6.911503 942919 974336 005751 037167 068583 099999 131415 162831 194247 30 7.225663 257079 288494 319910 351326 382742 414158 445574 476990 508406 40 7.539822 571238 602654 634070 665486 696901 728317 759733 791149 822565 50 7.853981 885397 916813 948229 979645 011061 042477 073893 105309 136724 60 8.168140 199556 230972 262388 293804 325220 356636 388052 419468 450884 70 8.482300 513716 545132 576547 607963 639379 670795 702211 733627 765043 80 8.796458 827875 859291 890707 922123 953539 984954 016370 047786 079202 90 9.110618 142034 173450 204866 236282 267798 299114 330530 361946 393362 3.00 9.424777

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0.045520 8.658211 1.31 0.046218 8.664816 1.33 0.046921 8.671372 1.34 0.047629 8.677878 1.35 0.048343 8.684336 1.36 0.049062 8.689746 1.37 0.049786 8.697110 1.38 0.050515 8.703427 1.39 0.051250 8.709698 1.40 0.051990 8 . 7●5924 1.41 0.052735 8.722107 1.42 0.053453 8.728245 1.43 0.054242 8.734340 1.44 0.054007 8.740393 1.45 0.055770 8.746404 1.46 0.056542 8.752374 1.47 0.057319 8.758303 1.48 0.058102 8.764192 1.49 0.058889 8.770071 1.50 0.059687 8.775851 1.51 0.060481 8.781622 1.52 0.061285 8.793356 1.53 0.062094 8.793051 1.54 0.062908 8.798710 1.55 0.063728 8.804332 1.56 0.064556 8.809918 1.57 0.065383 8.815468 1.58 0.066219 8.820983 1.59 0.067059 8.826463 1.60 0.067906 8.831908 1.61 0.068754 8.837320 1.62 0.069614 8.842698 1.63 0.070476 8.848044 1.64 0.071343 8.853356 1.65 0.072216 8.858636 1.66 0.073091 8.863885 1.67 0.073944 8.869101 1.68 2.074533 8.874595 1.69 0.075760 8.879442 1.70 0.076659 8.884566 1.71 0.077564 8.889661 1.72 0.078440 8.894725 1.73 0.079355 8.899761 1.74 0.080309 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1.796670 0.254468 8.24 1.801039 0.255523 8.25 1.805413 0.256576 8.26 1.809793 0.257629 8.27 1.814177 0.258679 8.28 1.818568 0.259729 8.29 1.822963 0.260777 8.30 1.827363 0.261825 8.31 1.831769 0.262870 8.32 1.836181 0.263915 8.33 1.840597 0.264958 8.34 1.845019 0.266001 8.35 1.849480 0.267041 8.36 1.853879 0.268081 8.37 2.858316 0.269119 8.38 1.862760 0.270156 8.39 1.867208 0.271192 8.40 1.871662 0.272227 8.41 1.876121 0.273250 8.42 1.880585 0.274293 8.43 1.885054 0.275324 8.44 1.889529 0.276353 8.45 1.894010 0.277382 8.46 1.898495 0.278409 8.47 1.902886 0.279435 8.48 1.907482 0.280450 8.49 1.911983 0.281484 8.50 1.916490 0.282506 8.51 1.921002 0.283528 8.52 1.925520 0.283748 8.53 1.930042 0.285566 8.54 1.934570 0.286584 8.55 1.939104 0.287601 8.56 1.943642 0.288616 8.57 1.948186 0.289630 8.58 1.952735 0.290643 8.59 1.957290 0.291655 8.60 1.961849 0.292665 8.61 1.966414 0.293675 8.62 1.970985 0.294683 8.63 1.975561 0.295690 8.64 1.980142 0.296696 8.65 1.984728 0.297701 8.66 1.989320 0.298704 8.67 1.993916 0.299707 8.68 1.998529 0.300708 8.69 2.003126 0.301708 8.70 2.007406 0.302707 8.71 2.012357 0.303705 8.72 2.016981 0.304701 8.73 2.021609 0.305697 8.74 2.026243 0.306691 8.75 2.030883 0.307685 8.76 2.035528 0.308677 8.77 2.040177 0.309668 8.78 2.044833 0.310657 8.79 2.049493 0.311646 8.80 2.054159 0.312634 8.81 2.058830 0.313620 8.82 2.063507 0.314606 8.83 2.068189 0.315590 8.84 2.072876 0.316573 8.85 2.077568 5.317555 8.86 2.082266 0.318536 8.87 2.086969 0.319516 8.88 2.091677 0.320494 8.89 2.096391 0.321472 8.90 2.101110 0.322448 8.91 2.105834 0.323424 8.92 2.110564 0.324498 8.93 2.115299 0.325371 8.94 2.120039 0.326343 8.95 2.124784 0.327314 8.96 2.129535 0.328284 8.97 2.134291 0.329253 8.98 2.139053 0.330221 8.99 2.143819 0.331178 9.00 2.148591 0.332153 9.01 2.153368 2.333118 9.02 2.158151 0.334081 9.03 2.162939 0.335044 9.04 2.167732 0.336005 9.05 2.172531 0.337966 9.06 2.177335 0.338925 9.07 2.182144 0.338883 9.08 2.186958 0.339840 9.09 2.191778 0.340796 9.10 2.196603 0.341751 9.11 2.201433 0.342705 9.12 2.206269 0.343658 9.13 2.211110 0.344610 9.14 2.215956 0.345561 9.15 2.220808 0.346511 9.16 2.225665 0.347459 9.17 2.230527 9.348407 9.18 2.235394 0.349354 9.19 2.240267 0.350299 9.20 2.245145 0.351244 9.21 2.250029 0.352881 9.22 2.254917 0.353130 9.23 2.259811 0.354072 9.24 2.264711 0.355012 9.25 2.269615 0.355872 9.26 2.274525 0.356890 9.27 2.279440 0.357828 9.28 2.284361 0.358764 9.29 2.289287 0.359700 9.30 2.294218 0.360634 9.31 2.299154 0.361568 9.32 2.304096 0.362500 9.33 2.309043 0.363432 9.34 2.313996 0.364362 9.35 2.318953 0.365292 9.36 2.323916 0.366220 9.37 2.328885 0.367148 9.38 2.333858 0.368074 9.39 2.338847 0.369000 9.40 2.343821 0.369924 9.41 2.348144 0.370848 9.42 2.353806 0.371770 9.43 2.358806 0.372692 9.44 2.363478 0.373612 9.45 2.368489 0.374532 9.46 2.373838 0.375451 9.48 2.378893 0.376368 9.48 2.383886 0.377285 9.49 2.388918 0.378201 9.50 2.393955 0 . 37911● 9.51 2.398998 0.380029 9.52 2.404045 0.380942 9.53 2.409099 0.381854 9.54 2.414157 0.382765 9.55 2.419221 0.383675 9.56 2.424290 0.384584 9.57 2.429364 0.385492 9.58 2.434444 0.386399 9.59 2.439529 0.387306 9.60 2.444629 0.388211 9.61 2.449715 0.389115 9.62 2.454816 0.390019 9.63 2.459922 0.390921 9.64 2.465034 0.391822 9.65 2.470150 0.392723 9.66 2.475273 0.393623 9.67 2.480400 0.394521 9.68 2.485533 0.395419 9.69 2.490671 0.396316 9.70 2.495814 0.397212 9.71 2.500963 0 . 39810● 9.72 2.506117 0.399001 9.73 2.511243 0.399894 9.74 2.516441 0.400786 9.75 2.521611 0 . 4●1678 9.76 2.526786 0.402568 9.77 2.531966 0.403458 9.78 2.537152 0.404346 9.79 2.542343 0.405234 9.80 2.547540 0.406121 9.81 2.552745 0.407006 9.82 2.557948 0.407891 9.83 2.563164 0.408775 9.84 2.568378 0.409659 9.85 2.573601 0.410541 9.86 2.578829 0.411422 9.87 2.584063 0.412303 9.88 2.589302 0.413182 9.89 2.594546 0.414061 9.90 2.599795 0.414939 9.91 2.605050 0.415816 9.92 2.610310 0.416692 9.93 2.615573 0.417567 9.94 2.620846 0.418441 9.95 2.626122 0.419315 9.96 2.631404 0.420187 9.97 2.636690 0.421059 9.98 2.641982 0.421929 9.99 2.647279 0.422789 10.00 2.652582 0.423668

A Table for the speedy finding of the Length or Circumference answering to any Arch in Degrees and Decimal Parts .

A Table for the speedy finding of the Length or Circumference answering to any Arch , in Degrees and Decimal Parts .

1 0.0174 5329 2519 2 0.0349 0658 5038 3 0.0523 5987 7557 4 0.0698 1317 0076 5 0.0872 6646 2595 6 0.1047 1975 5114 7 0.1221 7304 7633 8 0.1396 2634 0152 9 0.1570 7963 2671 10 0.1745 3292 5190 11 0.1919 8621 7709 12 0.2094 3951 0228 13 0.2268 9280 2747 14 0.2443 4609 5266 15 0.2617 9938 7785 16 0.2792 5268 0304 17 0.2967 0597 2823 18 0.3141 5926 5342 19 0.3316 1255 7861 20 0.3490 6585 0380 21 0.3665 1914 2899 22 0.3839 7245 5418 23 0.4014 2572 7937 24 0.4188 7902 0456 25 0.4363 3231 2975 26 0.4537 8560 5495 27 0.4712 3889 8013 28 0.4886 8219 0532 29 0.5061 4548 3051 30 0.5235 4877 5570 31 0.5410 5206 8089 32 0.5585 0536 0608 33 0.5759 5865 3127 34 0.5934 1194 5646 35 0.6108 6523 8165 36 0.6283 1853 0684 37 0.6457 7128 3203 38 0.6632 2511 5722 39 0.6806 7840 8241 40 0.6981 3170 0760 41 0.7155 8499 3279 42 0.7330 3828 5798 43 0.7504 9157 8317 44 0.7679 4487 0836 45 0.7853 9816 3355 46 0.8028 3145 5874 47 0.8203 0474 8393 48 0.8377 5804 0912 49 0.8552 1133 3431 50 0.8726 6462 4950 51 0.8901 1791 8469 52 0.9075 7121 0988 53 0.9250 2450 3507 54 0.9424 7779 6026 55 0.9599 3108 8545 56 0.9773 8438 1064 57 0.9948 3767 3583 58 1.0122 9096 6102 59 1.0297 4425 8621 60 1.0471 9755 1140 61 1.0646 5084 3659 62 1.0821 0413 6178 63 1.0995 5742 8697 64 1.1170 1072 1216 65 1.1344 6401 3735 66 1.1519 1730 6254 67 1.1693 7059 8773 68 1.1868 2389 1292 69 1.2042 7718 3811 70 1.2217 3047 6330 71 1.2391 8376 8849 72 1.2566 3706 1368 73 1.2740 9035 3887 74 1.2915 4364 6406 75 1.3089 9693 8925 76 1.3264 5023 1444 77 1.3439 0352 3963 78 1.3613 5681 6482 79 1.3788 1010 9001 80 1.3962 6340 1520 81 1.4137 1669 4039 82 1.4311 6998 6558 83 1.4486 2327 9057 84 1.4660 7657 1596 85 1.4835 2986 4115 86 1.5009 8315 6634 87 2.5184 3644 9153 88 1.5358 8974 1572 89 1.5533 4303 4191 90 1.5707 9632 6710 91 1.5882 4961 9229 92 1.6057 0291 1748 93 1.6231 5620 4267 94 1.6406 0949 6786 95 1.6580 6278 9305 96 1.6755 1608 1824 97 1.6929 6937 4343 98 1.7104 2266 6862 99 1.7278 7595 9381 100 1.7453 2925 1900

A Common Divisor for the speedy converting of the Table , shewing the Area of the Segments of a Circle whose Diameter is 2.0000 &c. into a Table shewing the Area of the Segment of any Circle whose Area is given .

1 0031 4159 2653 2 0062 8318 5306 3 0094 2477 7959 4 0125 6637 0612 5 0157 0796 3265 6 0188 4955 5918 7 0219 9114 8571 8 0251 3274 1224 9 0282 7433 3877 10 0314 1592 6530 11 0345 5751 9183 12 0376 9911 1836 13 0408 4070 4489 14 0439 8229 7142 15 0471 2388 9795 16 0502 6548 2448 17 0534 0707 5101 18 0565 4866 7754 19 0596 9026 0407 20 0628 3185 3060 21 0659 7344 5713 22 0691 1503 8366 23 0722 5663 1019 24 0753 9822 3672 25 0785 3981 6325 26 0816 8140 8978 27 0848 2300 1631 28 0889 6459 4284 29 0911 0618 6937 30 0942 4777 9590 21 0973 8937 2243 32 1005 3096 4896 33 1036 7255 7549 34 1068 1415 0202 35 1099 5574 2855 36 1130 9733 5508 37 1162 3892 8161 38 1193 8052 0814 39 1225 2211 3467 40 1256 6370 6120 41 1288 0529 8773 42 1319 4689 1426 43 1350 8848 4079 44 1382 3007 6732 45 1413 7166 9385 46 1445 1326 2038 47 1476 5485 4691 48 1507 9644 7344 49 1539 3803 9997 50 1570 7963 2650 51 1602 2122 5303 52 1633 6281 7956 53 1665 0441 0609 54 1696 4600 3262 55 1727 8759 5915 56 1759 2918 6568 57 1790 7078 1221 58 1822 1237 3874 59 1853 5396 6527 60 1884 9555 9180 61 1916 3715 1833 62 1947 7874 4486 63 1979 2033 7139 64 2010 6192 9792 65 2042 0352 2445 66 2073 4511 5098 67 2104 8670 7751 68 2136 2830 0404 69 2167 6989 3057 70 2199 1148 5710 71 2230 5307 8363 72 2261 9467 1016 53 2293 3626 3669 74 2324 7785 6322 75 2356 1944 8975 76 2387 6104 1628 77 2419 0263 4281 78 2450 4422 6934 79 2481 8581 9587 80 2513 2741 2240 81 2544 6900 4893 82 2576 1059 7546 83 2607 5219 0199 84 2638 9378 2852 85 2670 3537 5505 86 2701 7696 8158 87 2733 1856 0811 88 2764 6015 3464 89 2796 0174 6117 90 2827 4333 8770 91 2858 8493 1423 92 2890 2652 4076 93 2921 6811 6729 94 2953 0970 9382 95 2984 5130 2035 96 3015 9289 4688 97 3047 3448 7341 98 3078 7607 9994 99 3110 1767 2647 100 3141 5926 5300

A Table shewing the Ordinates , Arches and Areas of the Segments of a Circle , whose Diameter is 2000 , &c. to every Hundredth Part of the Radius .

Ordinates Deg. & Dec. p. Areas 100 10000000000 90.00000000 1.57079632 99 9999499971 89.42704196 1.55079682 98 9997999799 88.85400799 1.53079890 97 9995498987 88.28987110 1.51080538 96 99919967974 87.70756124 1.49081774 95 9987492177 87.13402020 1.47083808 94 9981983770 86.56018749 1.45086837 93 9975469913 85.98601581 1.43091081 92 9967948635 85.41143529 1.41096718 91 9959417653 84.83639513 1.39103966 90 9949874371 84.26083018 1.37113017 89 9939315871 83.68468641 1.35124084 88 9927738916 83.10789860 1.35137360 87 9915139938 82.53040793 1.31153053 86 9901515035 81.95215479 1.29171372 85 9886859966 81.37307468 1.27192518 84 9871170138 80.79310474 1.25216697 83 9854440623 80.21218180 1.23244118 82 9836666101 79.63024030 1.21274989 81 9817840903 79.04721672 1.19309522 80 9797958971 78.46304188 1.17347924 79 9777013859 77.87762112 1.15390361 78 9754998718 77.29096735 1.13437189 77 9731906288 76.70292903 1.11488481 76 9707728879 76.11243681 1.09544458 75 9682458365 75.52248845 1.07605462 74 9656086163 74.92996014 1.05671627 73 9628603221 74.33573392 1.03743102 72 9600000000 73.73979456 1.01820220 71 9570266454 73.14202474 0.99903143 70 9539392014 72.54239737 0.97992192 69 950●365565 71.94076969 0.96087497 68 9474175425 71.33707564 0.94189323 67 9439809319 70.73122476 0.92297905 67 9439809319 70.73122476 0.92297905 66 9404254356 70.12312662 0.90413479 65 9367496997 69.51268522 0.88536283 64 9329523031 68.89980401 0.86666560 63 9290317540 68.28438326 0.84804557 62 9249864864 67.66631784 0.82950517 61 9208148564 67.04550117 0.81104695 60 9165151389 66.42182324 0.79267345 59 9120855222 65.79516567 0.77438721 58 9075241043 65.16541298 0.75619089 57 9028288874 64.53244020 0.73808713 56 8979977728 63.89612058 0.72007866 55 8930285549 63.25631645 0.70216884 54 8879189152 62.61289754 0.68435845 53 8826664149 61.96570387 0.66665234 52 8772684879 61.31459838 0.64905275 51 8717224755 60.65941181 0.63156249 50 8660254037 60.00000000 0.61418485 49 8601744009 59.33617061 0.59692260 48 8541662601 58.66774875 0.57977892 47 8479976415 57.99454553 0.56275702 46 8416650165 57.31636147 0.54586011 45 8351646544 56.63307065 0.52909299 44 8284926070 55.94420256 0.51245467 43 8216446926 55.24977433 0.49595300 42 8146264741 54.54945742 0.47959008 41 8074032449 53.84299205 0.46336957 40 8000000000 53.13010237 0.44725221 39 7924014134 52.41049708 0.43137885 38 7846018098 51.68386597 0.41560051 37 7765951325 50.94987748 0.39998818 36 7683749084 50.20810657 0.38453683 35 7599342076 49.45831012 0.36925312 34 7512655988 48.70012721 0.35414227 34 7512655988 48.70012721 0.35414227 33 7423610981 47.93293539 0.33920561 32 7332121111 47.15635717 0.32444946 31 7238093671 46.36989113 0.30987884 30 7141428428 45.57299618 0.29549884 29 7042016756 44.76508489 0.28131493 28 6939740629 48.94551977 0.26733268 27 6834471449 43.11360613 0.25355796 26 6726068688 42.26858452 0.23999689 25 6614378277 41.40962595 0.22665594 24 6499230723 40.53580228 0.21354168 23 6380438856 39.64611132 0.20066138 22 6257795138 38.73942400 0.18802248 21 6131068422 37.81448867 0.17563291 20 6000000000 36.86989765 0.16350111 19 5864298764 35.90406873 0.15163601 18 5723635208 34.91520640 0.14004722 17 5577633906 33.90125515 0.12874491 16 5425863986 32.85988059 0.11774053 15 5267826876 31.78833069 0.10704574 14 5102940328 30.68341722 0.09667379 13 4930517214 29.54136121 0.08663902 12 4749736834 28.35773666 0.07695728 11 4559605246 27.12675321 0.06764629 10 4358898943 25.84193282 0.05872590 09 4146082488 24.49464857 0.05021866 08 3919183588 23.07391815 0.04215095 07 3675595189 21.56518547 0.03455313 06 3411744421 19.94844363 0.02746204 05 3122498999 18.19487244 0.02092302 04 2800000000 16 . 260204●1 0.01499411 03 2431049156 14.06986184 0.00975364 02 1989974874 11.47834097 0.00551730 01 1410673597 8.10961446 0.00188278 010 1410673597 8.10961446 0.00188278 009 1338618691 7.69281247 0.00160779 008 1262378707 7.25224680 0.00134761 007 1181143513 6.78328892 0.00110317 006 1093800713 6 . 279●8064 0.00087554 005 0998749217 5.73196797 0.00066616 004 0893532316 5.12640010 0.00047674 003 0774015503 4.43922228 0.00030969 002 0632139225 3.62430750 0.00016860 001 0447101778 2.56255874 0.00005961

1.57079632 999 199999 1.56879632 998 199999 1.56679632 199999 997 1.56479633 199998 996 1.56279634 199997 995 1.56079636 199996 994 1.55879639 199995 993 1.55679644 199994 992 1.55479649 199992 991 1.55299657 199991 990 1.55079666 199988 989 1.54879677 199986 988 1.54679690 199984 987 1.54479706 199981 986 1.54279724 199978 985 1.54079745 199976 984 1.53879769 199972 983 1.53679796 983 1.53679796 199969 982 1.53479827 199965 981 1.53279862 199962 980 1.53079899 199957 979 1.52879941 199953 978 1.52679988 199949 977 1.52480039 199944 976 1.52280095 199939 975 1.52080156 199934 974 1.51880222 199929 973 1.51680293 199924 972 1.51480369 199918 971 1.51280451 199912 970 1.51080539 199906 969 1.50880633 199909 968 1.50680733 199894 967 1.50480839 199887 966 1.50280952 966 1.50280952 199880 965 1.50081072 199873 964 1.49881199 199866 963 1.49681333 199859 962 1.49481474 199851 961 1.49281623 199843 960 1.49081774 199835 959 1.48881938 199827 958 1.48682110 199819 957 1.48482291 199810 956 1.48282480 199801 955 1.48082678 199792 954 1.47882885 199783 953 1.47683102 199774 952 1.47483328 199764 951 1.47283563 199754 950 1.47083808 199744 949 1.46884063 949 1.46884063 199734 948 1.46684328 199724 947 1.46484604 199713 946 1.46284890 199702 945 1.46085187 199691 944 1.45885496 199680 943 1.45685815 199669 942 1.45486146 199657 941 1.45286489 199645 940 1.45086837 199633 939 1.44887204 199621 938 1.44687583 199608 937 1.44487975 199596 936 1.44288379 199585 935 1.44088794 199570 934 1.43889224 199557 933 1.43689667 199543 932 1.43490124 932 1.43490124 199530 931 1.43290594 199516 930 1.43091078 199502 929 1.42891578 199488 928 1.42692090 199473 927 1.42492617 199459 926 1.42293158 199444 925 1.42093714 199429 924 1.41894305 199413 923 1.41694892 199398 922 1.41495494 199382 921 1.41296112 199366 920 1.41096746 199350 919 1.40897396 199334 918 1.40698062 199318 917 1.40498744 199301 916 1.40299443 199284 915 1.40100159 915 1.40100159 199267 914 1.39900892 199250 913 1.39701642 199232 912 1.39502410 199215 911 1.39303195 199197 910 1.39103998 199178 909 1.38904820 199160 908 1.38705660 199142 907 1.38506518 199123 906 1.38307395 199104 905 1.38108291 199085 904 1.37909206 199066 903 1.37710140 199047 902 1.37511093 199027 901 1.37312066 199007 900 1.37113059 900 1.37113017 198987 899 1.36914030 198967 898 1.36715063 198946 897 1.36516117 198925 896 1.36317192 198904 895 1.36118288 198883 894 1.35919405 198861 893 1.35720544 198839 892 1.35521705 198818 891 1.35322887 198797 890 1.35124090 198775 889 1.34925315 198752 888 1.34726563 198729 887 1.34527834 198707 886 1.34329127 198684 885 1.34130443 198661 884 1.33931782 198638 88● 1.33733144 883 1.33733144 198619 882 1.33534525 198590 881 1.33335935 198566 880 1.33137360 198541 879 1.32938819 198517 878 1.32740302 198499 877 1.32541803 198480 876 1.32343323 198449 875 1.32144874 198418 874 1.31946456 198393 873 1.31748063 198367 872 1.31549696 198341 871 1.31351355 198315 870 1.31153053 198289 869 1.30954764 198262 868 1.30756502 198235 867 1.30558267 198209 198209 866 1.30360058 198182 865 1.30161876 198154 864 1.29963722 198127 863 1.29765595 198100 862 1.29567495 198072 861 1.29369423 198044 860 1.29171379 198015 859 1.28973357 197986 858 1.28775371 197958 857 1 . 2857741● 197929 856 1.28379484 197900 855 1.28181584 197871 854 1.27983713 197841 853 1.27785872 197811 852 1.27588061 197781 851 1.27390280 197751 850 1.27192529 850 1.27192518 197721 849 1.26994797 197691 848 1.26797106 197660 847 1.26599446 197629 846 1.26401817 197598 845 1.26204219 197561 844 1.26006658 197534 843 1.25809124 197489 842 1.25611635 197457 841 1.25414178 197427 840 1.25216751 197395 839 1.25019356 197374 838 1.24821982 197341 837 1.24624641 197308 836 1.24427333 197275 835 1.24230058 197241 834 1.24032817 197212 197272 833 1.23835605 197173 832 1.23638432 197139 831 1.23441293 197105 830 1.23244118 197072 829 1.23047046 197036 828 1.22850010 197001 827 1.22653009 196966 826 1.22456043 196930 825 1.22259113 196895 824 1.22062218 196861 823 1.21865357 196825 822 1.21668532 196787 821 1.21471745 196750 820 1.21274989 196714 819 1.21078275 196677 818 1.20881598 196640 817 1.20684958 817 1.20684954 1966●3 816 1.20488355 196565 815 1.20291790 196527 814 1 . 20●95263 196479 813 1.19898774 196451 812 1.19702323 196413 811 1.19505910 196375 810 1.19309525 196347 809 1.19113254 196298 808 1.18916956 196258 807 1.18720698 196219 806 1.18524479 196188 805 1.18328291 196148 804 1.18132143 196100 803 1.17936043 196060 802 1.17739983 196019 801 1.17543964 195978 800 1.17347986 800 1.17347924 195938 799 1.17151986 195897 798 1.16956089 195855 797 1.16760234 195814 796 1.16564420 195773 795 1.16368647 195731 794 1.16172916 195689 793 1.15977227 195646 792 1.15781581 195603 791 1.15585978 195561 790 1.15390417 195518 789 1.15194899 195472 788 1.14999427 195429 787 1.14803998 195388 786 1.14608610 195344 785 1.14413266 195300 784 1.14217966 784 1.14217966 195256 783 1 . 14022●10 195211 782 1.13827499 195166 781 1.13632333 195122 780 1.13437211 195076 779 1.13242135 195031 778 1.13047102 194985 777 1.12852117 194939 776 1.12657178 194893 775 1.12462285 194847 774 1.12267438 194801 773 1.12072637 194755 772 1.11877882 194708 771 1.11683174 194661 770 1.11488487 194614 769 1.11293867 194566 768 1.11099301 194518 194518 767 1.10904783 194471 766 1.10710312 194423 765 1.10515889 194374 764 1.10321515 194325 763 1.10127190 194276 762 1.09932914 194227 761 1.09738687 194173 760 1.09544514 194129 759 1.09350385 194079 758 1.09156306 194029 757 1.08962277 193980 756 1.08768297 193930 755 1.08574367 193878 754 1.08380489 193827 753 1.08186662 193777 752 1.07992885 193726 751 1.07799159 751 1.07799159 193674 750 1.07605485 193622 749 1.07411863 193570 748 1.07218293 193518 747 1.07024775 193466 746 1.06831309 193414 745 1.06637895 193361 744 1.06444534 193308 743 1.06251226 193255 742 1.06057971 193201 741 1.05864770 193147 740 1.05671623 193093 739 1.05478530 193039 738 1.05285491 192985 737 1.05092506 192931 736 1.04899575 192876 735 1.04706699 192821 192821 734 1.04513878 192766 733 1.04321112 192710 732 1.04128402 192655 731 1.03935747 192600 730 1.03743147 192543 729 1.03550604 192486 728 1.03358118 192430 727 1.03165688 192373 726 1.02973115 192316 725 1.02780999 172259 724 1.02588740 192213 723 1.02396527 192155 722 1.02204372 192086 721 1.02012286 192029 720 1.01820221 191970 719 1.01628251 191911 718 1.01436340 718 1.01436340 191853 717 1.01244487 191794 716 1.01052693 191734 715 1.00860959 191674 714 1.00669285 191615 713 1.00477670 191556 712 1.00286114 191505 711 1.00094609 191444 710 0.99903165 191374 709 0.99711791 191313 708 0.99520478 191252 707 0.99329226 191191 706 0.99138035 191129 705 0.98946906 191067 704 0.98755839 191005 703 0.98564834 190943 702 0.98273891 190881 701 0.98183010 190818 700 0.97992192 190755 699 0.97801437 190692 698 0.97610745 190629 697 0.97420116 190566 696 0.97229550 190502 695 0.97039048 190438 694 0.96848610 190376 693 0.96658234 190304 692 0.96467930 190244 691 0.96277686 190179 690 0.96087497 190113 689 0.95897384 190048 688 0.95707336 189983 687 0.95517353 189917 686 0.95327436 189851 685 0.95137585 189784 684 0.94947801 189717 189717 683 0.94758084 189651 682 0.94568433 189584 681 0.94378848 189516 680 0.94189324 189448 679 0.93999876 189381 678 0.93810495 189313 677 0.93621182 189244 676 0.93431938 189176 675 0.93242762 189107 674 0.93053655 189038 673 0.92864617 188969 672 0.92675648 188899 671 0.92486749 188823 670 0.92297905 188769 669 0.92109136 188696 668 0.91920440 188619 667 0.91731821 188549 188549 666 0.91543272 188478 665 0.91354794 188407 664 0.91166387 188336 663 0.90978051 188264 662 0.90789787 188192 661 0.90601595 188120 660 0.90413479 188048 659 0.90225431 187973 658 0.90037458 187900 657 0.89849558 187829 656 0.89661729 187757 655 0.89473972 187685 654 0.89286287 187610 653 0.89098677 187535 652 0.88911142 187461 651 0.88723681 187386 650 0.88536295 650 0.88536284 187311 649 0.88348973 187237 648 0.88161736 187163 647 0.87974573 187087 646 0.87787486 187010 645 0.87600476 186934 644 0.87413542 186858 643 0.87226684 186782 642 0.87039902 186705 641 0.86853197 186628 640 0.86666560 186551 639 0.86480009 186473 638 0.86293536 186395 637 0.86107141 186317 636 0.85920824 186239 635 0.85734585 186161 634 0.85548424 186083 633 0.85362341 633 0.85362341 186004 632 0.85176337 185924 631 0.84990413 185845 630 0.84804557 185764 629 0.84618793 185684 628 0.84433109 185606 627 0.84247503 185525 626 0.84061978 185444 625 0.83876534 185363 624 0.83691171 185281 623 0.83505890 185200 622 0.83320690 185119 621 0.83135571 185038 620 0.82950517 184956 619 0.82765561 184873 618 0.82580688 184790 617 0.82395898 184707 184707 616 0.82211191 184624 615 0.82026567 184540 614 0.81842027 184456 613 0.81657571 184372 612 0.81473199 184288 611 0.81288911 184204 610 0.81104695 184119 609 0.80920576 184035 608 0.80736541 183949 607 0.80552592 183865 606 0.80368727 183780 605 0.80184947 183693 604 0.80001254 183606 603 0.79817548 183519 602 0.79634029 183433 601 0.79450596 183346 600 0.79267250 600 0.79267345 183258 599 0.79084087 183170 598 0.78900917 183082 597 0.78717835 182994 596 0.78534841 182906 595 0.78351935 182818 594 0.78169117 182729 593 0.77986388 182640 592 0.77803748 182551 591 0.77621197 182461 590 0.77438736 182371 589 0.77256365 182281 588 0.77074084 182191 587 0.76891893 182100 586 0.76709793 182009 585 0.76527784 181918 584 0.76345866 131826 583 0.76164040 181734 582 0.75982306 181639 581 0.75800667 181543 580 0.75619124 181458 579 0.75437670 181365 578 0.75256305 181271 577 0.75075934 181178 576 0.74893856 181085 575 0.74712771 180991 574 0.74531780 180897 573 0.74350883 180802 572 0.74170081 180707 571 0.73989374 180611 570 0.73708713 180516 569 0.73628197 180422 568 0.73447775 180326 567 0.73267449 180230 180230 566 0.73087219 180134 565 0.72907085 180037 564 0.72727048 179940 563 0.72547108 279843 562 0.72367265 179745 561 0.72187520 179647 560 0.72007866 179548 559 0.71828318 179450 558 0.71648868 179353 557 0.71469515 179254 556 0.71290261 179155 555 0.71111106 179056 554 0.70932050 178956 553 0.70753094 178856 552 0.70574238 178755 551 0.70395483 178654 550 0.70216829 550 0.70216834 178553 549 0.70038281 178452 548 0.69859829 178352 547 0.69681477 178250 546 0.69503227 178149 545 0.69325078 178048 544 0.69147030 177943 543 0.68969087 177841 542 0.68791246 177738 541 0.68613508 177634 540 0.68435845 177528 539 0.68258317 177423 538 0.68080894 177318 537 0.67903576 177218 536 0.67726358 177114 535 0.67549244 177009 534 0.67372235 176903 533 0.67195332 176799 532 0.67018533 176693 531 0.66841840 176585 530 0.66665234 176479 529 0.66488755 176372 528 0.66312383 176265 527 0.66136118 176158 526 0.65959960 176050 525 0.65783910 175942 524 0.65607968 175834 523 0.65432134 175725 522 0.65256409 175622 521 0.65080787 175512 520 0.64905275 175398 519 0.64729877 175289 518 0.64554588 175179 517 0.64379409 175068 175068 516 0.64204341 174957 515 0.64029384 174846 514 0.63854538 174735 513 0.63679803 174624 512 0.63505179 174512 511 0.63330667 174400 510 0.63156249 174287 509 0.62981962 174174 508 0.62807788 174062 507 0.62633726 173948 506 0.62459778 173835 505 0.62285943 173721 504 0.62112222 173607 503 0.61938615 173492 502 0.61765123 173377 501 0.61591746 173262 500 0.61418484 500 0.61418485 173147 499 0.61245338 173031 498 0.61072307 172914 497 0.60899393 172798 496 0.60726595 172681 495 0.60553914 172564 494 0.60381350 172447 493 0.60208903 172329 492 0.60036574 172211 491 0.59864363 172093 490 0.59692260 171975 489 0.59520285 171856 488 0.59348429 171736 487 0.59176693 171617 486 0.59005076 171498 485 0.58833578 171377 484 0.58662201 171256 483 0.58490948 171136 482 0.58319809 171015 481 0.58148794 170893 480 0.57977892 170771 479 0.57807121 170649 478 0.57636472 170527 477 0.57465945 170406 476 0.57295539 170281 475 0.57125258 170158 474 0.56955100 170034 473 0.56785066 169910 472 0.56615156 169786 471 0.56445370 169661 470 0.56275702 169536 469 0.56106166 169411 468 0.55936755 169285 467 0.55767470 169159 169159 466 0.55598311 169035 465 0.55429278 168901 464 0.55260377 168779 463 0.55091598 168652 462 0.54922946 168524 461 0.54754422 168397 460 0.54586011 168268 459 0.54417743 168139 458 0.54249604 168010 457 0.54081594 167881 456 0.53913713 167751 455 0.53745962 167621 454 0.53578341 167491 453 0.53410850 167360 452 0.53243490 167229 451 0.53076261 167098 450 0.52909163 450 0.52909299 166966 449 0 . 527423●3 166834 448 0.52575499 166702 447 0.52408797 166570 446 0.52242227 166437 445 0.52075790 166302 444 0.51909488 166168 443 0.51743320 166035 442 0.51577285 165900 441 0.51411385 165765 440 0.51245467 165634 439 0.51079833 165494 438 0.50914339 165358 437 0.50748981 165222 436 0.50583759 16508● 435 0.50418673 164949 434 0.50253724 164811 433 0.50088913 164673 432 0.49924240 164535 431 0.49759705 164397 430 0.49595308 164259 429 0.49431049 164120 428 0.49266929 163980 427 0.49102949 163835 426 0.48939114 163700 425 0.48775414 163560 424 0.48611854 163419 423 0.48448435 163277 422 0.48285158 163135 421 0.48122023 162998 420 0.47959025 162843 419 0.47796165 162708 418 0.47633457 162565 417 0.47470892 162422 162422 416 0.47308470 162278 415 0.47146192 162134 414 0.46984058 161989 413 0.46822069 161844 412 0.46660225 161699 411 0.46498526 161570 410 0.46336957 161410 409 0.46175547 161260 408 0.46014287 161113 407 0.45853174 160966 406 0.45692208 160818 405 0.45531390 160670 404 0.45370720 160522 403 0.45210198 160373 402 0.45049825 160223 401 0.44889602 160073 400 0.44729529 400 0.44729522 159923 399 0.44569599 159773 398 0.44409826 159623 397 0.44250203 159472 396 0.44090731 159320 395 0.43931411 159168 394 0.43772243 159016 393 0.43613227 158863 392 0.43454364 158710 391 0.43295654 158557 390 0.43137086 158403 389 0.42978683 158248 388 0.42820435 158093 387 0.42662342 157938 386 0.42504404 157782 385 0 , 42346622 157626 384 0.42188996 157470 383 0.42031526 157313 382 0.41874213 157156 381 0.41717057 156999 380 0.41560058 156841 379 0.41403217 156682 378 0.41246535 156522 377 0.41090013 156363 376 0.40933650 156204 375 0.40777446 156044 374 0.40621402 155883 373 0.40465519 155722 372 0.40309797 155561 371 0.40154236 155399 370 0.39998818 155238 369 0.39843580 155025 368 0.39688555 154911 367 0.39533644 154788 154788 366 0.39378896 154584 365 0.39224312 154419 364 0.39069893 154254 363 0.38915639 154089 362 0.38761550 153923 361 0.38607627 153757 360 0.38453683 153591 359 0.38300092 153424 358 0.38146668 153256 357 0.37993412 153088 356 0.37840324 152920 355 0.37687404 152751 354 0.37534653 152582 353 0.37382071 152443 352 0.37229658 152242 351 0.37077416 152075 350 0.36925315 350 0.36925312 151905 349 0.36773407 151728 348 0.36621679 151556 347 0.36470123 151384 346 0.36318739 151211 345 0.36167528 151038 344 0.36016490 150865 343 0.35865625 150690 342 0.35714935 150515 341 0.35564420 150340 340 0.35414227 150164 339 0.35264063 149988 338 0.35114075 149811 337 0.34964264 149634 336 0.34814630 149457 335 0.34665173 149279 334 0.34515894 149100 333 0.34366794 148921 332 0.34217873 148742 331 0.34069131 148562 330 0.33920561 148381 329 0.33772180 148200 328 0.33623980 148024 327 0.33475956 147842 326 0.33328114 147663 325 0.33180451 147480 324 0.33032971 147288 323 0.32885683 147104 322 0.32738579 146919 321 0.32591660 146735 320 2.32444946 146550 319 0.32298396 146362 318 0.32152034 146175 317 0.32005859 145990 145990 316 0.31859869 145803 315 0.31714066 145614 314 0.31568452 145425 313 0.31423027 145236 312 0.31277791 145047 311 0.31132744 144856 310 0.30987884 144665 309 2.30843219 144474 308 0.30698745 144282 307 0.30554463 144090 306 0.30410373 143897 303 0.30266476 143703 304 0.30122773 143508 303 0.29978265 143315 302 0.29835950 143120 301 0.29692830 142926 300 0.29549904 300 0.29549884 142730 299 0.29407154 142533 298 0.29264621 142335 297 0.29122286 142137 296 0.28980149 141939 295 0.28838210 141741 294 0.28696469 141460 293 0.28555009 141260 292 0.28413749 141191 291 0.28272558 140990 290 0.28131493 140730 289 0.27990763 140527 288 0.27850236 140331 287 0.27709905 140124 286 0.27569781 139920 285 0.27429861 139720 284 0.27290141 139517 283 0.27150624 139311 282 0.27011313 139105 281 0.26872208 138898 280 0.26733268 138690 279 0.26594578 138482 278 0.26456096 138273 277 0.26317823 138063 276 0.26179760 137853 275 0.26041907 137643 274 0.25904264 137432 273 0.25766832 137220 272 0.25629612 137008 271 0.25492604 136795 270 0.25355796 136583 269 0.25219213 136370 268 0.25082843 136153 267 0.24946690 135936 135936 266 0.24810754 135720 265 0.24675034 135504 264 0.24539530 135287 263 0.24404243 135069 262 0.24269174 134850 261 0.24134324 134553 260 0.23999689 134333 259 0.23865356 134189 258 0.23731165 133968 257 0.23597197 133746 256 0.23463451 133523 255 0.23329928 133300 254 0.23196628 133076 253 0.23063552 132801 252 0.22930751 132575 251 0.22798176 132399 250 0.22665777 250 0.22665594 132173 249 0.22533421 131946 248 0.22401475 131718 247 0.22269757 131488 246 0.22138269 131259 245 0.22007010 131029 244 0.21875981 130799 243 0.21745182 130567 242 0.21614615 130334 241 0 21484281 130101 240 0.21354168 129867 239 0.21224301 129632 238 0.21094669 129396 237 0.20965273 129160 236 0.20836113 128924 235 0.20707189 128688 234 0.20578501 128449 233 0.20450052 128208 232 0.20321844 123968 231 0.20193876 127729 230 0.20066138 127488 229 0.19938650 127245 228 0.19811405 127002 227 0.19684403 126758 226 0.19557645 126514 225 0.19431131 126269 224 0.19304862 126023 223 0.19178839 125776 222 0.19053063 125528 221 0.18927535 125279 220 0.18802248 125027 219 0.18677221 124777 218 0.18552444 124529 217 0.18427915 124278 124278 216 0.18303637 124025 215 0.18179612 123771 214 0.18055841 123517 213 0.17932324 123262 212 0.17809062 123006 211 0.17686056 122749 210 0.17563291 122490 209 0.17440801 122232 208 0.17318569 122974 207 0.17196595 121713 206 0.17074882 121451 205 0.16953431 121189 204 0.16832242 120926 203 0.16711316 120663 202 0.16590653 120399 201 0.16470254 120133 200 0.16350121 200 0.16350111 119866 199 0.16230245 119598 198 0.16110647 119329 197 0.15991318 119959 196 0.15872259 118789 195 0.15753470 118518 194 0.15634952 118246 193 0.15516706 117972 192 0.15398733 117698 191 0.15281035 117422 190 0.15163596 117146 189 0.15046450 116869 188 0.14929581 116591 187 0.14812990 116312 186 0.14696678 116032 185 0.14580646 115751 184 0.14464895 115468 183 0.14349427 115084 182 0.14234243 114900 181 0.14119343 114615 180 0.14004728 114328 179 0.13890400 114040 178 0.13776354 113752 177 0.13612602 163462 176 0.13549140 113164 175 0.13435926 112873 174 0.13323103 112587 173 0.13210516 112292 172 0.13098224 111996 171 0.12986228 111700 170 0.12874498 111403 169 0.21763088 111105 168 0.12651983 110805 167 0.12541178 110503 110503 166 0.12430675 110200 165 0.12320475 109896 164 0.12210579 109592 163 0.12100987 109287 162 0.11991700 108980 161 0.11882720 108671 160 0.11774053 108361 159 0.11665692 108047 158 0.11557645 107735 157 0.11449910 107425 156 0.11342485 107110 155 0.11235375 106794 154 0.11128581 106478 153 0.11022103 106159 152 0.10915944 105838 151 0.10810106 105517 150 0.10704589 150 0.10704589 105194 149 0.10599395 104870 148 0.10494525 104545 147 0.10389980 104218 146 0.10285762 105889 145 0.10181873 103560 144 0.10078313 103229 143 0.09975084 102895 142 0.09872199 102561 141 0.09769638 102213 140 0.09667379 101876 139 0.09565503 101550 138 0.09463953 101210 137 0.09362743 100869 136 0.09261874 100526 135 0.09161348 100181 134 0.09061167 99834 133 0.08961333 99461 132 0.08861872 99112 131 0.08762760 98786 130 0.08663902 98433 129 0.08565469 98078 128 0.08467391 97722 127 0.08369669 97364 126 0.08272305 97004 125 0.08175301 96643 124 0.08078658 96280 123 0.07982378 95915 122 0.07886463 95548 121 0.07790915 95179 120 0.07695736 94811 119 0.07600925 94438 118 0.07506487 94061 117 0.07412426 93685 93685 116 0.07318741 93307 115 0.07225434 92901 114 0.07132533 92524 113 0.07040009 92161 112 0.06947848 91774 111 0.06856074 91386 110 0.06764629 90944 109 0.06673685 90551 108 0.06583134 90208 107 0.06492926 89811 106 0.06403115 89412 105 0.06313703 89011 104 0.06224692 88608 103 0.06136084 88202 102 0.06047882 87793 101 0.05960089 87382 100 0.05872707 100 0.05872590 86969 99 0.05785621 86554 98 0.05699067 86137 97 0.05612930 85717 96 0.05527213 85293 95 0.05441920 84867 94 0.05357053 84440 93 0.05272613 84010 92 0.05188603 83666 91 0.05104937 83229 90 0.05121866 82700 89 0.04939166 82259 88 0.04856907 81814 87 0.04775093 81366 86 0.04693727 80916 85 0.04612811 89462 84 0.04532349 80005 83 0.04452344 79545 82 0.04372799 79083 81 0.04293716 78617 80 0.04215095 78147 79 0.04136948 77674 78 0.04058274 77197 77 0.03982077 76707 76 0.03905370 76224 75 0.03829146 75748 74 0.03753398 75250 73 0.03678140 74764 72 0.03603376 34265 71 0.03529111 73752 70 0.3455313 73246 69 0.03382067 72746 68 0.03309321 72232 67 0.03237089 71716 71716 66 0.03165373 71193 65 0.03094180 70664 64 0.03023516 70132 63 0.02953384 69995 62 0.02883789 69054 61 0.02814735 68508 60 0.02746204 67961 59 0.02928243 67405 58 0.02610838 66840 57 0.02543998 66273 56 0.02477725 65701 55 0.02412024 65123 54 0.02346901 64539 53 0.02282362 63950 52 0.02218412 63353 51 0.02155059 62750 50 0.02092309 50 0.02092302 62143 49 0.02030159 61528 48 0.01968631 60906 47 0.01907725 60277 46 0.01847448 59640 45 0.01787808 58996 44 0.01728812 58344 43 0.01670468 57683 42 0.01612784 57016 41 0.01555768 56340 40 0.01499411 55655 39 0.01443756 54960 38 0.01388796 54256 37 0.01334540 53540 36 0.01281000 52815 35 0.01228185 52079 34 0.01176106 51331 33 0.01124776 50572 32 0.01074204 49801 31 0.01024403 49016 30 0.00975364 48217 29 0.00927147 47405 28 0.00879742 46578 27 0.00833164 45734 26 0.00787430 44874 25 0.00742556 43997 24 0.00698559 43102 23 0.00655457 42185 22 0.00613272 41244 21 0.00572028 40273 20 0.00531730 39291 19 0.00492439 38297 18 0.00454142 37248 17 0.00416894 36176 36176 16 0.00380718 35071 15 0.00345647 33929 14 0.00311718 32746 13 0.00278972 31517 12 0.00247455 30236 11 0.00217219 28897 10 0.00188278 27442 9 0.00160836 25959 8 0.00134877 24434 7 0.00110443 22749 6 0.00087694 20925 5 0.00066769 18922 4 0.00047847 16675 3 0.00031172 14061 2 0.00017111 10792 1 0.00006319 6319 0 0.00000000

A TABLE SHEWING THE AREA OF THE SEGMENTS OF A CIRCLE WHOSE Whole Area is Unity , to the ten Thousandth part of the Diameter .

