Iohnsons Arithmatick in 2. bookes the first, of vulgare arithma: with diuers briefe and easye rules: to worke all the first 4. partes of arithmatick in whole numbers and fractions by the author newly invented the second, of decimall arithmatick wherby all fractionall operations are wrought, in whole numbers, in marchants accomptes without reduction; with interest, and annuityes by Iohn Iohnson survaighour; practitioner in the mattiematiqu Arithmetick Johnson, John, fl. 1602-1657. This text is an enriched version of the TCP digital transcription A04547 of text S107834 in the English Short Title Catalog (STC 14668.5). 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. 558 KB of XML-encoded text transcribed from 240 1-bit group-IV TIFF page images. EarlyPrint Project Evanston,IL, Notre Dame, IN, St. Louis, MO 2018 A04547 STC 14668.5 ESTC S107834 99843527 99843527 8266 2240868390

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Early English books online. (EEBO-TCP ; phase 1, no. A04547) Transcribed from: (Early English Books Online ; image set 8266) Images scanned from microfilm: (Early English books, 1475-1640 ; 1245:5) Iohnsons Arithmatick in 2. bookes the first, of vulgare arithma: with diuers briefe and easye rules: to worke all the first 4. partes of arithmatick in whole numbers and fractions by the author newly invented the second, of decimall arithmatick wherby all fractionall operations are wrought, in whole numbers, in marchants accomptes without reduction; with interest, and annuityes by Iohn Iohnson survaighour; practitioner in the mattiematiqu Arithmetick Johnson, John, fl. 1602-1657. Delaram, Francis, 1589 or 90-1627, engraver. [98], 368 p., folded table by Augustine Mathewes· and are to be sold at his hous in Stie. Brides Lane neere Fleetstre[t], Printed at london : 1623. Signatures: pi¹ A-2F. The title page is engraved and signed "FD sc", i.e. Francis Delaram. Hind, A. Engraving in England in the 16th and 17th cents., 1955; part II, pp. 236-7. Reproduction of the original in the Bodleian Library. Identified as STC 14670 on UMI microfilm. eng Arithmetic -- Early works to 1900. eebo Iohnsons Arithmatick in 2. bookes the first, of vulgare arithma: with diuers briefe and easye rules: to worke all the first 4. partes of arithmatick i Johnson, John 1623 478 59631 115 0 422 0 0 0 90.05 D The rate of 90.05 defects per 10,000 words puts this text in the D category of texts with between 35 and 100 defects per 10,000 words. 2005-11 Assigned for keying and markup 2006-01 Keyed and coded from ProQuest page images 2007-02 Sampled and proofread 2007-02 Text and markup reviewed and edited 2008-02 Batch review (QC) and XML conversion

Iohnsons Arithmatick

IN 2. Bookes

THE FIRST ▪ OF VULGARE ARITHMA with diuers Briefe and Easye rules : to worke all the first 4. partes of Arithmatick in whole numbers and fractions by the Author newly Invented

The Second , of Decimall Arithmatick wherby all fractionall operations 〈…〉 wrought in whole Numbers in Marchants accomptes without reduction ; with Interest , and Annuityes

By Iohn Iohnson Survaighour ; practitioner in the matticmatiqu

Printed at london by Augustine Mathewes and are to be sold at his hous in S tie . Brides Lane neere Fleetstrē 1623

TO THE RIGHT HONOVRABLE SIR EDWARD BARKSHAM , KNIGHT , Lord Maior of the City of London ; and to the right Worshipfull , the Shriefes and Aldermen his Brethren .

THe very Arts ( Right Honorable and Worshipfull ) which were wont to beare the attributions of [ honestae & liberales ] seeme now to temporize , and to haue learned the new-found skill of equiuocation . For , howsoeuer the former of these denominations adhereth censtantly vnto the Professors of Mathematick Sciences , yet the other , which was once deriued [ a liberalitate ] and then intimated , that they were anciently accustomed to performe liberall recompence to their louers and followers , hath now ( to spare cost ) purchased a different etymologie [ a libertate ] as properly accommodate to such as are [ liberi ] free-borne , or ( as our peculiar terme carrieth ) Free-men . Which being so , and seeing that in this particular , as well as in many other of greater consequence , Tempora mutantur — : I am thereby enforced to make vp the old verse , adding — e● nos mutamur in illis ; and to apply my long experience , together with tedious studies bestowed in this present art of numbers , to the vse and be hoofe of those persons , to whom by the generall appellation it properly belongeth , namely , to the studious thereof in this honorable Citie . Which is the cause that I presume ( without farther selfe-praise , of what I haue brought more vsefull , more easie , or more certaine and delightfull in the operations , then hath been seene before ) to present my Labours to your Hononrable and Worshipfull iudgements , to whom I owe of dutie , whatsoeuer can be of mee performed , to the furtherance of Art , and the honour of this noble Citie , and the worthie Companies therein .

Your Honours , and Worships deuoted in all humble respect , IOHN IOHNSON , Suruaighor .
THE EPISTLE TO THE READER .

GEntle and Curteous Reader ; hauing for many yeares past spent my time both in reading , practising , and conferring with others in , and about the studie of the Mathematicall Sciences , and through great paines and trauell , at the request of diuers Worshipfull Gentlemen . Merchants , and others of my very louing friends , haue at last collected and gathered together many excellent Rules and easie Abreuiations in the Science of Arithmatick , which at the entreaty , and by the meanes of the helpe of some of them , I haue at last made bold heere to present abroad vnto the worlds view , the first fruites of some idle houres studie , the most part whereof I doe acknowledge to haue gotten by the practice and vse of the most excellent Instrument , inuented by Master William Pratt , called , The Iewell of Arithmatick in which I haue done the best of my Indeauours , not to hide that Tallent in the earth , which God hath bestowed vpon me for the benefit of others , but rather to his great glory and prayse , and for the benefit of my Country , and for the furthering of all that are studious in the Art of Numbers , I haue laboured to set it forth in the most briefe , plaine , and easie manner that I could fit for the vnderstanding of the weakest and meanest capacitie . In which if any thing shall seeme obscure or doubtful to any man , I could wish my selfe were present to resolue his doubts , for I haue indeauoured to make the Rules as briefe , short and easie , as I could deuise .

In my first Book I haue intreated concerning vulgar Arithmatick , with new inuentions of my owne , in all the first foure parts of Arithmatick , viz in Addition and Subtraction , with two seuerall kinds of Multiplication , not charging of the memory , neuer extant before in any Author that I haue read , with foure seuerall kinds of Diuision , the latter of them bringing the proofe by Addition of the figures vnder the diuidend , without any multiplication , or casting away of nines , according to the accustomed manner .

Againe , in the worke of Fractions , I haue set them forth in plaine and perfit figures after another manner of my owne inuention , because the fractionall figures in most bookes of Arithmatick were so vnperfit , that they were scarce to be descerned , and in this manner they will performe all factionall operations , as well as if they were set out according to the vsuall manner . In the end of which Rules I haue shewed the reasons and proofes of fractions by the known parts of Coyne .

Thirdly , in the second part of the former booke , I haue set forth Reduction , both in Coyne sterling `waights , measures , time and motion ; the Tables whereof are in the first part of the book , with diuers Rules how to bring pence , or farthings at the first worke into pounds , shillings and pence ; with diuers questions wrought by Reduction , with Progression Arithmaticall and Geometricall , with examples .

And lastly , I haue shewed how to worke the Rule of 3 Direct and Conuerst , both in whole numbers and fractions , after diuers seuerall manners of workings , and how to find the diuisor in any question , as also diuers wayes to worke Fellowship , Barter , Exchange , Allegation , Interest , Position , and all other operations Arithmeticall , with examples and briefe Rules of euery part .

In my second booke of Decimall Arithmatick , I haue first described out the parts and vse of the decimall Table , and how to set forth any number giuen in Decimalls .

Secondly , I haue shewed how to worke all the seuerall parts of Arithmatick , viz. Numeration , Addition , Subtraction ▪ Multiplication and Diuision in Decimalls ; with examples and proofes of euery worke in the knowne parts of Coyne .

Thirdly , I haue handled in as briefe manner as I could , the Rule of 3 Fellowship , Barter , Exchange and Interest in Decimall Arithmatick , as before in vulgar , in which you may perceiue the great labour that is auoided , in vulgar Arithmatick , with diuers examples and proofes of the same .

Lastly , I haue added a small Treatice of Interest and Annuities ; with the manner how to calculate Tables or Breuiats at any rate , or yeares purchase giuen ; all which I haue drawne into a pocket vollume . If therefore any Gentleman , Merchant , or other , be desirous to haue further instruction , if they repaire to my lodging in Coleman-street , I shall be ready to giue them any satisfaction .

If therefore I shall find these my labours and indeauors to take that effect , which I do hope and wish for , I shall be thereby the sooner incouraged to hasten the comming forth of the third part of this volume concerning the extraction of Rootes , with many easie operations and rules showing the vse of the Square and ●ubicque rootes , concerning Mensurations of land Timber , Board , Glasse and Stone , and the reduction of Measures from one proportion to another by their squares giuen : and lastly , concerning Millitary affaires and Gunners Art ; concerning the Arithmeticall worke , with demonstration by examples , which I had intended to haue ioyned to this vollume , but that it would haue increased it far beyond a pocket book . And so hoping of your friendly censure & acceptance of these first fruits of my labours , I cease , hoping to haue my true indeauors and meaning well taken , and the faults in the Printing friendly amended , desiring a blessing from God vpon these my poore labours , I take my leaue ,