0 1 2 3 4 5 6 7 8 9 0 000000 000004 000007 000011 000014 000018 000025 000032 000039 000046 1 000053 000062 000071 000080 000089 000098 000108 000119 000130 000140 2 000151 000163 000175 000187 000200 000212 000225 000238 000251 000265 3 000278 000292 000307 000322 000336 000351 000366 000382 000397 000413 4 000428 000444 000461 000478 000494 000511 000529 000546 000564 000581 5 000599 000617 000636 000654 000673 000691 000710 000729 000748 000768 19 6 000787 000807 000827 000847 000867 000887 000908 000928 000949 000970 20 7 000991 001012 001034 001056 001077 001099 001121 001144 001166 001188 21 8 001211 001234 001257 001280 001304 001327 001350 001374 001398 001421 23 9 001445 001469 001494 001518 001542 001567 001592 001617 001642 001667 25 10 001692 001717 001743 001769 001794 001820 001846 001873 001899 001925 26 11 001952 001979 002005 002032 002059 002086 002113 002141 002168 002195 27 12 002223 002251 002279 002307 002335 002363 002392 002420 002449 002477 28 13 002506 002535 002564 002593 002623 002652 002681 002711 002741 002770 29 14 002700 002830 002860 002890 002921 002951 002982 003013 003043 003074 30 15 003105 003136 003167 003198 003229 003260 003291 003323 003355 003387 31 16 003419 003451 003483 003515 003548 003580 003612 003645 003678 003710 32 17 003743 003776 003809 003842 0038●6 003909 003942 003976 004009 004043 33 18 004077 004111 004145 004179 004213 004247 004281 004316 004351 004385 34 19 004420 004455 004490 004525 004560 004595 004630 004665 004700 004735 35 20 004770 004806 004843 004879 004915 004952 004988 005024 005061 005097 36 21 005133 005170 005206 005243 005280 005317 005354 005391 005428 005465 37 22 005502 005539 005●77 005615 005652 005690 005728 005766 005804 005842 38 23 005880 005918 005957 005995 006023 006072 006111 006150 006188 006227 39 24 006266 0063●5 006344 006383 006423 006462 006501 006541 006581 006620 39 25 006660 006700 006739 006779 006819 006859 006899 006940 006980 007021 40 26 007061 007102 007142 007183 007223 007264 007305 007346 007387 007429 41 27 007470 007511 007553 007594 007635 007677 007719 007761 007802 007844 42 28 007886 007928 007970 008012 008055 008097 008140 008182 008225 008267 42 29 008310 008353 008396 008439 008482 008525 008568 008611 008654 008698 43 30 008471 008785 008828 008872 008916 008959 009203 009047 009091 009135 44 31 009179 009223 009267 009312 009356 009400 009445 009490 009535 009579 45 32 009624 009669 009714 009759 009804 009849 009894 009939 009985 010030 45 33 010075 010121 010167 010212 010258 010303 010349 010395 010441 010487 46 33 010075 010121 010167 010212 010258 010303 010349 010395 010441 010487 46 34 010533 010580 010626 010672 010719 010765 010812 010858 010905 010952 47 35 010999 011045 011093 011139 011186 011233 011281 011328 011375 011422 47 36 011469 011517 011565 011612 011660 011707 011755 011803 011851 011899 47 37 011947 011995 012043 012092 012140 012188 012237 012285 012334 012382 48 38 012431 012479 012528 012577 012626 012675 012724 012773 012823 012872 49 39 012921 012970 0130●0 013069 013118 013168 013218 013267 013317 013367 50 40 013417 013467 013517 013567 013617 013667 013717 013767 013818 013868 50 41 013919 013969 014020 014071 014121 014172 014223 014274 014325 014375 51 42 014426 014478 014529 014580 014632 014683 014734 014786 014837 014889 51 43 014941 014992 015044 015096 015148 015199 015252 015304 015356 015408 52 44 015460 015512 015565 015617 015669 015721 015774 015827 015879 015932 52 45 015985 016038 016091 016144 016197 016249 016303 016356 0164●9 016462 53 46 016515 016569 016622 016676 016729 016783 016837 016891 016944 016998 54 47 017052 017906 017160 017214 017268 017322 017376 017431 017485 017539 54 48 017593 017648 017703 017757 017812 017866 017921 017976 018031 018086 55 49 018141 018196 018251 018306 018361 018416 018471 018527 018582 018638 55 50 018693 018749 018804 018860 018916 018971 019027 019083 019139 019195 56 51 019251 019307 019363 019419 019475 019531 019588 019644 019701 019757 56 52 019813 019870 019927 019984 020040 020097 020154 020211 020268 020325 57 53 020381 020439 020496 020553 020610 020667 020725 020782 020840 020897 57 54 020954 021012 021070 021128 021185 021243 021301 021359 021416 021474 57 55 021532 021590 021649 021707 021765 021823 021882 021940 021999 022057 58 56 022115 022174 022233 022292 022350 022409 022468 022527 022586 022645 59 57 022703 022763 022822 022881 022949 022999 023058 023118 023177 023237 59 58 023296 023356 023415 023475 023534 023594 023654 023714 023774 023834 60 59 023894 023954 024014 024074 024134 024194 024254 024315 024375 024436 60 60 024496 024557 024617 024678 024738 024799 024860 024921 024981 025042 60 61 025103 025164 025225 025286 025347 025408 025470 025531 025692 025654 61 62 025715 025776 025838 025899 025961 026022 026084 026146 026208 026270 62 63 026331 026393 026455 026517 026579 026641 026703 026766 026828 026890 62 64 026952 027015 027077 027140 027202 027264 027327 027390 027453 027515 63 65 027578 027641 027704 027767 027830 027892 027956 028019 028082 028145 63 66 028208 028271 028335 028398 028461 028524 028588 028652 028715 028779 63 67 028842 028906 028970 029034 029097 029161 029225 029289 02935● 029417 64 68 029481 029545 029610 029674 029738 029802 029867 029931 029996 030060 65 69 030184 030189 030253 030318 030383 030447 030512 030577 030642 030707 65 70 030772 030837 030902 030967 031032 031097 031163 031228 031293 031359 65 71 031424 031489 031555 031620 031686 031751 031817 031883 031949 032014 66 72 032080 032146 032212 032278 032344 032409 032476 032542 032608 032674 66 73 032741 032807 032873 032939 033006 033072 033139 033205 033272 033338 66 74 033405 033472 033538 033605 033672 033738 033875 033879 033939 034006 67 75 034073 034140 034208 034275 034342 034409 034477 034544 034612 034679 67 76 034746 034814 034881 034949 035016 035084 035152 035219 035287 035355 68 77 035423 035491 035559 035627 035695 035763 035831 035899 035968 036036 68 78 036104 036172 036249 036309 036378 036446 036515 036583 036651 036720 69 79 036789 036858 036927 036995 037064 037133 037202 037271 037339 037408 69 80 037477 037546 037615 037684 037752 037822 037891 037961 038030 038099 69 81 038169 038239 038308 038378 038447 038517 038587 038657 038727 038797 70 82 038867 038937 039007 039077 039147 039217 039287 039357 039428 039498 70 83 039568 039638 039709 039779 039849 039919 039990 040061 040131 040202 71 84 040272 040343 040414 040485 040555 040636 040707 040778 040849 040920 71 85 040981 041052 041123 041194 041265 041336 041407 041479 041550 041621 71 86 041692 041764 041835 041907 041978 042050 042122 042193 042265 042336 72 87 042408 042480 042552 042624 042696 042768 042840 042912 042984 043056 72 88 043128 043200 043272 043345 043417 043489 043562 043634 043706 043779 73 89 043852 043924 043697 044069 044142 044214 044287 044360 044433 044505 73 90 044578 044651 044724 044797 044870 044943 045016 045089 045163 045236 73 91 045309 045382 045456 045529 045603 045676 045749 045823 045896 045969 74 92 046043 046117 046190 046264 046338 046411 046485 046559 046633 046707 74 93 046781 046855 046929 047003 047077 047151 047●25 047299 047374 047448 74 94 047522 047596 047671 047745 047819 047894 047969 048043 048118 048192 75 95 048267 048342 048417 048491 048566 048641 048716 048791 048866 048941 75 96 049015 049091 049166 049241 049316 049391 049466 049542 049617 049692 75 97 049767 049843 049918 049994 050069 050144 050226 050296 050371 050447 75 98 050522 050598 050674 050750 050826 050901 050977 051053 051129 051205 76 99 051281 051358 051434 051510 051586 051662 051738 051815 051891 051968 76 100 052044 052120 052197 052273 052350 052426 052503 052579 052656 052733 76 0 1 2 3 4 5 6 7 8 9 D 100 052044 052120 052197 052273 052350 052426 052503 052579 052656 052733 076 101 052810 052886 052963 053040 053117 053193 053271 053348 053425 053502 077 102 053579 053656 053733 053810 053887 053964 054041 054119 054196 054273 077 103 054351 054428 054506 054583 054661 054738 054816 054893 054973 055049 078 104 055127 055204 055282 055360 055438 055516 055594 055672 055750 055828 078 105 055906 055984 056062 056140 056218 056296 056374 056453 006531 056610 078 106 056688 056766 056845 056923 057002 057080 057159 057237 057316 057395 079 107 057474 057552 057631 057710 057789 057868 057946 058025 058104 058183 079 108 058262 058341 058420 058499 058578 058658 058737 058816 058895 058975 079 109 059054 059133 059213 059292 059372 059451 059531 059610 059690 059769 079 110 059849 059929 060009 060088 060168 060248 060328 060408 060488 060560 080 111 060648 060728 060808 060888 060968 061048 061128 061208 061289 061369 080 112 061449 061529 061610 061690 061771 061851 061932 062012 062093 062173 080 113 062254 062334 062415 062496 062576 062657 062738 062819 062900 062981 081 114 063062 063143 063224 063305 063386 063467 063548 063629 063710 063791 081 115 063873 063954 064035 064116 064198 064279 064360 064442 064523 064605 081 116 064686 064768 064849 064931 065013 065095 065176 065258 065340 065421 082 117 065503 065585 065667 065749 065831 065913 065995 066077 066159 066241 082 118 066323 066405 066488 066570 066652 066735 066817 066899 066981 067064 082 119 067146 067229 067311 067393 067476 067559 067642 067724 067807 067889 083 120 067972 068055 068138 068221 068304 068387 068469 068552 068635 068718 083 121 068801 068884 068967 069051 069134 069217 069300 069384 069467 069550 083 122 069633 069717 069800 069884 069967 070051 070134 070218 070301 070385 083 123 070468 070552 070635 070719 070803 070887 070970 071054 071138 071222 084 124 071306 071390 071474 071558 071642 071726 071810 071895 071979 072063 084 125 072148 072232 072316 072400 072484 072569 072653 072733 072822 072906 084 126 072991 073075 073160 073244 073329 073414 073498 073583 073668 073752 084 127 073837 073922 074007 074092 074177 074262 074347 074432 074517 074602 085 128 074687 074772 074857 074942 075027 075112 075197 075283 075368 075453 085 129 075539 075624 075709 075795 075880 075966 076051 076137 076222 076908 085 130 076394 076479 076565 076651 076736 076822 076908 076994 077079 077165 086 131 077251 077337 077423 077509 077595 077681 077767 077853 077939 078025 086 132 078112 078198 078184 078370 078457 078543 078629 078716 078802 078889 086 133 078975 079062 079148 079235 079321 079408 079494 079581 079668 079754 086 133 078975 079062 079148 079235 079321 079408 079494 079581 079668 079754 086 134 079841 079928 080015 080101 080188 080275 080362 080449 080536 080623 087 135 080710 080797 080885 08g972 081059 081147 081234 081321 081408 081495 087 136 081582 081669 081756 081841 081931 032018 082106 082193 082281 082368 087 137 082456 082543 082631 082718 082806 082894 082981 083069 083157 083245 087 138 083333 083420 083508 083596 083684 083772 083860 083948 084036 084124 088 139 084212 084300 084388 084477 084565 084653 084741 084830 084918 085006 088 140 085095 085183 085271 085359 085448 085536 085625 085714 085802 085891 088 141 085979 086068 086157 086246 086334 086423 086512 086601 086689 086778 089 142 086867 086956 087045 087134 087223 087312 087401 087490 087579 087668 089 143 087757 087846 087935 088025 088114 088203 088292 088383 088471 088561 089 144 088650 088740 088829 088920 089009 089099 089188 089278 089366 089456 090 145 089545 089635 089725 089814 089904 089994 090084 090174 090264 090354 090 146 090444 090533 090623 090713 090803 090893 090983 091073 091163 091253 90 147 091344 091434 091524 091614 091704 091795 091885 091975 092066 092156 90 148 092247 092337 092427 092518 092608 092699 092790 092880 092971 093061 90 149 093152 093243 093333 093424 0●3515 093606 093696 093787 093878 093969 91 150 094060 094151 094242 094333 094424 094515 094606 094697 094788 094879 91 151 094971 095062 095153 095244 095335 095427 095518 095609 095701 095792 91 152 095884 095975 096067 096158 096249 096341 096433 096524 096616 096707 92 153 096799 096891 096982 097074 097166 097258 097349 097441 097533 097625 92 154 097717 097809 097901 097993 098085 098177 098269 098361 098453 098545 92 155 098637 098729 098822 098914 099006 099098 099191 099283 099375 099468 92 156 099569 099653 099745 099838 099830 100023 100115 100208 100300 100393 93 157 100486 100578 100671 100763 100856 100949 101042 101135 101227 101320 93 158 101413 101506 101599 101692 101785 101878 101971 102064 102157 101250 93 159 102343 102436 102529 102622 102715 102809 102902 102995 103088 103182 93 160 103275 103368 103462 103555 103649 103742 103836 103929 104023 104116 93 161 104210 104304 104397 104491 104584 104678 104772 104866 104959 105053 94 162 105147 105241 105335 105429 105523 105617 105711 105805 105899 105991 94 193 106087 106181 106275 106369 106463 106557 106651 106745 106840 106934 94 164 107028 107122 107217 107311 107406 107500 107594 107689 107783 107878 94 165 107972 108067 108161 108256 108350 108445 108540 108634 108727 108824 95 166 108919 109013 109108 109203 109298 109393 109488 109583 109678 109773 95 166 108919 109013 109108 109203 109298 109392 109487 109683 109677 109773 95 167 109867 109963 110058 110153 110248 110343 110438 110533 110628 110723 95 168 110818 110913 111009 111104 111295 111390 111485 111485 111581 111676 95 169 111772 111867 111963 112058 112154 112249 112344 112440 112535 112631 96 170 112726 112822 112918 113013 113109 113205 113301 113397 113492 113588 96 171 113684 113780 113876 113972 114068 114164 114260 114356 114452 114548 96 172 114644 114740 114836 114932 115028 115125 115221 115318 115414 115511 96 173 115607 115703 115799 115896 115992 116088 116184 116281 116377 116473 97 174 116570 116666 116763 116860 116956 117053 117150 117246 117343 117439 97 175 117536 117633 117730 117827 117924 118021 118118 118215 118312 118409 97 176 118506 118603 118700 118797 118894 118991 119088 119185 119282 119379 97 177 119476 119574 119671 119768 119866 119963 120060 120158 120255 120352 97 178 120449 120547 120644 120742 120839 120936 121034 121132 121229 121327 98 179 121424 121562 121620 121717 121815 121913 122010 122108 122206 122304 98 180 122402 122500 122597 122696 122794 122892 122990 123088 123186 123284 98 181 123382 123480 123578 123676 123774 123872 123970 124068 124167 124265 98 182 124363 124461 124560 124658 124756 124855 124953 125051 125150 125248 98 183 125347 125445 125544 125642 125741 125840 125938 126037 126136 126234 98 184 126333 126432 126530 126629 126728 126827 126925 127024 127123 127221 99 185 127320 127419 127518 127617 127716 127815 127914 128013 128112 128211 99 186 128310 128409 128508 128607 128706 128805 128905 129004 129103 129202 99 187 129302 129401 129500 129600 129699 129799 129897 139997 130096 130196 99 188 130296 130395 130495 130594 130694 130794 130893 130993 131093 131192 100 189 131292 131392 131491 131591 131691 131791 131890 131990 132090 132190 100 190 132290 132390 132490 132590 132690 132790 132890 132990 133090 133190 100 191 133290 133390 133490 133590 133690 133790 133890 133990 134090 134191 100 192 134292 134392 134492 134593 134693 134794 134894 134994 135095 135195 100 193 135296 135396 135497 135597 135697 135798 135899 135999 136100 136201 101 194 136302 136402 136503 136604 136704 136805 136906 137007 137108 137209 101 195 137310 137410 137511 137612 137713 137814 137915 138016 138117 138218 101 196 138320 138421 138522 138623 138724 138825 138926 139027 139129 139230 101 197 139331 139432 139534 139635 139737 139838 139939 140041 140142 140244 101 198 140345 140447 140548 140650 140751 140853 140954 141055 141157 141259 101 199 141361 141462 141564 141665 141767 141869 141970 142072 142174 142276 102 200 142378 142480 142582 142684 142786 142888 142990 143092 143194 143296 102 201 143398 142500 143602 143704 143806 143909 144011 144113 144215 144317 10● 202 144419 144521 144624 144726 144828 144931 145033 145135 145238 145340 102 203 145443 145545 145648 145750 145853 145955 146058 146160 146263 146365 102 204 146468 146571 146673 146776 146878 146981 147084 147187 147289 147392 102 205 147495 147598 147701 147803 147906 148009 148112 148215 148318 141421 103 206 148524 148627 148730 148833 148936 149039 149142 149245 149348 149451 103 207 149555 149658 149761 149864 149967 150071 150174 150277 150387 150484 103 208 150588 150691 150794 150898 151001 151105 151208 151312 151415 151519 103 709 151623 151726 151829 151933 151037 152141 152244 152348 152452 152555 103 210 152659 152762 152866 152970 153074 153178 153281 153385 153489 153592 103 211 153696 153800 153903 154009 154111 154215 154319 154423 154523 154632 103 212 154736 154840 154944 155048 155152 155257 155361 155465 155569 155674 104 213 155778 155882 155987 156092 156195 156300 156404 156508 156613 156717 104 214 156822 156926 157030 157135 157239 157344 157448 157553 157657 157762 104 215 157867 157971 158076 158180 158285 158390 158494 158599 158704 158809 105 216 158914 159018 159123 159228 159333 159438 159453 159648 159753 159858 105 217 159963 160068 160173 160278 160383 160488 160593 160698 160803 160908 105 218 161013 161118 161223 161329 161434 161539 161644 161749 161855 161959 105 219 162065 162170 162276 162387 162487 162592 162697 162803 162908 162014 105 220 163119 163225 163330 163436 163542 163648 163753 163859 163964 164070 106 221 164176 164281 164387 164493 164598 164704 164810 164916 165021 165127 106 222 165233 165339 165445 165550 165656 165762 165868 165974 166080 166186 106 223 166292 166398 166504 166610 166716 166822 166928 167034 167141 167247 106 224 167353 167459 167565 167672 167778 167884 167990 168096 168203 168309 106 225 168415 168521 168521 168734 168841 168947 169053 169160 169266 169373 107 226 169479 169586 169692 169799 169905 170012 170119 170225 120332 170438 107 227 170545 170652 170758 170865 170972 171079 171185 171292 171399 171506 107 228 171613 171719 171826 171933 172040 172147 172254 172361 172468 172575 107 229 172682 172789 172896 173003 173110 173217 173324 173431 173538 173645 107 230 173753 173860 173967 174074 174181 174289 174396 174503 174610 174718 107 231 174825 174932 175040 175147 175255 175362 175469 175577 175684 175792 107 232 175899 176007 176114 176222 176329 176437 176554 176652 176759 176867 108 233 176975 177082 177190 177298 177405 177513 177621 177729 177836 177944 108 0 1 2 3 4 5 6 7 8 9 233 176975 177082 177190 177298 177405 177513 177621 177729 177836 177944 108 234 178052 178160 178268 178375 178483 178591 178699 178807 178915 179027 108 235 179131 179239 179347 179455 179563 179672 179780 179888 179996 180104 108 236 180212 180320 180428 180536 180644 180752 180860 180969 181077 181085 108 237 181294 181402 181510 181618 181727 181835 181943 182052 182160 182269 108 238 182377 182486 182594 182703 182811 182920 183028 183137 183245 183354 108 239 183462 183571 183680 183788 183897 184006 184114 184223 184332 184440 109 240 184549 184658 184767 184875 184984 185093 185202 185311 185419 185528 109 241 185637 185746 185855 185964 186073 186182 186291 186400 186509 186618 109 242 186728 186837 186946 187055 187164 187273 187382 187491 187600 187709 109 243 187819 187928 188037 188146 188256 188365 188474 188584 188693 188802 109 244 188911 189021 189130 189240 189349 189458 189568 189677 189787 189896 109 245 199006 190116 190225 190335 190444 190554 190664 190773 190883 190992 110 246 191102 191212 191322 191431 191541 191651 191761 191871 191980 192090 110 247 192200 192310 192420 192529 192639 192749 192859 192969 193079 193189 110 248 193298 193408 193518 193628 193738 193848 193958 194068 194178 194288 110 249 194399 194509 194619 194729 194839 194949 195059 195169 195279 195389 110 250 195500 195611 195721 195831 195942 196052 196163 196273 196384 196494 110 251 196605 196715 196826 196936 197047 197157 197268 197378 197489 197599 110 252 197710 197821 197931 198041 198152 198262 198373 198484 198594 198705 110 253 198815 198926 199037 199148 199258 199369 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FINIS .

COSMOGRAPHIA , THE Second Part. OR , THE DOCTRINE OF THE PRIMUM MOBILE .

AN INTRODUCTION TO Astronomy . The First Part.

Of the Primum Mobile .

CHAP. I.

Of the General Subject of Astronomy .

AStronomy , is a Science concerning the Measure and Motion of the Spheres and Stars .

2. Astronomy hath two parts , the first is Absolute , and the other Comparative .

3. The Absolute part of Astronomy is that which treateth of the Measure and Motion of the Orbs and Stars absolutely without respect to any distinction of Time .

4. The Comparative part of Astronomy is that , which treateth 〈◊〉 the Motion of the Stars , in reference to some certain distinction of Time .

5. The Absolute part of Astronomy treateth of the Primum Mobile , or Diurnal Motion of all the Celestial Orbs or Spheres .

6. The Primum Mobile , or Diurnal Motion of the Heavens , is that Motion , by which the several Spheres are moved round the World in a Day 〈…〉 from East towards West , and ●o forward● from West towards East , and so continually returning to the same point from whence they began their Motion .

7. This first and common Motion of the Heavens , will be best understood , by help of an Instrument called a Globe , which is an Artificial representation of the Heavens , or the Earth and Waters under that Form and Figure of Roundness which they are supposed to have .

8. This Representation or Description of the Visible World is by Circles , great and small , some of which are expressed upon , and others are framed without the Globe .

9. The Circles without the Globe are chiefly two ; the Meridian and the Horizon , the one of Brass , and the other of Wood ▪ And these two Circles are variable or mutable ; for although there is but one Horizon and one Meridian in respect of the whole World , or in respect of the whole Heaven and Earth , yet in respect of the particular parts of Heaven , or rather in respect of the diverse Provinces , Countries and Cities on the Earth , there are diverse both Horizons and Meridians .

10. The Meridian then is a great Circle without the Globe , dividing the Globe , and consequently the Day and Night into two equal parts , from the North and South ends whereof a strong Wyre of Brass or Iron is drawn or supposed to be drawn through the Center of the Globe representing the Axis of the Earth , by means whereof the whole Globe turneth round within the said Circle , so that any part may be brought directly under this Brass Meridian at pleasure .

11. This Brass Meridian is divided into 4 equal parts or Quadrants , and each of them are subdivided into 90 Degrees , that is 360 for the whole Circle . The reason why this Circle is not divided in 360 Degrees throughout , but still stoping at 90 , beginneth again with 10. 20. 30 &c. is , for that the use of this Meridian , in reference to its Division in Degrees , requireth no more than that Number .

12. The Horizon is a great Circle without the Globe , which divides the upper part of Heaven from the lower , so that the one half is always above that Circle , and the other under it .

13. The Poles of this Circle are two , the one directly over our Heads , and is called the Zenith ; the other is under feet , and is called the Nadir .

14. The Horizon is either Rational or Sensible .

15. The Rational Horizon is that , which divideth the Heavens and the Earth into two equal parts , which though it cannot be perceived and distinguished by the eye , yet may be conceived i● our minds , in which respect all the Stars may be conceived to rise and set as in our view .

16. The Visible Horizon is that Circle which the eye doth make at its farthest extent of sight , when the body in any particular place doth turn it self round . Of these two Circles there needeth no more to be said at present , only we may observe , that it was ingeniously devised by those , who first thought upon it , to set one Meridian and one Horizon without the Globe , to avoid the confusion , if not the impossibility , of drawing a several Meridian and a several Horizon for every place , which must have been done if this or the like device had not been thought upon .

17. Besides these two great Circles without the Globe , there are 4 other great Circles drawn upon the Globe it self besides the Meridian . 1. The AEquator or Equinoctial Circle . 2. The Zodiack . 3. The AEquinoctial Colure . 4. Solstitial Colure . And these four Circles are imm●table , that is , in whatsoever part of the World you are , these Circles have no variation , as the other two have .

18. The AEquator is a great Circle drawn upon the Globe , in the middle between the two Poles ▪ and plainly dividing the Globe into two equal parts .

19. The AEquator is the measure of the Motion of the Primum Mobile , for 15 Degrees of this Circle do always arise in an hours time ; the which doth clearly shew , that the whole Heavens are turned round by equal intervals in the space of one day or 24 hours .

20. In this Circle the Declinations of the Stars are computed from the mid-Heaven towards the North or South .

21. This Circle gives denomination to the AEquinox , for the Sun doth twice in a Year and no more cross this Circle , to wit , when he enters the first points of Aries , and Libra , and then he maketh the Days and the Nights equal : His entrance into Aries is in March , and is called the Vernal Equinox ; and his entrance into Libra , is in September , and is called the Autumnal Equinox .

22. And from one certain point in this Circle , the Longitude of Places upon the Earth are reckoned ; and the Latitude of Places are reckoned from this Circle towards the North , or the South Poles .

23. The Zodiack is a great Circle drawn upon the Globe , cutting the AEquinoctial Points at Oblique Angles : for although it divides the whole World into two equal parts , in reference to its own Poles ; yet in reference to the Poles of the World , it hath an Oblique Motion .

24. The Poles of this Circle are as far distant from the Poles of the World , as the greatest Obliquity thereof is from the Equinoctial , that is 23 Degrees , and 31 Minutes or thereabouts .

25. This Circle doth differ from all other Circles upon the Globe in this : other Circles ( to speak properly ) have Longitude assigned them , but no Latitude ; but this hath both . Whereas other Circles are in reference to their Longitude or Rotundity only divided into 360 Degrees , this Circle in respect of its Latitude is supposed to be divided into 16 Degrees in Latitude .

26. The Zodiack then in respect of Longitude is commonly divided into 360 Degrees as other Circles are : but more peculiarly in respect of its self it is divided into 12 Parts called Signs , and each Sign into 30 Degrees , and 12 times 30 do make 360.

27. The 12 Signs into which the Zodiack is divided , have these Names and Characters . Aries ♈ . Taurus ♉ . Gemini ♊ . Cancer ♋ . Leo ♌ . Virgo ♍ . Libra ♎ . Scorpio ♏ . Sagittarius ♐ . Capricornus ♑ . Aquarius ♒ . and Pisces ♓ .

28. These two Circles of the Equator and Zodiack are crossed by two other great Circles , which are called Colures : They are drawn through the Poles of the World , and cut one another as well as the Equator at Right Angles . One of them passeth through the Intersections of the Equinoctial points , and is called the Equinoctial Colure . The other passeth through the points of the greatest distance of the Zodiack , from the Equator , and is called the Solstitial Colure .

29. The other great Circles described upon the Globe are the Meridians : Where we must not think much to hear of the Meridians again . That of Brass without the Globe is to serve all turns , and the Globe is framed to apply it self thereto . The Meridians upon the Globe , will easily be perceived to be of a new and another use .

30. The Meridians upon the Globe are either the great or the less : Not that the great are any greater than the less , for they have all one and the same center , and equally pass through the Poles of the Earth ; But those which are called less , are of less use than that , which is called the great .

31. The great is otherwise called the fixt and first Meridian , to which the less are second , and respectively moveable . The great Meridian is as it were the Landmark of the whole Sphere , from whence the Longitude of the Earth , or any part thereof is accounted . And it is the only Circle which passing through the Poles is graduated or divided into Degrees , not the whole Circle but the half , because the Longitude is to be reckoned round about the Earth .

32. The lesser Meridians are those black lines ; which you see to pass through the Poles and succeeding the great at 10 and 10 Degrees , as in most Globes ; or at 15 and 15 Degrees difference , as in some . Every place never so little more East or West than another , hath properly a several Meridian , yet because of the huge distance of the Earth from the Heavens , there is no sensible difference between the Meridians of places that are less than one Degree of Longitude asunder , and therefore the Geographers as well as the Astronomers allow a new Meridian to every Degree of the Equator ; which would be 180 in all : but except the Globes were made of an extream and an unusual Diameter , so many would stand too thick for the Description . Therefore most commonly they put down but 18 , that is , at 10 Degrees distance from one another ; the special use of the lesser Meridians being to make a quicker dispatch , in the account of the Longitudes . Others set down but 12 , at 15 Degrees difference ; aiming at this , That the Meridians might be distant from one another a full part of time , or an hour : for seeing that the Sun is carried 15 Degrees of the Equinoctial every hour , the Meridians set at that distance must make an hours difference in the rising or setting of the Sun in those places which differ 15 Degrees in Longitude .

And to this purpose also upon the North end of the Globe , without the Brass Meridian , there is a small Circle of Brass set , and divided into two equal parts , and each of them into twelve , that is , twenty four all ; to shew the hour of the Day and Night , in any place where the Day and Night exceed not 24 hours ; for which purpose it hath a little Brass Pin turning about upon the Pole , and pointing to the several hours , which is therefore the Index Horarius , or Hour Index .

33. Having described the great Circles framed without and drawn upon the Globe , we will now describe the lesser Circles also ; And these lesser Circles are called Parallels , that is , such as are in all places equally distant from the Equator ; and these Circles how little soever , are supposed to be divided into 360 Degrees : but these Degrees are not so large as in the great Circles , but do proportionably decrease according to the Radius by which they are drawn .

34. These lesser Circles are either the Tropicks or the Polar Circles .

35. The Tropicks are two small Circles drawn upon the Globe , one beyond the Equator towards the North Pole , and the other towards the South , Shewing the way which the Sun makes in his Diurnal Motion , when he is at his greatest distance from the Equator either North or South . These Circles are called Tropicks 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , that is , from the Suns returning : for the Sun coming to these Circles , he is at his greatest distance from the Equator , and in the same Moment of time sloping as it were his course , he returns nearer and nearer to the Equator again .

36. These Tropical Circles do shew the point of Heaven in which the Sun doth make either the longest Day , or the Shortest Day in the Year , according as he is in the Northern or the Southern Tropick : And are drawn at 23 Degrees and a half distant from the Equator .

37. The Polar Circles are two lesser Circles drawn upon the Globe at the Radius of 23 Degrees and a half distant from the Poles of the World , shewing thereby the Poles of the Zodiack , which is so many Degrees distant from the Equator on both sides thereof .

38. These Polar Circles are 66 Degrees and a half distant from the Equator , and 43 Degrees distant from his nearest Tropick . They are called the Arctick and Antarctick Circles .

39. The Arctick Circle is that which is described about the Arctick Pole , and passeth almost through the middle of the Head of the greater Bear . It is called the Arctick Circle 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 from the two conspicuous Stars towards the North ; called the greater and the lesser Bear .

40. The Antarctick Circle is that which is described about the Antarctick or South Pole . It is so called 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 that is , from being opposite to the greater and lesser Bear .

Having thus described the Globe or Astronomical Instrument by which the Frame of the World is represented to our view , I will proceed to shew the use for which it is intended .

CHAP. II.

Of the Distinctions and Affections of Spherical Lines or Arches .

THE uses of the Globe as to practice , are either such as concern the Heavens or the Earth , in either of which , if we should descend unto particulars , the uses would be more in number , than a short Treatise will contain : Seeing therefore that all Problems which concern the Globe , may be best and most accurately resolved by the Doctrine of Spherical Triangles , we will contract these uses of the Globe ( which otherwise might prove infinite ) to such Problems as come within the compass of the 28 Cases of Right and Oblique angled Spherical Triangles .

2. And that the nature of Spherical Triangles may be the better understood , and by which of the 28 Cases the particular Problems may be best resolved , I will set down some General Definitions and Affections , which do belong to such Lines or Arches of which the Triangle must be framed , with the Parts and Affections of those Triangles , and how the things given and required in them , may be represented and resolved upon and by the Globe , as also how they may be represented and resolved by the Projection of the Sphere , and by the Canon of Triangles .

3. A Spherical Triangle then is a Figure consisting of three Arches of the greatest Circles upon the Superficies of a Sphere or Globe , every one being less than a semicircle .

4. A great Circle is that which divideth the Sphere or Globe into two equal parts , and thus the Horizon , Equator , Zodiack and Meridians before described are all of them great Circles : And of these Circles or any other , there must be three Arches to make a Triangle , and every one of these Arches severally must be less than a semicircle : To make this plain .

In Fig. 1. The streight Line HAR doth represent the Horizon , PR the height of the Pole above the Horizon , PMS a Meridian , and these three Arches by their intersecting one another do visibly constitute four Spherical Triangles . 1. PMR . 2. PMH . 3. SHM. 4. SMR . And every Arch is less than a semicircle , as in the Triangle PMR , the Arch PR is less than the Semicircle PRS , the Arch MR is less than the Semicircle AMR , and the Arch PM is less than the Semicircle PMS , the like may be shewed in the other Triangles .

5. Spherical or circular Lines are Parallel or Angular .

6. Parallel Arches or Circles , are such as are drawn upon the same Center within , without , or equal to another Arch or Circle . Thus in Fig. 1. The Arches ♋ M ♋ and ♑ O ♑ are though lesser Circles , parallel to the Equinoctial AE A Q and do in that Scheme represent the Tropicks of Cancer and Capricorn . The manner of describing them or any other Parallel Circle is thus , set off their distance from the great Circle , to which you are to draw a parallel with your Compasses , by help of your Line of Chords , which in this Example is 23 Degrees and a half from AE to ♋ , then draw the Line A ♋ , and upon the point ♋ erect a Perpendicular , where that Perpendicular shall cut the Axis PAS extended , is the Center of that Parallel .

7. A Spherical Angle , is that which is conteined by two Arches of the greatest Circles upon the Superficies of the Globe intersecting one another : Angles made by the Intersection of two little Circles , or of a little Circle with a great , we take no notice of in the Doctrine of Spherical Triangles .

8. A Spherical Angle is either Right or Oblique .

9. A Spherical Right Angle is that which is conteined , by two Arches of the greatest Circles in the Superficies of the Sphere cutting one another at Right Angles , that is , the one being right or perpendicular to the other : thus the Brass Meridian cutteth the Horizon at right Angles ; and thus the Meridians drawn upon the Globe , as well as the Brass Meridian , do all of them cut the Equator at Right Angles .

10. An Oblique Spherical Angle , is that which is conteined by two Arches of the greatest Circles in the Superficies of the Sphere , not being right or perpendicular to one another .

11. An Oblique Spherical Angle is Obtuse , or Acute .

12. An Obtuse Spherical Angle , is that which is greater than a Right Angle . An Acute is that which is less than a Right Angle .

13. If two of the greatest Circles of the Sphere shall pass through one anothers Poles , those two great Circles shall cut one another at Right Angles : Thus the Brazen Meridian doth intersect the Equinoctial and Horizon .

14. If two of the greatest Circles of the Sphere shall intersect one another , and pass through each others Poles , they shall intersect one another at unequal or Oblique Angles , the Angle upon the one side of the intersection being Obtuse , or more than a Right , and the Angle upon the other side of the intersection being Acute or less than a Right . Thus in Fig. 1. The Arch PM doth intersect the Meridian and Horizon , but not in the Poles of either , therefore the Angle HPM upon one side of the intersection of that Arch with the Meridian , is more than a Right Angle ; And the Angle MPR upon the other side of the Intersection is less . And so likewise the Angle PMH upon the one side of the intersection of the Arch PM with the Horizon HR , is greater than a right Angle ; and the Angle RMP upon the other side of the Intersection is less than a Right .

15. A Spherical Angle is measured by the Arch of a great Circle described from the Angular point between the sides of the Angle , those sides being continued unto Quadrants . Thus the Arch of the Equator TQ in Fig. 1. is the measure of the Angle MPR , or TPQ , the sides PT and PQ being Quadrants .

And the measure thereof in the Projection may thus be found : lay a Ruler from P to T , and it will cut the Primitive Circle in V ; and the Arch VQ being taken in your Compasses and applyed to your Line of Chords , will give the Quantity of the Angle propounded .

16. The Complement of a Spherical Arch or Angle , is so much as it wanteth of a Quadrant , if the Arch or Angle given be less than a Quadrant ; or so much as it wanteth of a Semicircle , if it be more than a Quadrant .

17. An Arch of a great Circle cutting the Arch of another great Circle , shall intersect one another at Right Angles , or make two Angles ; which being taken together , shall be equal unto two Right . Thus in Fig. 1. The Axis PAS or Equinoctial Colure doth cut the Equator AE A Q at Right Angles ; but the Meridian PMS doth cut the Horizon HMR at Oblique Angles , making the Angle PMR less than a Right , and the Angle SMR more than a Right , and both together equal to a Semicircle .

18. From these general Definitions proper to Spherical Lines or Arches , the general Affections of these Arches may easily be discerned ; I mean the various Positions of the Globe of the Earth , in respect of all and singular the Inhabitants thereof .

19. And the whole Body of the Sphere or Globe , in respect of the Horizon , is looked upon by the Earths Inhabitants , either in a Parallel , a Right , or an Oblique Sphere .

20. A Parallel Sphere is , when one of the Poles of the World is elevated above the Horizon to the Zenith , the other depressed as low as the Nadir , and the Equinoctial Line joyned with the Horizon . They which there inhabite ( if any such be ) see not the Sun or other Star rising or setting , or higher or lower in their diurnal revolution . And seeing that the Sun traverseth the whole Zodiack in a Year , and that half the Zodiack , is above the Horizon and half under it , it cometh to pass , that the Sun setteth not with them , for the space of six Months , nor giveth them any Light for the space of other six Months , and so maketh but one Day and Night of the whole Year .

21. A Right Sphear is , when both the Poles of the World do lie in the Horizon , and the Equinoctial Circle is at his greatest distance from it , passing through the Zenith of the place . And in this position of the Sphere , all the Coelestiall Bodies , Sun , Moon , and other Planets , and fixed Stars , by the daily turning about of the Heaven , do directly ascend above , and also directly descend below the Horizon , because the Motions which they make in their Daily motion do cut the Horizon Perpendicularly , and as it were at Right Angles . In this Position of the Sphere , all the Stars may be observed to rise and set in an equal space of time , and to continue as long above the Horizon , as they do under it , the Day and Night to those Inhabitants , being always of an equal length .

22. An Oblique Sphere is , when the Axis of the World ( being neither Direct nor Parallel to the Horizon ) is inclined obliquely towards both sides of the Horizon , as in Fig. 1. Whence it cometh to pass ; that so much as one of the Poles is elevated above the Horizon , upon the one side ; so much is the other depressed under the Horizon , upon the other side .

And in this Position of Sphere , the Days are sometimes longer than the Nights , sometimes shorter , and sometimes of equal length . When the Sun is in either of the Equinoctial Points , the Days and Nights are equal ; but when he declineth from the Equator towards the elevated Pole , the Days are observed to encrease ; and when he declineth from the Equator towards the opposite Pole , or the Pole depressed , the Days do decrease ▪ as is manifest in Fig. 1. For when the Sun riseth at M , the Line M ♋ above the Horizon is the Semidiurnal Arch of the longest day . When he riseth at C , the Arch C ♑ above the Horizon , is the Semidiurnal Arch of the shortest Day : And when he riseth at A , the Days and Nights are of equal Length , the Semidiurnal Arch AAE , being equal to the Seminocturnal Arch AQ .

CHAP. III.

Of the kind and parts of Spherical Triangles ; and how to project the same upon the Plane of the Meridian .

HAving shewed what a Spherical Triangle is , and of what Circles it is composed , with the general Affections of such Lines : I will now shew how many several sorts of Triangles there are , of what Circular parts they do consist , and such Affections proper to them as will render the so●ition of them more clear and certain .

2. Spherical Triangles are either Right or Oblique .

3. A Right Angled Spherical Triangle , is that which hath one or more Right Angles .

4. A Spherical Triangle which hath three Right Angles , hath always his three sides Quadrants . As in Fig. 1. The Spherical Triangle AZR , the Angles ZRA , RAZ and AZR are right Angles , and the three sides AZ , ZR and AR are Quadrants also .

5. A Triangle that hath two right Angles , hath the sides opposite to those Angles Quadrants , and the third side is the measure of the third Angle . As in Fig. 1. The sides of the Spherical Triangle TPQ , namely TP and PQ are Quadrants , and the Angles opposite to these sides , to wit , PTQ and TQP are Quadrants also , and the third Angle TQ is the measure of the third Angle TPQ . But the Right Angled Triangle which hath one Right and two Acute Angles , is that which cometh most commonly to be resolved .

6. The Legs of a right Angled Spherical Triangle are of the same Affection with their opposite Angles ; as in the Triangle ZAR Fig. 1. The side ZA is a Quadrant , and the Angle at A is right , because Z is the Pole of the Arch AR and ZA is perpendicular thereunto . And in the Triangle RAAE the side RZAE being more then a Quadrant the Angle RAAE is more then a Quadrant also , being more then the Right Angle RAZ . And in the right Angled Spherical Triangle APR the side PR being less then a Quadrant , the Angle PAR is less then a Quadrant also , being less then the right Angle RAZ .

7. An Oblique angled Spherical Triangle is either acute or obtuse .

8. An Acute angled Spherical Triangle hath all his Angles Acute , and each Side less then a Quadrant ; As in the Triangles , ZFP . Fig. 2. The Angles at Z and P are acute , as appeareth by inspection ; and the Angle at F is acute also because the Measure thereof CD = EM is less then a Quadrant .

9. An Oblique Angled Spherical Triangle hath all his Angles either acute or obtuse : viz. Acute and mixt .

10. The Sides of a Spherical Triangle may be turned into Angles , and the Angles into Sides ; The Complement of the greatest Side or greatest Angle to a Semicircle being taken in each conversion . For Example . If it were required to turn the Angles of the Oblique Angled Spherical Triangle ZFP into sides in Fig. 3. EAE is the measure of the Angle at P , and AD in the Triangle ADC equal thereunto , AC is the Complement of FZP to a Semicircle , and KM the the Measure of the Angle at F is equal to DC , and so the Sides of the Spherical Triangle ADC are equal to the Angles of the Spherical Triangle FZP , making the side AC equal to the Complement of the Angle Z to a Semicircle .

11. In Right Angled Spherical Triangles the Sides intending the Right Angle we call the Legs ; The Side subtending it the Hypotenuse .

12. In every Spherical Triangle besides the Area or space contained , there are six parts . viz. Three Sides and three Angles and of these six there must be always three given to find the rest , but in right Angled Spherical Triangles there are but five of the six parts parts which come into question , because one of the Angles being right is allways known , and so any two of the other five being given , the three remaining parts whether Sides or Angles , may be found . But before I come to the solution of these Triangles whether right or oblique , I will first shew how they may be represented upon the Globe , and projected upon the plane of of the Meridian .

13. A right Angled Spherical Triangle may be represented upon the Globe in this manner : Elevate one of the Poles of the Globe above the Horizon , to the quantity of one of the given Legs , so shall the distance between the AEquinox and the Zenith be equal thereunto , and at the Zenith fasten the Quadrant of altitude , so shall there be delineated upon the Globe the right Angled Spherical Triangle AEZB as may be seen in Fig. 1. In which the outward Circle HZR doth represent the Brass Meridian , AEAQ the Equator , and ZC the Quadrant of altitude .

14. An Oblique Angled Spherical Triangle may be represented upon the Globe in this manner . Number one of the given sides from one of the Poles to the Zenith ; and there fasten the Quadrant of Altitude , upon which number another side , the third upon the great Meridian , from the Pole towards the Equinoctial , then turn the Globe till the Side numbred upon the Quadrant of Altitude , and the Side numbred upon the great Meridian shall intersect one another ; so shall there be delineated upon the Globe the Oblique Angled Spherical Triangle ZFP in Fig. 3. In which ZP is numbred upon the Brass Meridian from S the Pole of the World to Z the Zenith , ZF the Azimuth Circle represents the Quadrant of Altitude , and PF the great Meridian upon the Globe intersecting the Quadrant of Altitude at F.

15. A Right or Oblique Angled Spherical Triangle being thus delineated upon the Globe , there needs no further instructions , as to the measure of the sides , all that is wanting , is the laying down the Angles comprehended by those sides , and the finding out the measure of these Angles being so laid down . And that this may be the better understood , I will first shew ; how the several Circles upon the Globe before described , may be projected upon the Plane of the Meridian , and the several useful Triangles that are described by such Projection with such Astronomical Propositions as are conteined and resolvable by these Triangles .

16. The Circles in the first Figure are the Meridian , AEquator , Horizon , AEquinoctial Colure , and the Tropicks . The Brass Meridian without the Globe , is a perfect Circle described by taking 60 Degrees from your Line of Chords , as the Circle HZRN in Fig. 1. Within which all the other are projected . The Horizon , AEquator , AEquinoctial Colure , East and West Azimuths are all streight Lines . Thus the Diameter HAR represents the Horizon , AEAQ the Equator , PAS the Equinoctial Colure and ZAN the East and West Azimuths , in the drawing of these there is no difficulty , PMS is a Meridian , and ZCN an Azimuth Circle , for the drawing of which there are three points given and the Centers of the Meridians do always fall in the Equinoctial extended if need be , the Centers of the Azimuth Circles do fall in the Horizon extended if need be , and for the drawing of these Circles there needs no further direction , supposing the middle point given to be in the AEquator or Horizon , but yet the Centers of these Circles may be readily found , by the Lines of Tangents or Secants , for the Tangent of the Complement of AT set from A to D , or the Secant of the Complement set from A to D will give the Center of the Meridian PTS. The other two Circles in the 1. Fig. are the Tropicks whose Centers are thus found ; each Tropick is Deg. 23 ½ from the Equinoctial , which distance being set upon the Meridian from AE to ♋ and AE to ♑ , if you draw a Line from A to ♋ and another perpendicular thereunto from ♋ it will cut the Axis SAP extended in the Center of that Tropick , by which extent of the compasses the other Tropick may be drawn also . Or thus the Co-tangent of AE ♋ set from ♋ to the Axis extended will give the Center as before , and thus may all other Parallels be described .

17. In the second and third figures , the two extream points given in the Meridians are not eqnidistant from the third , for the drawing of which Circles , if the common way of bringing three points into a Circle be not liked ; you may do thus , from the given point at F and the Center A draw the Diameter TAS , and cross the same at Right Angles with the Diameter BAG , a Ruler laid from G to F will cut the primitive Circle in L , make EL = BL a Ruler laid from G to E will cut the Diameter SAT in V the Center of the Circle BDG . Which Circle doth cut the Diameter HAR in the Pole of ZF , and the Diameter AEAQ in D in the Pole of PFX , and a Ruler laid from Z to C will cut the Primitive Circle in Y , and making Y O equal to Y a Ruler laid from Z to O will cut the Diameter HAR , extended in the Center of the Circle ZF .

18. Having drawn the Circle ZFI , in Fig. 13. The Circle PEX , or any other passing through the point F , may easily be described . Draw AEQ at right Angles to PX , a Ruler laid from G unto ( e ) will cut the Primitive Circle in ( m ) make mn = Bn , a Ruler laid from G to n shall cut the Diameter TFS in p make Fq = Fp so shall FQ be the Radius , and the Center of the Circle PFX as was desired .

19. The preceeding directions are sufficient for the projecting of several Circles of the Globe before described upon the Plane of the Meridian , and the parts of those Circles so described may thus be measured . In Fig. 1. HZ = CZ = AZ 90 Degrees . Whence it followeth , that the Quadrant CZ is divided into Degrees from its Pole M , by the Degrees of the Quadrant HZ , that is a Ruler laid from M to any part of the Quadrant HZ will cut as many Degrees in CZ as it doth in the Quadrant HZ , and thus the Arch CF = HK the Arch CB = HL , and the Arch BF = LK .

20. That which is next to be considered is the projecting or laying down the Angles of a Triangle , and the measuring of them being projected , and the Angles of a Triangle are either such as are conteined between two right Lines as the Angle A in the Triangle PAR ; or such as are conteined by a streight and a Circular Line , as the Angle PMR . Fig. 1. Or such as are conteined by two circular Lines , as the Angles FZP or ZFP in Fig. 3. The projecting or measuring the first sort of these Angles , needs no direction .

21. To project an Angle conteined by a streight and a circular line as the Angle AEBZ in Fig. 1. Do thus , lay a Ruler from N to C , and it will cut the Primitive Circle in K make ZX = HK , a Ruler laid from N to X will cut the Diameter HAR in the point M the Pole of the Circle ZCN , a Ruler laid from M to B the Angular point propounded , will cut the primitive Circle in I , make NY = HL a Ruler laid from N to Y will cut the Circle ZCN in W , a Ruler laid from B to W will cut the Primitive Circle in A , make AQ equal to the Angle propounded , and draw the Diameter BAQ , then is the Angle AEBZ or NBQ = NQ as was required .

22. If the Angle had been projected and the measure required , a Ruler laid from M to B would give L and making NY = HL a Ruler laid from M to Y would give W , from B to W would give A , and AQ would be the measure of the Angle propounded .

23. To project an Angle conteined by two circular lines , one of them being an Arch of the Primitive Circle , as the Angle AEZB , Fig. 1. Do thus , set off the quantity of the Angle given from H to G , a Ruler laid from Z to G will cut the Diameter HAR in the point C , so may you draw the Circle ZCN and the Angle HZC will be equal to the Arch HG = HC as was required .

24. If the Angle had been projected and the measure required , a Ruler laid from Z to C would cut the Primitive Circle in G and HG would be the measure of the Angle propounded .

25. To project an Angle conteined by two oblique Arches of a Circle , as the Angle ZFP in Fig. 3. You must first find the Pole of one of the two Circles conteining the Angle propounded , suppose ZBI , a Ruler laid from C the Pole thereof to F , the Angular point propounded , will cut the Primitive Circle in a make ab equal to the Angle propounded , a Ruler laid from F to b will cut the Diameter AEAQ in D the Pole of the Circle PEX , a Ruler laid from G to e will cut the Primitive Circle in m , make mn = Bm ler laid from G to n will cut the Diameter TAS in p , make Aq = Ap so shall Fp be the Radius and the Center of the Circle PFX and the Angle ZFP = ab , as was propounded .

26. If the Angle had been projected and the measure required ; through the point F draw the Diameter TFS and the Diameter BAG at right Angles thereunto , a Ruler laid from G to F will cut the Primitive Circle in K , and making KE = BK a Line drawn from G to E will cut the Diameter TAS in the Center of the Circle GDB cutting the Diameter HAR in C the Pole of the Circle ZBI , and the Diameter AEAQ in D , the Pole of the Circle PEX and a Ruler laid from F to C and D will cut the Primitive Circle in a and b the measure of the Angle required .

Or a Ruler laid from F to K and M will cut the Primitive Circle in Deg. the measure of the Angle propounded as before .

Or thus a Ruler laid from C and D to F will cut the Primitive Circle in ae and h set 90 Degrees from e and h to f and l a Ruler laid from C to f will cut ZBI in M and a Ruler laid from D to l will cut PEX in K. This done a Ruler laid from F to K and M will cut the Primitive Circle in g and d the measure of the Angle as before .

And in Fig. 2. The quantity of the Angle ZEP may thus be found . A Ruler laid from C the Pole of the Circle ZFI to F the angular point will cut the Primitive Circle in a , set off a Quadrant from a to b , a Ruler laid from C to b will cut the Circle ZFI in the point M. In like manner a Ruler laid from D the Pole of the Circle PEX , will cut the Primitive Circle in D , set off a Quadrant from A to h , a Ruler laid from D to P will cut the Circle PFX in K : Lastly a Ruler laid from F to K , and M will cut the Primitive Circle in NS the measure of the Angle KFM or ZFP , as was propounded .

27. Having shewed how a right or oblique Angled Spherical Triangle may be projected upon the Plane of the Meridian , as well as delineated upon the Globe , we will now consider the several Triangles usually represented upon the Globe , with the several Astronomical and Geographical Problems conteined in them , and resolved by them .

28. The Spherical Triangles usually represented upon the Globe are eight , whereof there are five Right angled Triangles , have their Denomination from their Hypotenusas .

The first is called the Ecliptical Triangle , whose Hypotenusa is an Arch of the Ecliptick , the Legs thereof are Arches of the AEquator and Meridian , this is represented upon the Globe , by the Triangle ADF , in Fig. 1. In which the five Circular parts , besides the Right Angle are ;

1. The Hypotenuse or Arch of the Ecliptick AF.

2. The Leg or Arch of the AEquator , AD.

3. The Leg or Arch of the Meridian DF.

4. The Oblique Angle of the Equator with the Ecliptick and the Suns greatest Declination DAF .

5. The Oblique Angle of the Ecliptick and Meridian , or the Angle of the Suns position AFD .

The two next I call Meridional , because the Hypotenusas in them both , are Arches of a Meridian . One of these is noted with the Letters MPR in Fig. 1. In which the five Circular parts are ;

1. The Hypotenusa or Arch of a Meridian PM .

2. The Leg or Arch of the Horizon MR , the Suns Azimuth North.

3. The Leg or Arch of the Brass Meridian , representing the height of the Pole PR.

4. The Oblique Angle of the Meridian upon the Globe ; with the Brass Meridian , or Angle of the Hour from Midnight . P.

5. The Oblique Angle of the Suns Meridian with the Horizon , or the Complement of the Suns Angle of Position PMR .

The other Right Angled Meridional Triangle is noted with the Letters AEG in Fig. 1. In which the 5 Circular parts are .

1. The Hypotenusa or present Declination of the Sun , AE.

2. The Leg or Suns Amplitude at the hour of six , AG.

3. The other Leg or Suns height at the same time EG .

4. The Angle of the Meridian with the Horizon , or Angle of the Poles elevation , EAG .

5. The Angle of the Meridian with the Azimuth , or the Angle of the Suns position , AEG .

The fourth Right Angled Spherical Triangle , I call an Azimuth Triangle , because the Hypotenusa doth cut the Horizon in the East and West Azimuths , as is represented by the Triangle ADV. in Fig. 1. In which the 5 Circular parts are ,

1. The Hypotenusa , or Arch of the Sun or Stars Altitude AV.

2. The Leg or Declination of the Sun or Star , DV .

3. The other Leg , or Right Ascension of the Sun or Star , AD.

4. The Oblique Angle or Angle of the Poles elevation , DAV .

5. The other Oblique Angle or Angle of the Sun or Stars Position , DVA .

The fifth and last Right Angled Spherical Triangle , that I shall mention , I call an Horizontal Triangle , because the Hypotenusa thereof is an Arch of the Horizon , and is represented by the Triangle AMT in Fig. 1. In which the 5 Circular parts are ;

1. The Hypotenusa and Arch of the Horizon , or Amplitude of the Sun at his rising or setting , AM.

2. The Leg conteining the Sun or Stars Declination TM .

3. The other Leg or Ascensional difference AT , that is , the difference between DT the Right Ascension and DA the Oblique Angle .

4. The Oblique Angle of the Horizon and Equator , or height of the Equator TAM .

5. The other Oblique Angle , or Angle of the Horizon and Meridian AMT .

The Oblique Angled Spherical Triangles usually represented upon the Globe are three . The first I call the Complemental Triangle , because the sides thereof are all Complements , and this is represented by the Triangle FZP in Fig. 1. Whose Circular parts are ;

1. The Complement of the Poles elevation ZP .

2. The Complement of the Suns Declination , FP .

3. The Complement of the Suns Altitude or Almicantar FZ .

4. The Suns Azimuth or Distance from the North FZP .

5. The hour of the day or distance of the Sun from Noon ZPF .

6. The Angle of the Suns Position ZFP .

The second Oblique Angled Spherical Triangle , I call a Geographical or Nautical Triangle , because it serveth to resolve those Problems , which concern Geographie and Navigation , and this is also represented by the Triangle FZP in Fig. 1. Whose parts are .

1. The Complement of Latitude as before ZP .

2. The distance between the two places at Z and F or side FZ .

3. The Complement of the Latitude of the place at F or side FP .

4. The difference of Longitude between the two places at Z and F or the Angle FBZ.

5. The point of the compass leading from Z to F or Angle FZP .

6. The point of the Compass leading from F to Z , or Angle ZFP .

The third Oblique Angled Spherical Triangle is called a Polar Triangle , because one side thereof is the distance between the Poles of the World , and the Poles of the Zodiack . This Triangle is represented upon the Coelestial Globe , by the Triangle FSP in Fig. 4. In which the Circular parts are ;

1. The distance between the Pole of the World , and the Pole of the Ecliptick , or the Arch SP.

2. The Complement of the Stars Declination , FP .

3. The Complement of the Stars North Latitude , from the Ecliptick or the Arch FS .

4. The Angle of the Stars Right Ascension FPS .

5. The Complement of the Stars Longitude FSP .

6. The Angle of the Stars Position SFP .

29. And thus at length I have performed , what was proposed in the 15 of this Chapter , that is , I have shewed how the several Circles of the Globe , may be projected upon the Plane of the Meridian , the several useful Triangles that are described by such projection , with such Astronomical Propositions as are contained and resolveable by those Triangles ; And although the most accurate way of resolution is by the Doctrine of Trigonometry and the Canon of Lines and Tangents , yet it is not impertinent to do the same upon the Globe it self , which as to the sides is easie , but to measure or lay down the Angles is sometimes a little labourious .

In the Right Angled Spherical Triangle AEBZ in Fig. 1. The measure of the Angle AEZB is reckoned in the Horizon from H to C but to lay down or measure the Angle AEBZ the readiest way is to describe the Triangle again , making AEZ = AEB and AEB = AEZ , so will the Angle AEBZ stand where the Angle AEZB is , and may be measured in the Horizon as the other was .

And so in the Oblique Angled Spherical Triangle FZP in Fig. 1. The Angles at Z and P are easily measured or laid down upon the Globe , but to perform the same with the Angle ZFP , you may represent it at the Pole or Zenith and find the measure in the Equator or Horizon .

30. And now having , as I hope , sufficiently prepared the young Student for the first part of Astronomy , the Doctrine of the Primum Mobile , by shewing how the Heavens and the Earth are represented upon the Globe , or may be projected in Plane , I will now proceed to such Astronomical Propositions as are generally useful , and may be sufficient for an Introduction to this noble Science : to go through the several Triangles before propounded , will be very tedious , I will therefore shew the several Problems in one Right Angled and one Oblique Angled Spherical Triangle and the Canons by which they are to be resolved , and leave the rest for the Practice of my Reader . To this purpose I will next acquaint you with my Lord Nepiers Catholick Proposition for the solution of all Right and Oblique Angled Spherical Triangles .

CHAP. IV.

Of the solution of Spherical Triangles .

IN Spherical Triangles there are 28 Varieties or Cases , 16 in Rectangular , and 12 in Oblique , whereof all the Rectangular and ten of the Oblique may be resolved by the two Axioms following .

1. Axiom . In all Right Angled Spherical Triangles having the same Acute Angle at the Base , the Sines of the Hypotenusas are proportional to the Sines of their Perpendicular .

2. Axiom . In all right Angled Spherical Triangles , the Sines of the Bases and the Tangents of the Perpendicular are proportional .

That all the Cases of a Right Angled Spherical Triangle may be resolved by these two Axioms , the several parts of the Spherical Triangle proposed , that so the Angles may be turned into sides , the Hypotenusa , into Bases and Perpendiculars and the contrary . By which means the proportions as to the parts of the Triangle given , are sometimes changed into Co-sines instead of Sines , and into Co-tangents instead of Tangents . Which the Lord Nepier observing ; those parts of the Right Angled Spherical Triangle , which in conversion do change their proportion , he noteth by their Complements . viz. The Hypotenuse and the two Acute Angles : But the sides or Legs are not so noted , as in the Right Angled Spherical Triangle MPR in Fig. 1. And these five he calleth the Circular parts of the Triangle , amongst which the Right Angle is not reckoned .

2. Now if you reckon five Circulat parts in a Triangle , one of them must needs be in the middle , and of the other four , two are adjacent to that middle part , the other two are disjunct , and which soever of the five you call the middle part , for every one of them may by supposition be made so ; those two Circular parts which are on each side of the middle are called extreams adjunct , and the other two remaining parts , are called extream disjunct , as in the Triangle MPR if you make the Leg PR the middle part , then the other Leg MR and the Angle Comp. P. Are the extreams conjunct , the Hyp. Comp. MP and Comp. M , are the extreams disjunct , and so of the rest , as in the following Table .

Mid. Part Exctr. conj . Extr. disj● Leg. MR Comp. M Leg PR Comp. P Comp. MP Leg. PR Comp. MP Leg MR Comp. M Comp. P Leg. MR Comp. P Comp. M Comp. MP Leg. PR Comp. M Leg. PR Comp. MP Comp. P. Leg. MR Comp. MP Leg. MR Comp. P Leg. PR Comp. M

3. These things premised , the Ld. Nepier as a consectory from the two preceeding Axioms hath composed this Catholick and Universal Proposition .

The Rectangle made of the Sine of the middle part and Radius is equal to the Rectangle made of the Tangents of the Extremes conjunct or the cosines of the Extremes disjunct .

Therefore if the middle part be sought , the Radius must be in the first place ; if either of the extremes , the other extreme must be in the first place .

Only note that if the middle part , or either of the extremes propounded be noted with its Comp. in the circular parts of the Triangle , instead of the Sine or Tangent you must use the Cosine or Cotangent of such circular part or parts .

That these directions may be the better conceived , we have in the Table following set down the circular parts of a Triangle under their respective Titles , whether they be taken for the middle part , or for the extremes , conjunct or disjunct , and unto these parts , we have prefixed the Sine or Cosine , the Tangent or Co-tangent , as it ought to be by the former Rule .

Mid. Par. Extr. Conj . Ext. Disj . Tang. MR Sine M Sine PR Tang. P. Sine MP Tang. PR Sine MP Sine MR Cotang . M. Sine P Tang. MR Sine P Cosine M Cot. MP Cos. PR Cotang . M. Cos. PR Cos. MP Cotang . P Cos. MR Cot. MP Cos. MR Cosine P Tang. PR Sine M

Now then according to this Table and the former Rules . 1. Sine PR x Rad. = t MR x ct P. 2. Sine PR x Rad. = s M x s MP . 3. Sine MR x Rad. = t PR x ct M. 4. Sine MR x Rad. = s MP x s P. 5. Cos. M x Rad. = t MR x ct MP . 6. Cos. M x Rad. = s P x cs PR. 7 Cos. MP x Rad. = ct M x ct P. 8. Cos. MP x Rad. = cos. PR x cs MR. 9. Cos. P x Rad. = ct MP x t PR. 10. Cos. P x Rad. = cos. MR x s M.

By these 10 Rectangles may the 16 Cases of a Right angled Spherical Triangle be resolved , and some of them twice over ; for although there are but 16 varieties in all Right angled Spherical Triangles , yet 30 Astronomical Problems may be resolved by one Triangle , as by the following Examples shall more clearly appear .

Of Right angled Spherical Triangles .
CASE 1.

The Legs given , to find the Angles .

IN the Right angled Spherical Triangle MPR . The given Legs are MR and RP . The Angles at M and P are required .

By the first of the 10 equal Rectangles s PR x Rad. = t MR x ct P. in which P is fought , therefore putting MR in the first place . The proportion is . t MR. x Rad. ∷ s PR. ct P.

And by the third equal Rectangle . t PR. Rad. ∷ s MR. ct M.

CASE 2.

The Legs given , to find the Hypotenuse .

In the Right angled Spherical Triangle MPR . The given Legs are MR and PR. The Hypotenuse MP is required .

By the eighth of the 10 Rectangles cos. MP x Rad. = cos. PR x cos. MR in which MP the middle part is sought , therefore Radius must be put in the first place , and then the proportion is .

Rad: cos. PR ∷ cos. M. R. cos. MP .

CASE 3.

A Leg with an Angle opposite thereunto being given , to find the other Leg.

In the Right angled Spherical Triangle MPR , let there be given . The Leg MR. The Angle P. The Leg PR inquired .

By the first of the 10 Rectangles . Rad. tMR ∷ cot . P. Sine PR. or The Leg PR and the Angle M given , to ●ind MR.

By the 3 of the 10 Rectangles . Rad. tPR ∷ ct M. Sine MR.

CASE 4.

A Leg with an Angle conterminate therewith being given , to find the other Leg.

In the Right angled Spherical Triangle , MPR , The given Leg is MR , with the Angle M. The Leg PR is required .

By the 3 Rectangle . cot . M. Rad ∷ Sine MR. tPR .

The given Leg RP , and Angle P. The Leg MR is required .

By the 1. Rectangle . ctP. Rad ∷ sine RP . tang . MR.

CASE 5.

A Leg and an Angle conterminate therewith being given , to find the Hypotenuse .

In the Right angled Spherical Triangle MPR , let there be given , The Leg MR and the Angle M PR and the Angle P to find MP .

By the 5. Rectangle , t MR. Rad ∷ cos. M. ct MP .

By the 9. Rectangle . t PR. Rad. ∷ cos. P. ct MP .

CASE 6.

The Hypotenuse and a Leg given , to find the contained Angle .

In the Right angled Spherical Triangle MPR , let there be given , The Hypotenuse MP , and Leg MR. PR. To find M.

By the 5. Rectangle , Rad. ct MP ∷ t MR. cos. M.

By the 9. Rectangle , Rad. ct MR ∷ t PR. cos. P.

CASE 7.

The Hypotenuse and one Angle given , to find the other Angle .

In the Right angled Spherical Triangle MPR , let there be given , The Hypotenuse MP & Angle M P. To find the Angle P. M.

By the 7. Rectangle , cot . M. Rad ∷ cos. MP . cot . P.

By the 7. Rectangle cot . P. Rad ∷ cos. MP . cot . M.

CASE 8.

The Oblique A●gles given , to find the Hypotenuse .

In the Right angled Spherical Triangle MPR , let there be given The Angles at P and M , To find the Hypotenuse PM .

By the 7. Rectangle . Rad. ct P ∷ cot . M. cos. MP .

CASE 9.

The Hypotenuse and an Angle given , to find the Leg conterminate with the given Angle .

In the Right angled Spherical Triangle MPR , let there be given , The Hypotenuse PM Angle P. M. To find PR. MR.

By the 9. Rectangle , ct PM . Rad ∷ cos. P. t PR.

By the 5. Rectangle , ct PM . Rad ∷ cos. M. tMR .

CASE 10.

The Hypotenuse and an Angle given , to find the Leg opposite to the given Angle .

In the Right angled Spherical Triangle MPR , let there be given , The Hypotenuse PM and the Angle M. P. To find PR. MR.

By the 2. Rectangle , Rad. s MP ∷ s M. Sine PR.

By the 4. Rectangle , Rad. s MP ∷ s P. Sine MR ▪

CASE 11.

A Leg and an Angle opposite thereunto being given , to find the Hypotenuse .

In the Right angled Spherical Triangle MPR , let there be given , The Leg PR. MR. and the Angle M P to find the Hypotenuse PM .

By the 2. Rectangle , s M. Rad ∷ s PR. s MP .

By the 4. Rectangle , s P. Rad ∷ s MR. s PM .

CASE 12.

The Hypotenuse and a Leg given , to find the Angle opposite to the given Leg.

In the Right angled Spherical Triangle PMR , the Hypotenuse MP and the Leg MR are given , the Angle at P is required .

By the fourth Rectangle Sine MP to , Rad ∷ s MR. s P.

The Hypotenuse MP and Leg PR given , the Angle M is required .

By the second Rectangle . sMP. Rad ∷ s PR. s M.

CASE 13.

The Angle and Leg conterminate with it being given , to find the other Angle .

In the Right angled Spherical Triangle PMR , let there be given , The Angle M P and the Leg MR PR to find the Angle P. M.

By the tenth Rectangle , Rad. cs MR ∷ s M. cs P.

By the sixth Rectangle , Rad. s P ∷ cs PR. cs M.

CASE 14.

An Angle and a Leg opposite thereunto being given , to find the other Angle .

In the Right angled Spherical Triangle MPR , let there be given , The Angle P M and the Leg MR PR to find the Angle M. P.

By the 10. Rectangle , cs MR. Rad ∷ cs P. csM.

By the 6. Rectangle , cs PR. Rad ∷ cs M. sP.

CASE 15.

The Oblique Angles given , to find a Leg.

In the Right angled Spherical Triangle MPR , let there be given , the Angles at M and P , to find the Legs MR and PR.

By the 10. Rectangle , sM. Rad ∷ cs P. cs MR.

By the 6. Rectangle , s P. Rad ∷ cs M. cs PR.

CASE 16.

The Hypotenuse and one Leg given , to find the other Leg.

In the Right angled Spherical Triangle MPR , let there be given , The Hypotenuse MP and the Leg PR MR to find the Leg MR. PR.

By the 8. Rectangle , csPR. Rad ∷ csMP. csMR. csMR. Rad ∷ csMP. csPR.

Thus I have given you the Proportions by which the 16 Cases of a Right angled Spherical Triangle may be resolved , In which there are contained 30 Astronomical Problems . Two in every Case except the Second and the Eighth . In both which Cases there are but two Problems . And thus I have done with Right angled Spherical Triangles .

5. If the Angles at the Base be both acute or both obtuse , the Perpendicular shall fall within the Triangle ; but if one of the Angles of the Base be acute and the other obtuse , the Perpendicular shall fall without the Triangle .

6. However the Perpendicular falleth , it must be always opposite to a known Angle , for your better direction , take this General Rule . From the end of a Side given , being adjacent to an Angle given , let fall the Perpendicular .

As in the Triangle FPS in Fig. 4. If there were given the Side F S and the Angle at S , the Perpendicular by this Rule must fall from P upon the Side S P extended , if need require .

But if there were given the Side P S and the Angle at S , the Perpendicular must fall from F upon the Side F S.

7. To divide an Oblique angled Spherical Triangle into two Right , by letting fall a Perpendicular upon the Globe it self , is not necessary , because all the Cases may be resolved without it , but in projection it is convenient to inform the fancy : and seeing the reason by which it is done in projection doth depend upon the nature of the Globe , I will here shew it both ways , first upon the Globe , and then by projection .

An Oblique angled Spherical Triangle may be divided into two Right , by letting fall a Perpendicular upon the Globe it self , in this manner . In the Oblique angled Spherical Triangle FPS in Fig. 4. let it be required to let fall a Perpendicular from P upon the Side FS . Suppose the Point P to stand in the Zenith , where the Arch FS shall cut the Zodiack , which in this Figure is at K , make a mark , and from this Point of Intersection of the Circle upon which the Perpendicular is to fall with the Zodiack , reckon 90 Degrees , which suppose to be at P ; a thin Plate of Brass with a Nut at one end thereof , whereby to fasten it to the Meridian , as you do the Quadrant of Altitude , being graduated as that is , but of a larger extent ( for that a Quadrant in this case will not suffice ) being fastned at P and turned about till it cut the Point L in the Zodiack , will describe upon the Globe the Arch of a great Circle PEL , intersecting the Side F S at Right Angles in the Point E , because the Point L in the Zodiack is the Pole of the Circle SFK , now all great Circles which passing through the Point L , shall intersect the Circle SKG , shall intersect it at Right Angles ; by the 13. of the 2. Chapter .

9. And hence to divide an Oblique angled Spherical Triangle into two Right by projection is easie , as in the Triangle FPS , the Pole of the Circle SFK is L , therefore the Circle BLP shall cut the Arch FS at Right Angles in the Point E. And because the Point M is the Pole of the Circle BFP , therefore the Circle GMS shall cut the Circle BFP at Right Angles in the Point D , the Side F P being extended . Come we now to the several Cases which after this preparation may be resolved , by the Catholick Proposition .

CASE 1.

Two Sides with an Angle opposite to one of them being given , to find the Angle opposite to the other .

In the Oblique angled Spherical Triangle F P S , in Fig. 4. the Sides and Angles given and required will admit of six Varieties ; all which may be resolved by the Catholick Proposition , at two operations , but those two may be reduced to one , as by the following Analogies to every Variety will plainly appear . Given Required FP Rad. s PS ∷ s PSF . s PE 1. PS PFS s. PF . Rad ∷ s PE. s PFS PSF s PF . s PS ∷ s PSF . s PFS FP Rad. s FP ∷ s F. s PE 2. PS PSF s PS . Rad ∷ s PE. s PSF PFS s. PS. s FP ∷ s PFS . s PSF PS Rad. s SF ∷ s F. sDS 3. FS FPS s PS . Rad ∷ s DS. s SPD PFS s. PS. s SF ∷ s PFS . s PSF PS Rad. s PS ∷ s SPD . s DS 4. FS PFS s FS . Rad ∷ s DS. s SF FPS s FS . s PS ∷ s SPF. s SF FS Rad. s FS ∷ s S. s FC 5. FP FPS s. FP . Rad ∷ s FC . s FPC FSP s. FP . s FS ∷ s PSF . s FPS FS Rad. s FP ∷ s FPC . s FC 6. FP FSP s FS . Rad ∷ s FC . s S FPS s. FS . s FP ∷ FPS . s PSF .

CASE 2.

Two Sides with an Angle appo●ite to one of them being given , to find the contained Angle .

In this Case there are six Varieties , all which may be resolved by the Catholick Proposition , according to the Table following . Given Required FP 1 : cot PSF . Rad ∷ cs PS. ct EPS 1. PS FPS 2. ct PS. Rad ∷ cs EPS . t EP PSF 3. Rad. t EP ∷ ct FP . cs FPE EPS+EPF = FPS ct PS. cs EPS ∷ ct FP . ct FPE FP 1. cot PFS . Rad ∷ cs PF . ct EPF 2. PS FPS 2. ct PF . Rad ∷ cs EPF . t EP PFS 3. Rad. t EP ∷ cot PS. cs EPS EPS+EPF = FPS cot PF . cs EPF ∷ ct PS ct EPS PS 1. cot PFS . Rad ∷ cs FS . ct FSD 3. FS PSF 2. ct FS . cs FSD ∷ Rad. t DS PFS 3. Rad. t DS ∷ ct PS. cs PSD FSD-PSD = PSF ct FS . cs FSD ∷ ct PS. cs PSD PS 1. cot FPS . Rad ∷ cs PS. ct PSD 4. FS PSF 2. ct PS. cs PSD ∷ Rad. t DS FPS 3. Rad. t DS ∷ ct FS . cs FSD FSD-PSD = PSF ct PS. cs PSD ∷ ct FS . cs FSD FS 1. cot FSP . Rad ∷ cs FS . ct SFC 5. FP PFS 2. ct FS . cs SFC ∷ Rad. t FC FSP 3. Rad. t FC ∷ ct FP . cs PFC SFC-PFC = PFS ct FS . cs SFC ∷ ct FP . cs PFC FS 1. cot FPS . Rad ∷ cs PF . ct PFC 6. FF PFS 2. cot FP . cs PFC ∷ Rad. t FC FPS 3. Rad. t FC ∷ ct FS . cs SFC SFC-PFC = PFS ct FP . cs PFC ∷ ct FS . cs SFC .

CASE 3.

Two Sides and an Angle opposite to one of them being given , to find the third side .

The Varieties in this Case , with their resolution by the Catholick Proposition , are as followeth . Given Required FP 1. ct PS ∷ cs PSF . t ES 1. PS FS 2. cs ES. cs PS ∷ Rad. cs EP PSF 3. Rad. cs EP ∷ cs FP . cs FE ES+FE = FS cs ES. cs PS ∷ cs FP . cs FE FP 1. cot FP . Rad ∷ cos PFS . t FE 2. PS FS 2. cos FE . cos FP ∷ Rad. cos EP PFS 3. Rad. cos EP ∷ cos PS. cos SE SE+FE = FS cos FE . cos FP ∷ cos PS. cos SE PS 1. cot FS . Rad ∷ cos PFS . t FD 3. FS FP 2. cos FD. cos FS ∷ Rad. cs SD PFS 3. Rad. cos SD ∷ cos PS. cs PD FD-PD = FP cos FD. cos FS ∷ cs PS. cs PD PS 1. cot PS. Rad ∷ cos FPS . t PD 4. FS FP 2. cos PD . cos PS ∷ Rad. cos SD FPS 3. Rad. cos SD ∷ cos FS . cs FD FD-PD = FP cos PD . cos PS ∷ cos FS . cs FD FS 1. cot FS . Rad ∷ cos FSP . t SC 5. FP PS 2. cos SC. cos FS ∷ Rad. cos FC FSP 3. Rad. cos FC ∷ cos FP . cos PC SP-PC = PS cos SC. cos FS . cos FP . cos PC FS 1. cot FP . Rod ∷ cos FPS . t PC 6. FP PS 2. cos PC . cos FP ∷ Rad. cos FC FPS 3. Rad. cos FC ∷ cos FS . cos SC SC-PC = PS cos PC . cos FP ∷ cos FS . cos SC

CASE 4.

Two Angles with a Side opposite to one of them being given , to find the Side opposite unto the other .

The Varieties in this Case , with their Resolution by the Catholick Proposition , are as followeth . Given Required PFS Rad. s. PS ∷ s DPS. s SD 1. FPS FS S. FP . Rad ∷ s SD . s FS PS s. PFS . s PS ∷ s FPS . s FS PFS Rad. s FS ∷ s PFS . s. SD 2. FPS PS s. FPS . Rad ∷ s SD . s PS FS s. FPS . s FS ∷ s PFS . s PS FPS Rad. s FP ∷ s FPS . s FC 3. PSF FS s. PSF . Rad ∷ s FC . s FS FP s. PSF . s FP ∷ s FPS . s FS FPS Rad. s FS ∷ s PSF . s FC 4. PSF FP s. FPS . Rad ∷ s FC . s FP FS s. FPS s FS ∷ s PSF . s FP PSF Rad. s PS ∷ s PSF . s PE 5. SFP FP s. SFP . Rad ∷ s PE. s FP PS s. SFP . s PS ∷ s PSF . s FP PSF Rad. s FP ∷ s PFS . s PE 6. SFP PS s. PSF . Rad ∷ s PE. s PS FP s. PSF . s FP ∷ s PFS . s PS

CASE 5.

Two Angles and a side opposite to one of them being given , to find the Side between them .

The Varieties and Proportions , are as followeth . Given Required PFS 1. ct PS. Rad ∷ cs DPS. PD 1. FPS FP 2. ct DPS. s PD ∷ Rad. t DS PS 3. Rad. t DS ∷ ct PFS . s FD FD-PD = FP ct DPS. s PD ∷ ct PFS . s FD PFS 1. ct DFS. Rad ∷ cs PFS . t FD 2. FPS FP 2. cot PFS . s FD ∷ Rad. t DS FS 3. Rad. t DS ∷ ct FPS . s PD FD-PD = FP FPS 1. cot FP . Rad ∷ cs FPC . t PC 3. PSF PS 2. cot FPC . s PC ∷ Rad. t FC FP 3. Rad. t FC ∷ ct PSF . s SC SC-PC = PS cot FPC . s PC ∷ ct PSF . CS FPS 1. cot FS . Rad ∷ cs PSF . t SC 4. PSF PS 2. cot PSF . s SC ∷ Rad. t FC FS 3. Rad. t FC ∷ cot FPS . s PC SC-PC = PS cot PSF . s SC ∷ cot FPS . s PC PSF 1. cot PS. Rad ∷ cs PSF . t SE 5. SFP FS 2. cot PSF . s SE ∷ Rad. t PE PS 3. Rad. t PE ∷ cot SFP . s FE FE+SE = FS cot PSF . s SE ∷ cot SFP . s FE PSF 1. cot FP . Rad ∷ cs SFP . t FE 6. SFP FS 2. cot SFP . s FE ∷ Rad. t PE FP 3. Rad. t PE ∷ cos PSF . s SE FE+SE = FS cot . SFP . s FE ∷ cs PSF . s SE

CASE 6.

Two Angles and a Side opposite to one of them being given , to find the third Angle .

The Varieties and Proportions are as followeth . Given Required PFS 1. ct DPS. Rad ∷ cs PS. ct PSD 1. FPS PSF 2. s PSD . cs DPS ∷ Rad. cs DS PS 3. cs DS. Rad ∷ cs DFS. s FSD FSD-PSD = PSF cs DPS. s PSD ∷ cs DFS. s FSD PFS 1. ct PFS . Rad ∷ cs FS . ct FSD 2. FPS PSF 2. s FSD . cs PFS ∷ Rad. cs DS FS 3. cs PDS. Rad ∷ cs DPS. cs PSD FSD-PSD = PSF cs PFS . s FSD ∷ cs DPS. cs PSD FPS 1. ct FPC . Rad ∷ cs FP . ct PFC 3. PSF PFS 2. s PFC . cs FPG ∷ Rad. cs FC FP 3. cs FC . Rad ∷ cs PSF . s FC SFC-PFC = PFS cs FPC . s PFC ∷ cs PSF . s SFC FPS 1. cot PSF . Rad ∷ cos FS . ct SFC 4. PSF PFS 2. s SFC . cs PSF ∷ Rad. cs FC FS 3. cs FC . Rad ∷ cs CPF . s PFC SFC-PFC = PFS cs PSF . s SFC ∷ cs CPF . s PFC PSF 1. cot PSF . Rad ∷ cs PS. ct SPE 5. SFP FPS 2. s SPE . cs PSF ∷ Rad. cs PE PS 3. cs PE. Rad ∷ cs SFP . s FPE FPE+SPE = FPS cs PSF . s SPE ∷ cs SFP . s FPE PSF 1. cot SFP . Rad ∷ cs FP . ct FPE 6. SFP FPS 2. s FPE . cs SFP ∷ Rad. cs PE FP 3. cos PE Rad ∷ cs PSF . s SPE FPE+SPE = FPS cs SFP . s FPE ∷ cs PSF . s SPE

CASE 7.

Two Sides and their contained Angle being given , to find either of the other Angles .

The Varieties and Proportions are as followeth . Given Required FS 1. ct FP . Rad ∷ cs PFS . t FE 1. FP FSP 2. ct PFS . s FE ∷ Rad. t PE PFS 3. t PE. Rad. ∷ s ES. ct PSF FS-FE = ES s EF. ct PFS ∷ s ES. ct PSF FS 1. cot FS . Rad: : cs PFS . t DF 2. FP FPS 2. cot PFS . s DF : : Rad. t DS PFS 3. t DS. Rad: : s PD . ct SPD FD - FP = PD s DF. ct PFS : : s PD . ct SPD FP 1. cot FP . Rad: : cos FPC . t PC 3. PS PSF 2. cot FPC . s PC : : Rad. t FC FPS 3. t FC . Rad: : s CS . cot FSP PS+PC = CS s PC . ct FPC : : s CS . ct FSP FP 1. cot PS. Rad: : cos SPD . t PD 4. PS SFP 2. cot SPD . s PD : : Rad. t DS FPS 3. t DS Rad: : s FD. cot SFP FP+PD = FD s PD . ct SPD : : s FD. cot SFP PS 1. cot PS. Rad: : cs PSF . t SE 5. FS SFP 2. cot PSF . s SE: : Rad. t PE PSF 3. t PE. Rad: : s FE . cot SFP FS-SE = FE s SE. ct PSF : : s FE . ct SFP PS 1. cot FS . Rad: : cs PSF . t SC 6. FS FPS 2. cot PSF . s SC : : Rad. t FC PSF 3. t FC . Rad: : s PC . cot FPC SC-PS = PC s SC. cot PSF : : s PC . ct FPC

CASE 8.

Two Sides and their contained Angle being given , to find the third Side .

The Varieties and Proportions are as followeth . Given Required FS 1. ct FP . Rad: : cs PFS . t FE 1. FP PS 2. cs FE . cs FP : : Rad. cos PE PFS 3. Rad. cs PE : : cs ES. cs PS FS-FE = ES cs FE . cs FP : : cs ES. cs PS FP 1. ct PS. Rad: : cs SPD . t PD 2. SP FS 2. cs PD . cs PS: : Rad. cos DS FPS 3. Rad. cos DS : : cs FD. cs FS FP+PD = FD cs PD . cs PS: : cs FD. cs FS PS 1. ct PS. Rad: : cs PSF . t. ES 3. FS FP 2. cs ES. cs PS: : Rad. cos PE PSF 3. Rad. cos PE : : cos FE . cos FP FS-ES = FE cs ES. cs PS: : cos FE . cs FP

CASE 9.

Two Angles and their contained Side being given , to find one of the other Sides .

Given Required PFS 1. ct PFS . Rad: : cs FP ct FPE 1. FPS PS 2. ct FP . cs FPE : : Rad. t PE FP 3. t PE. Rad: : cs EPS . ct PS FPS-FPE = EPS cs FPE . ct FP : : cs EPS . ct PS PFS 1. cot FPC . Rad: : cs FP . t PFC 2. FPS FS 2. cot FP . cs PFC : : Rad. t FC FP 3. t FC . Rad: : cs SFC . ct SF SFP+PFC = SFC ct FP . cs PFC : : cs SFC . ct SF FPS 1. ct SPD . Rad: : cs PS. ct PSD 3. PSF SF 2. ct PS. cos PSD : : Rad. t DS PS 3. t DS. Rad: : cs FSD . ct SF PSF+PSD = FSD cs PSD . ct PS: : cs FSD . ct SF FPS 1. ct PSF . Rad: : cs PS. ct SPE 4. PSF FP 2. ct PS. cs SPE : : Rad. t PE PS 3. t PE. Rad: : cs PPE. ct FP FPS-EPS = FPE cs SPE . ct PS: : cs FPE . ct FP PSF 1. ct PSF . Rad: : cs SF . ct SFC 5. SFP FP 2. ct SF . cs SFC : : Rad. t FC SF 3. t FC . Rad: : cs CFP . ct FP SFC-SFP = CFP cs SFC . ct SF : : cs CFP . ct FP PSF 1. ct SFP . Rad: : cs FS . ct FSD 6. SFP PS 2. ct FS . cs FSD : : Rad. t SD SF 3. t SD . Rad: : cos PSD . ct PS FSD-FSP = PSD cs FSD . ct FS : : cs PSD . ct PS

CASE 10.

Two Angles and the Side between them being given , to find the third Angle .

The Varieties and Proportions are as followeth . Given Required SFP 1. ct SFP . Rad: : cs FP . ct FPE 1. FPS PSF 2. s FPE . cs F : : Rad. cs PE FP 3. Rad. cs PE : : s EPS . cs PST FPS-FPE = EPS s FPE . cs PFS : : s SPE . cs PSF FPS 1. ct SPD . Rad: : cs PS. ct PSD 2. PSF SFP 2. s PSD . cs SPD : : Rad. cs DS PS 3. Rad. cs DS : : s FSD . cs SFP PSF+PSD = FSD s PSD . cs SPD : : s FSD . cs SFP PSF 1. ct PSF . Rad: : cs SF . ct SFC 3. SFP FPS 2. s SFC . cs PSF : : Rad. cs FC SF 3. Rad. cs FC : : s PFC . cs FPS SFC-SFP = PFC s SFC . cs PSF : : s PFC . cs FPS

CASE 11.

The three Sides being given , to find an Angle .

This Case may be resolved by the Catholick Proposition also , according to the direction of the Lord Nepier , as I have shewed at large in the Second Book of my Trigonometria Britannica , Chap. 2. but may as I conceive be more conveniently solved , by this Proposition following .

As the Rectangle of the Square of the Sides containing the Angle inquired ;

Is to the Square of Radius : So is the Rectangle of the Square of the difference of each containing Side , and the half sum of the three Sides given .

To the Square of the Sine of half the Angle inquired .

In this Case there are three Varieties , as in the Triangle FZP . Fig. 3.

Given Required ZP s ZP x s PF . Rad. q. 1. PF ZPF s ½ Z-ZP x s ½ Z-PF . Q FZ s ½ ZPF ZP s PF x s PZ . Rad. q. 2. PF PFZ s ½ Z-PF x s ½ Z-FZ . Q FZ s ½ PFZ ZP s ZP x s FZ Rad. q. 3. PF FZP s. ½ Z-ZP x ½ Z-ZF . Q FZ s ½ FZP

CASE 12.

The three Angles given , to find a Side .

This is the Converse of the last , and to be resolved after the same manner , if so be we convert the Angles into Sides , by the tenth of the third Chapter : for so the Sides of the Triangle ACD will be equal to the Angles of the Triangle FZP n Fig. 3.

That is AD = AEE the measure of the Angle ZPF . DC = KM the measure of the Angle ZFP . AC = HB the Complement of FZP to a Semicircle . The Angle DAC = QR = ZP . ACD = rM = Hf = Zoe = ZF . ADM = sK = AEl = Ph = PF .

And thus the Sides of the Triangle ZPF are equal to the Angles of the Triangle ACD . The Complement of the greatest Side PF to a Semicircle being taken for the greatest Angle ADC .

And in this Case therefore , as in the preceding , there are three Varieties which make up sixty Problems in every Oblique angled Spherical Triangle ; which actually to resolve in so many Triangles , as have been mentioned , would be both tedious , and to little purpose ; I will therefore select some few , that are of most general use in the Doctrine of the Sphere , and leave the rest to thine own practice .

CHAP. V.

Of such Spherical Problems as are of most General Use in the Doctrine of the Primum Mobile or Diurnal Motion of the Sun and Stars .

PROBLEM 1.

The greatest Declination of the Sun being given , to find the Declination of any Point of the Ecliptick .

THe Declination of the Sun or other Star , is his or their distance from the Equator , and as they decline from thence either Northward or Southward ; so is their Declination reckoned North or South .

2. The Sun 's greatest Declination , which in this and many other Problems is supposed to be given , with the Distance of the Tropicks , Elevation of the Equator , and Latitude of the Place , may thus be found .

The Sun 's greatest Meridian H ♋ . 61.9916 least Altitude H ♑ . 14.9416
Their difference is the distance of the Tropicks ♋ . ♑ . 47. 050 Half that Difference , is the Sun 's greatest Declination AE ♋ . 23. 525 Which deduct from the Sun 's greatest Altitude , the remainer is the height of the Equator HAE. 38. 467 The Complement is the height of the Pole AEZ or PR. 51. 533

Now then in the Right angled Spherical Triangle ADF in Fig. 1. there being given .

1. The Angle of the Sun 's greatest Declination DAF . 23. 525.

2. The Sun 's supposed distance from ♈ to ♎ AF. 60 deg.

The Sun 's present Declination DF may be found , by the 10 Case of Right angled Spherical Triangles .

As the Radius 9.60113517 9.93753063 9.53866580

PROBLEM 2.

The Sun 's groatest Declination , with his Distance from the next AEquinoctial Point being given , to find his Right Ascension .

In the Right angled Spherical Triangle ADF in Fig. 1. Having the Angle of the Sun 's greatest Declination DAF . 23. 525. And his supposed distance from ♈ or ♎ , the Hypotenusa AF. 60. The Right Ascension of the Sun , or Arch of the AEquator , AD may be found , by the ninth Case of Right angled Spherical Triangles ,

9.76143937 10.00000000 9 ▪ 96231533 10.20087596
PROBLEM 3.

To find the Declination of a Planet or Fixed Star with Latitude .

In the Oblique angled Spherical Triangle FPS in Fig. 4. we have given , 1. PS = AE ♋ the greatest Declination of the Ecliptick , 2. The Side FS the Complement of the Stars Latitude from the Ecliptick at K. 3. The Angle PSF the Complement of the Stars Longitude . To find PF the Complement of Declination . By the eighth Case of Oblique angled Spherical Triangles , the Proportions are .

10.3611802 10.0000000 9.9729858 9.6118056 63. 75. 0.0336046 9.9623154 9.6457058 9.6416258

Whose Complement , is FT . 25. 99. the Declination sought .

PROBLEM 4.

To find the Right Ascension of a Planet , or other Star with Latitude .

The Declination being found by the last Problem , we have in the Oblique angled Spherical Triangle PFS in Fig. 4. All the Sides with the Angle FSP 20 deg. or the Complement of the Stars Longitude . Hence to find FPS by the first Case of Oblique angled Spherical Triangles , I say

0.0463059 9.5340516 9.9984407 9.5787982

Whose Complement 67. 72. is the Right Asc. of a Star II. 10. North Lat. 4.

PROBLEM 5.

The Poles Elevation , Sun's greatest Declination and Meridian Altitude being given , to find his true place in the Zodiack .

If the Meridian Altitude of the Sun be less than the height of the AEquator , deduct the Meridian Altitude from the height of the AEquator , the Remainer is the Sun's Declination towards the South Pole : but if the Meridian Altitude of the Sun be more than the height of the AEquator , deduct the height of the AEquator from the Meridian Altitude , what remaineth , is the Sun's Declination towards the North Pole , in these Northern Parts of the World : the contrary is to be observed in the Southern Parts .

Then in the Right angled Spherical Triangle ADF in Fig. 1. we have given the Angle FAD the Sun's greatest Declination .

The Leg DF the Sun's present Declination , To find AF the Sun's distance from the next Equinoctial Point .

Therefore by the Case of Right angled Spherical Triangles .

0.3988648 9.5945468 10.0009000 9.9934116
PROBLEM 6.

The Poles Elevation and Sun's Declination being given , to find his Amplitude .

The Amplitude of the Sun 's rising or setting is an Arch of the Horizon intercepted betwixt the AEquator and the place of the Sun 's rising or setting ; and it is either Northward or Southward , the Northward Amplitude is when he riseth or setteth on this Side of the AEquator towards the North Pole ; and the Southern when he riseth or setteth on that Side of the AEquator which is towards the South Pole : That we may then find the Sun's Amplitude or Distance from the East or West Point , at the time of his rising or setting . In the Right angled Spherical Triangle ATM , in Fig. 2. let there be given the Angle TAM . 38. 47. the Complement of the Poles Elevation ; and TM . 23. 15. the Sun 's present Declination : To find AM the Sun's Amplitude .

By the eleventh Case of Right angled Spherical Triangles .

0.2061365 10.0000000 9.5945468 9.8006833
PROBLEM 7.

To find the Ascensional Difference .

The Ascensional Difference is nothing else , but the Difference between the Ascension of any Point of the Ecliptick in a Right Sphere , and the Ascension of the same Point in an Oblique Sphere ; As in Fig. 1. AT is the Ascensional difference between DA the Sun's Ascension in a Right Sphere , and DT the Sun's Ascension in an Oblique Sphere . Now then in the Right angled Spherical Triangle AMT , we have given . The Angle MAT. 38. 47. the Complement of the Poles Elevation . And MT. 23. 15. To find AT the Ascensional difference .

10.0999136 9.6310051 9.7309187
PROBLEM 8.

Having the Right Ascension and Ascensional Difference , to find the Oblique Ascension and Descension .

In Fig. 1. DT represents the Right Ascension , AT the Ascensional Difference . DA the Oblique Ascension which is found by deducting the Ascensional Difference AT . from the Right Ascension DT . according to the Direction following .

If the Declination be N. North Subt. Add The Ascentional Difference from the Right , and it giveth the Oblique Ascension . The Ascensional Difference to the Right , and it giveth the Oblique Descension . South Add Subt. The Ascensional Difference to the Right , and it giveth the Oblique Ascension . The Ascensional Difference from the Right , and it giveth the Oblique Descension . 57.80 27.62 30.18 85.42
PROBLEM 9.

To find the time of the Sun 's rising and setting , with the length of the Day and Night .

The Ascensional Difference of the Sun being added to the Semidiurnal Arch in a Right Sphere , that is , to 90 Degrees in the Northern Signs , or substracted from it in the Southern , their Sum or Difference will be the Semidiurnal Arch , which doubled is the Right Arch , which bisected is the time of the Sun rising , and the Day Arch bisected is the time of his setting .

As when the Sun is in 0 deg. ♊ . his Ascensional Difference is 27. 62. which being added to 90 degrees , because the Declination is North , the Sum will be 117.62 the Semidiurnal Arch.

The double whereof is 235.22 the Diurnal Arch , which being converted into time makes 15 hours 41 minutes : for the length of the Day , whose Complement to 24 ; is 8 hours 19 minutes the length of the Night ; the half whereof is 4 hours 9 minutes 30 Seconds the time of the Sun 's rising .

PROBLEM 10.

The Poles Elevation and the Sun's Declination being given , to find his Altitude at any time assigned .

In this Problem there are three Varieties . 1. When the Sun is in the AEquator . , that is , in the beginning of ♈ and ♎ in which case supposing the Sun to be at B , 60 degrees or four hours distant from the Meridian , then in the Right angled Spherical Triangle BZ AE , in Fig. 1. we have given , AE Z , 51. 53. the Poles Elevation , and B AE 60 degrees , to find BZ .

Therefore by the 2 Case of Right angled Spherical Triangles .

9.7938635 9.6989700 9.4928335

Whose Complement BC. 18. 12. is the ☉ Altitude required .

The second Variety is when the Sun is in the Northern Signs , that is , in ♈ . ♉ . ♊ . ♋ . ♌ . ♍ . in which Case supposing the Sun to be at F in Fig. II Then in the Oblique angled Spherical Triangle FZP , we have given . 1. PZ 38. 47 the Complement of the Poles Elevation . 2. FP . 67. 97 the Complement of Declination . 3. ZPF . 45 the Distance of the ☉ from the Meridian , To find FZ .

Therefore by the eighth Case of Oblique angled Spherical Triangles .

10.0997059 10.0000000 9.8494850 9.7497791 67.97 29.33 38.64 0.0595768 9.8937251 9.8926982 9.8460001

Whose Complement FC . 44. 55 is the ☉ Altitude required .

The third Variety is when the Sun is in the Southern Signs as in ♎ . ♏ . ♐ . ♑ . ♒ . ♓ . And in this Case supposing the ☉ to be ♐ 10 degrees , and his Declination South Db 22. 03. and his Distance from the Meridian 45 as before , then in the Oblique angled Spherical Triangle Z bP in Fig. 1. we have given Z P. 38. 47. The Side bP 112. 03. and the Angle ZPb 45. To find Zb.

Therefore by the 8 Case of Oblique angled Spherical Triangles .

10.0997059 10.0000000 9.8494850 9.7497791 112.03 29.33 82.70 0.0595768 9.8937251 9.1040246 9.0573265

Whose Complement 6.55 is the ☉ Altitude required .

PROBLEM 11.

Having the Altitude of the Sun , his Distance from the Meridian , and Declination , to find his Azimuth .

The Azimuth of the Sun is an Arch of the Horizon intercepted between the Meridian and the Vertical Line passing by the Sun , being understood by the Angle HZC in Fig. 1. or Arch HC . And in all the Varieties of the last Problem , may be found , by the first Case of Oblique angled Spherical Triangles .

Thus in the Triangle ZBP. 0 . 022090● 9.9375306 10.0000000 9.9596209

In the Triangle ZFP . I say .

s. ZF . s. ZPF ▪ : : s. FP . s. FZP .

In the Triangle ZbP. I say .

Sine Zb. Sine ZPb : : Sine bP. Sine bZP.

PROBLEM 12.

The Poles Elevation , with the Sun's Altitude and Declination given , to find his Azimuth .

In the Oblique angled Spherical Triangle FZP in Fig. 1. let there be given .

1. FP . 67. 97 the Complement of the ☉ Declination .

2. ZP . 38. 47 the Complement of the Poles Elevation .

3. FZ . 45. 46 the Complement of the ☉ Altitude .

And let the Angle FZP the ☉ Azimuth be required .

By the 11 Case of Oblique Angled Spherical Triangles .

As the Sine ZP x Sine FZ , Is to the Square of Radius .

So is the Sine 1/2 Z of the Sides ZP x 1/2 Z cr — ZF .

To the Square of the Sine of half the Angle FZP .

The Sum of the three Sides is 151.89

The half Sum is 75.945 from which deduct PZ 38. 47. The difference is 37.475 And the Difference between 75.945 and FZ is 30. 495.

0.2061365 0.1471308 9.7842000 9.7054045 19.8428618 9.9214309

The double whereof is 115.88 the ☉ Azimuth from the North. And the Complement 64.12 , is the ☉ Azimuth from the South .

PROBLEM 13.

To find the Point of the Ecliptick Culminating , and its Altitude .

Before we can know what Sign and Degree of the Ecliptick is in the Medium Coeli ; we must find the Right Ascension thereof , to do which , we must add the Sun's Right Ascension to the time afternoon , being reduced into Degrees and Minutes of the AEquator , the Sum is the Right Ascension of the Medium Coeli .

Example . Let the time given , be March the 20. 1674. at one of the Clock in the Afternoon .

At which time the Sun's place is in ♈ . 10 deg. 23 Centesms .

To find the Right Ascension thereof , in the Right angled Spherical T●iangle ADF in Fig. 1. we have given ; The Angle of the Sun 's greatest Declination DAF 23. 525 and the Sun's distance from the next Equinoctial Point AF 10. 23.