London in Coloman-Streete , this 18 of August , 1622. Iohn Iohnson ▪
A Table of the Contents of the first Booke .
CHAP. I. OF Numeration , with examples . CHAP. II. 2. Of Addition in Coyne sterling , Waights , liquid , drie , and long measures of Time and Motion . 3. Examples and questions wrought by Addition , with two seuerall proofes of Addition . CHAP. III. 1. Of Subtraction , with examples of Coyne , Waights , Measures , Time and Motion , with the proofes of the same . 2. How to subtract from a vnite in any pl●●e , any numbers , and to shew the remainer at first sight of the worke . 3. The proofe of Subtraction two seuerall wayes . CHAP. IV. 1 Of Multiplication , with the Table , and the vse of the same . 2. Examples after the vsuall manner , with the exposition of the same . 3. A second way to multiply , not charging of the memorie with bearing any numbers in mind , to be added in the next place , with examples , and exposition of the same . 4. A third way to multiply and bring the product in the last line ; with examples and exposition of the same . CHAP. V. 1. Of Diuision vulgar , after the vsuall manner , with examples and exposition . 2. A second manner of Diuision , more easie and speedie , with lesse charge to the memory . 3. A third kind of Diuision , more easie and certaine , bringing the proofe by Addition , without multiplication , or making any new worke for the proofe . 8. How to diuide by a vnite with Cyphers . 6. Briefe Rules by Multiplication and Diuision .
The Table of the second part of the first Booke . The Rule of Reduction . 1 Reduction of Coyne vnto 6 To bring pence into pounds , shillings and pence at the first worke by Diuision . 6 To bring farthings into pounds , shillings and pence at the first worke by Diuision . 7 A second way to bring pence , or farthings , into pounds , shilling and pence . 10 Reduction of waights . 11 Reduction of measures . 12 Reduction of time . 14 Reduction of motion . 16 Questions of Reduction vnto the 26 Of Progression Arithmeticall . What Progression Arithmeticall is . 27 To find the summe of a Progression . 27 To find the latter terme of a Progression . 30 To find the number of Termes . 31 To find the excesse or difference . 32 To find any middle terme . 32 To find what number shall begin and finish any progression , with examples . 33 Of Progression Geometricall . What Geometricall Progression is . 37 To find any terme giuen in a Progression . 38 To find the summe of a Progression . 41 Examples of Progression and proofe . 44 Of Fractions . Of Fractions , and how to worke them according to my owne inuention . 47 How to reduce fractions of fractions . 48 How to reduce fractions of Integers . 50 How to proue a fraction by the parts of Coyne . 52 Addition in fractions . 55 Proofe of Addition by parts of Coyne . 57 Subtraction in fractions . 58 Proofe of Subtraction by the parts of Coyne . 59 Multiplication in fractions . 59 Proofe of Multiplication by parts of Coyne . 60 Diuision in fractions . 61 Proofe of Diuision by parts of Coyne . 63 How to worke whole numbers with fractions . 64 How to worke whole numbers and fractions with fractions . 67 How to abreuiate a fraction . 68 How to find the value of any fraction . 70 How to change the sirname of a fraction . 71 Questions of fractions vnto 76 Rules of Practice . Rules of Practice by the first Table . 77 The first and second Table . 78 Rules of Practice by the second Table . 84 How to proue questions in Practice . 89 How to proue one question in Practice , by the working of another . 91 The third and fourth Table of Practice . 94 Rules of Practice by the third Table : 95 Rules of Practice by the fourth Table . 102 Generall Rules of Practice without Tables . 104 Another way to worke Practice . 111 The Golden Rule . Of the Rule of Three Direct . 114 A second way to worke the Rule of Three . 116 To know if a question giuen be to be answered by the Rule Direct , or Conuerse . 120 To find if any number giuen bee proportionall , or not 132 The Rule of Three in fractions . 142 A generall Rule . 156 How to worke the double Rule of Three at one operation . 157 Fellowship without time . 159 Fellowship , with diuersitie of tim● . 164 Position Single . Position single , requiring one faigned number . 172 Position wrought a second way . 175 Double Position . The Rule of Double Position . 178 Barter or Exchange . 185 Of Gaine and Losse . 193 To worke Compound interest at any rate . 198 How to gaine any rate in the hundred . 204 Equation of Payment . 208 Alligation Mediall . 212 Alligation Alternate . 215
The Table of the second Booke of Decimall Arithmatick . THe declaration of the parts of the Decimall Table . 219 To find the value of a Decimall in the knowne parts of Coyne . 220 Numeration in Decimalls . 222 How to set out a penny in Decimalls . 223 How to breake a pound into his exact parts . 225 How to expresse any numbers in Decimalls . 226 How to remoue a Decimall from one place to another . 227 Addition in Decimalls . 230 Subtraction in Decimalls . 232 Multiplication in Decimalls . 234 To change any fraction into Decimalls . 238 Diuision in Decimalls . 241 To diuide the smaller number by the greater . 242 To find the prime line in any Diuision . 245 Reduction in Decimalls . 251 Rules of Practice in Decimall . 258 To find the price of a vnite in any place of 10 , 100 , 1000 , 10000 , &c. 260 The price of any number of yards , ells , or pounds giuen , to find the price of a vnite . 264 The Golden Rule in Decimalls . 268 Diners wayes to worke the Golden Rule in decimalls . 275 Briefe Rules of abreuiating your worke by proportions . 276 Questions wrought without Reduction in Decimalls a second way . 284 Position in Decimalls . 293 Gaine and Losse in Decimalls . 298 How to work gaine & losse in pence & fathings . 315 The proofe of many examples . 320 Exchange in Decimalls . 329 a general Rule for exchange in Decimalls . 335 Reduction of Measures . 339 Of Interest and Annuities . How to frame tables to worke Compound Interest at any rate in the hundred . 345 How to calculate the table of 10 li. per cent . 346 The table of 10 li. per cent compoundintrest . 349 How to calculate a Table at any other rate , vnder or about tenne pound in the hundred compound Interest . 350 The Bre●iate of 8 pound in the hundred . 352 The vse of the Breuiats , or Tables . 353 To find what I pound due at any number of yeares is worth at the end of the terme . 353 To find what any yearely annuitie will make to bee payd at the end of the terme . 355 To find what any debt due at the end of any number of yeares is worth in ready money . 358 To find what any yearely annuitie at the end of any number of yeares is worth in ready money . 362
The end of the Table .
IOHNSONS ARITHMETICK .
CAHP. I. Numeration .

NVmeration is the first part of Arithmaticke , which sheweth how to pronounce the value of any number of figures giuen ; which are expressed by tenne figures , whereof the tenth is a Cypher , signifying nothing of it selfe ; but being ioyned with figures , helpeth to increase the value : the figures are these ;

one , two , three , foure , fiue , six , seuen , eight , nine , cipher . 1. 2. 3. 4. 5. 6. 7. 8. 9. 0.

How to expresse the value of a number giuen .

If a number be giuen , whose value is to be expressed , you shall vnderstand , that the figure next the right hand is the least in valew , and signifieth simply his owne valew , as the figure of 1 doth signifie but one , and the figure of 2 doth signifie but two , and the figure of 8 signifies but eight , and so of any other . And in the second place towards the left hand , euery figure is in valew tenne , so that the figure of one there doth signifie tenne , the figure of 2 twenty , the figure of 8 eightie , and so of all other : in the third place towards the left hand , euery figure is in valew one hundred , so that the figure 1 in that place signifies one hundred , the 2 , two hundred , &c. In the fourth place , euery figure is in valew one thousand , so there the figure of one signifies one thousand , the figure 2 , two thousand , &c. In the fifth place , euery figure is in value ten thousand : in the sixth place , one hundred thousand ; and in the seuenth place , one thousand thousands , or one million : in the eight place , ten millions : in the ninth place , one hundred millions : in the tenth place , one thousand millions , or one milliot ; and so infinitely names may be giuen to the valew of euery pricke , as is vsuall in the second part of Arithmetick , of Number , Square , Cube , sursolid , &c. or in Astronomicall Arithmetick , Primes , Seconds , Thirds , Fourths and Fifths , &c.

Now to expresse the valew of any number giuen , set a prick with the pen ouer the fourth figure towards the left hand , and ouer the seuenth , and tenth ; and so ouer euery third figure towards the left hand , to the end of your figures , as in this Example :

Now begin and expresse the first foure figures towards the right hand , as if they stood alone , which are 2567 , or two thousand fiue hundred sixty seuen . Then reade the figures belonging to the second prick , which are 430 , as if they stood alone thus , foure millions three hundred two thousand fiue hundred sixty seuen : then take the three figures belonging to the third prick , which are 635 , or sixe millions three hundred fiftie foure millions three hundred and two thousand fiue hundred sixtie seuen : and so this whole summe is thus to bee read , two hundred thirty seuen thousand eight hundred fifty sixe milliots three hundred fiftie foure millions three hundred and two thousand fiue hundred and sixty seuen ; and so of any other summe .

CHAP. II. Addition .

ADdition is the second part of Arithmatick , and serueth to adde or collect diuers summes of seuerall denominations , and to expresse their totall value in one summe .

In Addition begin to adde your sums at the right hand with the smallest numbers or denominations first , and gathering of their totall mark how many of the smaller makes one of the next greater ; as if your addition be Farthings , for euery foure farthings carry one peny in mind to be added to the numbers in the place of pence , and for euery 12 put one shilling into the number of shillings , and for euery 20 shillings , one pound into the place of pounds ; and therefore to know how many of the smaller denominations , makes one of the next greater : I haue here added in this place the seuerall Tables of Coyne sterling , of Weights , of liquid Measures , and drie Measures , of long Measures , of Time and Motion ; which are very necessary to be knowne of euery Practicioner in Arithmatick , before he proceeds any further in the practice of Arithmatick , being vsed in euery particular Rule of Arithmatick more or lesse .

TheTableofCoyneSterling. Pence Farth . Foure farthings makes one peny — 1 4 One shilling is — 12 48 One pound Sterling is 20 shillings — 240 960 One hundred pound Sterling is ▪ — 24000 96000

Example . l. s. d. l. s. d. q. 785976. 17. 3 324. 8. 11. 1 80254. 10. 7. 222. 17. 3. 1 23547. 11. 0. 187. 10. 2. 0 7853. 12. 2. 354. 12. 1. 0 248. 00. 0. 1856. 00. 2. 0 93. 10. 1. 7859. 1. 11. 1 7. 11. 3. 3275. 1. 9. 0 Su. 897981. 12. 4. 14079. 12. 3. 3.

The explanation of these examples .

In the first example toward the left hand I begin with farthings , which are 3 , which I set downe : then next 9 pence and 11 is 20 , and 2 is 22 , and 1 makes 23 , and 2 makes 25 , and 3 makes 28 , and 11 makes 39 pence , or 3 shillings 3 pence ; I set downe the 3 pence , and carry in mind the 3 shillings to be added to the place of shillings . Then adde the seuerall summes of shillings , which are 1. 1. 2 7. 8 , the totall is 19 , and the 3 in mind makes 22 shillings ; set downe the 2 shillings , and keepe two tennes to be added to the tennes of shillings , which are 3 tens , which makes 5 tennes , or 50 shillings ; set downe the odde tenne to the 2 shillings , which makes 12 shillings , & carry 2 pound for the forty shillings to the next place of pounds , which are 5. 9. 6. 4. 7. 2. 4 , and the 2 in mind makes 39 ; leaue the 9 vnder the place of vnites , and carry 3 tennes in mind , and 7. 5. 5. 5. 8. 2. 2 , totall is 37 ; set downe the 7 vnder the place of tennes , and carry 3 in mind for the 30 tennes , which is 3 hundred : then 3 in mind , and 2. 8. 8. 3. 1. 2. 3 , totall is 30 ; set a cypher , or 0 in the place of hundreds , and carry 3 for the 30 into the place of thousands : then last of all , 3 in mind , and 3. 7. 1 makes 14 thousand , and because it is the last summe , you must set them all downe , placing the 4 vnder the place of thousands , and the 1 one place more towards the left hand , and then the Totall summe of those particulars will be 14079 pound , 12 shillings , 3 pence , 3 farthings , as appeareth in the example ; and in the like manner is the other example to bee cast vp into one Totall : and so I will here end with Addition of Coine , & put a seueral example of euery table for the full tables & perfect vnderstanding of the said table , which are of great vse in all the seuerall rules of Arithmetick .

TheTableofHaberdepoyseweight. Haberd . the pound . oun . Dra. Scruple Grain● . One pound is — 16 128 384 7680 One half pound is 8 64 192 3840 One quarter of a pound is — 4 32 96 1920 One eighth of a pound is — 2 16 48 960 One sixteenth of a pound is — 1 8 24 480

The Hundred . Po● . Ounc . Dra Scruple . One hundred is — 112 1792 14336 43008 One half hundred is — 56 896 7168 21504 One quarter hundred is — 28 448 3534 10752 One half quarter hundred is — 14 224 1792 5376

Example of Weights . C. qu. li. o●● . C. q. li. o●● . d● . 27. 3. 27. 6 ▪ 127. 3. 17. 8. 3. 18. 1. 17. 12. 118. 2. 10. 12. 1. 13. 2. 10. 3. 33. 0. 0. 0. 0. 73. 0. 0. 5. 17. 1. 12. 2. 3. 83. 2. 5. 12. 22. 3. 1. 7. 0. 2 2 2 17. 0. 10. 3. 0. 2 1 2 211. 2. 6. 6. 336. 2. 24. 00. 7.

The Explanation .

In the Haberdepoyse waight , 20 graines makes one scruple , 3 scruples one dram , 8 drams one ounce , 16 ounces one pound , 112 pound is one hundred of the Haberdepoyse weight , wherby is sold all kind of Merchandise vsuall in this Realme , and therefore in Addition of Waights Haberdepoyse , for euerie 3 scruples adde one dramme , and for euery 8 drams one ounce , and for 16 ounces 1 pound , for 28 pound one quarter of a hundred , and for euery 4 quarters one hundred . first , I begin with the drams in the first example to the right hand , which are 3. 1. 3 , totall is 7 drames , which I note downe vnderneath , because they are lesse then one ounce . Secondly , the ounces are 3. 7. 2. 12. 8. totall is 32 ounces , or 2 pound , because 16 ounces is one pound ; which two I set vnder the place of pounds with a light touch of the penne for to remember it the better , and place a Cypher in the place of ounces .