Therefore by the ninth Case of Right angled Spherical Triangles .

10.7435974 10.0000000 19.9623154 9.2187180

To which adding the Equinoctial Degrees answering to one hour , viz. 15. the Sum is 24.39 the Right Ascension of the Mid Heaven . Hence to find the Point culminating ; in the Right angled Spherical Triangle ADF in Fig. 1. we have given AD 24. 39 and DAF 23. 525 to find AF.

Therefore by the fifth Case of Right Angled Spherical Triangles .

10.6564908 10.0000000 9.9623154 10.3058246

Therefore the Point culminating is ♈ 26. 31.

To find the Altitude thereof above the Horizon we have given in the same Triangle DAF 23. 525. and AF 26. 31. to find DF.

Therefore by the tenth Case of Right angled Spherical Triangles .

10.0000000 9.6466268 9.6011352 9.2477628

Which is the North Declination of the Point of the Ecliptick culminating , and being added to the height of the AEquator at London 38. 47 the Sum is 48.66 the Altitude of the Mid Heaven as was required .

PROBLEM 14.

Having the greatest obliquity of the Ecliptick together with the Distance of the Point given from the Equinoctial , to find the Meridian Angle , or Intersection of the Meridian with the Ecliptick .

Having drawn the Primitive Circle HZRN in Fig. 5. representing the Meridian , and the two Diameters HAR , and ZAN , set off the height of the Pole from R to P. 51. 53 , and from N to S , and draw the Diameters PAS for the Axis of the World , and AE AQ for the AEquator ; this done , the Right Ascension of the Mid Heaven being given , as in the last Problem 24.39 with the Point culminating . ♈ . 26.31 , and the Declination thereof 10.19 , if you set 10 deg. 19 Centesmes from AE to F and e to X , you may draw the Diameters FAX and cAd at Right Angles thereunto , and because the Imum Coeli is directly opposite to the Point culminating , that is , in ♎ 26.31 , if you set 26.31 from X to b , a Ruler laid from c to b will cut the Diameter FX in G , and then making Xh Z Xb you have the three Points b G h , by which to draw that Circle , which will cut the AEquator AE AQ in ♎ , and so you have the three Points X ♎ F by which to describe the Arch of the Ecliptick ♈ F ♎ X.

And in the Right angled Spherical Triangle ♈ AEF we have given . The Angle AE ♈ F. 23.525 the Sun 's greatest Declination , and ♈ F. 26. 31. the Point culminating , to find the Angle ♈ F AE .

Therefore by the seventh Case of Right angled Spherical Triangles .

10.3611802 10.0000000 9.9525062 9.5913260

Which is the Angle of the Ecliptick with the Meridian .

PROBLEM 15.

To find the Angle Orient , or Altitude of the Nonagesime Degree of the Ecliptick .

In Fig. 5. the Pole of the Ecliptick ♈ F ♎ X is at m , and so you have the three Points Z m N to draw the Vertical Circle Z k N cutting the Ecliptick at Right Angles in the Point a : And then in the Right angled Spherical Triangle F a Z , we have given ; FZ 41. 34 the Complement of FH the Altitude of the Mid Heaven ; And the Angle a FZ 68. 68 the Angle of the Ecliptick with the Meridian . To find Z a.

Therefore by the tenth Case of Right angled Spherical Triangles .

As the Radius .

9 . 819889● 9.9691128 9.7891027

Whose Complement is ak the Measure of the Angle agk 52. 03 the Angle of the Ecliptick with the Horizon , or Altitude of the Nonagesime Degree .

PROBLEM 16.

To find the place of Nonagosime Degree of the Ecliptick .

In Fig. 5. F represents the Point of the Ecliptick in the Mid Heaven , which according to Problem 14 is ♈ . 26.31 which being known , in the Triangle FZa , we have also given , FZ 41. 34 and the Angle ZFa. 68. 68. To find Fa.

Therefore by the ninth Case of Right angled Spherical Triangles .

10.0556361 10.0000000 9.5605957 9.5049596

Which being added to ♈ F 26. 31 the sum is ♈ a. 44. 04 the place of the Nonagesime Degree of the Ecliptick at a.

PROBLEM 17.

The Mid Heaven being given , to find the Points of the Ecliptick Ascending and Descending .

Having found by the last Problem , the place of Nonagesime Degree of the Ecliptick at a to be in ♉ . 14.04 , if you add 90 Degrees or three Signs thereto , the Ascendant at g will be in ♌ 14. 04 , and the Point descending by adding of six Signs will be in ♒ 14. 02. But these with the Cusps of the other Houses of Heaven may be otherwise found in this manner .

To the Right Ascension of the Medium Coeli or the tenth House , add 30 , it giveth the Ascension of the eleventh House , to which adding 90 Degrees more , it giveth the Ascension of the twelfth House , &c. According to which direction , the Ascensions of the six Houses towards the Orient , are here set down in the following Table .

10. 24.39 11. 54.39 12. 84.39 1. 114.39 2. 144.39 3. 174.39

Now because the Circles of Position must according to these Directions cut the AEquator at 30 and 30 Degrees above the Horizon , if in Fig. 5. you set 30 Degrees from AE to n , and n to r. A Ruler laid from P to n and r , shall cut the AEquator at B and K , and then you may describe the Circles of Position HBR and HKR , make AT = AK and AV = AB , and so you may describe the Circles HTR and HVR , and where these Circles do cut the Arch of the Ecliptick ♈ F ♎ there are the Cusps of the Coelestial Houses .

Thus a Ruler laid from m. the Pole of the Ecliptick to the Intersections ct s. t. g. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . will cut the Primitive Circle in 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . and the Arches 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 = Fs. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 = Ft. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 = Fg. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 = 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 = 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 being added to 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 B will give you the Cusps of the 11. 12. 1. 2 and 3 Houses , the other six are the same Degrees and Parts in the Opposite Signs .

Thus a Figure in Heaven may be erected by Projection , the Arithmetical Computation now followeth ; In which the height of the Pole above each Circle of Position is required , the which in the Projection is easily found ; as the Pole of the Circle of Position HBR is at the Point D. and so you have the three Points S , D , P , to describe that Circle by , which will cut the Circle HBR at Right Angles in the Point C. and the Arch PC is the height of the Pole above that Circle of Position , and may be measured by the Directions given in the nineteenth of the third Chapter .

In like manner the height of the Pole above the Circle of Position HKR , will be the Arch PE.

To compute the same Arithmetically in the Right angled Spherical Triangle HAEB in Fig. 5. we have given AEH . 38. 47 the height of the Equator . AEB 30. the difference of Ascension between the 10 and 11 Houses , to find HBAE the Angle of that Equator with the Circle of Position .

Therefore by the first Case of Right angled Spherical Triangles .

9.90000652 10.00000000 9.69897000 9.79888348

Whose Measure in the Scheme is EC , and the Complement thereof is CP. 32. 18374 the height of the Pole required .

Therefore the height of the Pole above the Circle of Position HKR . In the Triangle HAEK , we have given , H AE as before , and AE K. 60 to find HKAE . Therefore .

9.90008652 10.00000000 9.93753063 10.03744411

Whose Measure in the Scheme is GL , and the Complement thereof is PL 47. 46692. the height of the Pole required .

The height of the Pole above HDR is the same with HBR , and the height of the Pole above HTR is the same with HKR .

Having found the Ascensions of the several Houses together with the Elevation of the Pole above their Circles of Position , in the Oblique angled Spherical Triangle ♈ BS , we have given .

1. The Angle ♈ BS the Complement of HBAE .

2. The Angle B ♈ S. 23. The Sun 's greatest Declination .

3. Their included Side ♈ B. 54. 39 the Ascension of the eleventh House . To find ♈ S the Point of the Ecliptick , which is resolvable by the ninth Case of Right angled Spherical Triangles .

But in my Trigonometria Britannica , Problem . 5. for the resolving of Oblique angled Spherical Triangles , I have shewed how this Case as to our present purpose may be resolved , by these Proportions following .

1. s 1/2 Z Ang. s 1/2 X Ang: : t 1/2 ♈ B. t 1/2 X Cru . 2. cs 1/2 Z Ang. cs 1/2 X Ang: : t 1/2 ♈ B. t 1/2 Z Cru . 1/2 Z Cru + 1/2 X Cru = ♈ S the Arch of the Ecliptick desired .
For the Cusp of the Eleventh House . T B Arch ♈ B. 4439 the half whereof is 27. 195. ♈ B S. 122. 18374. B ♈ S. 23. 525. Z 145.70874 — 1/2 Z 72. 85437. X. 198.65874 — 1/2 X. 49. 32937. 0.01977589 9.88000800 9.71081089 9.61059478
2. Operation . 0.53012277 9.81395860 9.71081089 10.05489226

1. Arch. 22. 192. Their Sum is 70.803 the Point of the Ecliptick .

0.88517901 9.71164750 10.19072348 10.68754999

1. Arch — 53. 296. Their Sum 121.693 is the Point of the Ecliptick for the Ascendant .

For the Cusp of the Second House .

In the Oblique angled Spherical Triangle ♈ T 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . we have given , The half whereof is 72. 195. To find ♈ 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . The Angles are the same with those of the Twelfth House . Therefore .

0.00601663 9.92351651 9.92953314 10.49327695 10.42281009 2. Operation . 0.78170174 9.73628614 10.51798788 10.49327695 11.01126483

1. Arch. 69. 306. Their Sum is 53.740 is the Point of the Ecliptick for the Second House .

For the Cusp of the Third House .

In the Oblique angled Spherical Triangle ♈ 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , we have ,

1. ♈ 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . 174. 39. The half whereof is 87. 195. The Angls ♈ 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ♈ 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 are the same with those of the Eleventh House .

0.01977580 6.88000800 9.89978389 11.30984054

For the Eleventh House .

For the Cusp of the Twelfth House .

In the Oblique angled Spherical Triangle ♈ KF , we have given . The half whereof is . 42. 195. To find ♈ t.

½ Z. 80.49596 ½ X. 56.97096 0.00601663 9.92351651 9.95740882 9.88694196 2. Operation . 0.78170174 9.73628614 9.95740882 10.47539670

1. Arch. 37. 625. Their Sum 113.6691 is the Point of the Ecliptick for the Twelfth House .

For the Cusp of the Ascendant .

In the Oblique angled Spherical Triangle ♈ AG we have , The half whereof is 57. 195. The Complement of HAAE 38. 46667.

½ Z. 82.51916 ½ X. 59.00416 0 . 0037162● 9.93313477 10.19072348 10.12757454 2. Operation . 11.20962043 2. Operation . 0.53012277 9.81395860 10.34408137 11.39984054 11.65392191

1. Arch. 86. 468. Their Sum 175.197 is the Point of the Ecliptick for the Third House .

And thus we have not only erected a Figure for the Time given , but composed a Table for the general erecting of a Figure in that Eatitude ; for by adding together the first and second Numbers in each Proportion for the first , second and third Houses there is composed two Numbers for each House , to each of which the Artificial Tangent of half the Ascension of each House being added , their Aggregates are the Tangents of two Arches , which being added together , do give the distance of the Cusp of the House , from the first Point of Aries , as in the preceding Operations hath been shewed .

Only note , That if the Ascension of any House be more than a Semicircle , you must take the Tangent of half the Complement to a whole Circle . And to find the Cusp of the House , you must also take the Complement of the Sum of the Arches added together .

The Numbers according to the former Operations which do constitute a Table of Houses for the Latitude of London . 51. 53 are as followeth .

11 and 3 Houses Ascendant 12 and 2 Houses 1. Oper. 9.89978389 9.93685106 9.92953314 2. Oper. 10.34408137 10.59682651 10.51798788

The Six Oriental Houses , by the preceding Operations .

The opposite Houses are in the opposite Signs and Degrees . 10 House ♈ 26.311 11 House ♊ 10.803 12 House ♋ 23.691 Ascendant ♌ 11.693 2 House ♏ 3.740 3 House ♏ 25.197 4 House ♎ 26.311 5 House ♐ 10.803 9 House ♑ 23.691 7 House ♒ 11.693 8 House ♓ 3.740 9 House ♓ 25.197

A Figure of the Twelve Coelestial Houses .

ASTRONOMY . THE Second Part : OR , AN ACCOUNT OF THE Civil Year , With the Reason of the Difference Between the JULIAN & GREGORIAN Calendars , And the manner of Computing the Places of the SVN and MOON .

LONDON , Printed for Thomas Passinger , at the Three Bibles on London-Bridge . 1679.

AN INTRODUCTION TO Astronomy . The Second Book .
CHAP. I.

Of the Year Civil and Astronomical .

HAving shewed the Motion of the Primum Mobile , or Doctrine of the Sphere , which I call the Absolute Part of Astronomy ; I come now unto the Comparative , that is , to shew the Motion of the Stars in reference to some certain Distinction of Time .

2. And the Distinction of Time is to be considered either according to Nature , or according to Institution .

3. The Distinction of Time according to Nature , is that space of Time , in which the Planets do finish their Periodical Revolutions from one certain Point in the Zodiack , to the same again , and this in reference to the Sun is called a Year , in reference to the Moon a Month.

4. The Sun doth pass through the Zodiack in 365 Days , 5 Hours , and 49 Minutes . And the Moon doth finish her course in the Zodiack , and return into Conjunction with the Sun , in 29 Days , 12 hours , 44 Minutes , and 4 Seconds . And from the Motion of these two Planets , the Civil Year in every Nation doth receive its Institution .

5. Twelve Moons or Moneths is the measure of the Common Year , in Turkey in every Moneth they have 29 or 30 Days , in the whole Year 354 Days , and in every third Year 355 Days .

6. The Persians and Egyptians do also account 12 Moneths to their Year ; but their moneths are proportioned to the Time of the Suns continuance in every of the Twelve Signs ; in their Year therefore which is Solar , there are always 365 Days , that is eleven Days more than the Lunar Year .

7. And the Iulian Year which is the Account of all Christendom , doth differ from the other in this ; that by reason of the Sun's Excess in Motion above 365 Days , which is 5 Hours , 49 Minutes , it hath a Day intercalated once in 4 Years , and by this intercalation , it is more agreeable to the Motion of the Sun , than the former , and yet there is a considerable difference between them , which hath occasioned the Church of Rome to make some further amendment of the Solar Year , but hath not brought it to that exactness , which might be wished .

8. This intercalation of one Day once in 4 Years , doth occasion the Sunday Letter still to alter till 28 Years be gone about ; The Days of the Week which use to be signed by the seven first Letters in the Alphabet , do not fall alike in every Common Year , but because the Year consisteth of 52 Weeks and one Day , Sunday this Year will fall out upon the next Year's Monday , and so forward for seven years , but every fourth year consisting of 52 weeks and two days , doth occasion the Sunday Letter to alter , till four times seven years , that is till 28 years be gone about . This Revolution is called the Cycle of the Sun , taking its name from the Sunday Letter , of which it sheweth all the Changes that it can have by reason of the Bissextile or Leap-year . To find which of the 28 the present is , add nine to the year of our Lord , ( because this Circle was so far gone about , at the time of Christs Birth ) and divide the whole by 28 , what remaineth is the present year , if nothing remain the Cycle is out , and that you must call the last year of the Cycle , or 28.

9. This Intercalation of one day in four years , doth occasion the Letter F to be twice repeated in February , in which Moneth the day is added , that is , the Letter F is set to the 24 and 25 days of that Moneth , and in such a year S. Matthias day is to be observed upon the 25 day , and the next Sunday doth change or alter his Letter , from which leaping or changing , such a year is called Leap-year , aud the number of days in each Moneth is well expressed by these old Verses .

Thirty days hath September , April , June and November .

February hath 28 alone , All the rest have thirty and one .

But when of Leap-year cometh the Time , Then days hath February twenty and nine .

That this year is somewhat too long , is acknowledged by the most skilful Astronomers , as for the number of days in a year the Emperours Mathematicians were in the right , for it is certain , that no year can consist of more than 365 days , but for the odd hours it is as certain that they cannot be fewer than five , nor yet so many as six ; so then the doubt is upon the minutes , 60 whereof do make an hour , a small matter one would think , but how great in the consequence we shall see . The Emperours year being more than 10 minutes greater than the Suns , will in 134 years rise to one whole day , and by this means the Vernal or Spring Equinox , which in Iulius Caesar's time was upon the 24 of March , is now in our time upon the 10 of March , 13 days backward , and somewhat more , and so if it be let alone will go back to the first of March , and first of February , and by degrees more and more backward still .

10. To reform this difference , some of the late Roman Bishops have earnestly endeavoured . And the thing was brought to that perfection it now standeth , by Gregory the Thirteenth , in the year 1582. His Mathematicians , whereof Lilius was the Chief , advised him thus : That considering there had been an Agitation in the Council of Nice somewhat concerned in this matter upon the motion of that Question , about the Celebration of Easter . And that the Fathers of the Assembly , after due deliberation with the Astronomers of that time , had fixed the Vernal Equinox at the 21 of March , and considering also that since that time a difference of ten whole days had past over in the Calendar , that is , that the Vernal Equinox , which began upon the 21 of March , had prevented so much , as to begin in Gregorie's days at the 10 of the same , they advised , that 10 days should be cut off from the Calendar , which was done , and the 10 days taken out of October in the year 1582. as being the moneth of that year in which that Pope was born ; so that when they came to the fifth of the moneth they reckoned the 15 , and so the Equinox was come up to its place again , and happened upon the 21 of March , as at the Council of Nice .

But that Lilius should bring back the beginning of the year to the time of the Nicene Council and no further , is to be marvelled at , he should have brought it back to the Emperours own time , where the mistake was first entered , and instead of 10 , cut off 13 days ; however this is the reason why these two Calendars differ the space of 10 days from one another . And thus I have given you an account of the year as it now stands with us in England , and with the rest of the Christian World in respect of the Sun , some other particulars there are between us and them which do depend upon the motion of the Moon , as well as of the Sun , and for the better underderstanding of them , I will also give you a brief account of her revolution . But first I will shew you , how the day of the moneth in any year propounded in one Couutry , may be reduced to its correspondent time in another .

11. Taking therefore the length of the year , to be in several Nations as hath been before declared , if we would find what day of the moneth in one Conntry is correspondent to the day of that moneth given in another , there must be some beginning to every one of these Accounts , and that beginning must be referred to some one , as to the common measure of the rest .

12. The most natural beginning of All Accounts , is the time of the Worlds Creation , but they who could not attain to the Worlds Beginning , have reckoned from their own , as the Romans from the building of Rome , the Greeks from their Olympicks , the Assyrians from Nabonassar , and all Christians from the Birth of Christ : the beginning of which and all other the most notable Epochaes , we have ascertained to their correspondent times in the Julian Period , which Scaliger contrived by the continual Multiplication of those Circles , all in former time of good use , and two of them do yet remain ; the Circles yet in use are those of the Sun and Moon , the one , to wit , the Sun , is a Circle of 28 years , and the Circle of the Moon is 19 , as shall be shewed hereafter . The third Circle which now serves for no other use than the constituting of the Julian Period , is the Roman Indiction , or a Circle of 15 years ; if you multiply 28 the Circle of the Sun , by 19 the Circle of the Moon , the Product is 532 , which being multiplied by 15 , the Circle of the Roman Indiction , the Product is 7980 , the Number of years in the Julian Period : whose admirable condition is to distinguish every year within the whole Circle by a several certain Character , the year of the Sun , Moon , and Indiction being never the same again until the revolution of 7980 years be gone about , the beginning of this Period was 764 Julian years before the most reputed time of the Worlds Creation ; which being premised , we will now by Example shew you how to reduce the years of Forreigners to our Julian years , and the contrary .

1. Example .

I desire to know at what time in the Turkish Account , the fifth of Iune in the year of our Lord 1640. doth fall .

The Julian years complete are 1648 , and are thus turned into days , by the Table of days in Julian years .

365250 219150 14610 2922 151 5 602088

Now because the Turkish Account began Iuly 16. Anno Christi . 622. you must convert these years into days also .

219150 7305 365 181 15 227016 602088 375072 318930 56142 53155 02987 2835 152 148 4

Therefore the fifth of Iune 1649. in our English Account doth fall in the year 1058. of Mahomet , or the Turkish Hegira , the fourth day of the moneth Giumadi . 11

2. Example .

I desire to know upon what day of our Julian year the 17 day of the moneth in the 1069 year complete of the Persian Account from Ieshagile doth fall .

230639 365000 21900 3285 90 16 620930 365250 255680 255675 5

Therefore it falls out in the Julian year from Christ 1700. the fifth day of Ianuary .

He that understands this may by the like method convert the years of other Epochas , into our Julian years and the contrary .

The Anticipation of the Gregorian Calendar is more easily obtained , for if you enter the Table with the years of Christ complete , you have the days to be added to the time in the Julian Account , to make it answer to the Gregorian , which will be but ten days difference till the year 1700. and then the difference will be a day more , until the year 1800. and so forward three days difference more in every 400 years to come , unless our year shall be reformed as well as theirs .

CHAP. II.

Of the Cycle of the Moon , what it is , how placed in the Calendar , and to what purpose .

THat the Civil Year in use with us and all Christians , doth consist of 365 days , and every fourth year of 366 , hath been already shewed , with the return of the Sunday Letter in 28 years . In which time the Moon doth finish her course in the Zodiack no less than twelve times , which twelve Moons , or 354 days , do fall short of the Sun's year , eleven days in every common year , and twelve in the Bissextile or Leap-year .

And by Observation of Meton an Athenian , it was found out about 432 years before Christ , that the Moon in nineteen years did return to be in Conjunction with the Sun on the self same day , and this Circle of nineteen years is called the Cycle of the Moon , which being written in the Calendar against the day in every Moneth , in which the Moon did change , in Letters of Gold , was also called the Golden Number , or from the excellent use thereof , which was at first , only to find the New Moons in every Moneth for ever , but amongst Christians it serveth for another purpose also , even the finding of the time when the Feast of Easter is to be observed . The New Moons by this Number are thus found . In the first year of the Circle , or when the Golden Number is 1 , where the Number 1 was set in the Calendar in any Moneth , that day is New Moon , in the second Year where you find the golden Number 2 , in the third Year where you find the golden Number 3 , and so forward till the whole Circle be expired ; then you must begin with one again , and run through the whole Circle as before .

2. And the reason why the Calendar begins with the golden Number 3 , not 1 , is this . The Christians in Alexandria had used this Circle of the Moon two Years before the Nicene Council . And in the first of these Years the new Moon next to the Vernal Equinox was upon the 27th Day of the Egyptian month Phamenoth answering to the 23d of our March , against that Day therefore they placed the golden Number 1. And because there are 29 Days and a half from one new Moon to another , they made the distance between the new Moons to be interchangeably 29 and 30 Days , and so they placed the same golden Number against the 26 Day of Phurmuthi the Month following , and against the 26 Day of the Month Pachon and so forward , and upon this ground by the like progression was the golden Number set in the Roman Calendar ; and so the golden Number 1 by their example was set against March 23. April 21. Iune 19. Iuly 19. August 17. September 16. October 15. November 14. December . 13. But then because in the following Year the golden Number was 2. reckoning 30 Days from the 13th of December , the golden Number 2 was set to Innuary 12. February 10. March 12. April 10. May 10. Iune 8. Iuly 8. August 6. September 5. October 4. November 3. December . 2. From whence reckoning 13 Days as before , the golden Number 3 comes in course for the third Year to be set against the first of Ianuary .

But that you may know how the golden Number comes to be distributed in the Calendar according to the form in which it now is , you must consider that in 19 Solar Years there are not only 228 Lunar Months or 12 times 19 Lunar Months but 235 for the 11 Days which the common Solar Year doth exceed the Lunar , do in 19 Years arise to 209 Days , out of which there may be appointed 7 Months , 6 whereof will contain 30 Days apiece , and one Month 29 days ; and these 7 Months are called Embolismical Months , because by a kind of injection or interposition they are reckoned in some of the 19 Years . And those Years in which they are reckoned are called Embolismical Years , to distinguish them from the common Years which always contain 354 Days , whereas 6 of these Embolismical Years do each of them contain 384 Days , and the seventh Embolismical Year in which the Month of 29 Days is reckoned , doth contain 383 Days .

3. The Embolismical Years in the Cycle of the Moon are properly these Seven . 3 , 6 , 9 , 11 , 14 , 17 , 19. because in the third Year 11 Days being thrice reckoned do amount to 33 Days , that is one Month of 30 Days and 3 Days over . Again in the sixth Year the 11 Days which the Solar exceed the Lunar , being thrice numbred , do amount to 33 Days . which with the 3 Days formerly reserved do make 36 Days , that is one Month of 30 Days and 6 Days over . Again in the Ninth Year there are also 33 Days , to which the 6 Days reserved being added , there will arise one Month more and 9 Days over . But in the Eleventh Year twice 11 Days being added to the 9 Days reserved , do make 31 Days , that is , one Month of 30 days and one day over , which being added to the supernumerary days in the fourteenth Year do make another Month of 30 Days and 4 Days over , and these being added to the supernumerary Days in the sevententh Year do make another Month of 30 and 7 Days over , and these 7 Days being added to the 22 supernumerary Days in the Ninteenth Year of the Moons Cycle do make another Month of 29 Days .

4. But because there are 6939 Days and 18 Hours in 19 Solar Years , that is , 4 Days 18 Hours more then in the common and Embolismical Lunar Years , in which the excess between the Lunar and the Solar Year is supposed to be no more then 11 Days in each Year , whereas in every fourth Year the excess is one Day more , that is , 12 Days , that is , in 16 Years 4 Days , and in the remaining 3 Years three fourths of a day more . And that the new Moons after 19 Lunar Years or 235 Lunations do not return to the same days again , but want almost 5 days , it is evident that the civil Lunations do not agree with the Astronomical and that there must be yet some kind of intercalation used .

5. Now therefore in distributing the golden Number throughout the Calendar . If the new Moons should interchangeably consist of 30 and 29 days , and so but 228 Lunations in 19 Years ; we might proceed in the same order in which we have begun , and by which as hath been shewed the third Year of the Golden Number falls upon the Calends of Ianuary . But for as much as there are first six Lunations of 30 days apiece and one of 29 days to be interposed , therefore there must be 6 times 2 Lunations together consisting of 30 days and once three Lunations of 29 days . And that respect may be also had to the Bissextile days , although they are not exprest in the Calendar , that Lunation which doth contain the Bissertile day , if it should have been 29 days , it must be 30 , if it should have consisted of 30 days it must consist of 31.

6. And because it was thought convenient , as hath been shewed , to begin with the third Year of the Cycle of the Moon , because the Golden Number 3 is set to the Calends of Ianuary , therefore in this Cycle the Embolismical Years are , 2 , 5 , 8 , 11 , 13 , 16 , 19. But yet that it may appear , that these Years are in effect the same , as if we had begun with the first Year of the Golden Number , save only that the eighth Year instead of the ninth is to be accounted Embolismical , I have added the Table follwing , in which it is apparent that the former Embolismical years do agree with these last mentioned .

7. But as I said before , it was thought more convenient to begin the account from the number 3 set to the Calends of Ianuary , because by so reckoning 30 and 29 days to each Lunation interchangeably , the same Number 3 falls upon Ianuary 31. March 1 , and 31. April 29. May 29. Iune 27. Iuly 27. August 25. September 14. October 23. November 22. December 21. As if the Lunar years were compleated upon the 20 of December there remain just 11 Days , which the Solar years doth exceed the Lunar .

8. And by ranking on and accounting 4 for the Golden Number of the next year , you will find it set on Ianuary 20 , February 18 , March 20 , April 18 , May 18 , Iune 16 , Iuly 16 , August 14 , September 13 , Octob. 12 , Novemb. 11 , Decemb. 10.

Cycle of the Moon . Cycle of the Moon . Embolismical Years . Number of Days . 1 3 354 2 4 354 3 5 Embol . 384 4 6 354 5 7 354 6 8 Embol . 384 7 9 354 8 10 354 9 11 Embol . 384 10 12 354 11 13 Embol . 384 12 14 354 13 15 354 14 16 Embol . 384 15 17 354 16 18 354 17 19 Embol . 384 18 1 354 19 2 Embol . 384

9. But in going on , and taking 5 for the Golden Number in the third year , we must remember that that is an Embolismical Year , and therefore that somewhere there must be 2 Months together of 30 days . And for this reason the Golden Number 5 , is set to Ianuary 9 , February 7 , March 9 , April 7 , May 7 , Iune 5 , Iuly 5 , August 3 , September 2 , as also upon the second day of October , and not upon the first , that so there may be 2 Lunations together of 30 , and the same Number 5 is also set to the thirty first of October , to make the Lunation to consist of 29 days , and to the thirtieth of November instead of the twenty ninth , that so a Lunation of 30 may again succeed as it ought .

10. In like manner in the sixth Year , having gone through the fourth and fifth as common years , you may see the Golden Number 8 set to the fifth of April , which should have been upon the fourth , and in the ninth Year the Golden Number 11 is set to the second of February which should have been upon the first .

And there is a particular reason , for which these numbers are otherwise placed from the eighth of March to the fifth of April , namely , that all the paschal Lunations may consist of 29 days : For thus from the eighth of March to the sixth of April , to both which days the Golden Number is 16 , there are but 29 days . And from the ninth of March to the seventh of April , to both which days the Golden Number is 5 , there are also 29 days , and so of the rest till you come to the fifth of April , which is the last Paschal Lunation , as the eighth of March is the first , but at any other time of the Year , the length of the Month in the Embolismical Year , may be fixed as you please .

12. And in this manner in the 17 years , in which the lunations of the whole Circle are finished , and in which the Golden Number is 19 , the Month of Iuly is taken at pleasure , to the thirtieth day whereof is set the Golden Number 19 , which should have been upon the thirty first , and the same Number being notwithstanding placed upon the twenty eighth of August , that by the two Lunations of 29 days together , it might be understood , that the seventh Embolismical Month consisting of 29 days is there inserted , instead of a Month of 30 days . In which place the Embolismical or leaping Year of the Moon may plainly be observed for that year is one day less than the rest , which the Moon doth as it were pass over . The which one day is again added to the 29 days of the last Month , that we may by that means come , as in other Years , to the Golden Number , which sheweth the New Moon in Ianuary following . And for this reason the Epact then doth not consist of 11 but of 12 days . And thus you see the reason , for which the Golden Numbers are thus set in the Calendar as here you see . In which we may also observe , that every following Number is made by adding 8 to the Number preceding , and every preceding Number is also made by adding 11 to the Number next following , and casting away 19 when the addition shall exceed it .

For Example , if you add 8 to the Golden Number 3 set against the first of Ianuary , it maketh 11 , to which add 8 more and it maketh 19 , to which adding 8 it maketh 27 , from which substracting 19 the remainer is 8 , to which again adding 8 , the sum is 19 , to which adding 8 the sum is 24 , from which deducting 19 the remainer is 5 , and so of the rest . In like manner receding backward , to the 5 add 11 they make 16 , to the 16 add 11 they make 27 , from which deducting 19 the remainer is 8 , to which 11 being added the same is 19 , to which 11 being added the sum is 30 , from which deducting 19 the remainer is 11 , to which 11 being added the sum is 22 , from which deducting 19 the remainer is 3. And by this we may see that every following number will be in use 8 years after the preceding , and every preceding Number will be in use 11 years after the following , that is , the same will return to be in use after 8 Years and 11 , and the other after 11 Years and 8 , or once in 19 years .

CHAP. III.

Of the Vse of the Golden Number in finding the Feast of Easter .

THe Cycle of the Moon or Golden Number is a circle of 19 years , as hath been said already , which being distributed in the Calendar as hath been shewn in the last Chapter , doth shew the day of the New Moon for ever ; though not exactly : But the use for which it was chiefly intended , was to find the Paschal New Moons , that is , those new Moons on which the Feast of Easter and other moveable Feasts depend . To this purpose we must remember ,

1. That the vernal Equinox is supposed to be fixed to the twenty first day of March.

2. That the fourteenth day of the Moon on which the Feast of Easter doth depend , can never happen before the Equinox ; though it may fall upon it or upon the day following .

3. That the Feast of Easter is never observed upon the fourteenth day of the Moon , but upon the Sunday following ; so that if the fourteenth day of the Moon be Sunday , the Sunday following is Easter day .

4. That the Feast of Easter may fall upon the fifteenth day of the Moon , or upon any other day unto the twenty first , inclusively .

5. That the Paschal Sunday is discovered by the proper and Dominical Letter for every Year The which may be found as hath been already declared , or by the proper Table for that purpose . Hence it followeth ,

1. That the New Moon immediately preceding the Feast of Easter , cannot be before the eighth day of March , for if you suppose it to be upon March 6 , the Moon will be 14 days old March 19 , which is before the Equinox , contrary to the second Rule before given , and upon the seventh day of March there is no Golden Number fixed ; and therefore the Golden Number 16 , which standeth against March 8 , is the first by which the Paschal New Moon may be discovered .

2. It followeth hence , That the last Paschal New Moon cannot happen beyond the fifth day of April , because all the 19 Golden Numbers are expressed from the eighth of March to that day . And if a New Moon should happen upon the sixth of April , there would be two Paschal New Moons that year , one upon the eighth of March and another upon the sixth of April , the same Golden Number 16 being proper to them both , but this is absurd because Easter cannot be observed twice in one year .

3. It followeth hence , That the Feast of Easter can never happen before the twenty second day of March , nor after the twenty fifth day of April : For if the first New Moon be upon the eighth of March , and that the Feast of Easter must be upon the Sunday following the fourteenth day of the Moon ; it is plain that the fourteenth day of the Moon must be March 21 at the soonest : So that supposing the next day to be Sunday , Easter cannot not be before March the twenty second . And because the fourteenth day of the last Moon falleth upon the eighteenth day of April , if that day be Saturday , and the Dominical Letter D , Easter shall be upon the nineteenth day , but if it be Sunday , Easter cannot be till the twenty fifth .

4. It followeth hence , That although there are but 19 days , on which the fourteenth day of the Moon can happen , as there are but 19 Golden Numbers , yet there are 35 days from the twenty second of March to the twenty fifth of April , on which the Feast of Easter may happen , because there is no day within those Limits , but may be the Sunday following the fourteenth day of the Moon . And although the Feast of Easter can never happen upon March 22 , but when the fourteenth day of the Moon is upon the twenty first , and the Sunday Letter D , nor upon the twenty fifth of April , but when the fourteenth day of the Moon is upon April 18 , and the Dominical Letter C. Yet Easter may fall upon March 23 , not only when the fourteenth day of the Moon is upon the twenty second day which is Saturday , but also if it fall upon the twenty first which is Friday . In like manner Easter may fall upon April 24 , not only when the fourteenth day of the Moon is upon the eighteenth day which is Monday , but also if it happen upon the seventeenth being Sunday . And for the same reason it may fall oftner upon other days that are further distant from the said twenty second of March and twenty fifth of April .

5. It followeth hence , That the Feast of Easter may be easily found in any Year propounded : For the Golden Number in any Year being given , if you look the same between the eighth of March and fifth of April both inclusively , and reckon 14 days from that day , which answereth to the Golden Number given , where your account doth end is the fourteenth day of the Moon : Then consider which is the Dominical Letter for that Year , and that which followeth next after the fourteenth day of the Moon is Easter day . Example , In the year 1674 the Golden Number is 3 , and the Sunday Letter D , which being sought in the Calendar between the aforesaid limits , the fourteenth day of the Moon is upon April the thirteenth , and the D next following is April 19. And therefore Easter day that Year is April 19. Otherwise thus .

In March after the first C , Look the Prime wherever it be , The third Sunday after Easter day shall be . And if the Prime on Sunday be , Reckon that for one of the Three .

6. Thus the Feast of Easter may be found in the Calendar , and from thence a brief Table shewing the same , may be extracted in this manner . Write in one Column the several Golden Numbers in the Calendar from the eighth of March to the fifth of April , in the same order observing the same distance . In the second Column set the Dominical Letters in number 35 so disposed , as that no Dominical Letter may stand against the Golden Number 16 , but setting the Letter D against the Golden Number 5 , write the rest in this order . E , F , G , A , B , &c. and when you come to the Golden Number 8 , set the Letter C , and there continue the Letters till you come to C again , because when the Golden Number is 16 , which in the Calendar is set to the eighth day of March , is new Moon , and the fourteenth day of that Moon doth fall upon the twenty first , to which the Dominical Letter is C , upon which the Feast of Easter cannot happen ; and therefore in the third Column containing the day in which the Feast of Easter is to be observed , is also void . But in the next place immediately following , to wit , against the letter D is set March 22 , because if the fourteenth day of the Moon shall fall upon the twenty first of March being Saturday , the next day being Sunday , shall be the Feast of Easter .

To the Letters following , E , F , G , A , B , &c. are set 23 , 24 , 25 , and so orderly to the last of March , and so forward till you come to the twenty fifth of April , by which Table thus made , the Feast of Easter may be found until the Calendar shall be reformed .

For having found the Golden Number in the first Column , the Dominical Letter for the Year next after it , doth shew the Feast of Easter , as in the former Example , the Golden Number is 3 and the Dominicall Letter D , therefore Easter day is upon April 19. The other moveable Feasts are thus found .

Advent Sunday is always the nearest Sunday to St. Andrews , whether before or after .

Septuagesima Sunday is Nine Weeks before Easter .

Sexagesima Sunday is Eight Weeks before Easter .

Qainquagesima Sunday is Seven Weeks before Easter .

Quadragesima Sunday is Six Weeks before Easter .

Rogation Sunday is five Weeks after Easter .

Ascension day is Forty Days after Easter .

Whitsunday is Seven Weeks after Easter .

Trinity Sunday is Eight Weeks after Easter .

G. N. D. L. Easter . XVI V D 22 March E 23 XIII F 24 II G 25 A 26 X B 27 C 28 XVIII D 29 VII E 30 F 31 XV G 1 April IV A 2 B 3 XII C 4 I D 5 E 6 IX F 7 G 8 XVII A 9 VI B 10 C 11 XIV D 12 III E 13 F 14 XI G 15 A 16 XIX B 17 VIII C 18 D 19 E 20 F 21 G 22 A 23 B 24 C 25

CHAP. IV.

Of the Reformation of the Calendar by Pope Gregory the Thirteenth ; and substituting a Cycle of Epacts in the room of the Golden Number .

HItherto we have spoken of the Calendar which is in use with us , we will now shew you for what reasons it is alter'd in the Church of Rome , and how the Feast of Easter is by them observed .

The Year by the appointment of Iulius Caesar consisting of 365 days 6 hours , whereas the Sun doth finish his course in the Zodiack , in 365 days 5 hours 49 minutes or thereabouts , it cometh to pass that in 134 Years or less , there is a whole day in the Calendar more than there ought ; in 268 years 2 days more ; in 4002 years 3 days : and so since Iulius Caesar's time the vernal Equinox hath gone backward 13 or 14 days , namely from the 24 of March to the tenth . Now because the Equinox was at the time of the Nicene Council upon the twenty first of March , when the time for the observing of Easter was first universally established , they thought it sufficient to bring the Equinox back to that time , by cutting off 10 days in the Calendar as hath been declared , and to prevent any anticipation for the time to come , have appointed , that the Leap-year shall be thrice omitted in every 400 Years to come , and for memory sake , appointed the first omission to be accounted from the Year 1600 , not from 1582 , in which the reformation was made , because it was not only near the time , in which the emendation was begun , but also because the Equinox has not fully made an anticipation of 10 days from the place thereof , at the time of the Nicene Council , which was March 21.

The Years then 1700 , 1800 , 1900 , which should have been Bissextile Years , are to he accounted common years , but the Year 2000 must be a Bissextile : In like manner the Years 2100 , 2200 , 2300 , shall be common years , and the Year 2400 Bissextile , and so forward .

2. Again , because it was supposed that the Cycle of the Moon , or Golden Number was so fixed , that the new and full Moons would in every 19 years return to the same days again ; whereas their not returning the same hours , but making an anticipation of one hour 27 minutes or thereabouts , it must needs be that in 17 Cycles or little more than 300 Years , there would be an anticipation of a whole day . And hence it is evident that in 1300 Years since the Nicene Council , the New and Full Moons do happen more than 4 days sooner than the Cycle of the Moon or Golden Number doth demonstrate : Whence also it comes to pass , that the fourteenth day of the Moon by the Cycle is in truth the eighteenth day , and so the Feast of Easter should be observed not from the fifteenth day of the Moon to the twenty first , but from the nineteenth to the twenty fifth .

3. That the Moon therefore being once brought into order , might not make any anticipation for the time to come , it is appointed that a Cycle of 30 Epacts should be placed in the Calendar instead of the Golden Number , answering to every day in the Year ; to shew the New Moons in these days , not only for 300 Years or thereabouts , but that there might be new Epacts without altering the Calendar , to perform the same thing upon other days as need shall require .

4. For the better understanding whereof , to the Calendar in use with us , we have annexed the Gregorian Calendar also : In the first Column whereof you have 30 numbers from 1 to 30 , save only that in the place of 30 you have this Asterisk * , But they begin with the Calends of Ianuary , and we continued and repeated after a Retrograde order in this manner , * 29 , 28 , 27 , &c. and that for this cause especially , that the number being given which sheweth the New Moons in every Month for one Year , you might by numbring 11 upwards exclusively find the number which will shew the New Moons the Year following , to wit , the Number which falleth in the eleventh place .

5. And these Numbers are called Epacts , because they do in order shew those 11 days , which are yearly to be added to the Lunar Year consisting of 354 days , that it may be in conformity with the Solar Year consisting of 365 days . To this purpose , as hath been said concerning the Golden Number , these Epacts being repeated 12 times , and ending upon the twentieth day of December , the same Numbers must be added to the 11 remaining days , which were added to the first 11 days in the Month of Ianuary .

6. And because 12 times 30 do make 360 , whereas from the first of Ianuary to the twentieth of December inclusively , there are but 354 days , you must know that to gain the other six days , the numbers 25 and 24 are in every other Month both placed against one day , namely , to February 5 , April 5 , Iune 3 , August 1 , September 29 , and November 27. But why these two Numbers are chosen rather then any other , and why in these 6 Months the number 25 is sometimes writ to XVI , sometimes to XXV in a common character , and why the number 19 is set to the last day of December in a common Character , shall be declared hereafter .

7. Here only note that this Asterisk * is set instead of the Epact 30 , because the Epact shewing the Number of days which do remain after the Lunation in the Month of December , it may sometimes fall out that 2 Lunations may so end , that the one may require 30 for the Epact , and the other 0 , which would , if both were written , cause some inconveniences , and therefore this * Asterisk is there set , that it might indifferently serve to both . And the Epact 29 is therefore set to the second day of Ianuary , because after the compleat Lunation in the second of December there are 29 days , and for the like reason the Epact 28 is set against the third of Ianuary , because after the compleat Lunation in the third of December there are then 28 days over , and so the rest in order till you come to the thirtieth of Ianuary , where you find the Epact 1. because after the compleat Lunation on the thirtieth day there is only one day over .

8. And besides the shewing of the New Moons in every Month , which is and may be done by the Golden Number , the Epacts have this advantage , that they may be perpetual and keep the same place in the Calendar in all future ages , which can hardly be effected with the Golden Number , for in little more then 700 years , the New Moons do make an anticipation of one day , and then it will be necessary to set the Golden Number one degree backward , and so the Golden Number which at the time of the Nicene Council was set to the first of Ianuary , should in 300 years be set to the last of December , and so of the rest , but the Epacts being once fixed shall not need any such retraction or commutation . For as often as the New Moons do change their day either by Anticipation or by Suppression of the Bissextile year , you shall not need to do any more than to take another rank of 19 Epacts , insteed of those which were before in use . For instance , the Epacts which are and have been in use in the Church of Rome since the year of reformation 1582 , and will continue till the year 1700 , are these 10 following 1. 12. 23. 4. 15. 26. 7. 18. 29. 10. 21. 2. 13. 24. 5. 16. 27. 8. 19. And from the year 1700 the Epacts which will be in use are these . * 11. 22. 3. 14. 25. 6. 17. 28. 9. 20. 1. 12. 23. 4. 15. 26. 7. 18. and shall continue not only to the year 1800 , but from thence until the year 1900 also ; and although in the year 1800 the Bissextile is to be suppressed , yet is there a compensation for that Suppression , by the Moons Anticipation . To make this a little more plain , the motion of the Moon , which doth occasion the change of the Epact , must be more fully considered .

CHAP. V.

Of the Moons mean Motion , and how the Anticipation of the New Moons may be discovered by the Epacts .

THe Moon according to her middle motion doth finish her course in the Zodiack in 29 days , 12 hours 44 minutes , three seconds or thereabout , and therefore a common Lunar year doth consist of 354 days , 8 hours , 48 minutes , 38 seconds and some few thirds , but an Embolismical year doth consist of 383 days , 21 hours , 32 minutes , 41 seconds and somewhat more ; and therefore in 19 years it doth exceed the motion of the Sun 1 hour , 27 minutes , 33 seconds feré .

2. Hence it cometh to pass , that although the New Moons do after 19 years return to the same days ; yet is there an Anticipation of 1 hour , 27 minutes , 33 seconds . And in twice 19 years , that is , in 38 years , there is an Anticipation of 2 hours , 55 minutes , 6 seconds , and after 312 years and a half , there is an Anticipation of one whole day and some few Minutes . And therefore after 312 years no new Moon can happen upon the same day it did 19 years before , but a day sooner . Hence it comes to pass that in the Julian Calendar , in which no regard is had to this Anticipation , the New Moons found out by the Golden Number must needs be erroneous , and from the time of the Nicene Council 4 days after the New Moons by a regular Computation .

3. And hence it follows also , that if the Golden Number , after 312 were upon due consideration removed a day forwarder or nearer the beginning of the Months , they would shew the New Moons for 312 years to come . And being again removed after those years , a day more would by the like reason do the same again . But it was thought more convenient so to dispose 30 Epacts , that they keeping their constant places , 19 of them should perform the work of the Golden Number , until by this means there should be an Anticipation of one day . And when such an Anticipation should happen , those 19 Epacts being let alone , other 19 should be used , which do belong to the preceding day , without making any alteration in the Calendar .

4. And if this Anticipation would do the whole work , nothing were more plain , then to make that commutation of the 19 Epact once in 312 years : but because the detraction of the Bissextile days doth variously interpose and cause the 19 Epacts sometimes to be changed into these that do precede , sometimes into these that follow , sometimes into neither , but to continue still the same ; therefore some Tables are to be made , by which we may know , when the commutation was to be made and into what Epacts .

4. First therefore there was made a Table called Tabula Epactarum Expansa , in this manner .

First on the top were placed the 19 Golden Numbers in order , beginning with the Number 3 , which in the old Calendar is placed against the Calends of Ianuary , and under every one of these Golden Numbers there are placed 30 Epacts all constituted from the lowest number in the first rank in which the Epact is 1 , and in that first rank the Golden Number is 3 , the rest from thence towards the right Hand are made by the constant addition of it , and the casting away of 30 , as often as they shall exceed that number , only when you come to the 27 , the Epact under the Golden Number 19 , there must be added 12 instead of 11 , that so the Epact following may be 9 not 8 , for the Reasons already given in this Discourse concerning the Golden Number and Embolismical years . And this rank being thus made , the other Epacts are disposed in their natural order ascending upwards , and the number once again resumed after the Epact 30 or rather this Asterisk * set in the place thereof : only observe that under the Golden Number 12. 13. 14. 15. 16. 17. 18. 19. in the place of XX there is yet 25 in the common Character . And to the Epacts under the Golden Number 19 , 12 must still be added to make that Epact under the Golden Number 1. As was said before concerning the lowest Rank .

5. And on the left hand of these Epacts before those under the Golden Number 3. are set 30 Letters of the Alphabet , 19 in a small Character , and 11 in a great , in which some are passed by , for no other reason save only this , that their similitude with some of the small Letters , should not occasion any mistake in their use , which shall be shewed in its place .

6. Besides this Table there was another Table made which is called Tabula AEquationis Epactarum , in which there is a series of years , in which the Moon , by reason of her mentioned anticipation doth need AEquation , and in which the number of Epacts signed with the letters of the Alphabet , are to be changed ; being otherwise AEquated where it needeth , by the suppression of the Bissextile days .

7. But it supposeth , that it was convenient to suppress the Bissextiles once only in 100 years ; and the Moon to be aequated , or as far as concerns her self , the rank of Epacts to be changed , once only in 300 years , and the 12 years and a half more , to be referred till after the years 2400 , they do amount unto 100 years , and then an aequation to be made : but then it must be made by reason of the interposing this hundred not in the three hundredth but the hundredth year . Moreover this aequation is to be made as in referece to the Moon only , because as the suppression of the Bissextiles intervene , the order of changing the ranks of Epacts is varied , as shall be shewed hereafter .

8. Again this Table supposeth , that seeing the New Moon at the time of the Nicene Council was upon the Calends of Ianuary , the golden Number 3 being there placed , that it would have been the same if the Epact * had been set to the same Calends , that is if the Epacts had been then in use . And therefore at that time the highest or last rank of Epacts was to be used , whose Index is P , and then after 300 years , the lowest or first rank should succeed , whose Index is a , ( for the letters return in a Circle ) and after 300 years more , the following rank whose Index is b and so forward ; but that it is conceived , that the New Moon in the Calends of Ianuary , is more agreeable to the year of Christ 500 , than the time of the Nicene Councel ; and therefore as if the rank of Epacts under the letter l were sutable to the year 500 , it seemed good to make use of that rank under the letter a in the year of Christ 800 , and those under the letter b , in the year 1100 , and those under the letter e in the year 1400.

9. Which being granted , because in the year 1582 , ten days were cut off from the Calendar , we must run backward , or in an inverted order count 10 series , designed , suppose , by the letters b. a. P. N. M. H. G. F. E. D. so that from the year 1582 the series of Epacts whose literal Index is D , is to be used , and this is that rank of Epacts which is now used in the Church of Rome .

10. And therefore as if this Table had its beginning from that year ; the first number in the second column is 1582 , and then in order under it . 1600. 1700. 1800. 1900. 2000. &c. And in the third Column every fourth hundred year is marked for a Bissextile , that is , 1600. 2400. 2800 , &c. and in the fourth Column to every three hundreth ▪ Year is set this Character C , to shew in what year the Moon by her Anticipation of one day , doth need aequation ; but in the year 1800 the double character is set CC , to signify that then another hundred years are gotten by the 12 years and a half reserved , besides and above the other 300 years ; and this character is also set to the years 4300. 6800 , and for the same reason .

But in the first Column , or on the left hand of these years are placed the Letters or Indices of those ranks of Epacts in the former Table , which are to be used in those years and when the Letters are charged . Thus against the year 1600 the Letter D is continued , to shew that from that year , to the year 1700 the rank of Epacts is still to be used , which do belong to that Letter . And for as much as the Letter C is set to the year 1700 , it sheweth that that rank of Epacts is then to be used , which do belong thereto , and so of the rest .

11. The reason why these Letters in the first Column are sometimes changed in 100 years , sometimes in 200 , sometimes not in less then 300 Years , and that they are sometimes taken forward , sometimes backward , according to the order of the Alphabet , is because the suppression of the Bissextiles do intervene with the lunar aequation : for if the Bissextile were only to be suppressed , in these 300 or sometimes 400 years , in which the Moon needeth aequation , the rank of Epacts in that case would need no commutation , but would continue the same for ever ; and the golden Number would have been sufficient , if the suppression of the Bissextile , and anticipation of the Moon , did by a perpetual compensation cause the new Moons still to return to the same days : but because the Bissextile is ofttimes suppressed , when the Moon hath no aequation , the Moon hath sometimes an aequation when the Bissextile is not suppressed , sometimes also both are to be done and sometimes neither ; all which varieties may yet be reduced to these three Rules .

1. As often as the Bissextile is suppressed without any aequation of the Moon , then the letter which served to that time shall be changed to the next below it contrary to the order of the Alphabet . And the new Moons shall be removed one day towards the end of the Year .

2. As often as the Moon needeth aequation , without suppression of the Bissextile , then the Letter which was in use to that time shall be changed to the next above it according to the order of the Alphabet , that the New Moons may again return one day towards the beginning of the year .

3. As often as there is a Suppression and an aequation both , or when there is neither , the Letter is not changed at all but that which served for the former Centenary , shall also continue in the succeeding ; because the compensation so made , the New Moons do neither go forward nor backward , but happen in the compass of the same days .

1. And this is enough to shew for what reason the letters are so placed in the Table , as there you see them : for in the year 1600 the Bissextile being neither suppressed , nor the Moon aequated , the letter D used in the former Centenary or in the latter part thereof from the year 1582 , is still the same .

In the year 1700 , because there is a suppression , but no aequation , the commutation is made to the Letter C descending .

In the Year 1800 , because there is both a suppression and an aequation , the same letter C doth still continue .

In the Year 2400 , because there is an aequation and no suppression , there is an ascension to the Letter A.

And thus you see not only the construction of this Table , but how it may be continued to any other Year , as long as the World shall last .

12. And by these two Tables we may easily know which rank of the 30 Epacts doth belong to , or is proper for any particular age : for as in our age , that is , from the Year 1600 to the Year 1700 exclusively , that series is proper whose Index is D. Namely , 23 , 4 , 15 , 26 , &c. so in the two Ages following , that is , from the Year 1700 to the Year 1900 exclusively , that series is proper whose Index is C , namely these , 22 , 3 , 14 , 25. and in the three ages following thence , that is from the Year 1900 to the Year 2100 exclusively , that series is proper whose Index is B , namely these , 21 , 2 , 13 , 24 , &c. And so for any other .

Hence also it may be known , which of the 19 doth belong to any particular Year , for which no more is necessary , than only to know the Golden Number for the year given , which being sought in the head of the Table , and the Index of that Age in the side , the common Angle , or meeting of these two , will shew you the Epact desired : As in the year 1674 the Golden Number is 3 and the Index D ; therefore in the common Angle I find 23 for the Epact that year , and sheweth the New Moons in every Month thereof .

And here it will not be unseasonable to give the reason , for which the Epact 25 not XXV is written under the Golden Numbers 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19. namely , because the ranks of Epacts , which under these greater Numbers hath this Epact 25 , hath also XXIV , it would follow that in these Ages in which any of these Ranks were in use , the New Moon in 19 years will happen twice upon the same days ; in those six Months in which the Epacts XXV and XXIV are set to the same day : Whereas the New Moons do not happen on the same day till 19 years be gone about . To avoid this inconvenience , the Epact 25 not XXV is set under these great numbers , and the Epact 25 is in the Calendar , in these Months set with the Epact XXVI , but in the other Months with the Epact XXV .

14. Hence it cometh to pass , 1. That in these Years the Epacts 25 and XXIV do never meet on the same day . 2. That there is no danger that the Epacts 25 and XXVI should in these 6 Months cause the same inconvenience , seeing that the Epacts 25 and XXVI are never both found in the same Rank . 3. That the Epact 25 may in other Months without inconvenience be set to the same day with the Epact XXVI , because in these there is no danger of their meeting with the Epact XXIV on the same days . 4. That there is no fear that the Epacts XXV and XXIV being set on the same days , should in future Ages cause the same inconvenience , because the Epacts XXV and XXIV are not found together in any of the other Ranks . But that either one or both of them are wanting . Besides , when one of these Epacts is in use , the other is not , and that only which is in use is proper to the day . As in this our Age until the Year 1700 the Epacts in use are those in the rank whose Index is D. In which these two XXIV and XXV are not both found . And in the two following Ages , because the rank of Epacts in use is that whose Index is C , in which there is the Epact XXV , not XXIV , the New Moons are shewed by the Epact XXV not by XXIV . But because in three following Ages , the rank of Epacts in use is that whose Index is B , in which 25 and XXIV are both found , the New Moons are shewed by the Epact XXIV when the golden Number is 6. And by the Epact 25 when the golden Number is 17 , and not by the Epact XXV .

15. And if it be asked why the Epact 19 in the common Character is set with the Epact XX against the last day of December ; know that for the reasons before declared , the last Embolismical Month within the space of 19 years , ought to be but 29 days and not 30 , as the rest are ; and therefore when the Epact 19 doth concur with the golden Number 19 , the last Month or last Lunation beginning the second of December , shall end upon the 30 and not upon the 31 of that Month , and the New Moon should be supposed to happen upon the 31 under the same Epact 19 , that 12 being added to 19 and not 11 , you may have one for the Epact of the year following , which may be found upon the 30 of Ianuary , as if the Lunation of 30 days had been accomplished the Day before .

CHAP. VI.

How to find the Dominical Letter and Feast of Easter according to the Gregorian account .

HAving shewed for what reason , and in what manner the Epacts are substituted in the place of the golden Number , and how the New Moons may be by them found in the Calendar for ever ; I shall now shew you how to find the Feast of Easter and the other moveable Feasts according to the Gregorian or new account ; and to this purpose I must first shew you how to find the Dominical Letter , for that the Cycle of 28 years will not serve the turn , because of the suppression of the Bissextile once in a hundred years , but doth require 7 Cycles of 28 years apeice . The first whereof begins with CB , and endeth in D. The second begins with DC , and endeth in E. The third begins with ED , and endeth in F &c. The first of these Cycles began to be in use 1582 , in which year the dominical Letter according to the Julian account was G , but upon the fifteenth day of October , that Year was changed to C : for the fifth of October being Friday and then called the fifteenth , the Letter A became Friday , B Saturday , and C Sunday , the remaining part of the year , in which the Cycle of the Sun was 23 , and the second after the Bissextile or leap Year , and so making C , which answereth to the fifteenth year of that Circle , to be 23 , the Circle will end at D ; and consequently CB , which in the old account doth belong to the 21 year of the Circle , hath ever since been called the first , and so shall continue until the year 1700 , in which the Bissextile being suppressed , the next Cycle will begin with DC as hath been said already . Under the first rank or order of Dominical Letters are written the years 1582 and 1600 , under the second 1700 , under the third 1800 , under the fourth 1900 and 2000 , under the fifth 2100 , under the sixth 2200 and under the seventh 2300 and 2400. And again under the first Order , 2500 , under the second 2600 , under the third 2700 and 2800 , and so forward as far as you please , always observing the same order , that the 100 Bissextile years may still be joyned with the not Bissextile immediately preceding .

1. And hence it appears , that the seven orders of Dominical Letters , are so many Tables , successively serving all future Generations . For as the first Order serveth from the year 1582 and 1600 to the year 1700 exclusively , and the second Order from thence to the year 1800 exclusively , so shall all the rest in like manner which here are set down , and to be set down at pleasure . And hence the Dominical Letter or Letters may be found for any year propounded , as if it were required to find the dominical Letter for the year 1674 , because the year given is contained in the centenary 1600. I find the Cycle of the Sun by the Rule already given to be 3. In the first order against the number 3 , I find G for the Sunday Letter of that year , in like manner because the year 1750 is contained under the Centenary 1700 , the Cycle of the Sun being 27 , I find in the second rank the Letter D answering to that Number , and that is the Dominical Letter for that year , and so of the rest .

3. Again for as much as the fifth Order is the same with that Table , which serves for the old account , therefore that order will serve the turn for ever where that Calendar is in use , and so this last will be of perpetual use to both the Calendars .

4. Now then to find the time in which the Feast of Easter is to be observed , there is but little to be added to that which hath been already said concerning the Julian Calendar . For the Paschal Limits are the same in both , the difference is only in the Epacts , which here are used instead of the golden Number .

5. For the terms of the Paschal New Moons are always the eighth of March and the fifth of April : but whereas there are 11 days within these Limits to which no golden Number is affixed , there is now one day to which an Epact is not appointed , because there is no day within those Limits , on which in process of time a New Moon may not happen . And the reason for which the two Epacts XXV and XXIV are both set to the fifth of April , is first general , which was shewed before , namly that by doing the same in 5 other Months , the 12 time 30 Epacts might be contracted to the Limits of the lunar Year which consists of 354 days : but there is a particular reason also for it , that the Antients having appointed that all the Paschal lunations should consist of 29 days , it was necessary that some two of the Epacts should be set to one of these days in which the Paschal lunation might happen , the Epacts being 30 in number . And it was thought convenient to choose the last day , to which the Epact XXV belonging , the Epact XXIV should also be set ; and hence by imitation it comes to pass , that these and not other Epacts are set to that day in other Months , in which two Epacts are to be set to the same days .

6. The use of these Epacts in finding the Feast of Easter , is the same with that which hath been shewed concerning the golden Numbers . For the Epact and the Sunday Letter for that year propounded being given , the Feast of Easter may be found in the Calendar after the same manner . Thus in the year 1674 , the Epact is 23 and the Sunday Letter G , and therefore reckoning fourteen days from the eighth of March to which the Epact is set , the Sunday following is March 25 , which is the day on which the Feast of Easter is observed .

7. And hence as hath been shewed in the third Chapter concerning the Julian Calendar , a brief table may be made to shew the feast of Easter and the other moveable Feasts for ever , in which there is no other difference , save only that the Epacts as they are in this new Calendar , are to be used as the golden Numbers are , which stand in the old Calendar . And a Table having the golden Numbers of the old Calendar set in one Column , and the Epacts as they are in the new Calendar set in another , will indifferently shew the movable Feasts in both accounts , as in the Year 1674 , the golden Number is 3 and the Sunday Letter according to the Julian account is D , according to the Gregorian G , and the Epact 23 , and therefore according to this Table our Easter is April 19 , and the other , to wit , the Gregorian , is March 25. The like may be done for any other year past or to come .

CHAP. VII .

How to reduce Sexagenary numbers into Decimal , and the Contrary .

EVery Circle hath antiently , and is yet generally supposed to be divided into 300 degrees , each degree into 60 Minutes , each Minute into 60 Seconds , and so forward as far as need shall require . But this partition is somewhat troublesom in Addition and Subtraction , much more in Multiplication and Division ; and the Tables hitherto contrived to ease that manner of computation , do scarce sufficiently perform the work , for which they are intended . And although the Canon published by the learned H. Gellibrand , in which the Division of the Circle into 360 degrees is retained , but every degree is divided into 100 parts , is much better than the old Sexagenary Canon , yet some are of opinion , that if the Antients had divided the whole circle into 100 or 1000 parts , it would have proved much better then either ; only they think Custome such a Tyrant , that the alteration of it now will not be perhaps so advantagious ; leaving them therefore to injoy their own opinions , they will not I hope be offended if others be of another mind : for their sakes therefore , that do rather like the Decimal way of calculation ▪ Having made a Canon of artificial Signs and Tangents for the degrees and parts of a Circle divided into 100 parts , I shall here also shew you , how to reduce sexagenary Numbers into Decimal , and the contrary , as well in time as motion .

2. The parts of a Circle consisting of 360 degrees , may be reduced into the parts of a circle divided into 100 degrees or parts , by the rule of Three in this manner .

As 360 is to 100 , so is any other Number of degrees , in the one , to the correspondent degrees and parts in the other .

But if the sexagenary degrees have Minutes and Seconds joyned with them , you must reduce the whole Circle as well as the parts propounded into the least Denomination , and so proceed according to the rule given .

Example . Let it be required to convert 125 degrees of the Sexagenary Circle , into their correspondent parts in the Decimal . I say , as 360 is to 100 , so is 125 to 34 , 722222 , &c. that is , 34 degrees and 722222 Parts .

2. Example . Let the Decimal of 238 degrees 47 Minutes be required . In a whole Circle there are 21600 Minutes , and in 238 degrees , there are 14280 Minutes , to which 47 being added the sum is 14327. Now then I say if 21600 give 100 , what shall 14327. The Answ. is 66 , 3287 &c. In like manner if it were required to convert the Hours and Minutes of a Day into decimal Parts , say thus , if 24 Hours give 100 , what shall any other number of Hours give . Thus if the Decimal of 18 hours were required , the answer would be 75 , and the Decimal answering to 16 Hours 30 Minutes is 68 , 75.

But if it be required to convert the Decimal Parts of a Circle into its correspondent Parts in Sexagenary . The proportion is ; as 100 is to the Decimal given , so is 360 to the Sexagenary degrees and parts required .

Example . Let the Decimal given be 349 722222 , if you multiply this Number given by 360 , the Product will be 1249999992 , that is cutting off 7 Figures , 124 degrees and 9999992 parts of a degree . If Minutes be required , multiply the Decimal parts by 60 , and from the product cut off as many Figures , as were in the Decimal parts given , the rest shall be the Minutes desired .

But to avoid this trouble , I have here exhibited two Tables , the one for converting sexagenary degrees and Minutes into Decimals , and the contrary . The other for converting Hours and Minutes into Decimals , and the contrary . The use of which Tables I will explain by example .

Let it be required to convert 258 degrees 34′ . 47″ , into the parts of a Circle decimally divided .

The Table for this purpose doth consist of two Leaves , the first Leaf is divided into 21 Columns , of which the 1. 3. 5. 7. 9. 11. 13. 15. 17. 19 doth contain the degrees in a sexagenary Circle , the 2. 4. 6. 8. 10. 12. 14. 16. 18 and 20 doth contain the degrees of a Circle Decimally divided , answering to the former , and the last Column doth contain the Decimal parts , to be annexed to the Decimal degrees . Thus the Decimal degrees answering to 26 Sexagenary are 7 , and the parts in the last Column are 22222222 and therefore the degrees and parts answering to 26 Sexagenary degrees are 7. 22222222.

In like manner the Decimal of 62 degrees , 17. 22222222. And the Decimal of 258 degrees , 34′ . 47″ , is thus found .

71.66666666 .15747040 .00362652 71.82776358

is the Decimal of 258 degrees , 34′ . 47″ as was required .

In like mauner the Decimal of any Hours and Minutes may be found by the Table for that purpose .

Example . Let the Decimal of 7 Hours 28′ be required .

29.16666667 1.94444444 31.11111111

is the Decimal Sought .

To find the degrees and Minutes in a sexagenary Circle , answering to the degrees and parts of a Circle Decimally divided , is but the contrary work .

As if it were required to find the Degrees and minutes answering to this decimal 71. 02776359 , the Degrees or Integers being sought in the 2. 4. 6 or 8 Columns &c. of the first Leaf of that Table , right against 71. I find 256 and in the last Column these parts 11111111 , which being less than the Decimal given , I proceed till I come to 6666667 , which being the nearest to my number given , I find against these parts under 71. Degrees 258 , so then 258 are the degrees answering to the Decimal given and ,

71.82776359 71.66666667 16109692 11747640 00362652

Which is the Decimal of 47 seconds , and so the degrees and Minutes answering to the Decimal given are 258 degrees 34′ and 47″ , the like may be done for any other .

CHAP. VIII .

Of the difference of Meridiens .

HAving in the first part shewed how the places of the Planets in the Zodiack may be found by observation , and how to reduce the time of an observation made in one Country , to the correspondent time in another , as to the day of the Month , by considering the several measures of the year in several Nations , there is yet onething wanting , which is , by an observation made of a Planets place in one Country to find when the Planet is in that place in reference to another ; as suppose the ☉ by observation was found at Vraniburg to be in ♈ . 3d. 13′ . 14″ . March the fourteenth 1583 at what time was the Sun in the same place at London ? To resolve this and the like questions , the Longitude of places from some certain Meridian must be known ; to which purpose I have here exhibited a Table shewing the difference of Meridians in Hours and Minutes , of most of the eminent places in England from the City of London , and of some places beyond the Seas also . The use whereof is either to reduce the time given under the Meridian of London to some other Meridian , or the time given in some other Meridian to the Meridian of London .

1. If it be required to reduce the time given under the Meridian of London to some other Meridian , seek the place desired in the Catalogue , and the difference of time there found , either add to or subtract from the times given at London , according as the Titles of Addition or Subtraction shew , so will the time be reduced to the Meridian of the other place as was required . Example . The same place at London was in the first Point of ♉ , 6 Hours P. M. and it is required to reduce the same to the Meridian of Vraniburg I therefore seek in Vraniburg in the Catalogue of places , against which I find 50′ with the Letter A annexed , therefore I conclude , that the Sun was that day at Vraniburg in the first point of ♉ , 6 Hours 50′ . P. M.

2. If the time given be under some other Meridian , and it be required to reduce the same to the Meridian of London , you must seek the place given in the Catalogue , and the difference of time there found , contrary to the Title is to be added or subtracted from the time there given .

Example . Suppose the place of the Sun had been at Vraniburg , at 6 Hours 50′ . P. M. and I would reduce the same to the Meridian of London ; against Vraniburg as before I find 50′ A. therefore contrary to the Title I Subtract 50′ and the remainder 6 Hours is the time of the Suns place in the Meridian of London .

CHAP. IX .

Of the Theory of the Sun 's or Earth's Motion .

IN the first part of this Treatise we have spoken of the primary Motion of the Planets and Stars , as they are wheeled about in their diurnal motion from East to West , but here we are to shew their own proper motions in their several Orbs from West to East , which we call their second motions .

1. And these Orbs are supposed to be Elliptical , as the ingenious Repler , by the help of Tycho's accurate observations , hath demonstrated in the Motions of Mars and Mercury , and may therefore be conceived to be the Figure in which the rest do move .

2. Here then we are to consider what an Ellipsis is , how it may be drawn , and by what Method the motions of the Planets according to that Figure may be computed .

3. What an Ellipsis is Apollonius Pergaeus in Conicis , Claudius Mydorgius and others have well defined and explained , but here I think it sufficient to tell the Reader , that it is a long Circle , or a circular Line drawn within or without a long Square ; or a circular Line drawn between two Circles of different Diameters .

4. The usual and Mechanical way of drawing this Ellipsis is thus ; first draw a line to that length which you would have the greatest Diameter to be , as the Line AP in Figure 8 , and from the middle of this Line at X , set off with your compasses the Equal distance XM and XH .

5. Then take a piece of thred of the same length with the Diameter AP and fasten one end thereof in the point M and the other in the point H , and with your Pen extend the thred thus fastened to the point A , and from thence towards P keeping the thread stiff upon your Pen , draw a line from A by B to P , the line so drawn shall be half an Ellipsis , and in like manner you may draw the other half from P by D to A. In which because the whole thred is equal to the Diameter AP. therefore the two Lines made by thred in drawing of the Ellipsis , must in every point of the said Ellipsis be also equal to the same Diameter AP. They that desire a demonstration thereof geometrically , may consult Apollonius Pergaeus , Claudius Mydorgius or others , in their treatises of Conical Sections , this is sufficient for our present purpose , and from the equality of these two Lines with the Diameter , a brief Method of calculation of the Planets place in an Ellipsis , is thus Demonstrated by Dr. Ward now Bishop of Salisbury .

6. In this Ellipsis H denotes the place of the Suns Center , to which the true motion of the Planet is referred , M the other Focus whereunto the equal or middle motion is numbred , A the Aphelion where the Planet is farthest distant from the Sun and slowest in motion , P the Perihelion where the Planet is nearest the Sun and slowest in motion . In the points A and P the Line of the mean and true motion do convene , and therefore in either of these places the Planet is from P in aequality , but in all other points the mean and true motion differ , and in D and C is the greatest elliptick AEquation .

8. Now suppose the Planet in B , the line of the middle motion according to this Figure is MB , the line of the true motion HB . The mean Anomaly AMB. The Eliptick aequation or Prosthaphaeresis MBH , which in this Example subtracted from AMB , the remainer AHB is the true Anomaly . And here note that in the right lined Triangle MBH , the side MH is always the same , being the distance of the Foci , the other two sides MB and HB are together equal to AP. Now then if you continue the side MB till BE be equal to BH and draw the line HE , in the right lined Triangle MEH , we have given ME = AD and MH with the Angle EMH , to find the Angles MEH and MHE which in this case are equal , because EB = BH by Contraction , and therefore the double of BEH or BHE = MBH , which is the Angle required .

And that which yet remaineth to be done , is the finding the place of the Aphelion , the true Excentricity or distance of the umbilique points , and the stating of the Planets middle motion .

CHAP. X.

Of the finding of the Suns Apogeon , quantity of Excentricity aend middle motion .

THe place of the Suns Apogaeon and quantity of Excentricity may from the observations of our countrey man Mr. Edward Wright be obtained in this manner , in the years 1596 , and 1497 , the Suns entrance into ♈ and ♎ and into the midst of ♉ . ♌ . ♍ . and ♒ were as in the Table following expressed .

1596 1597 D. H. M. D. H. M. Ianuary . 25. 00.07 24. 05.54 ♒ . 15 March . 9. 18.43 10. 00.37 ♈ . 0 April . 24. 21.47 25. 03.54 ♉ . 15 Iuly . 28. 01.43 28. 09.56 ♌ . 15 September . 12. 13.48 12. 19.15 ♎ . 0 October . 27. 15.23 27. 21.50 ♍ . 15

And hence the Suns continuance in the Northern Semicircle from ♈ to ♎ in the year 1596 being Leap year , was thus found . d. h. 256. 13. 48. 69. 18.43 186. 19.05 255. 19.15 69. 09.37 186. 18.38

And the difference of the Suns continuance in these Arks in the year 1596 and 1597 is 27′ . and therefore the mean time of his continuance in those Arks is days 186. hours 18. minutes 51. seconds 30. And by consequence his continuance in the Southern Semicircle that is from ♎ to ♈ is 178 days . 11 hours , 8 minutes and 30 seconds . 185. 17.36 185. 17.56

And to find the middle motion answering to days 186. hours 18. Minutes 51. seconds 30 I say .

As 365 days , 6 hours , the length of the Julian , year is to 360 , the degrees in a Circle .

So is 186 days , 18 hours , 51′ . 30″ to 184 degrees . 03′ . 56″ .

In like manner the mean motion answering to 185 days , 17 h. 46′ is 183 degrees , 02′ . 09 . 180. 00.00 184. 03.56 364. 03.56 182. 01.58

In 1596 from 15 ♒ to 15 ♌ there are days 185 , hours 01 , minutes 36. In 1597. days 135. hours 4. 02′ .

And the mean motion answering thereunto is . 182 d. 30′ . 36″ . 15 ♉ to 15 ♍ . 180. 185. 17. 56. 181. 04.53 183. 32. 26

From 15 ♒ to 15 ♌ Days . 185. 04 h. 02′ 180. 182. 30. 36 181. 15. 18

Now then in Fig. from PGC. 181. 32. 26 deduct NKD 180 , the Remainer is DC+NP . 1. 32. 26. Therefore DC or NP. 46. 13 , whose Sine is HA.

And from XPG. 181. 15. 18 deduct TNK 180 , the Remainer is KG+TX 1. 15. 18. Therefore KG or TX 37. 39 , whose Sine is HR .

Now then to find the Apogaeon . 5.12851105 15.03948202 9.91097097 45 95. 49. 56. Hence to find the excentricity AR. 9.80043756 15.03948202 5.23904446

Or thus ,

In the Triangle 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 we have given 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . 5.03948202 15.12851105 10.08902903 45.

Apogaeon 95 deg. 49′ . 56″ . as before . Then for the Excentricity RA. 9.88945938 5.12851105 5.23905167

And this agreeth with the excentricity , used by Mr. Street in his Astron. Carolina , Pag. 23. But Mr. Wing as well by observation in former ages , as our own , in his Astron. Instaur . Pag. 39. doth find it to be 1788 or 1791. The work by both observations as followeth .

2. And first in the time of Ptolemy , Anno Christi 139 by comparing many observations together , he sets down for the measure nearest truth , the interval between the vernal Equinox and the Tropick of Cancer to be days 93. hours 23. and minutes 03. And from the Vernal to the Autumnal Equinox , days 186. hours 13. and minutes 5. D. 90. 36.00 92. 36.42 91. 18.21 180. 00.00 183. 52.03 181. 56.02 90. 58.01 00. 20.20

Again from GEK 181. 56. 02. deduct the Semicircle FED 180. the remainer is the summ DK and FG. 1. 56. 2. and therefore DK = FG. 58′ . 01″ . whose sign is BC. 168755. L is the place of the Aphelion , and AB the Excentricity .

Now then in the Triangle ABC . in the Fig. 6 we have given the two sides AC and BC. To find the Angle BAC and the Hypotenuse AB . For which the proportions are . 4.77192538 10.00000000 5.22725665 10.45533127 Secondly for AB . 9.97484352 5.22725665 10.00000000 5.25241313

Therefore the Aphelion at that time was in II 10. 41. 10. And the excentricity . 1788.

3. Again Anno Christi 1652 the Suns place by observation was found to be as followeth .

April . 24. hours . 10. ♉ . 15 October . 27. hours . 7. 10′ ♍ . 15 Ianuary . 24. hours . 11 . 20′ ♒ . 15 Iuly . 27. hours . 16.30 , ♌ . 15

Hence it appeareth that the Sun is running through one Semicircle of the Ecliptick , that is from ♉ 15 to ♍ 15. 185 days 21 hours and 10′ . And through the other Semicircle from ♒ 15 to ♌ 15 , days 184. hours 5. therefore the Suns mean motion , according to the practice in the last example , from ♉ 15 to ♍ 15 is 181. 30. 26. and from ♒ 15 to ♌ 15. 181. 16. 30.

Now then in Fig. 7. if we subtract the semicircle of the Orb KMH . 180. from WPV 181. 36. 26. the remainer is the sum of KW and HV 1. 36. 26. the Sine of half thereof 48′ . 13″ is equal to AC . 140252.

Again the mean motion of the Sun in his Orb from ♒ 15 to ♉ 15 is the Arch SKP. 181. 16. 30. whose excess above the Semicircle being bisected is 38. 15. whose Sine CB. 111345. now then in the Triangle ABC to find the Angle BAC , the proportion is . 5.14690906 10.00000000 5.04667072 9.89966166

Which being deducted out of the Angle . 69 A ♌ . 45 it leaveth the Angle 69 AL 6. 33. 39. the place of the ☉ Aphelion sought , and this is the quantity which we retain . And for the excentricity BC. 9.79356702 10.00000000 5.04667072 5.25310370 96. 33.39 70. 41.10 25. 52.29

And the difference of time is 1513 Julian years .

Hence to find the motion of the Aphelion for 2. years , say I , if 1513 years give 25. 52.29 , what shall one year give , and the answer is 00 d. 01′ 01″ . 33‴ . 56 iv . 44v. that is in Decimal numbers . 0. 00475. 04447. 0555.

And the motion for . 1651 years . 7. 84298. 4208862 , which being deducted from the place of the Aphelion Anno Christi . 1652 — 26. 82245. 3703703. The remainer , viz. 18. 97946. 9494841 is the place thereof in the beginning of the Christian AEra , which being reduced is , 68 deg. 19. min. 33. sec. 56. thirds .

4. The Earths middle motion , Aphelion and Excentricity being thus found , we will now shew how the same may be stated to any particular time desired , and this must be done by help of the Sun or Earths place taken by observation . In the 178 year then from the death of Alexander , Mechir the 27 at 11 hours P. M. Hipparcus found in the Meridian of Alexand. that the Sun entered ♈ 0. the which Vernal Equinox happened in the Meridian of London according to Mr. Wings computation at 9 hours 14′ , and the Suns Aphelion then may thus be found .

The motion of the Aphelion for one year , was before found to be . 0. 00475. 04447. 0555. therefore the motion thereof for one day is 0. 00001. 501491722. The Christian AEra began in the 4713 year compleat of the Julian Period , in which there are days 1721423. The AEra Alexandri began November the twelfth , in the year 4390 of the Julian Period , in which there are 1603397 days . And from the death of Alexander to the 27 of Mechir 178 , there are days 64781 , therefore from the beginning of the Julian Period , to the 178 year of the AEra Alexandri , there are days 1668178 which being deducted from the days in the Christian AEra , 1721423 , the remainer is 53245 , the number of days between the 178 year after the death of Alexander , Mechir 27 , and the beginning of the Christian AEra .

Or thus . From the AEra Alexandri to the AEra Christi there are 323 Julian years , and 51 days , that is 118026 days . And from the AEra Alexandri to the time of the observation , there are 64781 days , which being deducted from the former , the remainer is 53245 as before . Now then if you multiply the motion of the Aphelion for one day , viz. 0. 00001. 3014917 by 53245 , the product is 0. 69297. 9255665 , which being deducted from the place of the Aphelion in the beginning of the Christian AEra , before found . 18. 97946. 9494841. the remainer 18. 28649. 0239176 is the place of the Aphelion at the time of the observation , that is in Sexagenary numbers . deg. 65. 49′ . 53″ .

5. The place of the Aphelion at the time of the observation being thus found to be deg. 65. 49′ . 53″ . The Suns mean Longitude at that time , may be thus computed .

In Fig. 8. In the Triangle EMH we have given the side ME 200000 , the side MH 3576 , the double excentricity before found , and the Angle EMH 114. 10′ . 07″ . the complement of the Aphelion to a Semicircle , to find the Angle MEH , for which the proportion is ,

As the Summ of the sides , is to the difference of the sides , so is the Tangent of the half Summ of the opposite Angles , to the Tangent of half their difference . 4.69127343 5.29321855 9.91111512 9.79560710

The double whereof is the Angle MBH 1. 51. 10. which being Subtracted from 360 the remainer 358. 08. 50. is the estimate middle motion of the Sun , from which subtracting the Aphelion before found , 65. 49. 53. the remainer 292. 18. 57. is the mean Anomaly by which the absolute AEquation may be found according to the former operation . 4.69127343 5.29321855 10.17359517 10.15808715

Doubled 1. 54. 20 , which added to the middle motion before found gives the ☉ true place ♈ . 00. 3′ . 10″ , which exceeds the observation 3′ . 10″ . therefore I deduct the same from the middle motion before found , and the remainer 358. 05. 50. is the middle motion at the time of the observation of Hipparchus , to which if you add the middle motion of the Sun for 53245 days , or for 323 AEgyptian years 131 days , 280. 46. 08′ the Summ , rejecting the whole Circles , is 278. 51. 48 the Suns mean Longitude in the beginning of the Christian AEra .

6. But one observation is not sufficient , whereby to state the middle motion for any desired Epocha , we will therefore examine the same by another observation made by Albategnius at Aracta in the year of Christ 882 , March : 15. hours 22. 21. but in the Meridian of London at 18 hours . 58′ .

The motion of the Aphelion for 881 years , 74 days is 3. 806068653737 , which being added to the place thereof in the beginning of the Christian AEra , the place at the time of the observation will be found to be 22. 785538148578 , that is reduced , Deg. 82. 01′ . 40″ . And hence the AEquation according to the former operations is Deg. 2. 01′ . 16″ which being deducted from a whole Circle , the remainer 357 d. 58′ . 44″ is the estimate middle motion at that time , from which deducting the Aphelion deg. 82. 01. 40. the remainer 275. 57. 04 is the mean anomaly , and the AEquation answering thereto is deg. 2. 02′ . 18″ which being added to the middle motion before found , gives the ☉ place ♈ . 00. 01′ . 02″ which exceeds the observation 01′ . 02″ . therefore deduct the same from the middle motion before found , the remainer 357. 57′ . 22″ is the middle motion of the ☉ at the time of the observation , from which deducting the middle motion for 881 years , 74 days , 18 hours , 58 minutes , viz. 80d. 06′ . 10″ . the remainer 277 deg. 51′ . 12″ . is the ☉ mean Longitude in the beginning of the Christian AEra . 278. 51′ . 48″ 277. 51. 12 1. 00. 36

He that desires the same to this or any other Epocha , to more exactness , must take the pains to compare the Collection thereof from sundry Observations , with one another , this is sufficient to shew how it is to be found . Here therefore I will only add the measures set down by some of our own Nation , and leave it to the Readers choice to make use of that which pleaseth him best .

The ☉ mean Longitude in the beginning of the Christian AEra according to . 9. 8d. 00′ . 31″ 9. 7. 55. 56 9. 7. 54. 39 9. 8. 51. 48 9. 7. 51. 12

In the Ensuing Tables of the ☉ mean Longitude , we have made use of that measure given by Mr. Flamsted , a little pains will fit the Tables to any other measure .

CHAP. XI .

Of the quantity of the Tropical and Sydereal Year .

THe year Natural or Tropical ( so called from the Greek word 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , ( which signifies to turn ) because the year doth still turn or return into it self ) is that part of time in which the ☉ doth finish his course in the Zodiack by coming to the same point from whence it began .

2. That we may determine the true quantity thereof , we must first find the time of the ☉ Ingress into the AEquinoctial Points , about which there is no small difference amongst Astronomers , and therefore an absolute exactness is not to be expected , it is well that we are arrived so near the Truth as we are . Leaving it therefore to the scrutiny of after Ages , to make and compare sundry Observations of the ☉ entrance into the AEquinoctial Points , it shall suffice to shew here how the quantity of the Tropical year may be determined , from these following observations .

3. Albategnius , Anno Christi 882 observed the ☉ entrance into the Autumnal AEquinox at Aracta in Syria to be Sept. 19. 1 hour 15′ in the Morning . But according to Mr. Wings correction in his Astron. Instaur . Page 44 , it was at 1 hour 43′ in the Morning , and therefore according to the ☉ middle motion , the mean time of this Autumnal AEquinox was Sept. 16. 12 h. 14′ . 25″ . that is at London at 8 h. 54′ . 25″ .

4. Again by sundry observations made in the year 1650. the second from Bissextile as that of Albategnius was , the true time of the ☉ ingress into ♎ was found to be Sept. 12. 14 h. 40′ . and therefore his ingress according to his middle motion was Sept. 10. 13 h. 02.

5. Now the interval of these two observations is the time of 768 years , in which space by subtracting the lesser from the greater , I find an anticipation of 5 days , 9 hours , 52′ . 25″ . which divided by 768 giveth in the quotient 10′ . 55″ . 39 which being subtracted for 365 days , 6 hours , the quantity of the Julian year , the true quantity of the Tropical year will be 365 days , 5 hours , 49′ . 04″ . 21‴ . Others from other observations have found it somewhat less , our worthy countryman Mr. Edward Wright takes it to be 365 d. 5 hours . 48′ .

Mr. Iohn Flamsted , 5 h. 29′ . Mr. Tho. Street 5 h. 49′ . 01″ . taking therefore the Tropical year to consist of 365 days , 5 hours , 49 Minutes , the Suns mean motion for one day is 0 deg. 59′ . 8″ . 19‴ . 43 iv . 47 v. 21 vi . 29 vii . 23 viii . or in decimal Numbers , the whole Circle being divided into 100 degrees , the ☉ daily motion is 0. 27379. 08048. 11873.

6. The Sydereal or Starry year is found from the Solar by adding the Annual Motion of the eighth Orb or praecession of the AEquinoctial Points thereunto , that praecession being first converted into time .

7. Now the motion of the fixed Stars is found to be about 50″ . in a years time , as Mr. Wing hath collected from the several observations of Timocharis , Hipparchus , Tycho and others ; and to shew the manner of this Collection , I will mention onely two , one in the time of Timocharis , and another in the time of Tycho .

8. Timocharis then as Ptolemy hath it in his Almagist , sets down the Virgins Spike more northwardly than the AEquinoctial , 1 deg. 24′ . the time of this observation is supposed to be about 291 years before Christ , the Latitude 1 deg. 59′ South , and therefore the place of the Star was in ♍ . 21 d. 59′ . And by the observation of Tycho 1601 current , it was in ♎ 18. 16′ . and therefore the motion in one year 50″ , which being divided by 365 days , 6 hours , the quotient is the motion thereof in a days time . 00′ . 8‴ . 12 iv . 48 v. 47 vi . 18 vii . 30 viii . 13 ix . and in decimal Numbers , the motion for a year is 00385. 80246. 91358. The motion for a day . 00001. 05626. 95938.

9. Now the time in which the Sun moveth 50″ , is 20′ . 17″ . 28‴ , therefore the length of the sydereal year is 365 days , 6 hours , 9′ . 17″ . 28‴ . And the Suns mean motion for a day 59′ . 8″ . 19‴ . 43 iv . 47 v. 21 vi . 29 vii . 23 viii . converted into time is 00. 03′ . 56″ . 33‴ . 18 iv . 55 v. 9 vi . 23 vii . 57 viii . which being added to the AEquinoctial day , 24 hours , giveth the mean solar day , 24 hours . 3. ′ 56″ . 33‴ . 18 iv . 55. 9. 23. 57.

10. And the daily motion of the fixed Stars , being converted into time is 32 iv . 51 v. 15 vi . 9 vii . 14 viii . 24 ix . and therefore the AEquinoctial day being 24 hours , the sydereal day is 24 hours , 00′ . 00″ . 00‴ . 32 iv . 51. 15. 9. 14 24.

11. Hence to find the praecession of the AEquinoctial Points , or Longitude of any fixed Star , you must add or subtract the motion thereof , from the time of the observation , to the time given , to or from the place given by observation , and you have your desire .

Example . The place of the first Star in Aries found by Tycho in the year 1601 current , was in ♈ . 27 d. 37′ . 00. and I would know the place thereof in the beginning of the Christian AEra .

22 d. 13′ . 20″ 27. 37. 00 5. 231. 40 is the place thereof in the beginning of the Christian AEra .

12. Having thus found the ☉ middle motion , the motion of the Aphelion and fixed Stars , with their places , in the beginning of the Christian AEra ; we will now set down the numbers here exhibited AEra Christi . Mr. Wing from the like observations , takes the ☉ motion to be as followeth .

9. 8. 00. 31 2. 8. 20. 03 06. 29. 40. 28 The which in decimal Numbers are 77. 22460. 86419 18. 98171. 29629 58. 24289. 56790 The mean motions for one year . 99. 93364. 37563. 34 00. 00475. 04447. 05 99. 92889. 33116. 29 The ☉ mean motions for one day . 00. 27379. 08048. 11 00. 00001. 30149. 17 00. 27377. 77898. 94

And according to these measures are the Tables made shewing the ☉ mean Longitude and Anomaly , for Years , Months , Days and Hours .

CHAP. XII .

The Suns mean motions otherwise stated .

SOme there are in our present age , that will not allow the Aphelion to have any motion , or alteration , but what proceeds from the motion of the fixed Stars , the which as hath been shewed , do move 50 seconds in a year , and hence the place of the first Star in Aries , in the beginning of the Christian AEra was found to be ♈ . 5. 23d. 40.

Now then , if from the place of the Aphelion Anno Christi . 1652 as was shewed in the tenth Chapter , deg. 96. 33′ . 39. we deduct the motion of the fixed Stars for that time . 28. 19. 12. the remainer 68. 14. 27 is the constant place of the Aphelion ; but Mr. Street in his Astronomia Carolina Page 23 , makes the constant place of the Aphelion to be 68d. 20. 00 , and the ☉ excentricity 1732.

♈ . 0. 33. 19 0. 27. 27. 22 5. 03. 05. 57 8. 08. 20. 00 8. 24. 45. 57 1. 58. 47 8. 22. 47. 10 1. 58. 27 8. 24. 45. 37 8. 08. 20. 00 0. 27. 27. 22 ♈ . 00. 32. 59 ♈ . 00. 33. 19 001. 001. 20 8. 22. 47. 10 8. 22. 47. 30 1. 58. 27 8. 24. 45. 57

And so the mean Anomaly AEra Christi is 6. 23. 19. 56. But Mr. Flamsted according to whose measure the ensuing Tables are composed , takes the mean Anomaly AEra Christi . to be 6. 24. 07. 091. The place of the Aphelion to be 8 , 08. 23. 50. And so the Praecession of the AEquinox and Aphelion in the beginning of the Christian AEra . 8 , 13. 47. 30. in decimal Numbers .

AEra Christi . 56. 69976. 85185 20. 49768. 51851 The ☉ mean motions for one Year . 99. 93364. 37563. 34 00385. 80246. 91 99. 92978. 57316. 43 The ☉ mean Motions for one Day . 00. 27379. 08048. 11 00. 00001. 05699. 30 00. 27378. 02348. 81
CHAP. XIII .

How to Calculate the Suns true place by either of the Tables of middle motion .

WRite out the Epocha next before the given time , and severally under that set the motions belonging to the years , months and days compleat , to the hours , scruples , current every one under his like ( only remember that in the Bissextile years after the end of Frebruary the days must be increased by an unite ) then adding all together , the sum shall be the ☉ mean motion for the time given .

Example .

Let the given time be Anno Christi 1672. February 23. hours 11. 34′ . 54″ . by the Tables of the ☉ mean Longitude and Anomaly , the numbers are as followeth .

M. Longitud . M. Anomal . The Epocha 1660 80. 67440. 53.79815 Years 11 99. 81766. 99.76526 Ianuary 08. 48751. 08.48711 Day . 23 06. 29718. 06.29688 Hours 11 00. 12548. 00.12548 34 00. 00646. 00.00646 54 00. 00017. 00.00017 95. 40886. 68.47951

By the Tables of the Suns mean Anomaly and praecession of the AEquinox , the numbers are these .

Anomaly . Praece . AEqui . The Epocha 1660 53. 76721. 26.90200 Years 11 99. 77520. 00.04243 Ianuary 8. 48718. 00032 Days 23 6. 29694. 00024 Hours 11 0. 12548. 26.94499 34′ .00646 . 68.45882 54″ .00035 . 95.40381 ☉ mean Anomaly 68.45882

There is no great difference between the ☉ mean Longitude and Anomaly found by the Tables of mean Longitude and Anomaly , and that found by the Tables of mean Anomaly and Precession of the AEquinox . The method of finding the Elliptical AEquation is the same in both , we will instance in the latter only , in which the ☉ mean Anomaly is Degrees 68. 45882. And the precession of AEquin . deg. 26. 94499.

But because there is no Canon of Sines and Tangents as yet published , suitable to this division of the Circle into an 100 deg. or parts : We must first convert the ☉ mean Anomaly , and prec . of of the AEquin , given , into the degrees and parts of the common Circle : And this may be done either into degrees and decimal parts of a degree , or into deg. and minutes : if it were required to be done into degrees and minutes , the Table here exhibited for that purpose will serve the turn , but if it be required to be done into degrees and decimal parts , I judge the following method to be more convenient .

Multiply the degrees and parts given by 36 , the Product , if you cut off one figure more towards the right hand than there are parts in the number given , shall be the degrees and parts of the common Circle .

Anomaly . 68. 45882 Praec . AEquinox . 26. 94499 36 36 41075292 16166994 20537646 8983497 Anom . 246. 451752 Prae. AEq. 97. 001964

And if you multiply the parts of these Products , you will convert them into minutes .

Otherwise thus . Multiply the degrees and parts given by 6 continually , the second Product , if you cut off one figure more towards the right hand than are parts in the number given , shall be the degrees and parts of the common Circle . The third Product of the parts only shall give minutes , the fourth seconds , and so forward as far as you please . Example .

Praec . AEq. 26. 94499 6 16166994 97.001964 6 0.11784 6 7.0704

And thus the mean Anom . is deg. 246. 451742 or 27′ . 06. The Prec . AEq. 97. 001964. or 00′ . 07″ .

Hence to find the Elliptical AEquation in degrees and decimal parts : In Fig. 8. we have given in the right lined plain Triangle EMH , the sides ME , and MH , and the Angle EMH , 66. 451742. the excess of the mean Anomaly above a Semicircle , to find the Angle MEH .

200000 3468

Zcru . 203468 Co. ar . 4.69150389 Xcru . 196532 5.29343327 t frac12 Zangle . 56.774129 10.18374097 t frac12 Xangle . 55.857087 10.16867813

MEH . 0. 917042 the double whereof is the Angle MBH . 1. 834084 or Elliptick AEquation sought , which being added to the mean Anomaly and praecession of the AEquinox , because the Anomaly is more than a Semicircle , the same is the Suns true place .

246.451742 97.001964 1.834084 345.287790

But because the Elliptick AEquation thus found doth not so exactly agree to observation as is desired , Bullialdus in Chap. 3. of his Book entituled Astronomiae Philolaicae fundamenta clarius explicata , Printed at Paris , 1657. shews how to correct the same by an Angle applied to the Focus of middle motion , subtended by the part of the ordinate line , intercepted between the Ellipsis and the Circle circumscribing it . This Mr. Street maketh use of in his Astronomia Carolina , and this I thought not amiss to add here .

XN . GX ∷ OB tang . OEB. OM tang . OEM.

And the Angle OEM-OEB = BEM = ETY , the variation to be deducted from the Elliptick AEquation ETH , the Remainer is the absolute AEquation YTS in the first Quadrant .

In the second and third Quadrants , the variation or difference between the mean and corrected Anomaly , must be added to the Elliptick AEquation , to find the true and absolute AEquation .

For XN . XG . QV. tang . QEV. the comp. m. Anom . QR . t. QER. and the Angle VER = ECO is the variation , and ECO+ECH = OCH is the absolute AEquation sought in the second Quadrant .

Again , XN . SG ∷ a D , tang. a ED. a b , tang . aEB . And aEB — aED = DEf the variation = EFO and EfO+EfH = OfH the absolute AEquation sought in the third Quadrant .

Lastly , in the fourth Quadrant of mean Anomaly it is .

XN . XG ∷ ch. tang . eEH. eg . tang . eFg. and hEg is the variation : And EFH — 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 = 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 the absolute AEquation sought in the fourth Quadrant .

And to find XN the conjugate Semi-diameter , in the right angled Triangle ENX , we have given , EN = AX and EX the semi-distance of the umbilick points . And Mr. Briggs in Chap. 19. of his Arithm. Logar . hath shewed , that the half Sum of the Logarith . of the sum and difference of the Hypotenuse , and the given leg . shall be the Logarith . of the other leg .

Now then EN = AX. 100000 The Leg EX . 1734 Their Sum 101734 5.00745001 Their difference 98266 4.99240328 The Z of the Logarithms , 9.99985329 frac12 ; Z. Logarith . XN . 99983 4.99992664

Now then in the former Example the mean Anomaly is 246 deg. 451741. and the excess above a semicircle is the ang. aED . 66. 451742. Therefore .

4.99992664 5.00000000 10.36069857 10.36077193

aEB — aED = DEf .003544 the variation , which being added to the Elliptick AEquation before found , the absolute AEquation is 1. 837628. and therefore the ☉ true place 345. 291334. that is X. 15. 17. 28.

CHAP. XIV .

To find the place of the fixed Stars .

Example . Let the given time be 1500 , the difference of time is 100 years , and the motion of the fixed Stars for 100 years is 0. 38580.

7.67129 0.38580 7.28549

2. Example .

1674. 7.67129 0.23148 0.05401 7.95678
CHAP. XV.

Of the Theory of the Moon , and the finding the place of her Apogaeon , quantity of excentricity and middle motion .

THe Moon is a secondary Planet , moving about the Earth , as the Earth and other Planets do about the Sun , and so not only the Earth but the whole System of the Moon , is also carried about the Sun in a year . And hence , according to Hipparchus , there arises a twofold , but according to Tycho a three-fold Inequality in the Moons Motion . The first is Periodical and is to be obtained after the same manner , as was the excentrick AEquation of the Sun or Earth : in order whereunto , we will first shew how the place of her Apogaeon and excentricity may be found .

At Bononia in Italy , whose Longitude is 13 degrees Eastward from the Meridian of London , Ricciolus and others observed the apparent times of the middle of three lunar Eclipses to be as followeth .

The first 1642. April the 4. at 14 hours and 4 Minutes . The second 1642 , September 27 at 16 hours and 46 minutes . The third 1643. September 17 at 7 hours and 31 Minutes .