Thirdly , the pounds are 2. 10. 1. 12. 10. 17 totall is 52 pound , which is one quarter of a hundred , and 24 pound , place 24 pound vnder the place of pounds , and put one quarter , as before in the place of quarters of hundreds .

Fourthly , 1. 3. 1. 2. 3 quarters , are 10 quarters , or 2 hundred and 2 quarters , or halfe a hundred ; place 2 quarters in the place of quarters , and put ouer 2 into the place of hundreds for the 8 quarters .

Then 2. 7. 2. 7. 3. 8. 7 makes 36 hundred , place 6 , and carry 3 for the 30 : then say , 3. 1. 2. 1. 3. 1. 2 , totall is 13 ; place 3 there , and carry one for the 10 , which one in mind , and 1. 1 makes 3 , which set downe , and the total is 336 hundred , 2 quarters , 24 pound , 0 ounces , 7 drammes ; and so the other example is in the same manner to be cast vp , and so of all other .

TheTableofLiquidMeasures. Pints . One pound or pint — 1 One quart — 2 One pottle — 4 One Gallon — 8 8 Gallons , a Firkin of Ale , Sope , or Herring — 64 One Firkin of Beere — 72 One Firkin of Salmon , or Eles — 85 2 Firkins , or one Kilderkin of Beere — 128 2 Kilderkins , or one Barrell — 250 One Tirce of Wine — 336 63 Gallons one Hogshead of Wine — 504 2 Hogsheads , or a Pipe or Butt — 1008 2 Pipes , Butts , or a Tunne of Wine — 2016

TheTableofDryMasures. Pints . One Pint — 1 One Quart — 2 One Pottle — 4 One Gallon — 8 One Pecke — 16 4 Pecks one Bushell Land-Measure — 64 5 Pecks , one Water-bushell — 80 8 Bushels one Quarter — 512 4 Quarters , on Chaulder — 2048 5 Quarters one Waye — 2560

TheTableofLongMeasures. Inch. Three Barley Cornes in length , one Inch — 1 One Foote — 12 One Yard , or 3 Foote — 36 Or 3 Foote 9 Inches , an English Ell — 45 Or 6 Foote one Fadom — 72 Or 5 Yards and half , a Pole or Perch — 198 Or one Perch in bredth , and 40 long , one Roode — 198 Or 4 Perches breadth , and 40 long , an Acre of land — 792 160 Square Perches , is one Acre — 792 40 Roddes in length is one Furlong , and 8 Furlongs is an English Mile .

TheTableofTime. Minut. One Minute — 1 One Hower — 60 One Day naturall , or 24 Howers — 1440 One Weeke , or 7 Dayes — 10080 One Moneth , or 4 Weeks , or 28 Dayes — 40320 13 Moneths one Day 6 Houres , or 365 Dayes , one Yeare — 525960

TheTableofMoti●●. 360 Degrees , 21600 Minutes , 129600 seconds — 12 Signes . 30 Deg. 1800 min. 108000 sec. — 1 Signe . 1 deg. 60 min. 3600 sec. — 1 Degree . 1 min. is 60 sec. — 1 Minute . 1 second — 1 Second . 7776000 thirds makes the 12 Signes — 1 Third . 466560000 fourths makes the 12 Signes — 1 Fourth . 27993600000 fifths is 12 signs 1 Fifth . 1679616000000 sixths is 12 Signes — 1 Sixth .

The explanation of these Tables , and the examples following .

First , in the example of Acres , Roods and Perches ; for 40 Perches put 1 Rood into the place of Roods , and for euery 4 Roods one Acre .

Secondly , for euery 4 quarters of Inch , take 1 Inch , and for euery 12 Inches 1 foot , and for euery 3 foote , one yard .

Thirdly , for 16 pints take one pecke , and for euery 4 peckes one Bushell , into the place of Bushels .

Fourthly , for euery 8 pints of liquid measure , take one Gallon , and for euery 63 Gallons one Hogshead .

Fifthly , in the example of time ; for 60 , minuts take one houre , and for 24 houres one day , and for 365 dayes , one yeere .

Sixthly , for 4 nayles take one quarter of a yard , and for 4 quarters one yard , &c.

Lastly , in the example of motion , for 60 thirds , take 1 second , & for 60 seconds take one minute , and for 60 minutes take one degree , and for 30 degrees take one Signe .

And this is the vse of these Tables in Addition and Subtraction ; for looke what you carry ouer in Addition , that you must borrow in Subtraction , I wil heare adde examples of euerie kind , leauing the Reader to exercise himselfe by the Rules before taught .

Example . Acres , Rood . per. Feet . Inch. quart. 127. 3. 21 124. 7. 3 246. 1. 12 246. 11. 4 17. 3. 22 134. 7. 2 27. 1. 8 120. 8. 0 37. 0. 17 72. 10. 2 2 2 3 2 456. 2. 00 ▪ 699. 9. 3.

Bushel , Peeks , Pints . Yard . Quar. Naile . 127. 3. 11. 127. 2. 3 256. 1. 7 359. 1. 4 345. 0. 0 152. 3. 0 184. 2. 10 16. 0. 0 1 1 1 1 913. 3. 12 ▪ 655. 3. 3.

Yeares , dayes , houres , min. seconds . 356. 245. 16. 35. 20. 249. 100. 12. 30. 00. 756. 12. 00. 10. 12. 140. 27. 30. 25. 02. 1618. 00. 20. 00. 00. 1 3 1 3120. 22. 07. 40. 34.

Signes , degrees , minutes , seconds , thirds . 11. 22. 32. 24. 18. 8. 19. 17. 20. 12. 10. 07. 00. 08. 15. 2. 17. 35. 50. 59. 3. 29. 30. 12. 00. 3 1 1 1 37. 05. 55. 55. 44.

The Proofe of Addition .

The proofe of Addition is made by Subtraction ; for if you subtract the numbers which you added from the totall of the Addition , there will remaine nothing , if the worke be truly done .

Example . l. s. d. q. 378567. 19. 10. 1. 240023. 10. 2. 0. 854●26 . 07. 1. 0. 785634. 13. 3. 2. 320500. 00. 11. 1. 2 2 Totall , 2579●52 . 11. 4. 0.

First , adde together the greatest Summes in valew in the place of hundred thousands , which makes 23 , which take from 25 , and there will remaine 2 : then the figures in the fifth place , 26 taken from 27 , there will remaine 1. Thirdly , the figures in the place of thousands , makes 17 , which taken from 19 , leaues 2 : then 19 in the place of hundreds taken from 20 , leaues 1 : and againe , 13 in the place of tennes from 15 , leaues 2 : and lastly , 20 in the place of vnites from 22 ponnd , leaues 2 pound : then 49 shillings from 2 pound 11 shillings , leaues 2 shillings ▪ also 2 shillings 3 pence in the place of pence , from 2 shillings 4 pence , leanes 1 : and last of all , 4 farthings from 1 penny , leaues nothing , which prooues the worke to be truly wrought .

The second proofe of Addition .

Cut of the vppermost numbers with a dash of the pen , and adde the remayner into one Totall ; and then subtract that sum from the whole totall , and the remayner will bee the numbers which you cut off , if the worke be true , else not .

Example . 378567. 19. 10. 1. 240023. 10. 2. 0. 854326. 7. 1. 0. 785634. 13. 3. 2. 320500. 00. 11. 1. 2 2 The total 2579052. 11. 4. 0. of all . Subt. 2200484. 11. 5. 3. the sum . The 378567. 19. 10. 1. proofe .

And so much shall suffice to haue spoken of Addition , and the proofe thereof .

Questions of Addition .

What number is that , to the which if you doe adde 45 , the totall will be 357.

Answer : Subtract 45 from 357 , remaines 312.

Example . 357 45 312

What three numbers are those , to which if you adde 27 , 36 , and 45 , their products shall be equall , and the summe arising shall be 120.

Proofe. 120 120 120 93 27 35 45 27 93 89 75 120

What number is that , to the which if you do adde 354 pound , 7 shsllings , 9 pence , the totall will bee 512 pound , 15 shillings , o penny . Answere : Subtract 354 pound , 7 shillings , 9 pence , from 512 pound , 15 shillings , o penny , and the remainder will bee 158 pound , 7 shillings , 3 pence , which is the number that you doe seeke .

Example . l. s. d. 512. 15. 0. 354. 7. 9. 158. 7. 3.

CHAP. III. Subtraction .

SVbtraction s●●●eth to deduct one sum from another , the lesser from the greater , and to shew the remaines .

Place your greater number , from which the Subtraction is to be made , in the vppermost part , and the number to be subtracted , or deducted right vnderneath euery figure vnder his like kind , or denomination , viz. pounds vnder pounds , shillings vnder shillings , and pence vnder pence , &c. in this manner .

l. s. d. q. Lent . 7756. 13. 10. 1. Payd 3949. 17. 11. 2. Rest . 3806. 15. 10. 3. Proofe . 7756. 13. 10. 1.

Then begin your subtraction at the left hand , at the smallest numbers ; but if the lowest figure of the vndermost numbers be the greatest , that it cannot be abated out of the number aboue it , then adde one of your next greater denomination , and make your subtraction from both , noting the remainer ; as if you haue 10 pence to take from 7 pence , adde one shilling , or 12 pence , vnto 7 pence , that maketh 19 pence ; then take 10 pence from 19 pence , and there will remaine 9 pence , which note downe vnder the 10 pence : and because you did borrow one shilling , therefore in the number of shillings you shall take away one more then it is , in the next place of shillings , and this rule is generall , in Coyne , Measure , Time , Motion , or any other thing else whatsoeuer .

1. Example of Subtraction of Coyne . l. s. d. q. Lent . 789786. 17. 11. 3. Paid . 692583. 19. 10. 1. Rest . 97202. 18. 1. 2. Proofe 789786. 17. 11. 3.

2. Example of Weights . C. que . l. oun . Lent . 127. 3. 27. 10. Paid . 38. 2. 24. 15. Rest . 89. 1. 2. 11. Proofe 127. 3. 27. 10.

3. Example of Time . Yeeres , daies , houres , min. Totall . 1618. 340. 20. 56. Deduct . 1581. 122. 15. 59. Rest . 0037. 218. 04. 57. Proofe . 1618. 340. 20. 56.

4. Example of Motion . Sig. Deg. Min. Second . Thirds . Totall . 11. 22. 36. 52. 40. Subt. 7. 29. 51. 42. 56. Rest . 3. 22. 45. 09. 44. Proofe . 11. 22. 36. 52. 40.

The explanation of these examples .

In the first example of Coyne , begin your subtraction at the right hand , saying ; 1 farthing from 3 farthings , leaues 2 farthings , which note downe vnder the 1 farthing . Then 10 pence from 11 pence , leaues 1 penny . Thirdly , 19 shillings from 17 shillings you cannot haue , therefore take one pound , or 20 shillings , and adde to 19 shillings , saying , 19 shillings from 37 shillings , rests 18 shillings , which note downe . Then 1 that you borrowed , & 3 pound , is 4 pound from 6 pound , leaues 2 pound to set downe vnder 3. Then 8 from 8 leaues nothing , place there a Cypher , or 0 vnder 8. Then 5 from 7 rests 2 ; then 2 from 9 leaues 7 , which also note againe ; 9 from 8 cannot be taken , then make it 10 more , and say 9 from 18 leaues 9 , which set downe : and last of all , 1 borrowed and 6 is 7 , from 7 leaues nothing , and the worke is ended , and the remayner will bee 97202 pound 18 shillings 1 penny 2 farthings , as appeareth in the example before going .