The equal times reduced to the Meridian of London , with the places of the Sun in these three observations , according to Mr. Street in the 25 Page of his Astronomia Carolina , are thus .

d. ♈ . 25. 6. 54 ♎ . 14. 50. 09 ♎ 4. 20. 20

Hence the place of the Moon in the first observation is in ♎ 25. 6′ . 54. in the second ♈ 14. 50. 9. in the third ♓ 4. 20. 20. Now then in Fig. 10. let the Circle BHDGFE denote the Moons AEquant T the Center of the Earth , the Semidiameters TD , TE and TF the apparent places of the Moon , in the first , second and third observations , C the Center of the Excentrick , CD , CE and CF the Lines of middle motion .

176 d. 2 h. 20′ 169. 43. 15″ 19. 37. 07 150. 06. 08 140. 42. 28 530 d. 17 h. 9. 159. 13. 26 159. 07. 32 100. 05. 54 93. 46. 45 50. 00. 14 46. 55. 43

Suppose 10.00000000 the Logarithm of DC , continue FC to H , and with the other right Lines compleat the Diagram .

1. In the Triangle DCH we have given the Angle DCH 86. 13. 15. the complement of DCF 93. 46. 45 to a Semicircle . The Angle DHC 50. 02. 57. The half of the Arch DF and the side CD 1000000. To find CH.

9.88456640 19.83964197 9.95507557

2 In the Triangle HCE we have given CH as before , the Angle CHE 25. 00. 07. The half of the Arch FE , the Angle HCE 133. 04. 17 the complement of FCE , and by consequence the Angle CEH 21. 55. 36 To find the Side CE.

9.57219707 19.95507557 9.62597986 19.58105543 10.00885836

3. In the Triangle DCE , we have given DC . CE and the Angle DCE 140. 42. 28. whose complement 39. 17. 32 is the Summ of the Angles , to find the Angle CED and DE ,

10.00885836 10.00000000 10.00000000 19.99114164 the remainer is the half . As the Radius . 8.01109962 9.55265735 7.56375697 9.52216126 9.80159290 10.00000000 10.27943164

4. In the Isosceles Triangle DTE we have given the Side DE , the angle DTE 150. 06. 08 whose complement 29. 53. 52 is the Summ of the other two angles , the half whereof is the angle TDE 14. 56. 56 which being subtracted from the angle CDE . 19. 51. 21 the remainer is the angle CDT . 4. 54. 25.

0.30237482 9.41154778 10.27943164 9.99335424

5. In the Triangle CDT we have given DC . DT and the angle CDT , to find CTD and CT.

9.99335424 10.00000000 10.00000000 10.00664576 Deduct 45. As the Radius . 7.88368672 11.36854996 9.25223668 is the angle CTD 0.00391693 10.00000000 8.93215746 8.93607439 s. d. 6. 25. 06. 54 3. 07. 41. 01 3. 17. 25. 53 6. 25. 06. 54 04. 54. 25 7. 00. 01. 19 3. 27. 25. 53 3. 12. 35. 26

And for the excentricity in such parts , as the Radius of the AEquant is 100000 the Proportion is .

9.99335424 8.93607439 5.00000000 3.94272015

And this is the Method for finding the place of the Moons Apogaeon and excentricity . And from these and many other Eclipses as well Solar as Lunar , Mr. Street limits the place of the ☽ Apogaeon to be at the time of the first observation 21′ . 04″ more , and the mean Anomaly 20. 41″ less , and the excentricity 8765 such parts as the Radius of the AEquant is 100000.

And by comparing sundry observations both antient and modern , he collects the middle motion of the Moon , from her Apogaeon , to be in the space of four Julian years or 146 days , 53 revolutions , 0 Signes , 7 degrees , 56 minutes , 45 Seconds . And the Apogaeon from the AEquinox 5 Signes , 12 degrees , 46 minutes . And hence the daily motion of her mean Anomaly will be found to be 13 d. 03′ . 53″ . 57‴ . 09 iv . 58 v. 46 vi . Of her Apogaeon 0. 06. 41. 04. 03. 25. 33.

And according to these Measures , if you deduct the motion of the ☽ mean Anomaly for 1641 years April

8. 22. 02. 00. 3. 121. 35. 26 6. 201. 33. 26
6. 05. 311. 57 3. 17. 25. 57 9. 11. 55. 56 21.04 91. 121. 15200
CHAP. XVI .

Of the finding of the place and motion of the Moons Nodes .

ANno Christi 1652 , March 28 , hour . 22. 16′ , the Sun and Moon being in conjunction , Mr. Street in Page 33 , computes the ☽ true place in the Meridian of London to be in ♈ . 19. 14. 18 with latitude North 46′ . 15″ .

And Anno Christi 1654 August 1. hour . 21. 19′ . 30″ was the middle of a Solar Eclipse at London . at which time the Moons true place was found to be in ♌ 18. 58′ . 12″ with North Latitude 32′ . 01″ .

1654 August 1. 21. 19′ . 30″ ☽ place ♌ 18. 58. 12 1652 March 28. 22. 16. 00 ☽ place ♈ 19. 14. 18

From the first observation to the second there are 27 years , 4 months , 5 days , 23 hours 03′ . 30″ .

45. 19. 41 119. 43. 54 165. 03. 35

Therefore in the oblique angled Spherical Triangle DPB we have given BP. 89. 13. 45 the complement of the Moons Latitude in the first Observation 2. PD 89. 27. 50 the complement of the Moons Latitude in the second observation , and the angle DPB 165. 03. 35 , whose complement to a Semicircle is DPF 14. 56. 25. The angle PBD is required .

1. Proportion . 9.97114485 10.00000000 9.98506483 12.01191998

Their Z is FPB 178. 40. 27. whose complement Is the Arch FG 1. 19. 33.

2. Proportion . 0.00002037 10.57376158 8.36418419 8.93796614

And in the right angled Spherical Triangle BA ☊ right angled at A we have given AB 046′ . 15″ the Latitude in the first observation , and the Angle AB ☊ = PBD 85. 02. 56. to find A ☊ the Longitude of the Moon from the ascending Node .

8.93796614 10.00000000 8.12882290 9.19085676
2. To find the Angle A ☊ B. 8.12886212 10.00000000 9.18569718 11.05682506 ♈ . 19. 14. 18 8. 49. 17 ♈ . 10. 25. 01

The retrograde motion whereof in 4 Julian years or 2461 days , is by other observations found to be Sign 2. deg. 17. 22′ . 06″ . and therefore the daily motion deg. 0. 03′ . 10″ . 38‴ . 11 iv . 35 v.

And the motion thereof for 1651 years , March 28. h. 22. 16′ , viz. Sign 8. deg. 18. 26′ . 58″ being added to the place of the Node before found Sig. 0. 10. 25 : 01. Their Sum is the place thereof in the beginning of the Christian AEra Sign 8. deg. 28. 51′ . 59″ .

But the Rudolphin Tables as they are corrected by Mr. Horron and reduced to the Meridian of London , do differ a little from these measures , for according to these Tables , the Moons mean motions are .

AEra Christi . Sign . 04. deg. 02. 25. 55 Sign . 09. deg. 13. 46. 59 Sign . 06. deg. 18. 38. 56 Sign . 08. deg. 28. 33. 16

And according to these measures , the Moons mean motions in decimal Numbers are .

AEra Christi . 34. 00887.345677 78. 82862.654320 55. 18024.691357 74. 69845.679010
The ☽ mean motion for one year . 35. 94001. 44893. 1 11. 29551. 126365 24. 64450. 322566 05. 36900. 781604 The ☽ mean motion for one day . 03. 66010. 962873 00. 03094. 660620 03. 62916. 302253 00. 01470. 961045

And according to these measures are the Tables made shewing the Moons mean Longitude , Apogaeon , Anomaly , and Node retrograde for Years , Months , Days and Hours .

And hence to compute the Moons true place in her Orbit , I shall make use of the Method , which Mr. Horron in his Posthumas works lastly published by Mr. Flamsted , in which from the Rudolphin Tables he sets down these Dimensions .

00. 15′ . 30″ 11. 47. 22 deg. 50. 53. 41 9.91000022 9.82000044

Greatest 6685. 44 The Moons Mean 5523. 69 Excentricity Least 4361. 94 And her greatest variation 00. 36′ . 27″ .

These things premised his directions for computing the Moons place , are as followeth .

CHAP. XVII .

How to Calculate the Moons true place in her Orbit .

TO the given time find the true place of the Sun , or his Longitude from the Vernal AEquinox , as hath been already shewed .

2. From the Tables of the Moons mean motions , write out the Epocha next before the given time , and severally under that set the motions , belonging to the years , months and days compleat , and to the hours and scruples current , every one under his like ( only remember that in the Bissextile years , after the end of February , the days must be increased by one Unite ) then adding them all together , the Summ shall be the Moons mean motions for the time given : But in her Node Retragrade you must leave out the Radix or first number , and the Summ of the remainer being deducted from the Radix , shall be the mean place of her Node required .

3. Deduct the Moons Apogaeon from the ☉ true place , the rest is the annual Augment , the tangent of whose Complement 180 or 360 , being added to the artificial Number given 9. 82000044. the Summ shall be the tangent of an Arch , which being deducted from the said Complement , giveth the Apogaeon AEquation to be added to the mean Apogaeon , in the first and third quadrants of the annual Augment , and Subtracted in the second and fourth , their Summ or difference is the true Apogaeon .

4. The true Apogaeon being Deducted from the ☽ mean Longitude gives the Moons mean Anomaly .

5. Double the annual Augment , and to the Cosine thereof add the Logarithm of 1161. 75. the difference between the Moons mean and extream Excentricity , viz. 3. 06511268 , the Summ shall be the Logarithm of a number which being added to the mean Excentricity , if the double annual Augment be in the first or fourth quadrants ; or Subtracted from it , if in the second or third quadrants ; the Summ or difference shall be the Moons true Excentricity .

6. The Moons true Excentricity being taken for a natural Sine , the Arch answering thereto shall be the ☽ greatest Physical AEquation .

7. To the half of the Moons greatest Physical AEquation add 45 deg. the cotagent of the Summ is the artificial Logarithm of the Excentrick . To the double whereof if you add the tang . of half the mean Anomaly , the Summ shall be the tangent of an Arch , which being added to half the mean Anomaly , shall give the Excentrick Anomaly .

8. To the Logarithm of the Excentrick , add the tangent of half the Excentrick Anomaly , the Summ shall be the tangent of an Arch , whose double shall be the Coequated Anomaly , and the difference between this and the mean Anomaly is the terrestrial Equation , which being added to the Moons mean Longitude , if the mean Anomaly be in the first Semicircle , or Subtracted from it , if in the latter , the Summ or difference shall be the place of the Moon first Equated .

9. From the place of the Moon first Equated , Deduct the true place of the Sun , and double the remainer , and to the Sine of the double add the Sine of the greatest variation 0. 36. 27 , viz. 8. 02541571 , the Summ shall be the Sine of the true variation , at that time , which being added to the Moons place first Equated , when her single distance from the Sun is in the first or third quadrants , or Subtracted when in the second or fourth , the Summ or difference shall be the Moons true place in her Orbit .

Example .

Let the given time be Anno Christi 1672. Feb. 23. h. 11. 34′ . 54″ at which time the Suns true place is in ♓ 15. 29133 and the Moons middle motions are as followeth .

☽ Longitude ☽ Apogaeon ☊ Retrograde 1660 13. 36650. 41. 78372. 55.85177 11. 02. 66032. 24. 31246. 59.08943 Ianuary . 13. 46339. 00.95934 .45599 D. 23 84. 18252. .71177 .33832 H. 11 1. 67755. .01418 .00674 34′ .08641 . .00072 .00054 54 .00228 . .00012 .00001 ☽ Longitude 15. 43897. 67.78229 59.89082 95.96094

55.580292 244.015956 345.29133 244.01595 101.27538 78.72462 10.70033391 9.82000044 10.52033435

Secondly .

55.58029 238.49421 177.08608

Or thus .

171.56434 5.52174 177.08608

And hence it appears that working by the mean Anomaly instead of the mean Longitude , the true Apogaeon Equation must be added to the mean Anomaly , in the second and fourth Quadrants of the ☽ Annual Augment , and subtracted from it in the first and third .

Thirdly .

9.96545577 3.06511268 3.03056845

Fourthly .

9.96161914 11.59455229 11.55620143

Fifthly .

9.98080957 11.57505878 11.55586835 is the coequated Anomaly . 177.086080 is the Equation sought to 55.580292 is the ☽ place first Equated .

Sixthly ,

55.308748 345.291330 70.017418 9.80775260 8.02541571 7.83316831
CHAP. XVIII .

To compute the true Latitude of the Moon , and to reduce her place , from her Orbit to the Ecliptick .

THe greatest Obliquity of the Moon 's Orb with the Ecliptick or Angle A ☊ B Fig. 11. is by many Observations confirmed to be 5 Degrees just , at the time of the Conjunction or Opposition of the Sun and Moon , but in her Quarters deg. 5. 18′ . Now then then find her Latitude at all times , the said Mr. Horrox refers us to pag. 87. in the Rudolphin Tables , to find from thence the Equation of the Nodes , and Inclination limitis menstrui , in this manner .

1. From the mean place of the Node , deduct the ☉ true place , the Remainer is the distance of the ☉ from the ☊ . with which entring the said Table , he finds the Equation of the Node and Inclination limitis menstrui , which being added to or subtracted from the Nodes mean place according to the title , the Sum or difference is the true place of the Node , which being deducted from the place of the Moon in her Orb , the Remainer shall be the Augment of Latitude or Distance of the Moon from the Node , or Leg A ☊ .

2. With the Augment of Latitude , enter the Table of the Moon 's Latitude , and take thence her Simple and Latitude and Increase answering to it . Then say , as the whole excess of Latitude 18′ , or in Decimals 30. is to the Inclination of the Monethly limit : So is the increase of Latitude to the Part Proportional ; which being added to the simple Latitude , will give you the true Latitude of the Moon .

3. With the same Augment of Latitude , enter the Table of Reduction , and take thence the Reduction and Inclination answering thereto : Then say again , as 18′ . 00″ . or 0. 30. is to the Inclination of the Monethly limit : So is the increase of Reduction , to the Part Proportional ; which being added to the simple Reduction , shall give the true , to be added to , or subtracted from the place of the Moon in the Ecliptick . 345.459384 345.291334 . 168050 with which entring the Table , Entituled Tabula AEquationis Nodorum Lunae . I find the Node to need no Equation , and the Inclination limitis menstrui to be deg. 00. 30.

55.698954 345.459384 70.239570 00.28234 4.98610 00.06955 00.00855 00.07810 55.69895 55.62085 8. 25. 37′ . 15″ .
CHAP. XIX .

To find the Mean Conjunction and Opposition of the Sun and Moon .

TO this purpose we have here exhibited a Table shewing the Moons mean motion from the Sun , the construction whereof is this : By the Tables of the Moons mean motions , her mean

34.0088734567 56.6997085185 20.4976851851 77.4973937036 56.8114797531

In like manner the Moons mean distance from the Sun in a year or a day is thus found .

99.9297857316 0038580246 99.9336437562 35.9400144893 36.0063707331

Moons distance from the Sun in a days time .

27378.02348 1.05699 27379.08047 03. 66010.96287 03. 38631.88240

And according to these measures are the Tables made , shewing the Moons mean motion from the Sun , by which the mean conjunction of the ☽ and Moon may be thus computed .

To the given year and Month gather the middle motions of the Moon from the Sun , and take the complement thereof to a whole Circle , from which subtracting continually the nearest lesser middle motions , the day , hour , and minute enfuing thereto is the mean time of the Conjunction .

Example , Anno Christi 1676. I would know the time of the mean Conjunction or New Moon in October .

32.697283 50.254463 24.465038 03.386318 10.803102 89.196898 88.044289 1.152609 1.128772 0.023837 0.023516 .00321

Therefore the mean Conjunction in October , 1676. was the 26 day , 10 min. 8 seconds after 8 at night .

And to find the mean opposition . To the complement of the middle motion , add a semicircle , and then subtract the nearest lesser middle motions as before , the day , hour , and minute ensuing thereto , shall be the mean opposition required .

Example , Anno Christi , 1676. I desire to know the mean opposition in November .

32.697283 50.254463 29.440922 03.386318 15.778986 84.221014 50. 34.221014 33.863188 .357826 .282193 .075633 .075251 .000382

Therefore the Full Moon or mean Opposition of the Sun and Moon was November the 10th , Hours 2 , 32′ 09″ . The like may be done for any other .

And here I should proceed to shew the manner of finding the true Conjunction or Opposition of the Sun and Moon , but there being no decimal Canon yet extant , suitable to the Tables of middle motions here exhibited , I chuse rather to refer my Reader to Mr. Street's Astronomia Carolina , for instructions in that particular , and what else shall be found wanting in this Subject .

AN INTRODUCTION TO Geography , OR , The Fourth Part of COSMOGRAPHY .
CHAP. I.

Of the Nature and Division of Geography .

GEOGRAPHY is a Science concerning the measure and distinction of the Earthly Globe , as it is a Spherical Body composed of Earth and Water , for that both these do together make but one Globe .

2. And hence the parts of Geography are two , the one concerns the Earthy part , and the other the Water .

3. The Earthy part of this Globe is commonly divided into Continents and Islands .

4. A Continent is a great quantity of Land not separated by any Sea from the rest of the World , as the whole Continent of Europe , Asia , and Africa , or the Continents of France , Spain , and Germany .

5. An Island is a part of Earth environed round about with some Sea or other ; as the Isle of Britain with the Ocean , the Isle of Sicily with the Mediterranean , and therefore in Latine it is called Insula , because it is scituate in Salo , in the Sea .

6. Both these are subdivided into Peninsula , Isthmus , Promontorium .

7. Peninsula , quasi pene insula , is a tract of land which being almost encompassed round by water , is joyned to the main land by some little part of Earth .

8. Isthmus is that narrow neck of Land which joyneth the Peninsula to the Continent .

9. Promontorium is a high mountain which shooteth it self into the Sea , the outmost end whereof is called a Cape or Foreland , as the Cape of Good Hope in Africk .

10. The Watry part of this Globe may be also distinguished by diverse Names , as Seas , Rivers , Ponds , Lakes , and such like .

11. And this Terrestrial Globe may be measured either in whole , or in any particular part .

12. The measure of this Earthly Globe in whole , is either in respect of its circumference , ●o its bulk and thickness .

13. For the measuring of the Earths circumference , it is supposed to be compassed with a great Circle , and this Circle in imitation of Astronomers , is divided into 360 degrees or parts , and each degree is supposed to be equal to 15 common German miles , or 60 miles with us in England , and hence the circumference of the Earth is found , by multiplying 360 by 15 , to be 5400 German miles , or multiplying 360 by 60 , the circumference is 21600 English miles .

14. The circumference of the Earth being thus obtained , the Diameter may be found by the common proportion between the Circumference and the Diameter of a Circle , the which according to Archimedes is as 22 to 7 , or according to Van Culen as 1 to 3. 14159. and to bring an Unite in the first place .

As the circumference 3. 14159. is to 1 the Diameter , so is 1 the circumference to 318308 the Diameter , which being multiplied by 5400 , the Earths Diameter will be found to be 1718 German miles and 8632 parts , but being multiplied by 21600 , the Diameter will be 6875 English miles , and parts 4528.

15. The measure of the Earth being thus found in respect of its whole circumference and Diameter , that which is next to be considered , is the distinction of it into convenient spaces .

16. And this is either Primary or Secondary .

17. The Primary distinction of the Earthly Globe into convenient spaces , is by Circles considered absolutely in themselves , dividing the Globe into several parts without any reference to one another .

Dutch Geographer inclines much to the bringing back the great Meridian to the Fortunate Islands , more particularly to the Peak a Mountain so called from the sharpness in the top , in the Isle Teneriff , which is believed to be the highest Mountain in the World ; therefore the same Iohnson in his greatest Globe of the year 1616 , hath drawn the great Meridian in that place , and it were to be wished , that this might be made the common and unchangeable practice .

25. The Horizon is a great Circle , designing so great a Part of the Earth , as a quick sight can discern in an open field ; it is twofold Rational and Sensible .

26. The Rational Horizon is that which is supposed to pass through the Center of the Earth , and is represented by the wooden Circle in the Frame , as well of the Celestial , as the Terrestrial Globe , this Rational Horizon belongeth more to Astronomy than Geography .

27. The Sensible Horizon is that before defined , the use of it is to discern the divers risings and settings of the Stars , in divers places of the Earth , and why the days are sometimes longer , and sometimes shorter .

28. The great but less principal Circle upon the Terrestrial Globe is the Zodiack , in which the Sun doth always move . This Circle is described upon Globes and Maps for ornament sake , and to discover under what part of the Zodiack the several Nations lie .

29. The lesser Circles are those which do not divide the Terrestial Globe into two equal , but into two unequal Parts , and these by a general name are called Parallels , or Circles aequidistant from the Equinoctial ; of which as many may be drawn , as there can Meridians , namely 180 if but to each degree , but they are usually drawn to every ten Degrees in each Quadrant from the AEquator to the Poles .

30. These Parallels are not of the same Magnitude , but are less and less as they are nearer and nearer to each Pole : and their use is to distinguish the Zones , Climates and Latitudes of all Countries , with the length of the Day and Night ▪ in any Part of the World .

31. Again , a Parallel is either named or unnamed .

32. An unnamed Parallel is that which is drawn with small black Circular Lines .

33. A named Parallel is that which is drawn upon the Globe with a more full ruddy and circular Line : such as are the Tropicks of Cancer and Capricorn , with the Arctick and Antarctick Circles , of which having spoken before in the general description of the Globe , there is no need of adding more concerning them now .

CHAP. II.

Of the Distinction or Dimension of the Earthly Globe by Zones and Climates .

HAving shewed the primary distinction of the Globe into convenient spaces by Circles considered absolutely in themselves , we come now to consider the secondary Dimension or distinction of convenient spaces in the Globe , by the same Circles compared with one another , and by the spaces contained between those Circles .

2. This secundary Dimension or Distinction of the terrestial Globe into Parts , is either a Zone or a Clime .

3. A Zone is a space of the Terrestial Globe included either between two of the lesser nominated Circles , or between one and either Pole . They are in Number five , one over hot , two over cold , and two temperate .

4. The over hot or Torrid Zone , is between the two Tropicks , continually scorched with the presence of the Sun .

5. The two over cold or Frigid Zones , are scituated between the two polar Circles and the very Poles , continually wanting the neighbour hood of the Sun .

6. The two temperate Zones , are one of them between the Tropick of Cancer and the Arctick Circles and the other between the Tropick of Capricorn and the Antarctick Circle , enjoyning an indifferency between Heat and Cold ; so that the parts next the Torrid Zone are the hotter , and the parts next the Frigid Zone are the Colder .

7. The Inhabitants of these Zones , in respect of the diversity of their noon Shadows are divided into three kinds , Amphiscii , Heteroscii and Periscii . Those that inhabit between the two Tropicks are called Amphiscii , because that their noon Shadows are diversly cast , sometimes towards the South as when the Sun is more Northward than their vertical point , and sometimes towards the North , as when the Sun declines Southward from the Zenith .

Those that live between the Tropick of Cancer and the Arctick Circle or between the Tropick of Capricorn and the Antarctick Circle are ●alled Heteroscii , because the Shadows at noon are cast one only way , and that either North or South . They that inhabit Northward of the Tropick of Cancer have their Shadows always towards the North , and they that inhabit Southward of the Tropick of Capricorn , have their noon Shadows always towards the South .

Those that inhabit between the Poles and the Arctick or Antarctick Circles are called Periscii , because that their Gnomons do cast their Shadows circulary , and the reason hereof is , for that the Sun is carried round about above their Horizon in his whole diurnal revolution .

8. The next secundary Dimension or distinction of the earthy Globe into convenient parts or spaces , is by Climes .

9. And a Clime or Climate is a space of Earth conteined between three Paralells , the middlemo● whereof divideth it into two equal parts , serving for the setting out the length and shortness of the days in every Country .

10. These Climates and the Parallels by which they are conteined are none of them of equal quantity , for the first Clime as also the Parallel beginning at the AEquator is larger than the second , and the second is likewise greater than the third .

11. The Antients reckoned but seven Climates at the first , to which Number there were afterward added two more , so that in the first of these Numbers were comprehended fourteen parallels , but in the latter eighteen .

12. Ptolemy accounted the Paralells 38 each way from the Equator , that is 38 towards the North , and as many towards the South , 24 of which he reckoned by the difference of one quarter of an hour , 4 by the difference of half an hour , 4 by an whole hours difference , and 6 by a Months difference , but now the parallels being reckoned by the difference of a quarter of an hour , the Climates are 24 in Number till you come to the Latitude of 66 degrees 31 Minutes , to which are afterwards added 6 Climates more unto the Pole it self , where the Artificial day is 6 Months in length .

13. The distances of all both Climates and Parallels , together with their Latitudes from the AEquator , and difference of the quantity of the longest days , are here fully exprest in the Table following .

ATableoftheClimatesbelongingtothethreesortsofInhabitants. Inhabitants belonging to the several Climes Climes Paralells Length of the Day Poles Elevation Bea of the Clime 0 12.0 0.0 0 4.18 1 12.15 4.18 2 12.30 8.34 1 8.25 Amphiscii 3 12.45 12.43 4 13.0 16.43 2 7.50 5 13.15 20.33 6 13.30 23.10 3 7.3 7 13.45 27.36 8 14.0 30.47 4 6.9 9 14.15 33.45 10 14.30 36.30 5 5.17 11 14.45 39.02 12 15.0 41.22 6 4.30 13 15.15 43.32 14 15.30 45.29 7 3.48 15 15.45 47.20 16 16.0 49.21 8 3.13 17 16.15 50.13 18 16.30 51.58 9 2.44 19 15.45 53.17
Climes Paralells Length of the Days Poles Elevation Breadth of the Clime 20 17.00 54.29 10 2.17 Heteroscii 21 17.15 55.34 22 17.30 56.37 11 2.0 23 17.45 57.34 24 18.00 58.26 12 1.40 25 18.15 59.14 26 18.30 59.59 13 1.26 27 18.45 60.40 28 19.00 61.18 14 1.13 29 19.15 61.53 30 19.30 62.25 15 1.0 31 19.45 62.54 32 20.00 63.22 16 0.52 33 20.15 63.46 34 20.30 64.06 17 0.44 35 20.45 64.30 36 21.00 64.49 18 0.36 37 21.15 65.06 38 21.30 65.21 19 0.29 39 21.45 65.35 40 22.00 65.47 20 0.22 41 22.15 65.57 42 22.30 66.00 21 0.17 43 22.45 66.14
Clime Paralells Length of the Day Poles Elevation Breadth of the Clime 44 23.00 66.20 22 0.11 45 23.15 66.25 46 23.30 66.28 23 0.5 47 23.45 66.30 24 48 24.00 66.31 0.0
Periscii Here the Climates begin to be accounted by Months , from 66. 31 where the day is 24 hours long ; unto the Pole it self , where it is 6 Months in length . 1 67.15 2 69.30 3 73.20 4 78.20 5 84.0 6 90.0

14. Hitherto we have considered the inhabitants of the Earth in respect of the several Zones and Climes into which the whole Globe is divided ; there is yet another distinction behind into which the inhabitants of the Earth are divided in respect of their site and position in reference to one another , and thus the inhabitants of the Earth are divided into the Perioeci , Antoec● and Antipodes .

15. The Perioeci are such as dwell in the same Parallel on the same side of the AEquator , how far distant soever they be East and West , the season of the year and the length of the days being to both alike , only the midnight of the one is the moon to the other .

16. The Antoeci are such as dwell under the same Meridian and in the same Latitude , or Parallel distance from the AEquator , the one Northward and the other Southward , the days in both places being of the same length , but differ in the Seasons of the year , for when it is Summer in the one it is Winter in the other .

17. The Antipodes are such as dwell Feet to Feet , so as a right Line drawn from the one unto the other , passeth from North to South through the Center of the World . These are distant 180 degrees or half the compass of the Earth , they differ in all things , as Seasons of the year , length of days , rising and setting of the Sun and such like . A matter reckoned so ridiculous and impossible in former times , that Boniface Arch-Bishop of Mentz seeing a Treatise concerning these Antipodes written by Virgilius Bishop of Salisburg , and not knowing what damnable Doctrine might be couched under that strange Name , made complaint first to the Duke of Bohemia , and after to Pope Zachary Anno 745 by whom the poor Bishop ( unfortunate only in being learned in such a time of Ignorance ) was condemned of Heresie , but God hath blest this latter age of the World with more understanding , whereby we clearly see those things , which either were unknown , or but blindly guessed at by the Antients .

18. The second part of the Terrestial Globe is the Water which is commonly divided into these parts , or distinguished by these Names , Oceanus , Mare , Fretum , Sinus , Lacus and Flumen .

19. And first Oceanus or the Ocean is that general Collection of all Waters , which encompasseth the Earth on every side .

20. Mare the Sea , is a part of the main Ocean , to which we cannot come but through some Fretum or Strait , as Mare Mediterraneum . And it takes it name first either from the adjacent Shore , as Mare Adriaticum , from the City of Adria ; or secondly from the first discoverer , as Mare Magellanicum , from Magellanus who first found it , or thirdly from some remarkable accident , as Mare Icarium from the drowning of Icarus the Son of Daedalus .

21. Fretum , a Strait is a part of the Ocean penned within some narrow Bounds , and opening a way into some Sea , or out of some Sea into the Ocean , as the Strait of Hellespont , Gibralter , &c.

22. Lacus , a Lake is a great body or collection of Water , which hath no visible Intercourse with the Sea , or influx into it ; as the Lake of Thrasymene in Italy , and Lacus Asphaltites , or the dead Sea in the Land of C●naan .

23. Flumen or Fluvius is a water-course continually running , ( whereby it differs frum Stagnum a standing Pool ) issuing from some Spring or Lake , and emptying it self into some part of the Sea , or some other great River , the mouth or outlet of which is called Ostium .

And thus we have gon over those particulars both of Earth and Water , which appertain to this Science of Geography in the general ; We will now proceed to a more particular Consideration of the several parts into which the Terrestial Globe is commonly divided .

CHAP. III.

Of Europe .

THe Terrestial Globe is divided into two parts , known or unknown .

2. The unknown or the parts of the World not fully discovered , are distinguished into North and South , the unknown parts of the World towards the North , are those which lie between the North part of Europe or America and the North Pole ; and the unknown parts of the World toward the South , are those which ly between the South part of America and the South Pole .

3. The known parts of the World were antiently these three , Europe , Asia and Africk , to which in latter ages a fourth hath been added which is called America .

4. Europe is bounded on the North with the Northern Ocean , and on the South with the Mediterranean Sea , on the East with the River Tanais , and on the West with the Western Ocean , and is contained between the Tropick of Cancer , and the Pole Arctick , or 44 degrees as most do say , taking its beginning Southward from Sicily where the Pole is elevated 36 degrees , and is thence continued to 80 degrees of North Latitude , and so the whole Latitude of Europe is in English miles 2640 , but some allow to Europe 45 degrees of Latitude , that is in English miles 2700.

5. The Longitude of Europe is reckoned from the furthest part of Spain and the Atlantick Ocean , to the River Tanais , which some reckon to be 60 Degrees , to one of which Degrees passing through the middle of Europe , they allow fifteen German miles almost , or sixty English , and so the Longitude in German miles is 900 , in English 3600.

6. Europe though the least of all the four Quarters of the World , is yet of most renown amongst us : First , because of the temperature of the Air , and fertility of the Soil : Secondly , from the study of Arts , both ingenuous and mechanical : Thirdly , of the Roman and Greek Monarchies : Fourthly , from the purity and sincerity of the Christian Faith : Fifthly , because we dwell in it , and so give it the first place .

7. Europe may be considered as it stands divided into the Continent and the Islands : the Continent lying all together , containeth these Countries . 1. Spain . 2. France . 3. Germany . 4. Italy , and the Alpes . 5. Belgium . 6. Denmark ▪ 7. Swethland . 8. Russia . 9. Poland . 10. Hungary . 11. Sclavonia , 12. Dacia , and 13. Greece . Of each of which I will give some short account ; as also of the chief Islands as they are dispersed , in the Greek , AEgaean , Cretan and Io●ian Seas , with those in the Adriatick , Mediterranean , and in the British and Northern Ocean .

8. Amongst these I give Spain the first place , as being the most Western Part of all the Continent of Europe environed on all sides with the Sea , except towards France ; from which it is separated by the Pyrenaean Hills : but more particularly , it is bounded upon the North with the Cantabrian , on the West with the Atlantick Ocean , on the South with the Straits of Gibralter , on the East with the Mediterranean , and on the North East with the said Pyren●ean Hills . The Figure of it is compared by Strabo to an Oxes hide spread upon the Ground ; the Neck whereof being that Isthmus which unites it to France .

9. The greatest length hereof , it reckoned at 800 miles , the breadth where it is broadest at 500 , the whole Circumforence 2480 Italian miles : but Mariana measuring the compass of it by the bendings of the Pyrenaean Hills , and the creeks and windings of the Sea , makes the full circuit of it to be 2816 miles of Italian measure .

10. It is situate in the more Southerly Part of the Northern temperate Zone , and almost in the midst of the fourth and sixth Climates ; the longest day being 15 hours and a quarter in length in the most Northern Parts hereof : but in the extream South near to Gibralter not above fourteen , which Situation of this Country , rendreth the Air here very clear and calm , seldom obscured with mists and vapours , and not so much subject to Diseases as the more Northern Regions are .

11. This Continent is subdivided into the Kingdoms of Navarr . 2. Biscay . 3. Guipusco● . 4. Lean and Oviodo . 5. Gallicia . 6. Corduba . 7. Granada . 8. Murcia . 9. Toledo . 10. Castile . 11. Portugal . 12. Valentia . 13. Catalonia . 14. Majorca . And 15. Aragon ; but all of them are now united in the Monarchy of Spain .

12. France according to the present dimensions of it , is bounded on the East with a Branch of the Alpes which divide Dauphine and Piemont , as also with the Countries of Savoy , Switzerland , and some Parts of Germany and the Netherlands . On the West with the Aquitanick Ocean , and a Branch of the Pyrenaean Mountains which divide it from Spain . On the North with the English Ocean , and some Parts of Belgium , and on the South with the rest of the Pyrenaean Mountains , and the Mideterranean .

13. The Figure of it is almost square , each side of the Quadrature being reckoned 600 miles in length , but they that go more exactly to work upon it , make the length thereof to be 660 Italian miles , the breadth 570 , the whole Circumference 2040. It is seated in the Northern temperate Zone , between the middle Parallel of the first Clime , where the longest day is 15 hours , and the middle Parallel of the eighth Clime , where the longest day is 16 hours and a half .

14. The Principal Provinces in this flourishing Country , are . 1. France specially so called . 2. Champagne . 3. Picardy . 4. Normandy . 5. Bretagne . 6. The Estates of Angiou . 7. La Beausio . 8. Nivernois . 9. The Dukedom of Bourbon . 10. Berry . 11. Poictou . 12. Limosin . 13. Piregort . 14. Quercu . 15. Aquitain . 16. Languedoc . 17. Provence . 18. Daulphine . 19. La Bresse . 20. Lionnois . 21. The Dutchy . 22. The County of Burgundy . 23. The Islands in the Aquitanick and Gallick Ocean : Those of most note are these six . 1. Oleron . 2. Re● . 3. Iarsey . 4. Gernsey . 5. Sarke . 6. Aldernay on the shores of Normandy , of which the four last are under the Kings of England .

15. Italy once the Empress of the greatest part of the then known World , is compassed with the Adriatick , Ionian and Tyrrhenian Seas , except it be towards France and Germany , from which it is parted by the Alpes ; so that it is in a manner , a Peninsula , or a Demi-Island . But more particularly it hath on the East the lower part of the Adriatick , and the Ionian Sea , by which it is divided from Greece ; on the West the River Varus , and some part of the Alpes , by which it is parted from France , on the North in some part the Alpes which divide it from Germany ; and on the other , part of the Adriatick , which divides it from Dalmaria ; and on the South the Tyrrhenian and Tuscan Seas , by which it is separated from the main Land of Africa .

16. It containeth in length from Augusta Praetoria , now called Aost , at the foot of the Alpes , unto Otranto in the most Eastern Point of the Kingdom of Naples 1020 miles ; in breadth from the River Varo , which parts it from Provence , to the mouth of the River Arsia in Friuli , where it is broadest , 410 miles ; about Otranto , where it is narrowest not above 25 miles ; and in the middle parts from the mouth of Peseara in the Adriatick or upper Sea to the mouth of Tiber in the Tuscan or lower Sea , 126 miles . The whole compass by Sea reckoning in the windings and turnings of the shore , comes to 3038 miles ; which added to the 410 which it hath by Land , make up in all 3448 miles : but if the Coasts on each side be reckoned by a straight Line , then as Castaldo computes it , it comes to no more than 2550 miles .

17. The whole Country lieth under the first and sixth Climates of the Northern temperate Zone , which it wholly taketh up : so that the longest day in the most Northern Parts is 15 hours and three first parts of an hour ; the longest in the Southern Parts , falling short a full hour of that length .

18. Italy as it stands now is divided into the Kingdoms of Naples , Sicily and Sardinia . 2. The Land or Patrimony of the Church . 3. The great Dukedom of Tuscany . 4. The Common-wealths of Venice , Genoa and Luca. 5. The Estates of Lombardy , that is the Dukedoms of 1. M●llain . 2. Mantua . 3. Modena . 4. Parma . 5. Montferrat , and the Principality of Piedmont .

19. To the Peninsula of Italy belong the Alpes , aridge of Hills , wherewith as with a strong and defensible Rampart Italy is assured against France and Germany . They are said to be five days Journey high , covered continually with Snow , from the whiteness whereof they took this name , it doth contain the Dukedom of Savoy ; the Seigniory of Geneva ; the Country of Wallisland , Switzerland and the Grisons .

20. Belgium , or the Netherlands , is bounded on the East with Westphalin , Gulick , Cleve , and the Land of Triers , Provinces of the higher Germany ; on the West with the main Ocean , which divides it from Britain ; on the North with the River Ems , which parts it from East-Friezeland ; on the South with Picardie and Campagne two French Provinces ; upon the South-East with the Dukedom of Lorrain .

21. It is in compass 1000 Italian or 280 German miles , and is situated in the Northern temperate Zone , under the seventh , eighth and ninth Climates : the longest day in the midst of the seventh Climate where it doth begin , being 16 hours , in the beginning of the ninth Climate increased to 16 hours 3 quarters , or near 17 hours .

22. It containeth those Provinces which in these later Ages were possessed by the House of Burgundy , that is the Lordship of West-Friezeland , given to the Earls of Holland by Charles the Bald ; the Earldom of Zutphen united unto that of Gelder by Earl Otho of Nassau , and finally the Estate of Groening , Over-Yssel , and some part of Vtrecht , by Charles the Fifth . As it stands now divided between the Spaniards and the States it containeth the Provinces of 1. Flanders . 2. Artois . 3. Hainault . 4. The Bishoprick of Cambray . 5. Namur . 6. Luxemburg . 7. Limbourg . 8. Luyckland , or the Bishoprick of Leige . 9. Brabant . 10. Marquisate . 11. Meohlin . The rest of the Netherlands which have now for sometime withdrawn their obedience from the Kings of Spain , are 1. Holland . 2. Zeland . 3. West-Friezeland . 4. Vtrecht . 5. Over-Yssel . 6. Gelderland . 7. Zutphen . 8. Groening .

23. Germany is bounded on the East with Prussia , Poland , and Hungary ; on the West with France , Switzerland and Belgium ; on the North with the Baltick Seas , the Ocean , and some part of Denmark ; on the South with the Alps which part it from Italy .

24. The length from East to West , that is from the Vistula or Weissel to the Rhine , is estimated at 840 Italian miles , the breadth from North to South , that is from the Ocean to the Town of Brixen in Tyrol , 740 of the same miles . So that the Figure of it being near a Square , it may take up 3160 miles in compass , or thereabouts . Situate in the Northern temperate Zone , between the middle Parallels of the sixth and tenth Climates ; the longest day in the most Southern Parts being 15 hours and an half , and in the most Northern 17 hours and a quarter .

25. The Principal Parts of this great Continent , are 1. Cleveland . 2. The Estates of the three spiritual Electors , Colen , ●●●ntz , and Triers . 3. The Palatinate of the Rhine . 4. Alsatia . 5. Lorrain . 6. Suevia or Schwaben . 7. Bavaria . 8. Austria and its Appendices . 9. The Confederation of Waderaw . 10. Franconia . 11. Wirtenberg . 12. Baden . 13. The Palatinate of Northgoia , or the Upper Palatinate . 14. Bohemia and the Incorporate Provinces . 15. Pomerania . 16. Mecklenburg . 17. The Marquisate of Brandenburg . 18. Saxony , and the Members of it . 19. The Dukedom of Brunswick and Lunenburg . 20. The Lantgravedom of Hassia . 21. Westphalen . 22. East-Friezeland .

26. Denmark or Danemark , reckoning in the Additions of the Dukedom of Holstein , and the great Continent of Norway , with the Isles thereof , now all united and incorporated into one Estate is bounded on the East with the Baltick Sea and some part of Sweden ; on the West with the main Western Ocean ; on the North-East with a part of Sweden ; full North with the main frozen Seas ; and on the South with Germany , from which it is divided on the South-West by the River Albis , and on the South-East by the Trave ; a little Isthmus or neck of Land uniting it to the Continent .

27. It lieth partly in the Northern temperate Zone , and partly within the Arctick Circle ; extending from the middle Parallel of the tenth Clime , or 55 degree of Latitude where it joyneth with Germany , as far as the 71 degree where it hath no other bound but the frozen Ocean ; by which account the longest day in the most Southern Parts is 17 hours and a quarter , but in the Parts extreamly North , they have no Night for two whole Moneths , three Weeks , one Day , and about seven hours ; as on the other side no day for the like quantity of time , when the Sun is most remote from them , in the other Tropick .

28. The whole Body of the Estate consisteth chiefly of three Members . viz. 1. The Dukedom of Holstein ; containing Waggerland , Dilmarsh , Starmaria , and Holstein , especially so called . 2. The Kingdom of Denmark ; comprehending both Iuitlands , part of Scandia , and the Hemodes , or Baltick Islands . 3. The Kingdom of Norway consisting of Norway it self , and the Islands of the Northern Ocean .

29. Swethland is bounded on the East with Muscovy , on the West with the Doferine Hills , which divide it from Norway ; on the North with the great frozen Ocean spoken of before ; on the South with Denmark , Liefland , and the Baltick Sea .

30. It is situate under the same Parallels and Degrees with Norway , that is , from the first Parallel of the 12 Clime , where the Pole is elevated 58 degrees 26 minutes , as far as to the 71 degree of Latitude , by which account the longest day in the Southern Point is but 18 hours , whereas on the farthest North of all the Countrey , they have no Night for almost three whole Moneths together .

31. The whole Kingdom is divided into two Parts , the one lying on the East , the other on the West of the Bay or Gulf of Bodener , being a large and spacious branch of the Baltick Sea , extending from the most Southerly Point of Gothland , as far as to Lapland on the North. According to which Division we have the Provinces of 1. Gothland . 2. Sweden lying on the West side of the Gulph . 3. Lapland shutting it up upon the North. 4. Bodia or Bodden . 5. Finland on the East side thereof . 6. The Sweedish Islands , where it mingleth with the rest of the Baltick Seas .

32. Russia is bounded on the East by Tartary , on the West with Livonia and Finland , from which it is divided by great mountains and the River Poln , on the North by the frozen Ocean , and some part of Lapland , and on the South by Lituania a Province of the Kingdom of Poland , and the Crim Tartar inhabiting on the Banks of Palus Maeotis , and the Euxine Sea . It standeth partly in Europe and partly in Asia , the River Tanais or Don running through it , the common boundary of those great and noted parts of the world .

33. It is scituate North within the Artick Circle so far , that the longest day in Summer will be full six months , whereas the longest day in the southern parts is but 16 hours and an half .

34. It is divided into the Provinces of 1. Moscovy specially so called . 2. Snol●usio , 3. Masaisky , 4. Plesco , 5. Novagrod the great , 6. Corelia , 7. Blarmia , 8. Petzora , 9. Condora , 10 Obdora , 11. Iugria , 12. Severia , 13. Permia , 14. Rozan , 15. Wiathka , 16. Casau , 17. Astracan , 18. Novogordia inferiour , 10. The Morduits or Mordua , 20. Worotime , 21. Tuba , 22. Wolodomir , 23. Duina , 24. the Russian Islands .

35. Poland is bounded on the East with Russia , and the Crim-Tartar , from whom it is parted by the River Borysthenes ; on the West with Germany , on the North with the Baltick Sea and some part of Russia , on the South with the Carpathian Mountains , which divide it from Hungary , Transilvania , and Moldavia . It is of figure round in compass 2600 miles , scituate under the 8 and 12 Climates , so that the longest day in the southern parts is but 16 hours , and about 18 hours in the parts most North.

36. The several Provinces of which this Kingdom doth consist , are 1. Livonia , 2. Samogitia , 3. Lituania , 4. Volkinia , 5. Podolia , 6. Russia nigra , 7. Massovia , 8. Podlassia , 9. Prussia , 10. Pomerellia , 11. Poland specially so called .

37. Hungary is bounded on the East with Transilvania and Walachia , on the West with Sterria , Austria and Moravia , on the North with the Carpathian mountains which divide it from Poland , and on the South with Sclavonia , and some part of Dacia : it extendeth in length from Presburg along the Danow to the borders of Transilvania , for the space of 300 English miles , and 190 of the same miles in breadth .

38. It lieth in the Northern temperate Zone , betwixt the middle parallels of the 7 and 9 Climates , so that the longest Summers day in the Southern parts is but 15 hours and an half , and not above 16 hours in the parts most North.

40. This Country is commonly divided into the upper Hungary and the lower , the upper lying on the North of the River Danow , the lower lying on the South of that River , comprehending all Pannonia inferior and part of Superior , and is now possessed by the King of Hungary and the Great Turk , who is Lord of the most part by Arms and Conquest .

04. Sclavonia is bounded on the East with Servia , Macedonia and Epirus , from which it is parted by the River Drinus , and a line drawn from thence unto the Adriatick , on the West with Carniola in Germany , and Istria in the Seigniory of Venice , from which last it is divided by the River Arsia ; on the North with Hungary , on the South with the Adriatick Sea .

41. It is scituate in the Northern temperate Zone , between the middle Parallels of the sixth and seventh Climates , so that the longest day in Summer is about 15 hours and an half .

42. This Country as it came at last to be divided , between the Kings of Hungary and the State of Venice ; is distinguished into 1. Windischland , 2. Croatia , 3. Bosnia , 4. Dalmatia , 5. Liburnia or Cantado di Zara , and 6. The Sclavonian Islands .

43. Dacia is bounded on the East with the Euxine Sea and some part of Thrace ; on the West with Hungary and Sclavonia ; on the North with Podolia , and some other members of the Realm of Poland , on the South with the rest of Thrace and Macedonia .

44. It lieth on both sides of the Danow fronting all along the upper and the lower Hungary , and some part of Sclavonia ; extended from the 7 Climate to the 10 ; so that the longest Summers day in the most northern parts thereof , is near 17 hours , and in the most southern 15 hours 3 quarters .

45. The several Provinces comprehended under the name of Dacia , are 1. Transilvania , 2. Moldavia , 3. Walachia , 7. Rascia , 5. Servia , 6. Bulgaria , the first four in old Dacia , on the North side of the Danow ; the two last in new Dacia , on the South thereof .

45. Greece in the present Latitude and extent thereof , is bounded on the East with the Propontick , Hellespont , and AEgean Seas , on the West with the Adriatick ; on the North with Mount Haemus which parteth it from Bulgaria , Servia and some part of Illyricum ; and on the South with the Sea - Ionian ; so that it is in a manner a Peninsula or Demi-Island , environed on three sides by the Sea , on the fourth only united to the rest of Europe .

46. It is scituate in the northern temperate Zone , under the fifth and sixth Climates , the longest day being 15 hours .

47. In this Country formerly so famous for learning and government , the several Provinces are 1. Peloponnesus , 2. Achaia , 3. Epirus , 4. Albania , 5. Macedon , 6. Thrace , 7. The Islands of the Propontick ; 8. AEgean , and 9. The Ionian Seas , and 10. finally the Isle of Crete .

And thus I have given you a brief description of those Countries which are comprehended in the Continent of Europe ; the Islands in this part of the world are many ; I will mention only some few . These two in the British and Northern Ocean , known by the names of Great Britain and Ireland are the most famous , to which may be added Greenland . In the Mediterranaen Sea you have the Islands of Sicilia , Sardinia , Corsica and Crete , which is now called Candia the greater and the less : As for the other Islands belonging to this part of the world , the Reader may expect a more particular description from them who have or shall write more largely of this subject : This we deem sufficient for our present purpose . Let this then suffice for the description of the first part of the World called Europe .

CHAP. IV.

Of Asia .

ASia is bound on the West with the Mediteranean and AEgaean Seas , the Hellespont , Propontis , Thracian Bosphorus and the Euxine Sea , the Palus Maeotis , the Rivers Tanais and Duina , a Line being drawn from the first of the two said Rivers unto the other , by all which it is parted from Europe ; on the North it hath the main Scythick Ocean ; but on the East the Indian Ocean , and Mare del Eur by which it is separated from America ; on the South the Mediterranean , or that part of it , which is called the Carpathian Sea , washing the shoars of Anatolia , and the main Southern Ocean , passing along the Indian , Persian and Arabian Coasts : and finally on the south-west , the red Sea or Bay of Arabia , by which it is parted from Affrick . Environed on all sides with the Sea , or some Sea like Rivers , except a narrow Isthmus in the south-west , which joyns it to Africk , and the space of ground ( whatsoever it be ) between Duina and Tanais , on the North-west which unites it to Europe .

2. It is situated East and West , from the 52 to the 169 degree of Longitude ; and North and South from the 82 degree of Latitude to the very AEquator ; some of the Islands only lying on the South of that Circle : so that the longest summers day in the southern parts , is but twelve hours , but in the most northern parts hereof almost four whole Months together .

3. This Country hath heretofore been had in special honour ; 1. For the creation of Man , who had his first making in this part of the World . 2. Because in this part of it stood the Garden of Eden , which he had for the first place of his habitation . 3. Because here flourished the four first great Monarchies of the Assyrians , Babylonians , Medes and Persians . 4. Because it was the Scene of almost all the memorable Actions which are recorded by the pen-men of the Scriptures . 5. Because our Saviour Christ was borne here , and here wrought his most divine Miracles , and accomplished the great work of our Redemption . 6. And finally , because from hence all Nations of the World had their first beginning , on the dispersion which was made by the Sons of Noah after their vain attempt at Babel .

4. This part of the World for the better understanding of the Greek and the Roman Stories and the estate of the Assyrian , Babylonian and the Persian Monarchies , to which the holy Scriptures do so much relate , we shall consider as divided into the Regions of 1. Anatolia or Asia minor . 2. Cyprus . 3. Syria . 4. Arabia . 5. Chaldea . 6. Assyria . 7. Mesopotamia . 8. Turcomania . 9. Media . 10. Persia. 11. Tartaria . 12. China . 13. India . and 14. the Oriental Islands .

Anatolia or Asia minor .

Anatolia or Asia minor , is bounded on the East with the River Euphrates , by which it is parted from the greater Asia ; on the West with the Thracian Bosphorus , Propontis , Hellespont , and the AEgean Sea , by which it is parted from Europe ; on the North with Pontus Euxinus , called also the black Sea , and Mare Maggiore , and on the South by the Rhodian , Lydian and Pamphilian Seas , several parts of the Mediterranean . So that it is a Demi-Island or Peninsula environed on all sides with water , excepting a small Isthmus or Neck of Land extending from the head of Euphrates to the Euxine Sea , by which it is joyned to the rest of Asia .

It reacheth from the 51 to the 72 degree of Longitude , and from the 36 to the 45 degree of Latitude , and lyeth almost in the same position with Italy , extending from the middle Parallel of the fourth Clime , to the middle Parallel of the sixth , so that the longest summers day in the Southern Parts , is about 14 hours and a half ; and one hour longer in those parts which lie most towards the North.

The Provinces into which it was divided before the Roman Conquest were 1. Bithynia . 2. Pontus . 3. Paphlagonia . 4. Galatia . 5. Cappadocia . 6. Armenia Major & Minor . 7. Phrygia minor . 8. Phrygia major . 9. Mysia the greater and the less . 10. Asia specially so called , comprehending AEolis and Ionia . 11. Lydia . 12. Caria . 13. Lycia . 14. Lycaonia . 15. Pisidia . 16. Pamphylia . 17. Isauria . 18. Cilicia . 19. The Province of the Asian Isles , whereof the most principal are 1. Tenedos . 2. Chios . 3. Samos . 4. Choos . 5. Icaria . 6. Lesbos . 7. Patmos . 8. Claros . 9 Carpathos . 10. Rhodes .

Cyprus .

Cyprus is situated in the Syrian and Cilician Seas , extended in length from East to West 200 miles , in breadth 60 the whole compass reckoned 550 , distant about 60 miles from the rocky Shores of Cilicia in Asia minor , and about one hundred from the main Land of Syria .

It is situated under the fourth Climate , so that the longest day in Summer is no more than 14 hours and a half .

Divided by Ptolemy into the 4 provinces of 1. Paphia . 2. Amathasia . 3. Lepathia . 4. Salamine .

Syria .

Syria is bounded on the East with the River Euphrates by which it is parted from Mesopotamia ; on the West with the Mediterranean Sea ; on the North with Cilicia and Armenia minor , parted from the last by mount Taurus ; and on the South with Palestine , and some parts of Arabia . The length hereof from Mount Taurus to the Edge of Arabia , is said to be 525 Miles ; the breadth from the Mediterranean to the River Euphrates 470 Miles , drawing somewhat near unto a Square .

The whole Country was antiently divided into these six parts . 1. Phoenicia . 2. Palestine . 3. Syria specially so called . 4. Comagena . 5. Palmyrene . and Caelosyria , or Syria Cava .

Arabia .

Arabia hath on the East Chaldaea and the Bay or Gulf of Persia ; on the West Palestine , some part of Egypt , and the whole course of the red Sea , on the North the River Euphrates with some parts of Syria and Palestine , and on the South the main southern Ocean . It is in circuit about 4000 Miles , but of so unequal and heteregeneous Composition , that no general Character can be given of it , and therefore we must look upon it as it stands divided into Arabia Deserta , 2. Arabia Petraea . 3. Arabia Felix and 4. The Arabick Islands .

Chaldea .

Chaldea is bounded on the East with Susiana a Province of Persia ; on the West with Arabia deserta ; on the North with Mesopotamia ; and on the South with the Persian Bay and the rest of Deserta .

Assyria .

Assyria is bounded on the East with Media , from which it is parted by the Mountain called Coathras ; on the West with Mesopotamia , from which it is divided by the River Tygris ; on the South with Susiana ; and on the North with some part of Turcomania ; it was antiently divided into six parts . 1. Arraphachitis . 2. Adiabene . 3. Calacine . 4. Aobelites . 5. Apolloniates .

Mesopotamia .

Mesopotamia is bounded on the East with the River Tygris by which it is parted from Assyria ; on the West with Euphrates which divides it from Comagena a Province of Syria ; on the North with Mount Taurus ; by which it is separated from Armenia major ; and on the South with Chaldea and Arabia deserta from which last it is parted by the bendings of Euphrates also . It was antiently divided into , 1. Anthemasia . 2. Chalcitis . 3. Caulanitis . 4. Acchabene . 5. Ancorabitis and 6. Ingine .

Turcomania .

Turcomania is bounded on the East with Media and the Caspian Sea ; on the West with the Euxine Sea , Cappadocia and Armenia minor ; on the North with Tartary , and on the South with Mesopotamia and Assyria . A Countrey which consisteth of four Provinces . 1. Armenia major or Turcomania properly and specially so called . 2. Colchis . 3. Iberia . 4. Albania .

Media .

Media is bounded on the East with Parthia , and some part of Otyrcania , Provinces of the Persian Empire ; on the West with Armenia major , and some part of Assyria ; on the North with the Caspian Sea and those parts of Armenia major , which now pass in the account of Iberia , Georgia ; and on the South with Persia. It is now divided into two Provinces . 1. Atropatia . 2. Media major .

Persia.

Persia is bounded on the East with India ; on the West with Media , Assyria , and Chaldea ; on the North with Tartary , on the South with the main Ocean .

It is divided into the particular Provinces of 1. Susiana . 2. Persis . 3. Ormur . 4. Carmania . 5. Gedrosia . 6. Drangiana . 7. Arachosia . 8. Paropamisus . 9. Aria . 10. Parthia . 11. Hyrcania . 12. Margiana and 13. Bactria .

Tartaria .

Tartaria is bounded on the East with China , the Oriental Ocean , and the Straits of Anian , by which it is parted from America , on the West with Russia and Podolia , a Province of the Realm of Poland ; on the North with the main Scythick or frozen Ocean ; and on the South with part of China , from which it is separated by a mighty Wall , some part of India , the River Oxus parting it from Bactria and Margiana , two Persian Provinces ; the Caspian Sea which separates it from Media and Hyrcania ; the Caucasian Mountains interposing between it and Turcomania ; and the Euxine Sea which divideth it from Anatolia and Thrace .

It reacheth from the 50 degree of Longitude to the 195 which is 145 degrees from West to East ; and from the 40 degree of Northern Latitude , unto the 80 , which is within 10 degrees of the Pole it self , By which accompt it lieth from the beginning of the sixth Clime , where the longest day in Summer is 15 hours , till they cease measuring the Climates , the longest day in the most Northen parts hereof being full six Months , and in the winter half of the Year , the night as long .

It is now divided into these five parts . 1. Tartaria Precopensis . 2. Asiatica . 3. Antiqua . 4. Zagathay . 5. Cathay .

China .

China is bounded on the North with Altay and the Eastern Tartars , from which it is separated by a continued Chain of Hills , part of those of Ararat , and where that chain is broken off or interrupted , with a great wall extended 400 Leagues in length ; on the South partly with Cauchin China a Province of India , partly with the Ocean ; on the East with the oriental Ocean , and on the West with part of India and Cathay .

It reacheth from the 130 to the 160 degree of Longitude , and from the Tropick of Cancer to the 53 degree of Latitude ; so that it lieth under all the Climes from the third to the ninth inclusively . The longest summers day in the southern parts being 13 hours and 40 Minutes increased in the most northern parts to 16 hours and 3 quarters .

It containeth no fewer than 15 Provinces . 1. Canton . 2. Foquien . 3. Olam . 4. Sisnam 5. Tolenchia . 6. Causay . 7. Minchian . 8. Ochian . 9. Honan . 10. Pagnia . 11. Taitan . 12. Quinchen . 13. Chagnian 14. Susnan . 15. Cunisay . Besides the provinces of Suehuen , the Island of Chorea and the Island of Cheaxan .

India .

India is bounded on the East with the Oriental Ocean and some part of China ; on the West with the Persian Empire ; on the North with some Branches of Mount Taurus , which divide it from Tartary ; on the South with the Indian Ocean .

Extended from 106 to 159 degrees of Longitude , and from the AEquator to the 44th degree of Northern Latitude , by which account it lieth from the beginning of the first to the end of the sixth Clime , the longest Summers day in the southern Parts being 12 hours onely , and in the parts most North 15 hours and a half .

The whole Country is divided into two main parts , India intra Gangem , and India extra Gangem .

The Oriental Islands .

The Oriental Islands are 1. Iapan . 2. The Philippine and Isles adjoyning . 3. The Islands of Bantam . 4. The Moluccoes . 5. Those called Sinda or the Celebes . 6. Iava . 7. Borneo . 8. Sumatra . 9. Ceilan . and 10. others of less note .

CHAP. V.

Of Africk .

AFrick is bounded on the East by the Red Sea , and Bay of Arabia , by which it is parted from Asia ; on the West by the main Atlantick Oceans interposing between it and America ; on the North by the Mediterranean Sea , which divides it from Europe and Anatolia ; and on the South with the AEthiopick Ocean , separating it from Terra Australis incognita or the southern continent , parted from all the rest of the World except Asia only , to which it is joyned by a narrow Isthmus not above 60 miles in length .

It is situate for the most part under the Torrid Zones , the AEquator crossing it almost in the midst . It is now commonly divided into these seven parts . 1. AEgypt . 2. Barbary or the Roman Africk . 3. Numidia . 4. Lybia . 5. Terra Nigritarum . 6. AEthiopia superior . and 7. AEthiopia rinferior .

AEgypt .

AEgypt is bounded on the East with Idumea , and the Bay of Arabia ; on the West with Barbary , Numidia , and part of Lybia ; on the North with the Mediterranean Sea ; on the South with AEthiopia superior , or the Abyssyn Emperor ; it is situate under the second and fifth Climates , so that the longest day in Summer is but thirteen hours and a half .

Barbary .

Barbary is bounded on the East with Cyrenaica ; on the West with the Atlantick Ocean ; on the North with the Mediterranean Sea , the Straits of Gibralter and some part of the Atlantick also ; on the South with Mount Atlas , by which it is separated from Lybia inferior or the Desarts of Lybia .

It is situated under the third and fourth Climates : so that the longest Summers day in the parts most South , amounteth to 13 hours and 3 quarters , and in the most northern parts it is 14 hours and a quarter . This country is now reduced to the Kingdoms of 1. Tunis . 2. Tremesch or Algiers . 3. Fesse and 4. Morocco .

Numidia .

Numidia is bounded on the East with Egypt , on the West with the Atlantick Ocean ; on the North with Mount Atlas , which parteth it from Barbary and Cyrene ; on the South with Lybia Deserta .

Lybia .

Lybia is either Interior or Deserta , Libia interior is bounded on the North with Mount Atlas by which it is parted from Barbary and Cyrenaica ; on the East with Lybia Marmarica interposed between it and Egypt , and part of AEthiopia superior , or the Habassine Empire ; on the South with AEthiopia inferior , and the Land of the Negroes ; and on the West with the main Atlantick Ocean .

Lybia deserta is bounded on the North with Numidia or Biledulgerid ; on the South with the Land of the Negroes ; and on the West with Gulata another Province of the Negroes interposed between it and the Atlantick .

Terra Nigritarum .

Terra Nigritarum or the Land of the Negroes is bounded on the East with AEthiopia Superior ; on the West with the Atlantick Ocean ; on the North with Lybia deserta and on the South with the Ethiopick Ocean , and part of AEthiopia Inferior .

AEthiopia Superior .

AEthiopia Superior is bounded on the East with the Red Sea and the Sinus Barbaricus ; on the West with Lybia Interior , the Realm of Nubia in the Land of the Negroes and part of the Kingdoms of Congo in the other AEthiopia ; on the North with Egypt and Lybia Marmarica , and on the South with the Mountains of the Moon , by which it is parted from the main Body of AEthiopia Inferior .

It is situate on both sides of the AEquinoctial , extending from the South Parallel of seven degrees , where it meeteth with some part of the other AEthiopia to the Northern end of the Isle of Meroz situated under the fifth Parallel on the North of that Circle .

AEthiopia Inferior .

AEthiopia inferior is bounded on the East with the Red Sea ; on the West with the Ethiopick Ocean ; on the North with Terra Nigritarum , and the higher AEthiopia ; and on the South where it endeth , is a point of a Conus , with the main Ocean parting it from the Southern undiscovered Continent . This in Ptolemyes time went under the name of Terra incognita .

CHAP. IV.

Of America .

AMerica the fourth and last part of the World is bounded on the East with the Atlantick Ocean and the Vergivian Seas , by which it is parted from Europe and Africa ; on the West with the Pacifick Ocean , which divides it from Asia ; on the South with some part of Terra Australis incognita , from which it is separated by a long , but narrow Strait , called the Straits of Magellan ; the North bounds of it hither to not so well discovered , as that we can certainly affirm it to be Island or Continent .

It is called by some and that most aptly , The new World ; New for the late discovery , and World for the vast greatness of it . The whole is naturally divided into two great Peninsules , whereof that towards the North is called Mexicana . That towards the South hath the name of Peruana : the Isthmus which joyneth these two together is very long , but narrow in some places not above 120 miles from Sea to Sea , in many not above seventeen .

The Northern Peninsula called Mexicana , may be most properly divided into the Continent and Islands : The Continent again into the several Provinces of 1. Estotiland , 2. Nova Francia , 3. Virginia , 4. Florida , 5. California , 6. Nova Gallicia , 7. Nova Hispania , 8. Guntimala . The Southern Peninsula called Peruana , taking in some part of the Isthmus , hath on the Continent the Provinces of 1. Castella Aurea , 2. Nova Granada , 3. Peru , 4. Chile , 5. Paraguay , 6. Brasil , 7. Guiana , and 8. Paria . The Islands which belong to both , are dispersed either in the Southern Ocean called Mare del Zur , where there is not any one of Note but those called Los Ladrones and the Islands of Solomon . Or in the Northern Ocean called Mare del Noords , reduced unto the Caribes , Porto-Rico , Hispaniola , Cuba and Iamaica . And thus much concerning the real and known parts of the Terrestrial Globe .

CHAP. XV.

Of the Description of the Terrestrial Globe by Maps Vniversal and Particular .

HItherto we have spoken of the true and real Terrestrial Globe , and of the measure thereof by Circles , Zones , and Climates , as it is usually represented by a Sphere or Globe ; which must be confessed to be the nearest and the most ▪ commensurable to nature : Yet it may also be described upon a plain , in whole or in part many several ways : But those which are most useful and artificial are these two , by Parallelogram and by Planisphere .

2. The description thereof by Parallelogram is thus , the Parallelogram is divided in the midst by a line drawn from North to South , passing by the Azores or Canaries for the great Meridian . Cross to this and at eight Angles , another line is drawn from East to West for the AEquator ; then two parallels to each to comprehend the figure , in the squares whereof there are set down four parts of the world rather than the whole : And this way of description though not exact or near to the natural , hath yet been followed by such as ought still to be accounted excellent , and is the form of our plain Charts , and in places near the AEquinoctial may be used without committing any great error ; because the Meridians about the AEquinoctial are equi-distant , but as they draw up towards the Pole , they do upon the Globe come nearer and nearer together , to shew that their distance is proportionably diminished till it come to a concurrence , and answerably the Parallels as they are deeper in latitude , so they grow less and less with the Sphere ; so that at 60 degrees , the Equinoctial is double to the parallel of Latitude , and so proportionably of the rest .

3. Hence it followeth , that if the picture of the earth be drawn upon a Parallelogram , so that the Meridians be equally distant throughout , and the Parallels equally extended , the Parellel of 60 degrees shall be as great as the line of the AEquator it self is , and he that coasteth about the world in the latitude of 60 degrees , shall have as far to go by this Map , as he that doth it in the AEquator , though the way be but half as long . For the longitude of the Earth in the AEquator it self , is 21600 ; but in the Parallel of 60 but 10800 miles . So two Cities under the same parallel of 60 , shall be of equal Longitude to other two under the Line , and yet the first two shall be but 50 , the other two an hundred miles distant . So two Ships departing from the AEquator at 60 miles distance , and coming up to the Parallel of 60 , shall be thirty miles nearer , and yet each of them keep the same Meridians and sail by this Card upon the very points of the Compass at which they set forth . This was complained of by Martin Cortez and others , and the learned Mercator considering well of it , caused the degrees of the Parallel to increase by a proportion towards the Pole . The Mathematical Generation whereof , Mr. Wright in the second Chapter of his Correction of Errors in Navigation , hath sought by the inscription of a Planisphere into a Concave Cylinder . And this description of the Earth upon a Parallelogram , may indeed be so ordered by Art , as to give a true account of the scituation and distance of the parts , but cannot be fitted to represent the figure of the whole .

4. The description therefore of the whole by Planisphere is much better , because it represents the face of the Earth upon a plain , in its own proper Spherical Figure as upon the Globe it self . This description cannot well be contrived upon so few as one Circle or more than two .

Suppose then the Globe to be divided into two equal parts or Hemispheres , which cannot be done but by a great Circle : And therefore it must be done by the AEquator or Meridian ( for the Colure is all one with the Meridian ) the Horizon cannot fix , and the Zodiack hath nothing to do here .

5. Suppose then the Globe to be flatted upon the plain of the AEquator , and you have the first way of projection dividing the Globe into the North and South Hemispheres .

In this projection the Pole is the Centre , the AEquator is the Circumference divided into 360 degrees of Longitude , the Paralles are whole Circles , the Meridians are streight lines , the Parallels are Parallels indeed , and the Meridians equi-distantly concur , and therefore all the degrees are equal . After this way of projection , Ptolemy describes that part of the habitable world which was discovered to his time .

6. Suppose the Globe to be flatted upon the plain of the Meridian , and you have the other way of projection ; the AEquator here is a streight line , the great Meridian a whole Circle , in this Section the Meridians do not equi-distantly concur , the Parallels are not Parallels indeed , and therefore the degrees are all unequal .

However , this latter way is that which is now most and indeed altogether in use .

7. Particular Maps are but limbs of the Globe , and therefore though they are drawn asunder , yet are they still to be done with that proportion , as a remembring eye may suddenly acknowledge , and joyn them to the whole Body .

The Projection is most commonly upon a Parallelogram , in which the Latitude is to be expressed by Paralles from North to South , and the Longtitude by Meridians from West to East at 10 or 15 degrees distance , as you please , and may be drawn either by circle or right Lines ; but if they be right Lines , the Meridians are not to be drawn parallel , but inclining and concurring , to shew the nature of the whole , whereof they are such parts . For the Graduation ; the degrees of Longitude are most commonly divided upon the North and South sides of the Parallelogram ; the degrees of Latitude upon the East and West sides , or otherwise upon the most Eastern or Western Meridian of the Map , within the square . But it hath seemed good to some in these particular descriptions to make no graduation or projection at all ; but to put the matter off to a scale of Miles , and leave the rest to be believed .

The difference of Miles in several Countries is great , but it will be enough to know that the Italian and English , are reckoned for all one , and four of these do make a German Mile ; two a French League . The Swedish or Danish Mile consisteth of 5 Miles English and somewhat more . Sixty common English and Italian Miles answer to a degree of a great Circle .

Now as the Miles of several Countries do very much differ , so those of the same do not very much agree : and therefore the scales are commonly written upon with Magna , Mediocria and Parva , to shew the difference . In some Maps you shall find the Miles thus hiddenly set down , and the meaning is , that you should measure the Milliaria magna upon the lowermost Line , the Parva upon the uppermost , and the Mediocria upon the middlemost .

Scala Milliarium .

The use of the Scale is for the measuring the distances of places in the Map , by setting one foot of your Compasses in the little circle representing one place , and the other foot in the like little circle representing another , the Compasses kept at that distance being applied to the Scale , will shew the number of great or middle Miles according as the inhabitants of those places are known to reckon .

Soli Deo Gloria .

AViewofthemoreNotableEpochae Epochae . Years of the Julian Period . Months The Julian Period 1 Ian. 1 Creation of the World 765 Ian. 1 AEra of the Olympiades 3938 Iuly 8 The building of Rome 4961 Ap. 21 Epochae of Nabonasser 3667 Feb. 26 The beginning of Metons Cyrcle . 4281 Iune 26 The beginning of the periods of Calippus 4384 Iune 28 The Death of Alexander the great 4390 No. 12 AEra of the Caldees 4403 Oct. 15 The AEra of Dionysrus 4429 Mar. 25
The beginning of the Christian AEra falls in the 4713 year of the Julian Period . Years of Christ Month The Dioclesian AEra 284 Aug. 29 The Turkish AEra or Hegyra 622 Iuly 16 The Persian AEra from Iesdagird 632 Iune 16 The AEra from the Persian Sultan 1079 Mar. 14
DaysintheYearof Julian Accompt AEgypt and Persian Accompt 1 0 0 0 365 2 5 0 1 0 0 0 365 0 0 0 2 0 0 0 730 5 0 0 2 0 0 0 730 0 0 0 3 0 0 0 1095 7 5 0 3 0 0 0 1095 0 0 0 4 0 0 0 1461 0 0 0 4 0 0 0 1460 0 0 0 5 0 0 0 1826 2 5 0 5 0 0 0 1825 0 0 0 6 0 0 0 2191 5 0 0 6 0 0 0 2190 0 0 0 7 0 0 0 2556 7 5 0 7 0 0 0 2555 0 0 0 8 0 0 0 2922 0 0 0 8 0 0 0 2920 0 0 0 9 0 0 0 3287 2 5 0 9 0 0 0 3285 0 0 0 10 0 0 0 3652 5 0 0 10 0 0 0 3650 0 0 0
Days in Julian Months Days in AEgyptian Months Days in Persian Months Comon Bissex Thoth 30 Pharvadin 30 Ianuary 31 30 Paophi 60 Aripehast 60 February 59 60 Athyr 90 Chortat 90 March 90 91 Chaeae 120 Tirma 120 April 120 121 Tybi 150 Mertat 150 May 151 152 Michir 180 Sachriur 180 Iune 181 182 Phamenoth 210 Macherma 210 Iuly 212 213 Pharmuthi ; 240 Apenina Wahak 245 August 243 244 Pachon 270 September 273 274 Payny 300 Aderma 275 October 304 305 Ephephi 330 Dima 305 November 334 335 Mesori 330 Pechmam 335 December 365 366 Epagomena 365 Aphander 365
Days in Turkish or Arabical Years Days in Turkish Months 1 354 Muharran 30 2 709 Sapher 59 3 .1063 Rabie 1. 89 4 .1417 Rabie 2. 118 5 .1772 Giumadi 1. 148 6 .2126 Giumadi 2. 177 7 .2480 Regeb 207 8 .2835 Sahahen 236 9 .3189 Ramaddan 266 10 .3543 Scheval 295 11 .3898 Dulkadati 325 12 .4252 Dulhajati Dsilhittsche true 354 13 .4607 14 .4961 15 .5315 In anno Abundanti 355 16 .5670 17 .6024 18 .6378 19 .6733 20 .7087 21 7442 22 7796 23 8150 24 8505 25 8859 26 9213 27 9568 28 9922 29 10276 30 0 10631 0 60 0 21262 0 90 0 31893 0 120 0 42524 0 150 0 53155 0 180 0 63786 0 210 0 74417 0 240 0 05048 0 270 0 95679 0 300 0 106310 0
Ianuary February March 1 3 A Circumcis . D Purificat 3 D 2 B 11 E E 3 11 C 19 F 11 F 4 D 8 G G 5 19 E A 19 A 6 8 F Epiphany 16 B 8 B 7 G 5 C C 8 16 A D 16 D 9 5 B 13 E 5 E 10 C 2 F F 11 13 D G 13 G 12 2 E 10 A 2 A 13 F B B 14 10 G 18 C 10 C 15 A 7 D D 16 18 B E 18 E 17 7 C 15 F 7 F 18 D 4 G G 19 15 E A 15 A 20 4 F 12 B 4 B 21 G 1 C C 22 12 A D 12 D 23 1 B 9 E 1 E 24 C F F 25 9 D Conv. S. Paul 17 G S. Matthias 9 G Anunc . 26 E 6 A A 27 17 F B 17 B 28 6 G 14 C 6 C 29 A D 30 14 B 14 E 31 3 C 3 F
April May Iune 1 G 11 B Phil. & Jac. E 2 11 A C 19 F 3 B 19 D 8 G 4 19 C 8 E 16 A 5 8 D F 5 B 6 16 E 16 G C 7 5 F 5 A 13 D 8 G B 2 E 9 13 A 14 C F 10 2 B 2 D 10 G 11 C E A S. Barnaby 12 10 D 10 F 18 B 13 E G 7 C 14 18 F 18 A D 15 7 G 7 B 15 E 16 A C 4 F 17 15 B 15 D G 18 4 C 4 E 12 A 19 D F 1 B 20 12 E 12 G C 21 1 F 1 A 9 D 22 G B E 23 9 A 9 C 17 F 24 B D 6 G S. John B. 25 17 C Mark Evang. 17 E A 26 6 D 6 F 14 B 27 E G 3 C 28 14 F 14 A D 29 5 G 3 B 11 E Pet. Ap. 30 A C F 31 11 D
Iuly August September 1 19 G 8 C 16 F 2 8 A 16 D 5 G 3 B 5 E A 4 16 C F 13 B 5 5 D 13 G 2 C 6 E 2 A D 7 13 F B 10 E 8 2 G 10 C F 9 A D 18 G 10 10 B 18 E 7 A 11 C 7 F B 12 18 D G 15 C 13 7 E 15 A 4 D 14 F 4 B E 15 15 G C 12 F 16 4 A 12 D 1 G 17 B 1 E A 18 12 C F 9 B 19 1 D 9 G C 20 E Margaret A 17 D 21 9 F 17 B 6 E S. Matth 22 G 6 C F 23 17 A D 14 G 24 6 B 14 E Barthol . 3 A 25 C 3 F B 26 14 D 11 G 11 C 27 3 E 19 A 19 D 28 F B 8 E 29 11 G 8 C F S. Mich. 30 9 A D G 31 B E
October November December 1 16 A D All Saints 13 F 2 5 B 13 E All Souls 2 G 3 13 C 2 F A 4 2 D G 10 B 5 E 10 A P. Conspir . C 6 10 F B 18 D 7 G 18 C 7 E 8 18 A 7 D F 9 7 B E 15 G 10 C 15 F 4 A 11 15 D 4 G B 12 4 E A 12 C 13 F 12 B 1 D 14 12 G 1 C E 15 13 A D 9 F 16 B 9 E G 17 9 C F 17 A 18 D Luke Evang. 17 G 6 B 19 17 E 6 A C 20 6 F B 14 D 21 G 14 C 3 E S. Thomas 22 14 A 3 D F 23 3 B E 11 G 24 C 11 F 19 A 25 11 D 19 G B Chri. Nat. 26 19 E A 8 C S. Steph. 27 F 8 B D S. John 28 8 G Sim. & Jude C 16 E Innocents 29 A 16 D 5 F 30 16 B 5 E S. Andrew G 31 5 C 13 A Sylvester
Ianuary February March 1 * A XXIX D * D 2 XXIX B XXVIII E XXIX E 3 XXVIII C XXVII F XXVIII F 4 XXVII D 25. XXVI G XXVII G 5 XXVI E XXV . XXIV A XXVI A 6 25. XXV F XXIII B 25. XXV B 7 XXIV G XXII C XXIV C 8 XXIII A XXI D XXIII D 9 XXII B XX E XXII E 10 XXI C XIX F XXI F 11 XX D XVIII G XX G 12 XIX E XVII A XIX A 13 XVIII F XVI B XVIII B 14 XVII G XV C XVII C 15 XVI A XIV D XVI D 16 XV B XIII E XV E 17 XIV C XII F XIV F 18 XIII D XI G XIII G 19 XII E X A XII A 20 XI F IX B XI B 21 X G VIII C X C 22 IX A VII D IX D 23 VIII B VI E VIII E 24 VII C V F VII F 25 VI D IV G VI G 26 V E III A V A 27 IV F II B IV B 28 III G I C III C 29 II A II D 30 I B I E 31 * C * F
April May Iune 1 XXIX G XXVIII B XXVII E 2 XXVIII A XXVII C 25. XXVI F 3 XXVII B XXVI D XXV . XXIV G 4 25. XXVI C 25. XXV E XXIII A 5 XXV . XXIV D XXIV F XXII B 6 XXIII E XXIII G XXI C 7 XXII F XXII A XX D 8 XXI G XXI B XIX E 9 XX A XX C XVIII F 10 XIX B XIX D XVII G 11 XVIII C XVIII E XVI A 12 XVII D XVII F XV B 13 XVI E XVI G XIV C 14 XV F XV A XIII D 15 XIV G XIV B XII E 16 XIII A XIII C XI F 17 XII B XII D X G 18 XI C XI E IX A 19 X D X F VIII B 20 IX E IX G VII C 21 VIII F VIII A VI D 22 VII G VII B V E 23 VI A VI C IV F 24 V B V D III G 25 IV C IV E II A 26 III D III F I B 27 II E II G * C 28 I F I A XXIX D 29 * G * B XXVIII E 30 XXIX A XXIX C XXVII F 31 XXVIII D
Iuly August September 1 XXVI G XXV . XXIV C XXIII F 2 25. XXV A XXIII D XXII G 3 XXIV B XXII E XXI A 4 XXIII C XXI F XX B 5 XXII D XX G XIX C 6 XXI E XIX A XVIII D 7 XX F XVIII B XVII E 8 XIX G XVII C XVI F 9 XVIII A XVI D XV G 10 XVII B XV E XIV A 11 XVI C XIV F XIII B 12 XV D XIII G XII C 13 XIV E XII A XI D 14 XIII F XI B X E 15 XII G X C IX F 16 XI A IX D VIII G 17 X B VIII E VII A 18 IX C VII F VI B 19 VIII D VI G V C 20 VII E V A IV D 21 VI F IV B III E 22 V G III C II F 23 IV A II D I G 24 III B I E * A 25 II C * F XXIX B 26 I D XXIX G XXVIII C 27 * E XXVIII A XXVII D 28 XXIX F XXVII B 25. XXVI E 29 XXVIII G XXVI C XXV . XXIV F 30 XXVII A 25. XXV D XXIII G 31 25. XXVI B XXIV E
October November December 1 XXII A XXI D XX F 2 XXI B XX E XIX G 3 XX C XIX F XVIII A 4 XIX D XVIII G XVII B 5 XVIII E XVII A XVI C 6 XVII F XVI B XV D 7 XVI G XV C XIV E 8 XV A XIV D XIII F 9 XIV B XIII E XII G 10 XIII C XII F XI A 11 XII D XI G X B 12 XI E X A IX C 13 X F IX B VIII D 14 IX G VIII C VII E 15 VIII A VII D VI F 16 VII B VI E V G 17 VI C V F IV A 18 V D IV G III B 19 IV E III A II C 20 III F II B I D 21 II G I C * E 22 I A * D XXIX F 23 * B XXIX E XXVIII G 24 XXIX C XXVIII F XXVII A 25 XXVIII D XXVII G XXVI B 26 XXVII E 25. XXVI A 25. XXV C 27 XXVI F XXV . XXIV B XXIV D 28 25. XXV G XXIII C XXIII E 29 XXIV A XXII D XXII F 30 XXIII B XXI E XXI G 31 XXII C XX A
A Table shewing the Dominical Letter , Golden Number and Epact , according to the Julian account for ever , and in the Gregorian , till the Year 1700. 1672 1 GF CB 1673 2 E A 1674 3 D G 1675 4 C F 1676 5 BA ED 1677 6 G C 1678 7 F B 1679 8 E A 1680 9 DC GF 1681 10 B E 1682 11 A D 1683 12 G C 1684 13 FE BA 1685 14 D G 1686 15 C F 1687 16 B E 1688 17 AG DC 1689 18 F B 1690 19 E A 1691 20 D G 1692 21 CB FE 1693 22 A D 1694 23 G C 1695 24 F B 1696 25 ED AG 1697 26 C F 1698 27 B E 1699 28 A D
Year G Julian Gregor. N Epact Epact 1672 1 11 1 1673 2 22 12 1674 3 3 23 1675 4 14 4 1676 5 25 15 1677 6 6 26 1678 7 17 7 1679 8 28 18 1680 9 9 29 1681 10 20 10 1682 11 1 21 1683 12 12 2 1684 13 23 13 1685 14 4 24 1686 15 15 5 1687 16 26 16 1688 17 7 17 1689 18 18 8 1690 19 29 19

The anticipation of the Gregorian Calender . From 5 October 1582 D. 10 From 24 Feb. 1700 D. 11 From 24 Feb. 1800 D. 12 From 24 Feb. 1900 D. 13 From 24 Feb. 2100 D. 14 From 24 Feb. 2200 D. 15 From 24 Feb. 2320 D. 16

III IV V VI VII VIII 1 P * XI XXII III XIV XXV 2 N XXIX X XXI II XIII XXIV 3 M XXVIII IX XX I XII XXIII 4 H XXVII VIII XIX * XI XXII 5 G XXVI VII XVIII XXIX X XXI 6 F XXV VI XVII XXVIII IX XX 7 E XXIV V XVI XXVII VIII XIX 8 D XXIII IV XV XXVI VII XVIII 9 C XXII III XIV XXV VI XVII 10 B XXI II XIII XXIV V XVI 11 A XX I XII XXIII IV XV 12 u XIX * XI XXII III XIV 13 t XVIII XXIX X XXI II XIII 14 s XVII XXVIII IX XX I XII 15 r XVI XXVII VIII XIX * XI 16 q XV XXVI VII XVIII XXIX X 17 p XIV XXV VI XVII XXVIII IX 18 n XIII XXIV V XVI XXVII VIII 19 m XII XXIII IV XV XXVI VII 20 l XI XXII III XIV XXV VI 21 k X XXI II XIII XXIV V 22 i IX XX I XII XXIII IV 23 h VIII XIX * XI XXII III 24 g VII XVIII XXIX X XXI II 25 f VI XVII XXVIII IX XX I 26 e V XVI XXVII VIII XIX * 27 d IV XV XXVI VII XVIII XXIX 28 c III XIV XXV VI XVII XXVIII 29 b II XIII XXIV V XVI XXVII 30 a I XII XXIII IV XV XXVI
IX X XI XII XIII XIV XV VI XVII XXVIII IX XX I XII V XVI XXVII VIII XIX * XI IV XV XXVI VII XVIII XXIX III XIV XXV VI XVII XXVIII IX II XIII XXIV V XVI XXVII VIII I XII XXIII IV XV XXVI VII * XI XXII III XIV 25 VI XXIX X XXI II XIII XXIV V XXVIII IX XX I XII XXIII IV XXVII VIII XIX * XI XXII III XXVI VII XVIII XXIX X XXI II XXV VI XVII XXVIII IX XX I XXIV V XVI XXVII VIII XIX * XXIII IV XV XXVI VII XVIII XXIX XXII III XIV XXV VI XVII XXVIII XXI II XIII XXIV V XVI XXVI XX I XII XXIII IV XV XXVII XIX * XI XXII III XIV 25 XVIII XXIX X XXI II XIII XXIV XVII XXVIII IX XX I XII XXIII XVI XXVII VIII XIX * XI XXII XV XXVI VII XVIII XXIX X XXI XIV XV VI XVII XXVIII IX XX XIII XXIV V XVI XXVII VIII XIX XII XXIII IV XV XXVI VII XVIII XI XXII III XIV 25 VI XVII X XXI II XIII XXIV V XVI IX XX I XII XXIII IV XV VIII XIX * XI XXII III XIV VII XVIII XIX X XXI II XIII
XVI XVII XVIII XIX I II P XXIII IV XV XXVI VIII XIX N XXII III XIV 25 VII XVIII M XXI II XIII XXIV VI XVII H XX I XII XXIII V XVI G XIX * XI XXII IV XV F XVIII XXIX X XXI III XIV E XVII XXVIII IX XX II XIII D XVI XXVII VIII XIX I XII C XV XXVI VII XVIII * XI B XIV 25 VI XVII XXIX X A XIII XXIV V XVI XXVIII IX u XII XXIII IV XV XXVII VIII t XI XXII III XIV XXVI VII t X XXI II XIII 25 VI r IX XX I XII XXIV V q VIII XIX * XI XXIII IV p VII XVIII XXIX X XXII III n VI XVII XXVIII IX XXI II m V XVI XXVII VIII XX I l IV XV XXVI VII XIX * k III XIV 25 VI XVIII XXIX i II XXIII XXIV V XVII XXVIII h I XII XXIII IV XVI XXVII g * XI XXII III XV XXVI f XXIX X XXI II XIV 25 e XXVIII IX XX I XIII XXIV d XXVII VIII XIX * XII XXIII c XXVI VII XVIII XXIX XI XXII b 25 VI XVII XXVIII X XXI a XXIV V XVI XXVII IX XX
Anni Christi . N I P 320 P 580 Biss. a 800 Biss. C b 1100 Biss. C c 1400 Biss. C
Detractisdecemdiebus. D 1484 D 1600 Biss. C 1700 C 1800 CC B 1900 B 2000 Biss. B 2100 C A 2200 u 2300 A 2409 Biss. C u 2500 t 2600 t 2700 C t 2800 Biss. s 2900 s 3000 C r 3100 r 3200 Biss. r 3300 C q 3400 p 3500
AnniChristi. q 3600 Biss. C p 3700 n 3800 n 3900 n 4000 Biss. C m 4100 l 4200 l 4300 CC l 4400 Biss. k 4500 k 4600 C i 4700 i 4800 Biss. i 4900 C h 5000 g 5100 h 5200 Biss. C g 5300 f 5400 f 5500 C f 5600 Biss. e 5700 e 5800 C d 5900 d 6000 Biss. d 6100 C c 6200 b 6300 c 6400 Biss. C b 6500
A Table shewing the Dominical Letter both in the Julian and the Gregorian account for ever . Cy. ☉ 1 2 3 4 5 6 7 1 C B D C E D F E G F A G B A 2 A B C D E F G 3 G A B C D E F 4 F G A B C D E 5 E D F E G F A G B A C B D C 6 C D E F G A B 7 B C D E F G A 8 A B C D E F G 9 G F A G B A C B D C E D F E 10 E F G A B C D 11 D E F G A B C 12 C D E F G A B 13 B A C B D C E D F E G F A G 14 G A B C D E F 15 F G A B C D E 16 E F G A B C D 17 D C E D F E G F A G B A C B 18 B C D E F G A 19 A B C D E F G 20 G A B C D E F 21 F E G F A G B A C B D C E D 22 D E F G A B C 23 C D E F G A B 24 B C D E F G A 25 A G B A C B D C E D F E G F 26 F G A B C D E 27 E F G A B C D 28 D E F G A B C Anni 1582 1900 2300 1600 1700 1800 2000 2100 2200 2400 2700 3100 Chr. 2500 2600 2900 3000 2800 3200 3300
LXX Ash. East . Asci . Pent. Corp. Christi . Adv. 16 XXIII Ian. Feb. Mar. Apr. May. May. Nov. 5 XXII d 18 4 22 30 10 21 29 XXI e 19 5 23 Ma. 1 11 22 30 13 XX f 20 6 24 2 12 23 De. 1 2 XIX g 21 7 25 3 13 24 2 XVIII a 22 8 26 4 14 25 3 10 XVII b 23 9 27 5 15 26 No. 27 XVI c 24 10 28 6 16 27 28 18 XV d 25 11 29 7 17 28 29 7 XIV e 26 12 30 8 18 29 30 XIII f 27 13 31 9 19 30 Dec. 1 15 XII g 28 14 Ap. 1 10 20 31 2 4 XI a 29 15 2 11 21 Iun. 1 3 X b 30 16 3 12 22 2 No. 27 12 IX c 31 17 4 13 23 3 28 1 VIII d Feb. 1 18 5 14 24 4 29 VII e 2 19 6 15 25 5 30 VI f 3 20 7 16 26 6 Dec. 1 9 V g 4 21 8 17 27 7 2 17 IV a 5 22 9 18 28 8 3 6 III b 6 23 10 19 29 9 No. 27 II c 7 24 11 20 30 10 28 14 I d 8 25 12 21 31 11 29 3 * e 9 26 13 22 Iun. 1 12 30 XXIX f 10 27 14 23 2 13 Dec. 1 11 XXVIII g 11 28 15 24 3 14 2 XXVII a 12 Ma. 1 16 25 4 15 3 19 25. XXVI b 13 2 17 26 5 16 No. 27 8 XXV . XXIV c 14 3 18 27 6 17 28 d 15 4 19 28 7 18 29 e 16 5 20 29 8 19 30 f 17 6 21 30 9 20 Dec. 1 g 18 7 22 31 10 21 2 a 19 8 23 Iun. 1 11 22 3 b 20 9 24 2 12 23 No. 27 c 21 10 25 3 13 24 28
A Table to convert Sexagenary Degrees and Minutes into Decimals and the contrary . 1 00 37 10 73 20 109 30 145 40 181 50 2 38 74 110 146 182 3 39 75 111 147 183 4 01 40 11 76 21 112 31 148 41 184 51 5 41 77 113 149 185 6 42 78 114 150 186 7 43 79 115 151 187 8 02 44 12 80 22 116 32 152 42 188 52 9 45 81 117 153 189 10 46 82 118 154 190 11 03 47 13 83 23 119 33 155 43 191 53 12 48 84 120 156 192 13 49 85 121 157 193 14 50 86 122 158 194 15 04 51 14 87 24 123 35 159 44 195 54 16 52 88 124 160 196 17 53 89 125 161 197 18 05 54 15 90 25 126 35 162 45 198 55 19 55 91 127 163 199 20 56 92 128 164 200 21 57 93 129 165 201 22 06 58 16 94 26 130 36 166 46 202 56 23 59 95 131 167 203 24 60 96 132 168 204 25 61 97 133 169 205 26 07 62 17 98 27 134 37 170 47 206 57 27 63 99 135 171 207 28 64 100 136 172 208 29 08 65 18 101 28 137 38 173 48 209 58 30 66 102 138 174 210 31 67 103 139 175 211 32 68 104 140 176 212 33 09 69 19 105 29 141 39 177 49 213 59 34 70 106 142 178 214 35 71 107 143 179 215 36 10 72 20 108 30 144 40 180 50 216 60 217 60 253 70 289 80 325 90 277777778 218 254 290 326 555555555 219 255 291 327 833333333 220 61 256 71 292 81 328 91 111111111 221 257 293 329 388888889 222 258 294 330 666666667 223 259 295 331 944444444 224 62 260 72 296 82 332 92 222222222 225 261 297 333 500000000 226 262 298 334 777777778 227 63 263 73 299 83 335 93 055555555 228 264 300 336 333333333 229 265 301 337 511111111 230 266 302 338 888888889 231 64 267 74 303 84 339 94 166666667 232 268 304 340 444444444 233 269 305 341 722222222 234 65 270 75 306 85 342 95 000000000 235 271 307 343 277777778 236 272 308 344 555555555 237 273 309 345 833333333 238 66 274 76 310 86 346 96 111111111 239 275 311 347 388888889 240 276 312 348 666666667 241 277 313 349 944444444 242 67 278 77 314 87 350 97 222222222 243 279 315 351 500000000 244 280 316 352 777777778 245 68 281 78 317 88 353 98 055555555 246 282 318 354 333333333 247 283 319 355 611111111 248 284 320 356 888888889 249 69 285 79 321 89 357 99 166666667 250 286 322 358 444444444 251 287 323 359 722222222 252 70 288 80 324 90 360 100 000000000
A Table to Convert Sexagenary Minutes into Decimals and the contrary . Minutes Seconds Thirds 1 00462962 00007716 00000128 2 00925925 15432 257 3 01388889 23148 385 4 01851851 30864 515 5 02314814 00038580 00000643 6 02777778 46296 771 7 03240740 54012 900 8 03703703 61728 1028 9 04166667 69444 1157 10 04629629 00077160 00001286 11 05092592 084876 1414 12 05555555 092592 1543 13 06018518 100308 1671 14 06481480 108024 1800 15 06944444 00115740 1929 16 07409407 123450 2057 17 07870370 131172 2186 18 08333333 138889 2314 19 08796296 146604 2443 20 09259259 00154320 2572 21 00722222 162036 2700 22 10185185 169752 2829 23 10648148 177468 2957 24 11111111 185184 3086 25 11574074 00192900 3215 26 12037037 200616 3343 27 12500000 208332 3472 28 12962962 216048 3600 29 13425926 223764 3729 30 13888889 00231481 00003858 31 14351852 00239670 00003986 32 14814814 246913 4115 33 15277777 254629 4243 34 15747040 262345 4372 35 16203703 270061 4581 36 16666666 00277777 00004629 37 17129629 285493 4758 38 17592592 293209 4886 39 18055555 300925 5015 40 18518518 308640 5144 41 18981481 00316356 00005272 42 19444444 324072 5401 43 19907407 331788 5529 44 20370370 339504 5658 45 20833333 347220 5787 46 21296296 00354936 00005915 47 21759259 362652 6044 48 22222222 370●70 6172 49 22685185 378084 6301 50 23148148 385802 6430 51 23611111 00393518 00006558 52 24074074 401234 6687 53 24537037 408950 6815 54 25000000 416666 6944 55 25462963 424382 7073 56 25925926 00432098 00007201 57 26388888 439814 7330 58 26851852 447530 7458 59 27314814 455256 7587 60 27777777 00462962 00007716
A Table Converting Hours and Minutes into Degrees and Minutes of the AEquator , and into Hours . 1 04.16666667 2 08.33333333 3 12.5 4 16.16666667 5 20.83333333 6 25.0 7 29.16666667 8 33.33333333 9 37.5 10 41.66666667 11 45.83333333 12 50. 13 54.16666667 14 58.33333333 15 62.5 16 66.66606667 17 70.83333333 18 75.00 19 79.16660667 20 83.33333333 21 87.5 22 91.66666667 23 95.83333333 24 100.00000000
Minutes 1 0.06944444 2 0.13888888 3 0.20833333 4 0.27777777 5 0.34722222 6 0.41666666 7 0.48611111 8 0.55555555 9 0.625 10 0.69444444 11 0.76388888 12 0.83333333 13 0.90277777 14 0.97222222 15 1.04166666 16 1.11111111 17 1.18055555 18 1.25 19 1.31944444 20 1.38888888 21 1.45833333 22 1.52777777 23 1.59722222 24 1.66666666 25 1.73611111 26 1.80555555 27 1.875 28 1.94444444 29 2.01388888 30 2.08333333
The Decimal parts of a Day and the contrary . Seconds Minutes Seconds .00115740 31 2.15277777 .03587963 .00231481 32 2.22222222 .03703704 .00347222 33 2.29166666 .03819444 .00462962 34 2.36111111 .03935185 .00578703 35 2.43055555 .04050926 .00694444 36 2.5 .04166666 .00810184 37 2.56944444 .04282407 .00925925 38 2.63888888 .04398148 .01041660 39 2.70833333 .04513888 .01157405 40 2.77777777 .04629629 .01273148 41 2.84722222 .04745370 .01388888 42 2.91666666 .04861111 .01504630 43 2.98611111 . 0497685● .01620371 44 3.05555555 .05092592 .01736111 45 3.125 .05208333 .01851853 46 3.19444444 .05324074 .01967593 47 3.26388888 .05439814 .02083333 48 3.33333333 .05555555 .02199074 49 3.40277777 .05671296 .02314810 50 3.47222222 .05787037 .02430555 51 3.54166666 .05902777 .02546295 52 3.61111111 .06018518 .02662037 53 3.68055555 . 06134●59 .02777777 54 3.75 .0625 .02893518 55 3.81944444 .06365741 .03009259 56 3.88888888 .06481481 .03125000 57 3.95833333 .06597222 .03240741 58 4.02777777 .06712963 .03356482 59 4.09722222 .06828704 .03472222 60 4.16666666 .06944444
A Catalogue of some of the most eminent Cittes and Towns in England and Ireland wherein is shewed the difference of their Meridian from London , with the hight of the Pole . Names of Citties Differ . Merid. Hight Pole St. Albons 0 1 s 55.55 Barwick 0 6 s 55.49 Bedford 0 2 s 52.18 Bristol 0 11 s 51.32 Boston 0 0 53.2 Cambridge 0 1 a 52.17 Canterbury 0 5 a 51.27 Carlile 0 10 s 54.57 Chester 0 11 s 53.20 Coventry 0 4 s 52.30 Carmarthen 0 17 s 52.2 Chichester 0 3 s 50.56 Colchester 0 5 a 52.4 Darby 0 5 s 53.6 Dublin in Ireland 0 26 s 53.11 Duresm● 0 5 s 54.45 Dartmouth 0 15 s 50.32 Eely 0 1 a 52.20 Grantha● 0 2 s 52.58 Glocester 0 9 s 52.00 Halefax 0 6 s 52.49 Hartford 0 1 s 52.50 Hereford 0 11 s 52.14 Huntington 0 1 s 52.19 Hull 0 1 s 53.58 Lancaster 0 11 s 54.08 Leicester 0 4 s 52.40 Lincoln 0 1 s 53.12 Middle of the Isle of Man 0 17 s 54.22 Nottingham 0 4 s 53.03 Newark 0 3 s 53.02 Newcastle 0 6 s 54.58 N. Luffingham 0 3 s 52.41 Norwich 0 4 a 52.44 Northampton 0 4 s 52.18 Oxford 0 5 s 51.54 Okenham 0 3 s 52.44 Peterborough 0 2 s 52.35 Richmond 0 6 s 54.26 Rochester 0 3 a 51.28 Ross 0 10 s 52.07 St. Michaels Mount in Cornwal 0 23 s 50.38 Stafford 0 8 s 52.55 Stamford 0 2 s 52.41 Shrewsbury 0 11 s 52.48 Tredah in Ireland 0 27 s 53.28 uppingham in Rutland 0 3 s 52.40 Warwick 0 6 s 52.25 Winchester 0 5 s 50.10 Waterford in Ireland 0 27 s 52.22 Worcester 0 9 s 52.20 Yarmouth in Suffolk 0 6 a 52.45 York 0 4 s 54.00 London 0 00 51.32
The Suns mean Longitude and mean Anomaly in AEgyptian Years . ☉ Mean Longitude ☉ Mean Anomaly 1 99.9336437563 99.9288933116 2 99.8672875126 99.8577866232 3 99.8009312690 99.7866799348 4 99.7345750253 99.7155732465 5 99.6682187816 99.6444665581 6 99.6018625380 99.5733598697 7 99.5355062943 99.5022531814 8 99.4691500506 99.4211464930 9 99.4027938070 99.3600398046 10 99.3364375633 99.2889331162 100 93.3643756334 92.8893311628 1000 33.6437563341 28.8933116289
TheSunsMeanAnomalyandPraecessionoftheAEquinox8in.1AEgyptianYears. Year . ☉ Mean Anomaly Praecession AEquinox . 1 99.9297857316 00.0038580246 2 99.8595714632 00.0077160493 3 99.7893571949 00.0115740740 4 99.7191429265 00.0154320987 5 99.6489286582 00.0192901234 6 99.5787143898 00.0231481481 7 66.5085001114 00.0270061728 8 99.2978573164 00.0308641975 9 99.3680715847 00.0347222221 10 99.2978573164 00.0385802469 100 92.9785731642 00.3858024691 1000 99.7857316427 03.8580246913
The Suns mean Longitude and mean Anomaly in Julian Years . ☉ Mean Longitude ☉ Mean Anomaly 1 99.9336437563 99.9288933116 2 99.8672875126 99.8577866232 3 99.8009312689 99.7866799348 B 4 00.008365830 99.9892901234 5 99.9420095864 99.9181834350 6 99.875633427 99.8470767466 7 99.8092970990 99.7759700583 B 8 00.0167316602 99.9785802468 9 99.9503754165 99.9074735584 10 99.8840191728 99.8363668700 11 99.8176629291 99.7652591816 B 12 00.0250974903 99.9678703702 13 99.9587412466 99.8967636818 14 99.8923850029 99.8256569934 15 99.8260287592 99.7545503050 B 16 00.0334633205 99.9571604936 17 99.9671070768 99.8860548052 18 99.9007508331 99.8149481168 19 99.8343945894 99.7438414284 B 20 00.0418291506 99.9164506171 40 00.0836583012 99.8929012342 60 00.1254874518 99.8393518513 80 00.1673166024 99.7858024684 100 00.2091457530 99. 73225308●5 200 00.4182015060 99.4645061710 300 00.6274372590 99.1967592565 400 00.8365830120 98.9290123420 500 01.0457287650 98.6612654275 600 01.2548745180 98.3935185130 700 01. 4640●02710 98.1257715985
The ☉ mean Longitude and Anomaly AEra ☉ mean Longitude ☉ mean Anomaly Chr. 77. 22400.86419 58. 24289.56790 1600 80. 54891.97529 53. 95880.62961 1620 80. 59074.89035 53. 90525.69132 1640 80. 63257.80541 53. 85170.75303 1660 80. 67440.72047 53. 79815.81474 1680 80. 71623.63553 53. 74460.87645 1700 80. 75806.55059 53. 69105.93816 1720 80. 79989.46665 53. 63750.99987 1740 80. 84172.38171 53. 58396.06158 1760 80. 88265.29677 53. 53041.12329 ☉ mean Lon. in Mon. ☉ mean Ano. in Mo. Ianu. 08. 48751.49488 08. 48711.14867 Febr. 16. 15365.74832 16. 15288.96037 Mar. 24. 64117.24320 24. 64000.10904 April 32. 85489.65760 32. 85333.47872 May 41. 34241.15248 41. 34044.62739 Iune 49. 55613.56688 49. 55377.99708 Iuly 58. 04365.06176 58. 04089.14575 Aug. 66. 53116.55664 66. 52800.29442 Sept. 74. 74488.97104 74. 74133.66410 Octo. 83. 23240.46592 85. 22844.81277 Nov. 91. 44612.88032 91. 44178.18245 Dec. 99. 93364.37563 99. 92889.33116

In Anno ●issentili ; post Februarium adde unum diem & unius dies motum .