The exposition of the second example .

First , take 15 ounces from 10 , which cannot be , then adde 1 pound , or 16 ounces to 10 , makes 26 ; then say , 15 from 26 leaues 11 ounces , which note downe : then 1 borrowed and 24 is 25 , from 27 pound leaues 2 pound remaining ; then 2 quarters from 3 quarters , leaues 1 quarter remaining ; then 8 from 7 cannot bee , therefore take 8 from 17 , rest 9 , which note downe : then one borrowed and 3 makes 4 , from 12 rests 8 , and the worke is done , and the remaine is 89 hundred 1 quarter 2 pound 11 ounces .

3 Example .

First , take 59 minutes from 56 minuts cannot be , but then take 59 minuts from 60 minutes , or one houre , and there will remaine 1 minute , which adde to 56 minutes , and that will make 57 minutes , which note downe in the place of minutes : then 1 borrowed and 15 houres makes 16 houres , which taken from 20 houres leaues 4 , which note vnder the 15 ; and then 2 dayes from 0 cannot be , but 2 from 10 , and there will remaine 8 , which note downe : then 1 borrowed and 2 makes 3 , from 4 leaues 1 ; also 1 from 3 leaues 2 : lastly , 1 from 8 leaues 7 , and 8 from 11 leaues 3 ; then 1 borrowed and 15 makes 16 , from 16 leaues nothing , and the remayner will bee 37 yeares 218 dayes 4 houres 57 minutes ; the like is done in the other example of Motion , and therefore here needlesse to be rehearsed .

To subtract from a Vnite .

Set downe with your pen a vnite in any place , adding Cyphers vnto it , and the seuerall numbers which you will subtract from it of pounds , shillings and pence right vnderneath : then note what each seuerall number of your lowest numbers doth want of 9 vnto the place of vnites , and set that right vnder for the remayner : and lastly , note what your shillings and pence doth want of 20 shillings , and set that downe for your remaynor , and the worke is ended .

Example . l. s. d. Lent . 1000000. 00. 00. Payd . 232864. 17. 03. Rest . 767135. 2. 9. Proofe . 1000000. 00. 0.

The proofe of Subtraction .

The surest proofe of Subtraction is made by Addition : for if you doe adde the numbers remaining , vnto the numbers deducted , they will returne your former Summe , if the worke be truly wrought , as wil appeare in the proofe of all the seueral examples before going , and therefore here againe in this place needlesse to be rehearsed . Only I will adde one for examples sake .

In the last example , the numbers which did remaine , were 767135 pound 2 shillings 9 pence , and the numbers deducted , 232864 pound 17 shillings 3 pence ; these two numbers added together , ought to make a vnite in the seuenth place ; wherefore I adde 9 pence to 3 pence , makes 1 shilling ; and 1 shilling to 17 shillings , makes 18 shillings , and 2 shillings makes 20 shillings ; then 1 and 4 is 5 , and 5 is 10 , which is one in the next place : then 1 and 3 , and 6 is 10 ; and 1 1.8 makes 10 , and 1.7.2 makes 10 , and 1.6.3 is 10 , and lastly 1.7.2 makes 10 , or one vnite .

Multiplication . CHAP. IIII.

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

THis Table of Multiplication must bee learned perfectly by heart , for to know readily what the multiplication of any two digit numbers vnder nine , or vnto nine doe make , and then Multiplication will be very easie : for Multiplication is a number of additions speedily performed ; as if you should say , How many in number is 8 times 7 , if you should set downe 7 eight times one vnder another , and adde them together , the totall will be 56 : but if you looke in the Table for 8 in the head , and 7 in the side , you shall finde vnder 8 , right against 7 in the the same paralell 56 ; or if you find 8 in the side , and 7 in the head , the like number will appeare , and these numbers in the table are to be fit in memory .

1. Example according to the vsuall way . 87968. The multiplicand . 987. The multiplier .

First , begin your multiplication at the right hand , saying , 7 times 8 make 56 , place 6 vnder the 7 , and keepe 5 in mind , to bee added to the product of the multiplication of 7 by 6 , saying , 7 by 6 makes 42 , and 5 in mind is 47 ; set 7 downe vnder the 6 , and keepe 4 in mind : then 7 by 9 is 63 , and 4 makes 67 ; set 7 downe , and keepe 6 in mind : then 7 by 7 is 49 , and 6 is 55 ; place 5 and keepe 5 in mind : lastly , 7 by 8 is 56 , and 5 is 61 , which set downe the 1 first , and the 6 one place more towards the right hand ; and so the multiplication by the first figure 7 is done , then cancell the 7 of your multiplyer , and your worke will stand , as in this example .

Secondly , begin with 8 , the second figure of your multiplier , saying , 8 times 8 is 64 ; place the 4 vnder the said 8 , and keepe the 6 in mind : then 8 by 6 is 48 , and 6 makes 54 ; set downe 4 in the next place , and keepe 5 in mind : then 8 by 9 is 72 , and 5 makes 77 ; set downe 7 , and keepe 7 in mind : then 8 by 7 is 56 , and 7 makes 63 ; set downe 3 , and keepe 6. Lastly , 8 by 8 is 64 , and 6 makes 70 ; set the 0 first , and the 7 one place more towards the left hand , and cancell the 8 of your multiplier , and the worke will stand thus .

Thirdly , begin with 9 , the last figure of your multiplyer , saying , 9 by 8 is 72 ; place the 2 vnder the said 9 , and keepe 7 : then 9 by 6 is 54 , and 7 is 61 ; place 1 , and keepe 6 : then 9 by 9 is 81 , and 6 is 87 ; place 7 , and keep 8 : then 9 by 7 is 63 , and 8 is 71 ▪ place 1 , and keepe 7 : last of all , 9 by 8 is 72 , and 7 is 79 ; place the 9 first , and the 7 one place more towards the right hand , and the whole worke is ended , then gather the totall by addition .

1. Example .

2. Examples with Cyphers .

The exposition of this example .

First , 7 by 1 is 7 , which note downe : then 7 by 0 is nothing , set down a 0 in that place : and next 7 by 5 is 35 , set 5 , and carry 3 : then 7 by 6 is 42 , and 3 is 45 , place 5 , & carry 4 : then 7 by 8 is 56 , and 4 is 60 , set downe a 0 , and carry 6 againe : 7 by 2 is 14 , and 6 makes 20 , set downe a 0 , and carry 2 : then 7 by 3 is 21 , and 2 makes 23 , place 3 , and carry 2 : then 7 by 0 is 0 , leaue the 2 in that place : then lastly , 7 by 7 is 49 , being the last number set downe all the 9 vnder 7 , and the 4 one place more to the left hand , and the worke will then stand thus .

Secondly , cancell 7 , and then say ; 5 by 1 makes 5 , place that 5 vnder the 0 ; and then 5 by 0 is 0 , place a 0 vnder the 5 in the next place ; and then 5 by 5 is 25 , set downe 5 , and carry 2 : then 5 by 6 is 30 , and 2 makes 32 , set downe 2 , and carry 3 : then 5 by 8 is 40 , and 3 makes 43 , place 3 , and carry 4 : also 5 by 2 is 10 , and 4 makes 14 , set downe 4 , and carry 1 : then 5 by 3 is 15 , and 1 makes 16 , set downe 6 , and carry 1 : then 5 by 0 is 0 , set downe the 1 there : last of all , 5 by 7 is 35 , set them all downe , and the worke will then stand thus .

Thirdly , cancell the 5 , and then say , 0 by 1 is 0 , place a 0 vnder the 0 of your multiplier , & then proceed to the next figure of your multiplier , which is 2 , saying , 2 by 1 is 2 , place the 2 vnder the said 2 of your multiplier : then 2 by 0 is 0 , which set downe : then 2 by 5 , makes 10 , set downe a 0 , and carry 1 : then 2 by 6 is 12 , and 1 is 13 , set downe 3 , and carry 〈…〉 : also 2 by 8 is 16 , and 1 is 17 , set downe 7 , and carry 1 : also by ● is 4 , and 1 makes 5 , which set downe : againe , 2 by 3 is 6 , which set downe : lastly , by 7 is 14 , which set downe , and the worke will stand , as in this example .

Fourthly , cancell the 2 , and say , 3 by 1 is 3 , which place right vnder the said 3 : then 3 by 0 is 0 , and worke in all respects as before , and the worke being ended , will stand thus .

I will here adde some few examples to be wrought by the pen , without any troubling of the memory with bearing ought in mind .

Example .

2. Example .

The explanation of the worke by the pen , without charging the memorie .

The first example .

First , I multiply al the figures of my multiplicand by 7 , the lowest figure of my multiplier , saying , 7 by 8 is 56 , put 6 vnder the 7 , and 5 vnder the 8 : then 7 by 6 is 42 , leaue the 2 vnder 5 last placed , and set the 4 one place more towards the left hand vnder the 9 : then 7 by 9 is 63 , leaue 3 vnder the 4 last placed , and set 6 one place more to the left hand vnder 7 : then 7 by 7 is 49 , leaue 9 vnder the 6 last placed , and the 4 , set one place more to the left hand vnder the 8 : lastly , 7 by 8 , makes 56 , leaue 6 vnder the 4 , & place 5 one space more to the left hand , as before , then cancell 7 of your multiplier , and the multiplication by the first figure is ended , and the worke will stand thus .

Example .

Then for the second worke , say , 8 by 8 is 64 , place 4 vnder the said 8 , and put 6 vnder the next figure 3 : then 8 by 6 makes 48 , leaue 8 vnder 6 , and put 4 vnder the next 9 : and so working in all respects as at the first , and your second worke will stand thus , as in this example .

Lastly , cancell 8 your multiplier , and then multiply by 9 , as is before taught , placing the first figure of your product vnder the figure multiplying , and the worke being ended , it will stand thus ; and lastly , gathering the totall by addition , it is 86824416 ; as in this example .

There is no difficulty in this kind of working , but onely when there falls a 0 in in the multiplicand , or multiplier ; for if there be a cypher then you must fill vp the places as you worke , either with pricks , or cyphers , as if you had figures to set in their places , and the rest of the work is as before , is taught in the third example ; but I will here adde one example , hauing all the difficulties that may happen , for the better vnderstanding hereof .

Example .

Example .

How to multiply , and to bring the product in the last line .

Place your numbers right one vnder the other , as in the common way ; then make a right line somewhat distant from the first numbers with your pen , as in the example following .

Then begin and say , 7 by 8 is 56 , place the 6 vnder the line vnder the 7 , and the 5 aboue the line in a smaller figure in the next place towards the left hand : then 7 by 6 is 42 , and the 5 aboue the line makes 47 , leaue 7 vnder the 8 , and set the 4 againe aboue the line : then 7 by 9 is 63 , and the 4 aboue the line makes 67 , place 7 there , and set the 6 in the next place aboue the line : then 7 by 7 is 49 , and 6 aboue the line makes 55 , leaue 5 there , and put 5 againe ouer the line : lastly , 7 by 8 makes 56 , and the 5 last placed makes 61 , place that whole summe vnder the line , and the worke wil stand , as aboue in the example .

Secondly , draw a line againe a little disstant , as before from the last product , as in the example following .

Example .

Then say , 8 by 8 is 64 , and 7 makes 71 , place 1 vnder the 7 , and set 7 aboue the line : then 8 by 6 is 48 , and the two seauens betweene lines makes 62 , place 2 vnder the 7 , and set 6 againe ouer the line : then 8 by 9 is 72 , and 6 makes 78 , and 5 makes 83 , place 3 vnder the line , and 8 aboue the line ; then 8 by 7 is 56 , and 8 makes 64 , and 1 makes 65 ; place 5 vnder the line , and set 6 aboue : lastly , 8 by 8 is 64 , and 6 makes 70 , and 6 makes 76 , place them both downe ; and the worke will stand as aboue in the example .

Thirdly , draw a line againe , as before , a little distant from the last product , as in this example .

Example .

Thirdly , say 9 by 8 is 72 , and 2 makes 74 , place 4 , and put 7 ouer the line : then 9 by 6 is 54 , and 10 makes 64 , place 4 vnder the line , and put 6 aboue : then 9 by 9 is 81 , and 11 aboue makes 92 , leaue 2 vnder the line , and 9 ouer the line : then 9 by 8 is 63 , and 15 makes 78 , leaue 8 vnder the line , and 7 aboue : lastly , 9 by 8 is 72 , and 14 makes 86 , place them both vnder the line , and then bring downe the two figures which are cut off by two right downe lines , which are 1 and 6 , and the worke is ended , and the worke will stand , as appeareth in the example aboue , and the totall product is in the last line , 86824416 ; and this doth not charge the memory , for al the figures are set downe in view , and to bee seene at the first sight , and this is the second kind of multiplication , without charging of the memory .

CHAP. V. Diuision .

SEt your Diuidend , which is the number to be diuided in the vpper part , and the Diuisor next to the left hand , vnder the greatest figures in value of your Diuidend : If the vpper numbers bee greater then the lower , or else place your diuisor one place more towards your right hand , as in this example .

If you would diuide 78567 by 84 , place them as aboue ; for because you cannot haue 8 out of 7 in the Diuidend , therefore place your 8 one place more towards the right hand , and the 4 next to it , and your quotient you must place at the right side of your numbers behind a crooked line . But I will first giue an example of Diuision by one figure : I would diuide 65490 pound amongst 5 men ; place your numbers thus .

Example .

First , I seeke how oft 5 is in 6 , this I may haue but once ; then put 1 in the quotient beyond the crooked line , and take 5 out of 6 , and there will rest 1 , set that ouer 6 , and then remoue your diuisor one place more to the right hand , and then seeke you how many times 5 may be had in 15 , and the answer is , thrice , therefore place ; in the quotient , and by it multiply your diuisor 5 , makes 15 , which taken out of 15 , leaues nothing , place a 0 ouer the 5 , and remoue your diuisor , and seeke how ost you may haue 5 in the 4 ouer it , but you cannot haue it once ; wherefore put a 0 in the quotient , and remoue your diuisor , and seeke how many times you may haue 5 in the figures ouer and behind it , which are 49 , and you may haue it nine times , put 9 in the quotient , and by it multiply your diuisor 5 , makes 45 , which taken from 49 , leaues 4 , which place aboue the 9. And lastly , remoue againe your Diuisor 5 vnder the 0 , and seeke how many times 5 is in 40 , and you shall ●nd it 8 times , place 8 in the quotient , and by it multiply 5 , makes 40 which taken from 40 , leaues nothing remaining and the worke is ended , and will stand thus , as in the example , and I find , if I diuide 65490 pound amongst 5 men , euerie man shall haue for his part 13098 pound .

And this is the order of Diuision for one figure : but if your Diuisor doe consist of more figures then one , then you must take the first figure of your Diuisor no oftner out of the Diuidend , then you can also take euery seuerall figure of your Diuisor , out of the same figures of the Diuidend standing aboue them , as for example .

If you would diuide 86824416 by 987 , which was one of the products of the multiplications in the rules before going , for a triall of your former worke , then place your numbers , as in the example following .

Example .

Then I seeke how oft I may haue 9 in 86 , I find I may haue it 9 times ; but if I consider the next figure 8 of my Diuisor , I cannot haue also 9 times 8 out of the numbers remaining ; if I take 9 times 9 , which is 81 , out of 86 , there will remaine but 5 ; and then 9 times 8 , the next figure of my diuisor , makes 72 , which cannot be taken out of 58 which will remaine ; therefore I place 8 in the quotient , and by that I multiply all the figures of my Diuisor , 987 makes 7896 , which taken from 8682 , leaues 786 aboue them : and the worke will stand thus .

Secondly , I remoue my Diuisor 987 one place nearer the right hand , and then I seeke how oft I may haue 9 in 78 , which I see I can haue but 7 times , so I put 7 in the Quotient , and by that 7 , I multiply my Diuisor 987 , makes 6909 , which taken from 7864 , the numbers aboue them there will remaine 955 , and the worke will stand thus .

Example .

Thirdly , againe I remoue my diuisor 987 one place nearer the right hand , and seeke how many times I may haue 9 in 95 , and I find I may haue it 9 times , which 9 I set into the Quotient , and by it multiply 987 , makes 8883 , which taken from 9554 leaues 671 , and the worke will stand thus .

Example .

Fourthly , I remoue my Diuisor againe , and seeke how oft I may haue 9 in 67 , and I see I can haue it but 6 times , then I put 6 in the quotient , and by it multiplie 987 , makes 5922 , which taken from 6711 , leaues 789 , and the worke will stand in the example following .

Example .

Lastly , I remoue my Diuisor againe , and seeke how oft I may haue 9 in 78 , and I find I may haue it 8 times , which 8 I put into the quotient , and by it I multiply my Diuisor 987 makes 7896 , equal vnto the numbers aboue ; and so being taken away , leaues nothing remaining , and proues the multiplication to be truly wrought , as appeareth in the example following .

Example .

The third Example of Diuision .

The second kind of Diuision is this : first , place your diuidend & diuisor as in the former Examples , & then hauing found out the figure of your quotient , begin with the least figure of your diuisor towards the right hand first , and multiply that by the figure of the quotient found , and then subtract the sum of the multiplication of that figure from the figure aboue the same , if it exceed not 9 ; but if the product be aboue 9 , then for euery 10 beare one in mind to bee added to the product of the multiplication of the second figure of your Diuisor by the quotient ; and so in all respects worke for euery other figure , and you shall need make no more figures aboue your Diuidend then necessitie shall require , as for example .

I would diuide the product of the multiplication in the former Chapter of 79648039 by 8976 , which was found to be as followeth , viz. 714920798064 by 8976 : first , I place my Diuidend and Diuisor as followeth .

Then first I seeke how often I may haue 8 in 71 , I find by triall I can haue it but 7 times ▪ then hauing placed 7 in the Quotient , I first multiply 6 , the least , or smallest figure in value by 7 , makes 42 ; then I say , 42 from 42 , rest 0 , and carry 4 for the fortie in mind ; then I cancell the 2 ouer the 6 , and placea 0 in the roome ouer it . Secondly , I say , 7 by 7 is 49 , and 4 in mind makes 53 , from 59 leaues 6 , and carry 5● cancell the 9 , and place 6 ouer it . Thirdly , 7 by 9 is 63 , and 5 in mind is 68 , from 74 leaues 6 , and carry 7 , cancell the 4 , and place 6 aboue it : also 7 by 8 is 56 , and 7 makes 63 , which taken from 71 , leaues 8 remaining , which 8 place ouer the 1 , and cancell the 71 , and the first worke will stand thus .

Secondly , I remoue my Diuisor 8976 , and seeke , how many times I may haue 8 in 86 , ● find 9 times ; then I multiply 6 by 9 placed in the Quotient , makes 54 , which taken from 60 , leaues 6 ; place 6 aboue the first 0 , and carry 6 for the 60 : then say , 9 by 7 is 63 , and 6 in mind makes 69 , from 70 leaues 1 , and carry 7 in mind ; cancell the 0 ouer the 7 , and place the 1 ouer the 0. Againe , 9 by 9 is 81 , and 7 in mind is 88 , which taken from 96 , leaues 8 to bee placed aboue the first 6 , and carry 9 in mind : lastly , 9 by 8 is 72 , and 9 makes 81 , which taken from 86 , leaues 5 to bee placed aboue the 6 , and the worke will stand as followeth .

Example .

Thirdly , againe I remoue my Diuisor , and seeke how many times 8 is in 58 , and I find I can haue it but 6 times , which I place in the Quotient : then I say , 6 by 6 makes 36 , from 37 leaues 1 aboue 7 , and carry 3 : then 6 by 7 is 42 , and 3 is 45 , from 46 leaues 1 , aboue the 6 , and carry 4 : againe , 6 by 9 is 54 , and 4 makes 58 , from 61 leaues 3 aboue the 1 , and carry 6. Lastly , 6 by 8 is 48 , and 6 makes 54 , from 58 leaues 4 , and the worke stands thus , as in this example .

Fourthly , I remoue my Diuisor , and seeke how oft I may haue 8 in 43 , and I find but 4 times , I place 4 in the Quotient . Then 4 by 6 makes 24 , from 29 , leaues 5 , and carry 2 , set 5 ouer the 9 : then 4 by 7 is 28 , and 2 makes 30 , from 31 , leaues 1 , and carry 3. Againe , 4 by 9 is 36 , and 3 makes 39 , from 41 leaues 2 , and carry 4. Lastly , 4 by 8 is 32 , and 4 is 36 , from 43 , leaues 7 , and the work will then stand thus .

Example .

Fifthly , I remoue my Diuisor , and seeke how oft I may haue 8 in 72 ; I find 8 times , which placed in the quotient , I multiply 6 by 8 , makes 48 , from 48 , leaues 0 , and carry 4● then 8 by 7 makes 56 , and 4 is 60 , from 65 leaues 5 , and carry 6 : then 8 by 9 is 72 , and 6 makes 78 , from 81 , leaues 3 , and carry 8 : then 8 by 8 makes 64 , and 8 , is 72 , from 72 leaues 0 remaining , and the worke will stand thus .

Example .

Sixthly , I remoue my Diuisor , and seeke how oft I may haue 8 in 3 , which I find not once ; I place a 0 in the Quotient , and remoue my Diuisor one place more and seeke how many times 8 is in 35 ; I find I can haue it but 3 times , I place 3 in the Quotient beyond the 0 last placed , and say , 3 by 6 is 18 , from 26 rests 8 , and carry 2 : then 3 by 7 is 21 , and 2 is 23 , from 30 leaues 7 , and carry 3 : againe , 3 by 9 is 27 , and 3 is 30 , from 30 leaues a 0 , and carry 3 : also 3 by 8 is 24 , and 3 is 27 , from 35 leaues 8 ; and the worke will stand thus .

Example .

Lastly , I remoue my Diuisor , and seeke how oft I may haue 8 in 80 ; I find 9 times , I place 9 in the Quotient , and say , 9 by 6 is 54 , from 54 leaues 0 , & carry 5 : then 9 by 7 is 63 , and 5 is 68 , From 68 leaues 0 , & carry 6 : Then 9 by 9 is 81 , and 6 is 87 , from 87 leaues 0 , and carry 8 : last of all , 9 by 8 is 72 , and 8 makes 80 , from 80 there will remaine nothing but cyphers , and the worke is quite ended , and will stand , as in the example following .

Example .

The fourth and last kind of Diuision , is the most absolute , speedy , and easie , not charging the memory at all , with keeping any numbers in mind ; and also the proofe of your work is made by Addition , and not by multiplication , as hath heretofore been commonly vsed , but the figures of your worke are by Addition , the proofe of your worke , as shall appeare by examples following .

The third Worke .

First , place your Diuidend betweene two paralel lines , and your Quotient at the right side of your Diuidend , behind a crooked line , as before ; then place your Diuisor next to the left hand of your Diuidend , behind a perpendicular line : and lastly , marke how many figures your Diuisor hath , and in the roome of those figures place ciphers vnder the figures of your Diuidend , so many as your Diuisor hath figures , as in the last example ; which I will againe repeate in this place , and work it by this kind of Diuision , making the proofe of the work by Addition of the same figures .

Example .