TheSunsmeanLongitudeandmeanAnomalyinDays. ☉ mean Longitude ☉ mean Anomaly 1 0.2737908048 0.2737777898 2 0.5475816096 0.5475555796 3 0.8213724144 0.8213333694 4 1.0951632192 1.0951111592 5 1 . 3●89540240 1.3688889490 6 1.6427448288 1.6426667388 7 1.9165356336 1.9164445286 8 2.1903264384 2.1902223184 9 2.4641172432 2.4640001082 10 2.7379080480 2.7377778980 11 3.0116988528 3.0115556878 12 3.2854896576 3.2853334776 13 3.5592804624 3.5591112674 14 3.8330712672 3.8328890572 15 4.1068620720 4.1066668470 16 4.3806428768 4.3804446368 17 4.6544436816 4.6542224266 18 4.9282344864 4.9280002164 19 5.2020252912 5.2077780062 20 5.4758160960 5.4755557960 21 5.7496069008 5.7493335858 22 6.0233977056 6.0231113756 23 6.2971885104 6.2968891654 24 6.5709793152 6.5706669552 25 6.8447701200 6.8444447450 26 7.1185609248 7.1182225348 27 7.3923517296 7.3920003246 28 7.6661425344 7.6957781144 29 7.9399333392 7.9395559042 30 8.2137241440 8.2133336940 31 8.4875149488 8.4871114838

TheSunsmeanLongitudeandmeanAnomalyinDays ☉ Mean Longitude ☉ Mean Anomaly . 1 0.0114079502 0.0114074079 2 0.0228159004 0.0228148158 3 0.0342238506 0.0342222237 4 0.0456318008 0.0456296316 5 0.0570397510 0.0570370395 6 0.0684477012 0.0684444474 7 0.0798556514 0.0798518553 8 0.0912636016 0.0912592632 9 0.1026715518 0.1026666711 10 0.1140795020 0.1140740790 11 0.1254874522 0.1254814869 12 0.1368954024 0.1368888948 13 0.1483033526 0.1482963027 14 0.1597113028 0.1597037106 15 0.1711192530 0.1711111185 16 0.1825272032 0.1825185264 17 0.1939351534 0.1939259343 18 0.2053431036 0.2053333422 19 0.2167510538 0.2167407501 20 0.2281590040 0.2281481580 21 0.2395669542 0.2395555659 22 0.2509749044 0.2509629738 23 0.2623828546 0.2623703817 24 0.2737777048 0.2737777896

TheSunsmeanAnomalyandPraecessionoftheAEquinox. AEra ☉ Anomaly . Praecess . AEquinox Chr. 56. 69976.85185 20. 49768.51851 1600 53. 87323.10751 26. 67052.46907 1620 53. 83789.15687 26. 74768.51845 1640 53. 80255.20623 26. 82484.56783 1660 53. 76721.25559 26. 90200.61721 1680 53. 73187.30495 26. 97916.66659 1700 53.69653 ▪ 35431 27. 05632.71597 1720 53. 66119.40367 27. 13348.76535 1740 53. 65585.45303 27. 21064.81473 1760 53. 59051.50230 27. 28780.86411 ☉ Anomaly in Months Praecess . AEquinox in Months Ianu 08. 48718.72813 0. 00032.76678 Febr. 16. 15303.38579 0. 00062.36258 Mar. 24. 64022.11392 0. 00095.12937 April 32. 85362.81857 0. 00126.83916 May 41. 34081.54670 0. 00159.60594 Iune 49. 55422.25134 0. 00191.31573 Iuly 58. 04140.97947 0. 00224.08251 Aug. 66. 52859.70760 0. 00256.84929 Sept. 74. 74200.41225 0. 00288.55908 Octo. 83. 22919.14038 0. 00321.32587 Nov. 91. 44259.84502 0. 00353.03566 Dec. 99. 92978.57315 0. 00385.80244

The☉meanAnomaly,andPraecessionoftheAEquinoxinJulianYears. ☉ mean Anomaly Praecess . AEquinox 1 99.9297857316 00.0038580246 2 99.8595714612 00.0077160493 3 99.7893571949 00.0115740740 B 4 99.9929231686 00.0154320987 5 99.9227089002 00.0192901233 6 99.8524946318 00.0231481479 7 99.7822803634 00.0270061725 B 8 99.9858463372 00.0308641974 9 99.9156320688 00.0347222220 10 99.8454178004 00.0385802466 11 99.7752035321 00.0424382714 B 12 99.9787695058 00.0462962961 13 99.9085552374 00.0501543207 14 99.8383409690 00.0540123453 15 99.7681266066 00.0578703699 B 16 99.9716926744 00.0617283948 17 99.9014784060 00.0655864194 18 99.8312647376 00.0694444440 19 99.7610498692 00.0733024686 B 20 99.9646158434 00.0771604938 40 99.9292306868 00.1543209876 60 99.8938465302 00.2314814814 80 99.8584623736 00.3086419752 100 99.8270782170 00.3858024690 200 99.6461564340 00.7716049380 300 99.4692346510 01.1574074070 400 99.2923128680 01.5432098760 500 99.1153910850 01.9290123450 600 98.9384693020 02.3148148140 700 98.7615475190 02.7006172830

TheSunsmeanAnomalyandPraec.oftheAEqui.inDays. D ☉ Anomaly Praecess . AEquinox 1 0.2737802348 0.0000105699 2 0.5475604697 0.0000211398 3 0.8213407046 0.0000317097 4 1.0951209395 0.0000422797 5 1.3689011744 0.0000528496 6 1.6426814092 0.0000634195 7 1.9164616441 0.0000739894 8 2.1902418790 0.0000845593 9 2.4640221139 0.0000951292 10 2.7378023488 0.0001056993 11 3.0115825836 0.0001162692 12 3.2853628184 0.0001268391 13 3.5591430532 0.0001374090 14 3.8329232880 0.0001479789 15 4.1067035228 0.0001585488 16 4.3804837576 0.0001691187 17 4.6542639924 0.0001796886 18 4.9280442272 0.0001902585 19 4.2018244620 0.0002008284 20 5.4756046976 0.0002113986 21 5.7493849324 0.0002219685 22 6.0231651672 0.0002325384 23 6.2969454020 0.0002431083 24 6.5707256368 0.0002536782 25 6.8445058716 0.0002642481 26 7.1182861064 0.0002748180 27 7.3920663412 0.0002853879 28 7.6658455766 0.0002959580 29 7.9396258115 0.0003065279 30 8.2134070464 0.0003170979 31 8.4871872813 0.0003276678

TheSunsmeanAnomalyandPraec.oftheAEqui.inHours D ☉ mean Anomaly Praecess . AEquinox 1 0.0114075097 0.0000004404 2 0.0228150195 08808 3 0.0342225293 13212 4 0.0456300391 17616 5 0.0570375489 22020 6 0.0684450587 0.0000026424 7 0.0798525684 30828 8 0.0912600782 35232 9 0.1026675881 39636 10 0.1140750978 44041 11 0.1254826075 0.0000048445 12 0.1368901174 0.0000052849 13 0.1482976271 57253 14 0.1597051368 61657 15 0.1711126465 66061 16 0.1825201562 70465 17 0.1939276659 74869 18 0.2053351761 0.0000079272 19 0.2167426858 83677 20 0.2281501955 88081 21 0.2395577052 92485 22 0.2509652149 96889 23 0.2623727246 101293 24 0.2737802348 0.0000105698

THE TABLES OF THE MOONS MEAN MOTIONS .

TheMoonsmeanLongitudeandApogeon AEra ☽ Mean Longitude ☽ Apogaeon Chr. 34.0088734567 78.8286265432 1600 02.0644290122 63.5892746911 1620 39.1651134566 89.6540895059 1640 76.2658079010 15.7189033207 1660 13.3665023454 41.7837191355 1680 50.4671967898 67.6485339503 1700 87.5675912342 93.9133487651 1620 29.6685801230 19.9781635799 1740 61.7692801230 46.0429783947 1760 98.8699745674 72.1077932095 ☽ Mean Long. in Mon. ☽ Apogaeon in Mont. Ianu. 13.4633984897 00.9593447922 Febr. 15.9464670933 01.8258497658 Mar. 29.4098665830 02.7851945580 April 39.2131554440 03.7135927440 May. 52.6765539337 04.6729375362 Iune 62.4798427947 05.6013357222 Iuly 75.9432412844 06.5606805144 Aug. 89.4066397741 07.5200253066 Sept. 99.2099286451 08.4484234926 Octo. 12.6733271348 09.4077682848 Nov. 22.4766159958 10.3361664708 Dec. 35.9400144893 11.2955112636

TheMoonsmeanAnomalyandNodeRetrograde AEra ☽ Mean Anomaly ☽ Node Retrograde Chr. 55.1802469135 74.6984567901 1600 38.4751543211 78.2198302468 1620 49.5110239507 70.7638117283 1640 60.5469035803 63.3077932098 1660 71.5827832099 55.8517746913 1680 82.6186628395 48.3957561728 1700 93.6545424691 40.9397376543 1720 04.6904220987 33.4837191358 1740 15.7263017283 26.0277006173 1760 26.7621813579 18.5716820988 ☽ Mean Ano. in Mon. Node Ret. in Mont. Ianu. 12.5040536975 00.4559979224 Febr. 14.7206183275 00.8678670136 Mar. 27.2246720250 01.3238649360 April 35.4995627000 01.7651532480 May. 48.0036163975 02.2211511704 Iune 56.8785070725 02.6624394824 Iuly 69.3825607700 03.1184374048 Aug. 81.8866144675 03.5744353272 Sept. 90.7615051425 04.0157236392 Octo. 03.2655588400 04.4717215616 Nov. 12.1404495150 04.9130098736 Dec. 24.6445032256 05.3690078260
The Moons mean Motions in Julian Years . ☽ Mean Longitude ☽ Apogaeon 1 35.9400144893 11.2955112636 2 71.8800289786 22.5910225272 3 07.8200434679 33.8865337908 B 4 47.4201388888 45.2129629629 5 83.3601533781 56.5084742265 6 19.3001678674 67.8039854901 7 55.2401823567 79.0994967537 B 8 94.8402777777 90.4259259258 9 30.7802922670 01●7214371894 10 66.7203067563 13.0169484530 11 02.6603212456 24.3124597166 B 12 42.2604166666 35.6388888888 13 78. ●004311559 46.9344001524 14 14.1404456652 58 . 2299114●60 15 50.0804601545 69. ●2●4226706 B 16 89.6805555555 80.8518518518 17 25.6205700448 91.1473631154 18 61.5605845341 02.4428743790 19 97.5005990234 13.7383856426 B 20 37.1006944404 26 . 0648●48148 40 74.2013888888 52.1296296296 60 11.3020833333 78.1944444444 80 48.4027777777 04.2592592592 100 85.5034722222 30.3240740740 200 71.0069444444 60 . 648●●81●81 300 56.5104166666 90.9722222222 400 42.0138888888 21.2962962962 500 27 . 517361111● 51.6003703700 600 13.0208333333 91.9444444442 700 98.5243055555 12.2685185182

TheMoonsmeanMotionsinJulianYears ☽ Mean Anomaly ☽ Nodes Retrograde 1 24.6445032256 05.3690078260 2 49.2890064512 10.7380156520 3 73.9335096768 16.1070234780 B 4 02.2071759259 21.4912037037 5 26.8516791515 26.7602115297 6 5● . 4951823771 32.1292193557 7 76.1396856027 37.4982271817 B 8 04.4143518518 42.9824074074 9 29.0588550774 48.3514152334 10 53.7033583030 53.7204230594 11 78.3478615286 59.0894308854 B 12 06.6215277777 64.4736111111 13 37.2660310033 69.8426189371 14 55.9105342289 74.2116267631 15 80.5550374545 79.5806345891 B 16 08.8287037037 85.9648148148 17 33.4732069293 91.3338226408 18 58.1177101549 96.7028304668 19 82. ●622133805 02.0718382928 B 20 11 . 0●58796●97 07.4560185185 40 22.0717592594 14.9120370370 60 33.1076388891 22.3680555555 80 44.1435185188 29.8040740740 100 55.1793981487 37.2800925925 200 10.3587062074 74.5601851850 300 65.5381944461 11.8402777775 400 20.7175925948 49.1203703700 500 75.8969907435 86.4004629629 600 31.0763888922 23.6805555555 700 86.2557870409 60.9606481480

TheMoonsmeanMotionsinDays. Days ☽ Mean Longitude ☽ Apogaeon 1 03.6601096287 00.0309466062 2 07.3202192574 00.0618932124 3 10.9803288861 00.0928398186 4 14.6404385148 00.1237864248 5 18.3005481435 00.1547330310 6 21.9606577722 00.1856796372 7 25.6207674009 00.2166262434 8 29.2808770296 00.2475728496 9 32.9409866583 00.2785194558 10 36.6010962870 00.3094660620 11 40.2612059157 00.3404126682 12 43.9213155444 00.3713592744 13 47.5814251731 00.4023058806 14 51.2415348018 00.4332524868 15 54.9016444305 00.4641990930 16 58.5617540592 00.4951456992 17 62.2218636879 00.5260923054 18 65.8819733166 00.5570389116 19 69.5420829453 00.5879855178 20 73.2021925740 00.6189321240 21 76.8623022037 00.6498787302 22 80.5224118314 00.6808233364 23 84.1825214601 00.7117719426 24 87.8426310898 00.7427185488 25 91.5027407175 00.7736651550 26 95.1628503462 00.8046117612 27 98.8229599749 00.8355583674 28 02.4830696036 00.8665049736 29 06.1431792323 00.8974515798 30 09.8032888610 00.9283981860 31 13.4633984897 00.9593447922

TheMoonsmeanMotionsinDays. Days ☽ Mean Anomaly ☽ Node Retrograde 1 03.6291630225 00.0147096104 2 07.2583260450 00.0294192208 3 10.8874890675 00.0441288312 4 14.5166520900 00.0588384416 5 18.1458151125 00.0735480520 6 21.7749781350 00.0882576624 7 25.4041411575 00.1029672728 8 29.0333041800 00.1176768832 9 32.6624672025 00.1323864936 10 36.2916302250 00.1470961040 11 39.9207932475 00.1618057144 12 43.5499562700 00.1765153248 13 47.1791192925 00.1912249352 14 50.8082823150 00 . 20●9345456 15 54.4374453375 00.2206441560 16 58.0666083600 00.2353537664 17 61.6957713825 00.2500633768 18 65.3249344050 00.2647729872 19 68.9540974275 00.2794825976 20 72.5832604500 00.2941922080 21 76.2124234725 00.3089018184 22 79.8415864950 00.3236114288 23 83.4707495175 00.3383210392 24 87.0999125400 00.3530306496 25 90.7290755625 00.3677402600 26 94.3582385850 00.3824498704 27 97.9874016075 00.3971594808 28 01.6165646300 00.4118690912 29 05.2457276525 00.4265787016 30 08.8748906750 00.4412883120 31 12.5040536975 00.4559979224

TheMoonsmeanMotionsinHours. Hours ☽ Mean Longitude ☽ Apogaeon 1 00.1525045678 00.0012894419 2 00.3050091357 00.0025788838 3 00.4575137035 00.0038683257 4 00.6100182713 00.0041577676 5 00. ●625228391 00.0064172095 6 00.9150274071 00.0077366515 7 01.0675319749 00.0090260934 8 01.2200365427 00.0103155353 9 01.3725411105 00.0116049772 10 01.5250456786 00.0128044192 11 01.6775502464 00.0141838611 12 01.8300548143 00.0154733031 13 01.9825593821 00.0167627450 14 02.1350639499 00.0180521869 15 02.2875685177 00.0193416288 16 02.4400730855 00.0206310707 17 02.5925776533 00.0219205126 18 02.7450822211 00.0232099545 19 02.8975867891 00.0244993964 20 03 . 0500●13560 00.0257888384 21 03.2025959250 00.0270782803 22 03.3551004928 00.0283677222 23 03.5076050607 00.0296571642 24 03.6601096285 00.0309466061

TheMoonsmeanMotionsinHours. Hours ☽ Mean Anomaly ☽ Node Retrograde 1 00.1512151259 00.0006129004 2 00.3024302518 00.0012258008 3 00.4536453778 00.0018387013 4 00.6048605037 00.0024516017 5 00.7560756296 00.0030645021 6 00.9072907556 00.0036774026 7 01.0585058815 00.0042903030 8 01.2097210074 00.0049032034 9 01.3609361333 00.0055161038 10 01.5121512593 00.0061290043 11 01.6633663852 00.0067419047 12 01.8145815112 00.0073548052 13 01.9657066371 00.0079677056 14 01.1170117630 00.0085806060 15 02.2682068889 00.0091935064 16 02.4194420148 00 . 009●064068 17 02.5706571407 00 . 0●04193072 18 02.7218722666 00.0110722076 19 02.8730873926 00.0116451081 20 03 . 9243025●85 00.0122580085 21 03.1755176445 00.0128709090 22 03.3267327704 00.0134838004 23 03.4779478964 00.0140967099 24 03.6291630223 00.0147096103

TheMoonsmeanMotionsinMinutesofanHour M. ☽ M. Long. ☽ Apog . ☽ M. Au. ● Retrog . 1 .0025414 .0000214 .0025202 .0000102 2 .0050828 .0000429 .0050405 .0000204 3 .0076242 .0000643 .0075607 .0000306 4 .0101656 .0000859 .0100810 .0000408 5 .0127070 .0001074 .0126012 .0000510 6 .0152484 .0001288 .0151214 .0000612 7 .0177898 .0001502 .0176416 .0000714 8 .0203312 .0001716 .0201618 .0000816 9 .0228726 .0001930 .0226820 .0000918 10 .0254141 .0002149 .0252025 .0001021 11 .0279555 .0002363 .0277227 .0001123 12 .0304969 .0002577 .0302429 .0001225 13 .0330383 .0002791 .0327631 .0001327 14 .0355797 .0003004 .0352833 .0001429 15 .0381211 .0003218 .0378035 .0001531 16 .0406624 .0003432 .0403237 .0001633 17 .0432038 .0003646 .0428439 .0001735 18 .0457452 .0003860 .0453641 .0001837 19 .0482867 .0004079 .0478843 .0001939 20 .0508284 .0004298 .0504045 .0002041 21 .0533696 .0004512 .0529247 .0002143 22 .0559110 .0004726 .0554449 .0002245 23 .0584524 .0004940 .0579651 .0002347 24 .0609938 .0005154 .0604853 .0002442 25 .0635352 .0005368 .0630055 .0002544 26 .0660766 .0005582 .0655257 .0002642 27 .0686180 .0005795 .0680459 .0002744 28 .0711594 .0006008 .0705661 .0002846 29 .0737008 .0006222 .0730863 .0092948 30 .0762422 .0006437 .0756075 .0003064

TheMoonsmeanMotionsinSeconds. ☽ M. Long. ☽ Apog . ☽ M. Au. ☊ Retrog . 1 0000423 0000003 0000420 0000002 2 0000847 0000007 0000840 0000003 3 0001270 0000010 0001260 0000005 4 0001693 0000013 0001680 0000006 5 0002116 0000016 0002100 0000009 6 0002539 0000019 0002520 0000010 7 0002969 0000022 0002940 0000012 8 0003392 0000025 0003360 0000013 9 0003815 0000028 0003780 0000015 10 0004275 0000035 0004200 0000017 11 0004658 0000038 0004620 0000019 12 0005078 0000041 0005040 0000020 13 0005504 0000044 0005460 0000022 14 0005930 0000047 0005880 0000023 15 0006357 0000050 0006300 0000025 16 0006784 0000053 0006720 0000027 17 0007207 0000056 0007140 0000028 18 0007630 0000059 0007560 0000029 19 0008050 0000062 0007980 0000031 20 0008470 0000065 0008400 0000033 21 0008893 0000068 0008820 0000035 22 0009316 0000071 0009240 0000036 23 0009736 0000074 0009660 0000038 24 0010156 0000077 0010080 0000039 25 0010582 0000080 0010500 0000041 26 0011008 0000083 0010920 0000043 27 0011434 0000086 0011340 0000044 28 0011860 0000089 0011760 0000047 29 0012287 0000092 0012180 0000049 30 0012714 0000095 0012600 0000051

☉ Sig. o. & . 6 1 & 7 2 & 8 ☉ a AEqu. ☊ Inclin . AEqu. ☊ Inclin . AEqu. ☊ Inclin . a ☊ Addi limetis Addi limitis Addi limitis ☊ 0 0.00000 30000 1.06500 20000 1.08500 15000 30 1 0.00000 30000 1.12888 25722 1.01888 14527 29 2 0.00055 30000 1.19277 25472 0.95305 14055 28 3 0.00194 29972 1.25222 25166 0.88666 13583 27 4 0 . 0041● 29944 1.30833 24888 0.82055 13138 26 5 0.00888 29888 1.36166 24583 0.55333 12666 25 6 0.01472 29833 1.41055 24277 0.68694 12194 24 7 0.02305 29777 1.45666 23972 0.62416 11●22 23 8 0.02416 29722 1.49916 23638 0.56444 11250 22 9 0.04805 29638 1.53666 23305 0.50555 10750 21 10 0.06500 29555 1.58000 22972 0.44666 10250 20 11 0.08555 29444 1.60027 22638 0.38888 09750 19 12 0.10944 29361 1.62527 22277 0.34000 09250 18 13 0.13666 29250 1.64472 219●6 0.28972 08750 17 14 0.16833 29111 1.65771 21●55 0.25083 08250 16 15 0.20250 28972 1.66277 21222 0.20972 07750 15 16 0 . 2411● 28833 1.65805 20833 0.17388 07250 14 17 0.27472 28667 1.64527 20444 0.14138 06750 13 18 0. ●2944 28567 1.62638 20055 0.11305 06250 12 19 0.37916 28361 1.60194 19666 0.08805 05722 11 20 0.43277 28194 1.58222 19279 0.06722 05222 10 21 0.48888 18027 1.53972 18861 0.04916 04694 9 22 0.54833 27833 1.50333 18444 0.03416 04166 8 23 0.60694 27611 1.46222 18027 0.02333 03638 7 24 0.66833 27416 1.41799 17614 0.01500 03138 6 25 0.73555 27104 1.37027 17194 0.00888 02611 5 26 0.79805 26972 1 . 3186● 14750 0.00416 02083 4 27 0 . 8641● 26722 1.26527 16333 0 . 001●4 01555 3 28 0.93083 26500 1.20833 15888 0.00055 01027 2 29 0.99611 26250 1.14750 15444 0.00000 ●0527 1 30 1.06500 26600 1.08500 14000 0.00000 00000 0 Sntract Subtract . Subtract . Sig. 5. & 11 4 & 10 3 & 9

Aug. Sig. o. N. Incr. 1. North Incr. 2. North I ncr . Sig. 6. S. 7. South 8. South Lat. or Exc. or Exc. or Exc. Latit . Latit . Latit . 0 0.00000 00000 2.49750 15000 4.32888 26000 30 1 0.08722 00527 2.57277 15444 4.37166 26250 29 2 0.17444 01027 2.64722 15888 4.41361 26500 28 3 0.26166 01555 2.72083 16333 4.45388 26722 27 4 0.34861 02083 2.79361 16416 4.49277 26972 26 5 0.43555 02611 2.86555 17194 4.53055 27194 25 6 0.52222 03138 2.93638 17611 4.56666 27416 24 7 0.60888 03638 3.00666 18027 4.60166 27611 23 8 0.69527 04166 3. ●7583 18444 4.63500 27833 22 9 0.78138 04694 3.14416 18861 4.66722 28027 21 10 0.86722 05222 3.21166 19277 4.69777 28194 20 11 0.95277 05722 3.27805 19666 4.72694 28361 19 12 1.03833 06250 3.34333 20055 4.75472 28527 18 13 1.12333 06750 3.41055 20444 4.78111 28666 17 14 1.10805 07250 3.47111 20833 4.80583 28833 16 15 1.29250 07750 3.53333 21222 4.82916 28972 15 16 1.37666 08250 3.59444 21555 4.85111 29111 14 17 1.46027 08750 3.65472 21916 4.87166 29250 13 18 1.54333 09250 3.71361 22277 4.89055 29361 12 19 1.62611 09750 3.77138 22638 4.90777 29444 11 20 1.70805 10250 3.82833 22972 4.92388 29555 10 21 1.78972 10750 3.88388 23305 4.93833 29638 9 22 1.87111 11250 3.93805 23638 4.95111 29722 8 23 1.95166 11722 3.99138 23972 4.96250 29777 7 24 2.03166 12194 4.04333 24277 4.97250 29833 6 25 2.11083 12666 4.29416 24583 4.98083 29888 5 26 2.18972 13138 4.14361 24888 4.98777 29944 4 27 2.26777 13583 4.19166 25166 4.99301 29972 3 28 2.34500 14055 4.23861 25472 4.99694 29972 2 29 2.42166 14527 4.28444 25722 4.99916 18000 1 30 2.49750 15000 4.32888 26000 5.00000 18000 0 Sig. 11. S. Sig. 19. S. 9. South Sig. 5. N. 4 N. 3. North
A Table of the Moons Red. to the El. Subt As Lat. Sig. 0.6 S. 1.7 S. 2.8 As. Lat. Incr. Incr. Incr. Red. Red. Red. 0 .00000 00000 09444 01166 09472 01166 30 1 .00388 00055 09638 01194 09277 01138 29 2 .00750 00111 09805 01222 09055 01111 28 3 .01138 00166 09972 01250 08833 01083 27 4 .01527 00222 10111 01277 08611 01055 26 5 .01888 00250 10250 01277 08388 01027 25 6 .02277 00305 10388 01277 08138 01000 24 7 .02638 00333 10500 01305 07861 00972 23 8 .03000 00361 10583 01305 07611 00944 22 9 .03361 00416 10666 01305 07305 00916 21 10 .03722 00472 10750 01333 07027 00861 20 11 .04083 00527 10805 01333 06722 00833 19 12 .04444 00555 10861 01333 06416 00805 18 13 .04777 00611 10888 01333 06111 00777 17 14 .05111 00638 10916 01361 05805 00722 16 15 .05444 00666 10916 01361 05472 00666 15 16 .05777 00722 10916 01361 05138 00638 14 17 .06111 00777 10862 01361 04805 00611 13 18 .06416 00805 10861 01361 04444 00555 12 19 .06722 00833 10805 0133● 04111 00527 11 20 .07000 00861 10750 01333 03750 00472 10 21 .07305 00916 10694 01305 03388 00416 9 22 .07583 00944 10611 01305 03027 00361 8 23 .07888 00972 10500 01305 02092 00333 7 24 .08111 01000 10388 01277 02611 00305 6 25 .08361 01027 10277 01277 02222 00250 5 26 .08583 01055 10138 01277 01527 00222 4 27 .08823 01083 10000 01250 01138 00166 ●3 28 .09055 01111 09833 01222 00750 00111 2 29 .09250 01138 09638 01194 00388 00055 1 30 .09444 01166 09472 01166 00000 00006 0 11.5 10.4 9.3

ATableshewingthemeanMotionoftheMoonfromtheSuninYearsandMonths. AEra ☽ à ☉ in Years ☽ à ☉ in Years Chr. 56.8114797531 1 36.0063707331 1600 21.5206732464 2 72 . 012741466● 1620 58.5795367034 3 08.0191121993 1640 95.6384101604 4 47.4117836215 1660 32.6972836174 5 83.4181543546 1680 69.7561560744 6 19.4245250877 1700 06.8150305314 7 55.4308958208 1720 43.8739039884 8 94.8235672430 1740 80.9027774454 9 40.8298379761 1760 17.9916509024 10 76.8362087092 11 02.8426794423 Motion of the 12 42.2353508645 Moon from the 13 78.2417215976 Sun in Months . 14 14.2480923307 15 50.2544630638 16 89.6471344860 Ian. 04.9758835440 17 25.6535052191 Feb. 99.7928106160 18 61.6598759522 Mar. 04.7686941600 19 97.6662466853 April 06.3582588800 20 37.0589181075 40 74.1178362150 May 11.3341424240 60 11.1767543225 Iune 12.9237071440 80 48.2356724300 Iuly 17.8995906880 100 85.2945905375 Aug. 22.8754742320 200 70.5891810750 300 55.8837716125 Sept. 24.4650389520 400 41.1783621500 Octo. 29.4409224960 500 26.4729526875 Nov. 31.0304872160 600 11.7675432250 Dec. 36.0063707331 700 97.0621337625

ATableshewingthemeanMotionoftheMoonfromtheSuninDaysandHours. ☽ à ☉ in Days . ☽ à ☉ in Hours . 1 03.3863188240 1 00.1410966176 2 06.7726376480 2 00.2821932352 3 10.1589564720 3 00.4232898530 4 13.5452752960 4 00.5643864706 5 16.9315941200 5 00.7054830882 6 20.3179129440 6 00.8465797060 7 23.7042317680 7 00. ●876763236 8 27.0905505920 8 01.1287729412 9 30.4768694160 9 01.2698695588 10 33.8631882400 10 01.4109661766 11 37.2495070640 11 01.5520627942 12 47.6358258880 12 01.6931594120 13 44.0221447720 13 01.8342560296 14 47.4084635360 14 01.9753526472 15 50. ●947823600 15 0● . 1164492648 16 54.1811011840 16 02.2575458824 17 57.5674200080 17 02.3986425000 18 60.9537388320 18 02.5397391176 19 64.3400576560 19 02.6808357354 20 67.7263764800 20 02.8219323520 21 71.1126953040 21 02.9630289708 22 74.4990141280 22 03.1041255884 23 77.8853329520 23 03.2452222062 24 81.2716517760 24 03.3863188240 25 84.6579706000 26 88.0442804240 27 91 4306082480 28 94.8169270720 29 98.2032458960 30 01.5895647200 31 04.9758835440

ATableshewingthemeanMotionoftheMoonfromtheSuninMinutes. ☽ à ☉ in Minutes . 1 00.0023516102 2 00.0047032205 3 00.0070548308 4 00 . 00●4064411 5 00.0117580513 6 00.0141096617 7 00.0164612719 8 00.0188128822 9 00.0211644924 10 00.0235161029 11 00.0258677131 12 00.0262193233 13 00.0305709335 14 00.0329225437 15 00.0352741539 16 00.0376257644 17 00.0399773746 18 00.0423289848 19 00 . 044680●950 20 00.0170322052 21 00.0493838154 22 00.0517354256 23 00 . 0540870●58 24 00.0564386460 25 00.0587902562 26 00.0611418664 27 00.0634934766 28 00.0658450868 29 00.0681966970 30 00.0705483080 ☽ à ☉ in Minutes . 31 00.0728999183 32 00.0752515088 33 00.0776031390 34 00.0799547492 35 00.0823063594 36 00.0846579696 37 00.0870095798 38 00.0893611900 39 00.0917128002 40 00.0940644104 41 00.0964160206 42 00.0997676308 43 00.1011192410 44 00.1034708512 45 00.1058224614 46 00.1081740716 47 00.1105256818 48 00.1128772920 49 00.1152289022 50 00.1175805124 51 00.1199321226 52 00.1222837328 53 00.1246353430 54 00.1269869532 55 00.1293385634 56 00.1316901736 57 00.1340417838 58 00.1363933940 59 00.1387450050 60 00.1410966152

ATableshewingthemeanMotionoftheMoonfromtheSuninSeconds. ☽ à ☉ in Seconds 1 00.0000391935 2 00.0000783870 3 00.0001175805 4 00.0001567740 5 00.0001959675 6 00.0002351610 7 00.0002743545 8 00.0003135480 9 00.0003527415 10 60.0003919350 11 00.0004811285 12 00.0004703220 13 00.0005995155 14 00.0005487090 15 00.0005879025 16 00 . 00●6270960 17 00.0006662895 18 00.0007954830 19 00.0007446765 20 00.0007838700 21 00.0008230635 22 00.0008622570 23 00.0009014505 24 00.0009406440 25 00 . 0009798●75 26 00.0010190310 27 00.0010582245 28 00.0010974180 29 00.0011366115 30 00.0011758050 ☽ à ☉ in Seconds 31 00.0012149985 32 00.0012541920 33 00.0012933855 34 00.0013325790 35 00.0013717725 36 00.0014109660 37 00.0014501595 38 00.0014893530 39 00.0015285465 40 00.0015677400 41 00.0016069335 42 00.0016461270 43 00.0016853205 44 00.0017245140 45 00.0917637075 46 00.0018029010 47 00.0018420945 48 00.0018812880 49 00.0019204815 50 00.0019596750 51 00 0019988685 52 00.0020380620 53 00.0020772555 54 00.0021164490 55 00.0021556425 56 00.0021948360 57 00.0022340295 58 00.0022732230 59 00.0023124165 60 00.0023516100

ACatalogueofsomeofthemostnotablefixedStarsaccordingtotheobservationsofTychoBrahe,andbyhimrectifiedtothebeginningoftheYearofMansRedemption,1601. The Names of the Stars Longit. Latit . The first Star of Aries . 07.671 ♈ 7. 8. N 4 The bright Star in the top of the head of Aries . 00.583 ♉ 9. 57. N 3 The South Eye of Taurus . 01.169 ● 5. 31. S 1 The North Eye of Taurus . 00.801 ● 5. 31. S 1 The bright Star of the Pleiades . 06.620 ♉ 2. 6. S 3 The higher head of Gemini . 04.078 ● 4. 11. N 5 The lower head of Gemini . 04.921 ♋ 10. 2. N 2 The bright foot of Gemini . 01.069 ♋ 6. 38. N 2 In the South Arm of Cancer . 02.238 ♌ 6. 48. S 2 The bright Star in the neck of Leo. 06.662 ♌ 5. 8. S 3 The heart of Leo. 06.745 ♌ ● . 47 . N 2 In the extream of the tail of Leo. 04.458 ♍ 0. 26. N 1 In Virgo's Wing ; Vindemiatrix . 01.217 ♎ 12. 18. N 1 Virgins Spike . 05.074 ♎ 16. 15. N 3 South Ballance . 02.643 ♏ 1. 59. S 1 North Ballance . 03.833 ♏ 0. 26. N 2 The highest in the Forehead of Scorpio . 07.388 ♏ 8. 35. N 2 The Scorpions heart . 01.171 ● 1. 05. N 3 Former of the 3 in the head of Sagittarius . 02.203 ● 4. 27. S 1 Northern in the former horn of Capricorn . 07.861 ● 1. 24. N 4 The left Shoulder of Aquarius . 04.949 ♒ 7. 22. N 3 In the mouth of the South Fish . 03.620 ♓ 8. 42. N 3 The Polar Star or last Star in the ●ail of the lesser Bear . 9. 4. N 5 06.400 ● 66. 02. N 2 The last Star in the tail of the great Bear , 05.888 ♍ 54. 25. N 2 The Tongu● of the Dragon . 05.259 ♍ 76. 17. N 4 Arcturus in the skirt of his Garment . 05.181 ♎ 31. 2. N 1 The bright Star of the North Crown . 01.845 ♏ 44. 23. N 2 The Head of Hercules ▪ 02.921 ● 37. 23. N ● The bright S●●r of the H●rp . 0● . 699 ● 61. 47. N ● The Head of Medusa . 05.727 ♉ 22. 22. N 3 The bright Star in the Goa●s left Shoulder . 04.518 ♊ 22. 50. N 1 The middle of the Serp●nts Neck . 04.583 ♍ 25. 35. N 2 The bright Star in the ●agles Shoulder . 07.264 ♑ 29. 21. N 2 The bright Star in the 〈◊〉 Tail . 02.370 ♒ 29. 8. N 3 The mouth of Pegas●s . 07 . 3●4 ♒ 22. 7. N 3 The head of And●omeda . 0● . 4●0 ♈ ●5 . 42. N 2 In the top of the Triangle . 00.366 ♉ 16. 49. N 4 In the Snout of the Whale . 02.643 ♉ 7. 50. S The bright Star in the Whales Tail . 07.481 ♓ 20. 47. S 2 Bright Shoulder of Orion . 06.444 ♊ 16.06 S 2 Middlemost in the belt of Orion . 04.972 ♊ 24. 33. S 2 The last in the tail of the Har● . 0● . 324 ♊ 38. 26. S 4 The great Dogs mouth Sirius . 02.386 ● 38. 30. S 1 The lesser Dog Procyon . 05.641 ● 1● . 57 . S 2 In the top of the Ships Stern . 01.636 ♌ 43. 18. S 3 Brightest in Hydra's Heart . 06.044 ♌ 22. 24. S 1

FINIS
THE CONTENTS OF THE First Part , CONTAINING The Practical Geometry or the Art of Surveying . CHapter 1. Of the Definition and Division of Geometry . Chap. 2. Of Figures in the General , more particularly of a Circle and the Affections thereof . Chap. 3. Of Triangles . Chap. 4. Of Quadrangular and Multangular Figures . Chap. 5. Solid Bodies . Chap. 6. Of the measuring of Lines both Right and Circular Chap. 7. Of the measuring of a Circle . Chap. 8. Of the measuring of plain Triangles . Chap. 9. Of the measuring of Heights and Distances . Chap. 10. Of the taking of Distances . Chap. 11. How to take the Plot of a Field at one Station , &c. Chap. 12. How to take the Plot of a Wood , Park , or other Champian Plane , &c. Chap. 13. The Plot of a Field being taken by an Instrument , how to compute the Content thereof in Acres , Roods , and Perches . Chap. 14. How to take the Plot of mountainous and uneven Ground , &c. Chap. 15. To reduce Statute measure into Customary , and the contrary . Chap. 16 ▪ Of the measuring of solid Bodies . Tables . A Table of Squares . Page . 99 A Table for the Gauging of Wine Vessels . 114 A Table for the Gauging of Beer and Ale Vessels . 120 A Table shewing the third part of the Areas of Circles , in Foot measure and Deoimal parts of a Foot . 132 A Table shewing the third part of the Area of any Circle in Foot measure , not exceeding 10 f. circumf . 136 A Table for the speedy finding of the length or Circumference answering to any Arch in Degrees and Decimal parts . 151 A Common Divisor for the speedy converting of the Table , shewing the Areas of the Segments of a Circle whose Diameter is 2 &c. 154 A Table shewing the Ordinates , Arches , and A rea● of the Segments of a Circle , whose Diameter is 〈◊〉 &c. 156
The Contents of the Second Part of this Treatise , of the Doctrine of the PRIMUM MOBILE . CHap. 1. Of the General Subject of Astronomy . Chap. 2. Of the Distinctions and Affections of Spherical Lines and Arches . Chap. 3. Of the kind and parts of Spherical Triangles , and how to project the same upon the Plane of the Meridian . Chap. 4. Of the solution of Spherical Triangles . Chap. 5. Of such Spherical Problems as are of most general Vse in the Doctrine of the Primum Mobile , &c.
The Contents of the Third Part of this Treatise being an Account of the Civil Year with the reason of the difference between the Julian and Gregorian Calendars , and the manner of Computing the Places of the Sun and Moon . CHap. 1. Of the Year Civil and Astronomical . Chap. 2. Of the Cycle of the Moon , what it is , how placed in the Calendar , and to what purpose . Chap. 3. Of the use of the Golden Number in finding the Feast of Easter . Chap. 4. Of the Reformation of the Calendar by Pope Gregory the Thirteenth , &c. Chap. 5. Of the Moons mean Motion and how the Anticipation of the New Moons may be discovered by the Ep●●●ts . Chap. 6. To find the Dominical Letter and Feast of Easter according to the Gregorian Account . Chap. 7. How to reduce Sexagenary Numbers into Decimals , and the contrary . Chap. 8. Of the difference of Meridians . Chap. 9. Of the Theory of the Suns or Earths motion . Chap ▪ 10. Of the finding of the Suns Apogaeon , quantity of Excentricity and middle Motion . Chap. 11. Of the quantity of the tropical and sydereal Year . Chap. 12. Of the Suns mean Motion otherwise stated . Chap. 13. How to calculate the Suns true place by either of the Tables of 〈◊〉 middle Motion . I 〈…〉 Chap. 14. To find the place of the fixed Stars . Chap. 15. Of the Theory of the Moon and the finding the place of her Apogaeon , quantity of Excentricity , and middle motion . Chap. 16. Of the finding of the place and motion of the Moons Nodes . Chap. 17. How to calculate the Moons true place in her Orbs. Chap. 18. To compute the true Latitude of the Moon , and to reduce her place from her Orbit to the Ecliptick . Chap. 19. To find the mean Conjunctions and Opposition of the Sun and Moon ▪
The Fourth Part , or an Introduction to Geography . CHap. 1. Of the Nature and Division of Geography . Chap. 2. Of the Distinction or Dimension of the Earthly Globe by Zones and Climates . Chap. 3. Of Europe . Chap. 4. Of Asia . Chap. 5. Of Africk . Chap. 6. Of America . Chap. 7. Of the description of the Terrestrial Globe , by Maps Vniversal and Particular .
A Table of the view of the most notable Epochas . The Iulian Calendar . Page . 461 The Gregorian Calendar . 466 A Table to convert Sexagenary Degrees and Minutes into Decimals and the contrary . 476 A Table converting hours and minutes into degrees and minutes of the AEquator . 480 A Table of the Longitudes and Latitudes of some of the most eminent Cities and Towns in England and Ireland . 482 A Table of the Suns mean Longitude and Anomaly in both AEgyptian and Iulian Years , Months , Days , Hours and Minutes . 484 Tables of the Moons mean motion . 493 A Catalogue of some of the most notable fixed Stars , according to the observation of Tycho Brahe , rectified to the year 1601. 511
Books Printed for and sold by Thomas Passinger at the Three Bibles on the middle of London-Bridge .

THe Elements of the Mathematical Art , commonly called Algebra , expounded in four Books by Iohn Kersey , in two Vol. fol. A mirror or Looking-glass for Saints and Sinners , shewing the Justice of God on the one , and his Mercy towards the other , set forth in some thousands of Examples by Sam. Clark , in two Vol. fol.

The Mariners Magazine by Capt. Sam. Sturmy , fol.

Military and Maritime Discipline in three Books , by Capt. Tho. Kent , fol.

Dr. Cudworth's universal Systeme .

The Triumphs of Gods Revenge against the Crying and Execrable sin of wilful and premeditated Murther , by Iohn Reynolds , fol.

Royal and Practical Chymistry by Oswaldus Crollius and Iohn Hartman , faithfully rendred into English , fol.

Practical Navigation by Iohn Seller . Quarto .

The History of the Church of Great Britain from the Birth of our Saviour until the Year of our Lord 1667. quarto .

The Ecclesiastical History of France from the first plantation of Christianity there unto this time , quarto .

The book of Architecture by Andrea Palladio , quarto .

The mirror of Architecture or the ground Rules of the Art of Building , by Vincent Scammozi quarto .

Trigonometry , on the Doctrine of Triangles , by Rich. Norwood , quarto .

Markham's Master-piece Revived , containing all knowledge belonging to the Smith , Farrier , or Horse-Leach , touching the curing of all Diseases in Horses , quarto .

Collins Sector on a Quadrant , quarto .

The famous History of the destruction of Troy , in three books , quarto .

Safeguard of Sailers , quarto .

Norwood's Seamans Companion , quarto .

Geometrical Seaman , quarto .

A plain and familiar Exposition of the Ten Commandments , by Iohn Dod , quarto .

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Complete Shipwright , quarto .

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Thesaurus Astrologiae , or an Astrological Treasury by Iohn Gadbury , oct.

Gellibrand' s Epitome , oct.

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Monastieon Fevershamiense ; or a description of the Abby of Feversham , oct .

Scarborough ' s Spaw , oct.

French Schoolmaster , oct.

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Newton's Introduction to the Art of Logick , twelves .

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By Iames Atkinson . FINIS .