First , I point to the first cipher towards the left hand , and seeke how oft I may haue 8 , the greatest figure in value of my Diuisor , hauing respect to the other figures of my Diuisor , to take them also as often , out of the figures aboue , and I find I can haue it but 7 times , which 7 I place in the Quotient , and by that 7 I multiply my Diuisor 8976 , saying first , 7 by 6 is 42 , place the 2 vnder the lowest cypher towards the right hand , and carry 4 : then 7 by 7 is 49 , and 4 is 53 , set 3 vnder the next place to the left hand , and carry 5 : then 7 by 9 is 63 , and 5 is 68 , place the 8 in the next place , and carry 6. Lastly , 7 by 8 is 56 , and 6 in mind makes 62 , which place downe in their places , and the totall is 62832 , to be subtracted from 71492 , and there will remaine 8660 ; and the worke will stand thus .

Example .

Secondly , I cancell the first cypher to the left hand , and place one cypher more towards the right hand , vnder the 0 , and then I point againe to the first cypher , and see how oft I may haue 8 in 86 ; I find 9 times , and placing 9 in the Quotient , by it I multiplie 8976 my Diuisor , placing the lowest figure in value vnder the lowest cipher to the right hand , and the rest in order , and I find the product to bee 80784 , which taken from 86600 , leaues 5816 remaining , and then your worke will stand , as in this .

Example .

Thirdly , I cancell my Diuisor , or one Cypher , and place one cypher more vnder 7 , and then seeke how oft I may haue 8 in 58 , which I find 6 times , and by it I multiply my Diuisor 8976 makes 53856 , which taken from 58167 , leaues 4311 , and the work will stand as followeth .

Example .

Fourthly , I cancell one cypher , and place a cypher vnder 9 , and then seeke how oft I may haue 8 in 43 , which I find but 4 times , which place in the Quotient , and by it I multiply my Diuisor 8976 , makes 35904 , which taken from 43119 , leaues 7215.

Example .

Fifthly , I cancell one cypher , and place a cypher vnder 8 , and fee●e how oft ● is in 72 ; ● find 8 times , which placed in the Quotient , I multiply my Diuisor 8976 by it makes 71808 , which taken from 72158 , leaues 350 , and the worke stands , as in the example following .

Example .

Sixthly , I cancell one cypher , and place another vnder the 0 , and seeking I find I cannot haue 8 in 3 ; therefore I place a 0 in the Quotient . Seuenthly , I cancell one cypher , and place one other vnder the 6 , and seeke how oft I may haue 8 in 35 ; I find but 3 times , and placing 3 in the Quotient , by it I multiply 8976 , makes 26928 , which taken from 35006 , leaues remaining 8078.

Lastly , I cancell the next cypher , and doe place another vnder the last figure of my Diuidend 4 , and seeke how oft I may haue 8 in 80 ; I find 9 times , and then placing 9 in the Quotient , I multiply my Diuisor 8976 , and the Quotient is 80784 , equall vnto the numbers aboue , and so being subtracted from the numbers aboue , leaues o remaining , and the worke is ended , and will stand thus .

Example .

The proofe of this Diuision is made by Addition of the figures , vnder the line or Diuidend , for if they returne your former Diuidend , the worke is true wrought ; or otherwise be sure some error is in your work , if there remaines any fraction after your worke is ended , then it is to bee added into the lower figures in their seuerall places , as shall appeare by examples following .

Here in this example following , working according to this latter forme of worke , there is aduantage to be taken ; if the figures of the Quotient bee well noted , as here the fourth figure of the Quotient is 7 , the Product of the Diuisor multiplied by it is 1438816 , and also the eleuenth figure of the Quotient is 7 , so that comming to multiply the Diuisor againe by that 7 , I neede but take the Product of the first multiplication by 7 , which is 1438816 , and so place them in their seuerall places , as in the example , and so likewise there is 3 in the Quotient two times , so that for the latter multiplication , I take the first product 539556 , and sa●e that labour of multiplication of the Diuisor by 3 : and so of any other figure comming into the Quotient more times then once , as by the example before going will appeare .

Place the great example following , in this place .

Example ▪

Example .

Example .

How to diuide by a Vnite with Cyphers .

If you wil diuide by 10 , or by 100 , or 1000 , or with any other vnite with cyphers , one or more ; doe but cut off so many figures from the right hand of your Diuidend , as there are cyphers in your Diuisor , and the remaines is your Quotient .

Example .

If you would diuide 786589 by 10 , cut off the last figure 9 , and the residue is your Quotient 78658 2/10 ; or if you will diuide by 100 , cut off two figures , and the Quotient will be 7865 ●2/100 ; or by 1000 , and the Quotient will be 786 ●●2/1000 ; and so of all other .

If you will diuide the Product of 1999 squared ; that is to say , multiplied in it selfe , which is 3996001 by 1999 , for expedition of worke , after you haue found the first figure of the Quotient 1 , and taken that out , I find the next figure will be 9 , which taken out , the third and fourth figures are also found to be 9 , and so you need not make multiplication for euery seuerall 9 , but the first will serue for all , as in the example following .

Example .

Example .

Briefe Rules by Multiplication and Diuision .

If you multiply any number of nines ; as if you will multiply , or square 5 times 9 by 5 times 9 , then place your nines in this order following .

Example .

Then say , 9 times 9 is 81 , place the 1 vnder the first 9 to the right hand , and then subtract the 1 from the first 9 to the left hand , and adde the cyphers betweene , and the Product is ended , and is 9999800001 , as appeareth .

The proofe of the worke after the ordinary way .

To multiply any number by 9.

Adde a o to the number you intend to multiply , and then set the same numbers vnder them , and subtract them from the vppermost , and the remaines is the Product of that multiplication by 9.

Example .

To multiply by ½ , or ⅓ , or ¼ , or ⅕ .

If you will multiply 856 by 24½ , first , multiply 856 by 24 , makes 20544 ; and then for one halfe , take halfe 856 , which is 428 , and adde into the former summe , makes the totall 20972.

Example .

What number is that , which being diuided by 24 , the Quotient will be 856. Answer , multiply 856 by 24 , makes 20544 for the number that you seeke .

Example .

There is a plot of land containing 848 Perches , the one side is 24 , what must the other be . Answere , Diuide 848 by 24 , the Quotient is 35 ⅓ for the other side .

If you will diuide the Product of 5 times 9 squared , which is 9999800001 , by 5 nines , then set the Diuisor right vnderneath the Diuidend , and adde them together , and cut off the 5 cyphers from the Product , and the residue is the Quotient .

Example .

What number is that , which being multiplyed by 15 , the totall will be 756. Answ. diuide 756 by 15 , and the Quotient is 50 6/159 , or 2/5 , for the answer , or number you do seeke .

Example .

There are 825 men , to march 15 in one ranke , how many files will they make . Deuide 825 by 15 , it makes 55 files .

Example .

There is 948 pound of powder to bee imployed in an Assault of Battery with 6 pieces of Ordinance ; the first piece shooteth 4 pound , the second 5 , the third 6 , the fourth 7 , the fifth 8 , the sixth 10 pound , the question is , how many Shootes each piece may make , to make an equall number of Shotts . Answere ; diuide 948 by 40 , and it makes 23 Shootes , and there will remaine 28 pound .

Example .

THE RVLE OF REDVCTION .

TO reduce any great number into a smaller denomination it is done by multiplication , and to reduce small denominations into greater it is done by diuision : in this manner marke how many of the smaller denominations is contained in one of the next greater , and by that number you must multiply the greater : or of the contrary , if you would bring small denominations into greater , marke how many of the smaller denominations makes one of the next greater , and that number shall be your diuisor .

Example .

If you would reduce pounds starling into pence , multiply your pounds by 240 pence , because so many pence maketh pound starling , and the totall will be the number of pence in the summe of pounds giuen . And contrariwise , if you would bring pence into pounds starling : diuide your number of pence by 240 pence , which are the pence in one pound , and the Quotient will shew the number of pounds , in the summe of pence giuen : but in this operation the Tables in the beginning of this book will help much , for the speedy reducing of pounds , shillings pence , yards , ells , bushels , pecks , pints , &c. into smaller or greater denominations ; for if you search in the said Tables , you shal find your multiplier , or diuider , wherby you are to multiply , or diuide your number giuen , to performe the worke , as shall appeare by the seuerall examples following .

Reduction of Coyne .

In 87652 pound , how many pence : in the Table of Coyne I find 240 pence makes one pound , so that in multiplying 87652 pound by 240 , makes the summe of pence desired .

1. Example .

2. Example .

In ●759 pound , 17 shillings , 8 pence , how many shillings , pence , and farthings .

3. Example .

In 3785437289 farthings , how many pounds , shillings , and pence : diuide by 960 farthings , because 960 farthings makes one pound starling , & the remainer is farthings , which diuided by 48 , the farthings in one shilling , makes 3943163 pound , 16 shillings 10 pence , ¼ .

How to bring pounds , shillings , and pence at : he first worke by Diuision .

To bring pence at the first worke into pounds , shillings , and pence : adde a 0 to your number of pence , and diuide that sum by 240 , makes pounds , and the last figure will be primes , euery vnite in value 2 shillings , and the remainer alwaies lesse then 24 pence , or one prime .

Example .

In 902372 pence , how many pounds , shillings , and pence ; adde a 0 , makes 9023720 , which diuided by 240 pence , makes &c.

2. Example .

In 75000837504 pence , how many pounds , shillings , and pence : adde a cypher , or 0.

How to bring farthings into pounds , shillings , and pence at the first worke .

To bring farthings into pounds , shillings and pence at one worke : adde a 0 to your number of farthings , and diuide the summe by 960 , the number of farthings in one pound sterling , makes pounds ; and the last figure of your Quotient will bee primes euery one in value 2 shillings : and if there remaine 48 , it is one shilling , or take 48 from the remainer for one shilling , the rest are farthings lesse then 48.

Example .

Totall is 756. l. 12. s. 58. q. or 13. s. 2. d. 1. ob .

In 3785437248 farthings , how many pounds , shillings , and pence , adde a 0 , and diuide by 960 , makes ●943163 pound , 8 primes , or 16 shillings , 0 pence ▪

How to bring pence into pounds , shillings , and pence another way .

Diuide your number of pence by 4 , and the remainer is pence , then that Quotient by 6 , and the remainer is groats , alwaies lesse then 6 groates , or one prime , or 2 shillings ; and the latter Quotient , cutting off your Primes , is pounds , and so you haue pounds , shilling , and pence .

Example .

In 785697 pence , how many pounds , shillings , and pence , makes 3273 pound , 14 shillings , 9 pence .

If you will bring farthings into pounds , shillings , and pence : diuide first by 16 , and the remainer is farthings , alwaies lesse then 16 , or one groate ; and then againe by 6 , makes pounds , shillings , and pence , as before , cutting off the prime line .

Example .

In 8735672 farthings , how many pounds Shilling , and pence .

Reduction of Waights .

In 8756 hundred , 3 quarters , 24 pound , 12 ounces Haberde poyce , 16 ounces to the pound , and 112 pound to the hundred , how many pounds and ounces .

Example .

In 1569●492 ounces Haberdepoyse , how many hundreds , quarters , pounds , and ounces ; finde how many ounces makes 112 pound , in multiplying 112 pound by 16 ounces , makes 1792 ounces ; by which diuide , makes , as in the example following .

Reduction of Measures .

In 2356 Acres , 3 Roodes , 27 Perches , how many Perches in all .

Example .

In 765437 Perches , how many Acres , Roodes , and Perches : diuide by 160.

Example .

Reduction of Time .

In 356 yeares , 24 dayes , 36 houres , and 22 minuts ; how many dayes , houres and minutes .

Example .

The Proofe .

In 187150342 minuts , how many houres , dayes , yeares , and minutes .

Reduction of Motion .

In 11 Signes , 34 degrees , 25 minutes , 36 seconds , 24 thirds ; how many fourths .

Example .

The proofe .

In 4722971040 fourths , how many signes degrees , minutes , seconds , thirds , & fourths .

Example .

Questions by Reduction .
1. Question .

In 389 pound Starling , how many Dollars of 4 shillings 8 pence , or 14 groates a piece . Reduce 389 pound into groats , in multiplying them by 60 , makes 23340 groats ; which diuide by 14 groats , makes 1667 pound , and 8 pence .

Example .

2. Question .

In 300 pound starling , how many Angels at a 11 shillings a piece Reduce 300 pound into shillings , makes 6000 shillings ; which diuide by a 11 , makes 545 angels , and there will remaine 5 shillings .

Example .

3. Question .

In 3012 pound , how many Ryals of plate at 7 pence a Ryall . Reduce 3012 pound into pence , makes 722880 pence ; which diuided by 7 , makes , as in the example .

Example .

4. Question .

If one Dollar be worth 4 shillings 8 pence , how many Dollars is in 108579 pound , 16 shillings starling . Multiply your pounds by 60 , makes 6514740 ; then reduce 16 shillings into groates by 3 , makes 48 groates ; which added into one total , makes 6514788 which diuided by 14 , makes , as in the example .

Example .

In 465342 Dollars of 14 groats a piece , how much starling money : multiply your Dollars by 14 , makes 6514788 groates ; which diuide by 60 , makes 108579 pound , 16 shillings .

Example .

5. Questions .

If I receiue 8060 French Crownes at 6 shillings a piece in France , how much Starling must I pay for them at 6 shillings , 1 penny a piece : multiply 8060 by 73 pence , the number of pence in one French crowne , makes 588380 pence : which diuided by 240 pence , makes 2451 pound , 11 shillings , 8 pence .

Example .

6. Question .

If 564 yards of cloth cost 124 pound , 12 shillings , how may I sell a yard to gaine 22 pound , 7 shillings , by the whole Summe . Answere , adde 22 pound , 7 shillings , to 124 pound , 12 shillings , makes 146 pound 19 shillings : which reduce into pence , makes 35268 pence : which diuided by 564 , makes 5 s. 2 d. ½ 6/47 of a farthing for the price to sell one yard , for to gaine 22 pound 7 shillings by the bargaine .

Example .

7 Question .

If 156 ells of cloth cost 124 pound , what will one ell cost . Reduce 124 pound into shillings , makes 2480 shillings ; which diuide by 156 , makes 15 shillings , 4 pence 26/156 q.

Example .

8. Question .

If I sell 342 yards of Veluet for 241 pound , 17 shillings , how doe I sell one yard : reduce your 241 pound , 17 shillings , into shillings , makes 4837 shillings ; which diuided by 342 yards , makes 14 shillings , 1 penny , 43/●57 of a penny .

Example .

9. Question .

A certaine Nobleman sent his seruant to the Tower of London , with the Kings Maiesties Warrant to the Mint-master for 3408 pound , 15 shillings , willing him to bring it in pieces of 12 d. of 9 d. of 6 d. of 3 d. of 2 d. of ● d of 1 oh . commanding him to bring him of each sort a like quantity , or number of pieces ; the question is to know , how many of each sort hee shall bring vnto his master , to make the said sum of 3408 li. 15 s. reduce your mony into half pence , and also your seueral pieces of Coyne into half pence , and diuide the greater by the lesser , as in the example .

Example .

What Progression Arithmaticall is , and the Rule .

PRogression Arithmeticall is nothing else but a briefe summing , colecting , or gathering together of diuers numbers , increasing by equall proportion , into one totall summe . As for example : 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. &c. or also , 3. 4. 5. 6. 7. 8. &c. or , 2. 4 6. 8. 10 , 12. &c. or else by 3 , as , 5. 8. 11. 14 17. 20. 23. 26. &c. or of all such like kinds of Progrission , which doe increase equally by 2. 3. 4. 5 , or 6 , or any other greater increase , and such kind of Progression is called , Arithmeticall .

2. To find the summe of a Progression .

Marke first how many seuerall places there be in your Progression , and note that downe ; then adde the first number of the Progression to the last : then multiply halfe those two numbers by the whole number of the places , or else halfe the number of the places , by the whole number of the first and last terme added into one summe , and both waies will produce the totall summe of that Progression .

Example .

There is a Progression beginning at 4 , and is continued vnto 44 , increasing by 4. First , set downe the numbers of that Progression , beginning at 4 , and ending at 44.

Termes . 4. 8. 12. 16. 20. 24. 28. 32. 36. 40. 44. Places . 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Here the first terme is 4 , and the last is terme is 44 , which added together , makes 48 , the one halfe , which is 24 , multiplied by a 11 , the whole number of places makes 264 the totall .

Example .

First Question .

A certaine man gaue to his daughter in marriage the first day of Ianuary 1 pound , and the second day 2 pound , the third day 3 pound and so increasing euery day 1 pound , vntill 31 dayes were expired ; the question is , what he should receiue in the whole sum . First , 31 dayes is the number of places , and 31 li. is the last payment : adde the first terme 1 to the last terme 31 , makes 32 ; which multiplied by 15 one halfe , which is halfe 31 ; or take 31 and halfe 32 , and the product wil be the totall Summe of his wiues portion .

Example .

How to find the latter terme of a Progression .

If you would know the latter terme of a Progression of 100 termes , increasing by 3 , and beginning at 10 ; take one terme from 100 termes , & there will remaine 99 ; which multiply by 3 , the excesse or difference of the increase , makes 297 ; to the which if you adde the first terme 10 , makes 307 for the 100 terme of that Progression .

Example .

Or otherwise take the Excesse 3 from the first terme 10 , and there will rest 7 , which note a part , then multiply , the number of places 100 by the excesse 3 , makes 300 ; to which adde the 7 , makes 307 , as before .

Example .

Second Question .

A certaine Merchant bought 78 pieces of Exetor Carsies , to pay 2 shillings for the first piece 4 shillings , for the second 6 shilling , for the third , 8 s. & so forth increasing his price vnto 78 pieces , 2 shillings in euery piece ; the question is , what the Clothier had for his Carseys .

First , find the latter terme , taking one from 78 , makes 77 ; which multiply by 2 , makes 154 ; to which adde the first terme 2 , makes 156 for the 78 , or last terme : then adde 2 , the first terme , to 156 , the last , makes 158 ; which multiply by 39 , half of the number of places , makes 6162 shillings for the summe of money , the Clothier shall receiue for his 78 Carseys .

Example .

To find the number of termes .

There is a Progression , whose first terme is 2 , the last terme , 156 ; and the excesse was 2 , I would find the number of termes .

Subtract the first terme from the last , and diuide the remayner by the excesse , the quotient is the number of termes , wanting but one . Example : 2 , the first terme from 156 , the last leaues 154 ; which diuided by 2 , makes 77 ; to which adde 1 , makes 78 , the number of termes .

How to find the Excesse , or difference .

Subtract the first terme from the last , and diuide the remainer by one lesse , then the number of the Termes , and the Quotient will be the Excesse or difference .

Example .

Subtract 10 , the first terme , from 307 the last terme , there will remaine 297 ; which diuide by 99 , one lesse then the number of termes , which are 100 , makes 3 the excesse .

To find any middle terme .

Subtract a vnite from the number of the terme you would know , and multiply the remainer by the difference , and to that product adde the first terme , and the totall is the terme you doe seeke .

Example .

To find the 30 terme in the last example of 100 termes , subtract 1 , rests 29 ; which multiply by 3 , the Excesse makes 87 ; to which adde the first terme 10 , makes 97 for the 30 terme of that Progression .

Example .

How to find what number shall begin and finish a Progression .

To the number of termes adde one , which multiply by halfe the number of termes , and by the product diuide the sum of the progression , and the quotient will be the first terme , and excesse of that progression .

Example .

At 16 payments 353 pound , 12 shillings is to be paid , the question is , what number must begin , and continue the progression .

First , the money 7072 shillings ; then to 16 , the number of termes , adde 1 , makes 17 ; which multiply by 8 , halfe the number of termes , makes 136 for Diuisor ; by which diuide 7072 , and the quotient is 52 shillings for the first paiment and excesse , and by the same the other payments are found .

Example .

Example .

What Geometricall Progression is , and the Rule .

THe termes being 3 , to find a third proportionall betweene two extreames : diuide the Roote of the greater by the lesser extreame , and the quotient is your desire .

Example .

First , 8 and 12 are two extreames giuen , it is required to find a proportionall number betweene those two numbers giuen ; square 12 , it is 144 , which diuide by 8 , makes 18 for the third proportionall number . Secondly , multiply your extreames together , and extract the square roote for the meane proportionall , betweene two numbers giuen ; as let 4 and 9 bee two extreames , 4 by 9 is 36 , the square roote is 6 , for a meane proportionall number to those two numbers giuen .

Betweene 2 and 54 , let 2 meane proportionals be desired by the square of 2 , which is 4 ; multiply 54 , it makes 216 , the Cube roote whereof is 6 for the least of the two Meanes : Againe , by 2 multiply 2916 , which is the square of 54 , makes 5832 , of which the Cube roote is 18 , for the greater meane proportionall sought . But if the termes exceede 4 , hauing all one excesse , it is then called Geometricall Progression .

To find any middle Terme , or any other Terme in a Geometricall Progression .

Increase your Progression by the excesse , and the square of the terme when you cease , or the number multiplied in it selfe squarely , is the double of your Terme saue 1 , if the progression begin with an vnite .

But it the first terme be not an vnite , then the square of any terme is the double number of the said terme : as if you should square the sixth terme , then the product would bee the twelfth terme : & so of any other terme .

Example .

A Gentleman comming into a Market to buy a Horse , was asked 30 pounds for him . Nay ( said the Gentleman ) his price is ouer great . Then said the owner ( hauing more craft and subteltie then the Gentleman , as commonly the old Prouerb is true amongst Horse-coursers ) ; My Gelding ha●h fower shooes vpon his fower feete ( quoth he ) , you shall giue me for the first nayle ( there being 28 in all ) one farthing ▪ and for the second nayle 2 farthings , and for the third 4 farthings , and for the fourth 8 farthings ; and so double at euery nayle , you shall haue him . Whereat the Gentleman smiled , saying ; I will haue him . And so they bargained , and then went to an Arithmatician to cast vp the Summe : but how this Gentleman was able to pay for this Horse , shall appeare by the Worke , which I haue put for an example , because I would not haue any man ignorant in Arithmatick , to make any such blind matches without aduice , as I know many haue done to their cost .

1. Example .

Now according to the rule , I increase this progression vnto the seuenth Terme thus , 1. 2. 4. 8. 16. 32. 64 ; which 64 I multiply by it selfe squarely , the product is 4096 , which by the rule is the thirteenth Terme , which is one Terme lesse then the double of 7 : then multiply that 4096 by 2 , it makes 8192 , which is the fourteenth Terme . Then multiply 8192 by 8192 , and the product is 67108864 , which is the twentie seuenth Terme : the which being doubled , makes the last Terme 134217728.

Example .

The Extreames and Excesse of a Progression giuen to find the summe .

Multiply the last terme by the Excesse , and from the Product abate the first terme , and diuide the remayner by a vnite lesse then the excesse , and the Quotient is the summe of the Progression desired .

Example .

In the last examples , the excesse was 2 , by which I multiply 134217728 , and the Product is 268435456 , from which abate 1 , the first terme , and the remayner is 268435455 , which should bee diuided by one vnite lesse then the excesse , which is 2 , and ● lesse is but 1 ; therefore seeing 1 doth neither multiply , nor diuide , I conclude the price of the horse to be 268435455 farthings ; which I diuide by 960 , the farthings in one pound starling , and the quotient is ●79620 pound , 5 shillings , 3 pence , 3 farthings , the price of the Horse , as in this example .

Example .

I haue inserted in the next page the triall of this work , by increasing the Termes from 1 to 28 , and also the Addition of the totall , which shewes the answere to be true .

Example . 1 1 2 2 4 3 8 4 16 5 32 6 64 7 128 8 256 9 512 10 1024 11 2048 12 4096 13 8192 14 16384 15 32768 16 65536 17 131072 18 262144 19 524288 20 1048576 21 2097152 22 4194304 23 8388608 24 16777216 25 33554432 26 67108864 27 134217728 28 268435455 The Totall .

Otherwise , subtract the first terme from the last , and diuide the remayner by one ace lesse then the Excesse , and to the quotient adde the last Terme , and the totall is the summe .

Example .

To 12 men a summe of money is giuen to the eldest ½ , to the second ½ , the remayner , and so to euery one of the rest , and the last portion was found to bee 4 pound , and the last halfe being also 4 pound , was giuen to a friend to see the money to bee equally distributed ; what was each mans portion , and the summe giuen ?

Let 4 bee the last portion , and twelfth Terme , and so double vntill you come to the first terme , and you shall find euery mans portion Then by this second rule , you shal find the totall to be 16380 pound ; to which adde the Ezecutors part 4 pound , makes 16384 pound .

3. Example .

A Gentleman bought a Mannor , with all the appurtenances for a summe of money vnknowne ; but hee was to pay at seuerall dayes of payment by continuall triplation , of euery payment , from the first payment which was 4 pound , and the last 8748 li. the question is , what he paid for the said Mannor and lands .

Example .

Subtract the first terme 4 , from the last terme 8748 , there will remaine 8744 ; which diuide by the Excesse , one lesse , viz by 2 , and the quotient will bee 4372 ; to the which adde the latter terme 8748 , and the totall is 13120 pound , for the summe which the said Mannor and lands cost .

Fractions .

YOu shall vnderstand , that in the worke of Fractions hereafter ; in the next page following I haue vsed another forme of working , then heretofore hath been vsed : as when you will set forth any fraction , as ¾ thus heretofore vsed , set them out thus , 3 ∶ 4 ; or ⅞ , place thus 7 ∶ 8 with a double prick betweene them : and so of any other , as 25/20 of a pound , thus , 15 ∶ 20 of one pound : or fractions of fractions , thus , ⅔ of ¾ of ⅚ of a pound , set them thus , 2 ∶ 3 of 3 ∶ 4 of 5 ∶ 6 of a pound : and so of all other fractions , as shall appeare afterward in the operations following ; and so being placed , they are more apter and fitter for al the seuerall operations of Arithmatick , then being placed after the ordinary forme of working . And thus much I thought good to expresse for the better vnderstanding of the Rules hereafter following , in all fractionall operations . And now I will proceede vnto the seuerall rules of Fractions , with their Examples .

How to reduce Fractions of Fractions . First Rule .

Multiple their topps one into another for a new numerator , and likewise their Basses for a new denominator , and the worke is ended .

Example .

If you would reduce 3 ∶ 4 of 2 ∶ 3 of 7 ∶ 8 of one li starling ; multiply 3 by 2 , makes 6 , and then 6 by 7 makes 42 for the new numerator to your fraction : then 4 by 3 makes 12 , and 12 by 8 makes 96 for new denominator , and the fraction is 42 ∶ 96 of a pound .

3 ∶ 4 of 2 ∶ 3 of 7 ∶ 8 of 1 li. makes 42 ∶ 96 of a li.

2. Example .

Againe , 3 ∶ 5 of 9 ∶ 8 of 7 ∶ 10 of 11 ∶ 12 of a pound , makes 2079 ∶ 4800.

3. Example .

What is 1 ∶ 2 of 2 ∶ 3 of 3 ∶ 4 of 4 ∶ 5 of 5 ∶ 6 of 6 ∶ 7 of one pound . Answer : crosse al the by as equall termes , and set the vnequall termes 1 ∶ 7 of a pound for the totall summe : but after the other forme of worke , it would haue brought out 720 ∶ 5040 of a pound , which by abreuiation makes 1 ∶ 7

The Proofe .

72 ∶ 504 , 36 ∶ 252 , 18 ∶ 126 , 9 ∶ 63 , 3 ∶ 21 , 1 ∶ 7.

2. Rule : How to reduce Fractions of Integers .

Multiply all the Denominators of your seuerall fractions for the new , or common Denominator to all your giuen fractions .

Then to find new numerators to each of your giuen fractions ; multiply each fractions top into the basses , of each seuerall fraction , excepting his owne basse , for the new numerators , as in this Example .

Example .

If you would reduce 3 ∶ 4 and 5 ∶ 6 and 7 ∶ 8 of a pound into one Denomination : multiply all the basses together , saying ; 4 by 6 makes 24 , and 24 by 8 makes 192 for the common Denominator to all the giuen fractions .

Then multiply 3 , the numerator of the first fraction , by 6 , the denominator of the second fraction , makes 18 , & 18 by 8 makes 144 for his numerator .

Secondly , multiply 5 , the numerator of the second fraction by 4 and 8 , the Denominators of the other two fractions , makes 160 for the new Numerator of the second fraction .

Thirdly , multiply 7 , the numerator of the third fraction , by 6 and 4 , makes 168.

Example .

2. Example .

If you would reduce 2 ∶ 3 , and 3 ∶ 5 , and 8 ∶ 9 of a pound .

3. Example .

If you would reduce 7 ∶ 8 , 1 ∶ 3 , 2 ∶ 3 , 4 ∶ 5 , and 6 ∶ 7 of a pound ▪

How to proue a Fraction by the knowne parts of Coyne .

In the first example of fractions of fractions , I find that 3 ∶ 4 of 2 ∶ 3 of 7 ∶ 8 of a pound Sterling to bee 42 ∶ 96 parts of a pound : for triall whereof , take 7 ∶ 8 of a pound , which is 17 shillings 6 pence , or 210 pence , the 2 ∶ 3 of that number is 140 pence ; and ●●4 of 140 pence is 105 pence : now multiply 42 the numerator of your fraction by 240 d. and diuide by 96 , the denominator , makes 105 pence , the proofe , as followeth .

4. Example .

2. Example .

In the first example of fractions of integers , there was 3 ∶ 4 , 5 ∶ 6 and 7 ∶ 8 of a pound , reduced into one denomination , and the totall by Addition was 472 ∶ 192 of a pound . Now for the proofe of the worke , multiply the numerator 472 by 240 , makes 113280 ; which diuided by 192 the denominator , makes 590 pence ; which diuided by 12. pence , makes 2 pound , 9 shillings , 2 pence . The proofe of this triall in the parts of a pound , take first for 3 ∶ 4 of a pound , or 15 shillings ; then 5 ∶ 6 of a pound is 16 shillings 8 pence ; also 7 ∶ 8 of a pound is 17 shillings 6 pence ; and the totall added together , is 2 pound , 9 shillings , 2 pence , which prooues the worke to be true .

Example .

3. Rule . Addition in Fractions .

If your fractions be of one denomination , then adde all your numerators together , subscribing the common denominator vnder the line .

Example .

The second Rule .

If your Fractions be not of one denomition , then reduce them by the second rule of Reduction to one denomination , and then adde them into one summe subscribing vnder the common denominator .

Example .

If you would adde 40 ∶ 80 , 30 ∶ 200 , and 50 ∶ 90 ; cut off a cypher from each numerator and denominator , and the fractions remaining will be of the same with the giuen fractions , and then worke as before .

Example .

The proofe of Addition by parts of Coyne .

In the second Example , 2 ∶ 3 , 3 ∶ 4 , and 4 ∶ 5 of a pound , are found to be 133 ∶ 60 ; therfore diuide 133 by 60 , makes 2 pound and 13 ∶ 60 or 13 groats remaining , which is 2 pound , 4 shillings , 4 pence .

The proofe : adde 2 ∶ 3 of a pound , which is 13 shillings , 4 pence ; and 3 ∶ 4 of a pound , which is 15 shillings , and 4 ∶ 5 of a pound , which is 16 shillings , into one totall , makes 2 pound , 4 shillings , 4 pence , as before .

Example .

Rule 4. Subtraction in fractions .

As before in Addition , so also in Subtraction , reduce your fractions to one common denomination , then subtract the smaller numerator from the greater , and subscribe the common denominator vnder the remainer .

1. Example .

If you will subtract 3 ∶ 4 from 7 ∶ 4 , there will remaine 4 ∶ 4 , or one integer .

Also , 7 ∶ 12 from 13 ∶ 12 , leaues 6 ∶ 12 , or 1 ∶ 2 remaining . But if you will subtract 2 ∶ 3 from 7 ∶ 8 , then reduce them to one denomination , by the second rule of Reduction , and worke , as in this example .

Example .

2. Example .

Againe , 3 ∶ 8 from 15 ∶ 16 , leaues 72 ∶ 128 , remaines .

The proofe of Subtraction by the parts of Coyne .

In the example before , where I take 2 ∶ 3 from 7 ∶ 8 , the remayner was 5 ∶ 24 of a pound , which is 5 times 10 pence , or 4 shillings 2 pence . Also for proofe , take 13 shillings 4 pence , which is 2 ∶ 3 of a pound , from 7 ∶ 8 , which is 17 shillings 6 pence , there will remaine 4 shillings 2 pence , as before .

Rule 5. Multiplication in Fractions .

Multiply Numerator by Numerator , and Denominator by Denominator , to make the new Numerator , and new Denominator , and the worke is ended .

1. Example .

If you will multiply 2 ∶ 3 , by 3 ∶ 4 , the product of that multiplication will be 6 ∶ 12 , or 1 ∶ 2

The proofe of Multiplication by the parts of Coyne .

In the first example , 2 ∶ 3 is multiplied by 3 ∶ 4 , and the product makes 6 ∶ 12 of a pound or 10 shillings : for proofe whereof , multiply 13 shillings 4 pence , or 160 pence , which is 2 ∶ 3 of a pound by 15 shillings , or 180 pence , which is 3 ∶ 4 of a pound , and the product will be 28800 , which being diuided by 240 pence , the pence in one pound will yeeld in the quotient 120 pence , or to shillings .

Example .

6. Rule . Diuision in Fractions .

Multiply the numerator of the diuidend by the denominator of the diuisor for a new numerator ; and secondly the denominator of the diuidend by the numerator of the diuisor , for new denominator , and the diuision is ended : or otherwise place your diuidend fi●st aboue , and the diuisor vnderneath , after my manner , and multiply crosse , and place them ●● in these examples .

If you will diuide 6 ∶ 12 by 2 ∶ 3 , which was the product of 2 ∶ 3 by 3 ∶ 4 in the last example , then it will bring out 18 ∶ 24 , or 3 ∶ 4 , the other number , which prooues the worke good .

1. Example .

If the denominators of the fractions bee both alike , then diuide their numerators one by another ; as 27 ∶ 32 diuided by 3 ∶ 32 , makes the quotient to be 9 ∶ 32 , or Integers .

Example .

If the numerators bee alike , then set the denominator of the diuisor aboue , the denonominator of the diuidend , as 3 ∶ 4 by 3 ∶ 8 , makes the quotient 8 ∶ 4 , or two Integers , and contrariwise 3 ∶ 8 by 3 ∶ 4 , makes the quotient 4 ∶ 8 , or 1 ∶ 2

Example .

The proofe of Diuision by the parts of Coyne .

In the second of the first example , where I diuide 2 ∶ 3 by 4 ∶ 5 , the quotient is 10 ∶ 12 , which in coyne is 16 shillings 8 pence : for proofe , I do multiply 2 ∶ 3 of a pound , which is 160 pence , by 240 , makes 38400 ; which diuide by 4 ∶ 5 , or 192 pence , makes 200 pence , which is 16 shillings 8 pence , the proofe .

Example .

7. Rule . How to worke whole numbers with Fractions .

If you would adde , subtract , multiply , or diuide whole numbers with fractions , set the whole numbers fraction wise , and put 1 after for denominator , and then worke as in the Rules before , as if they were all fractions , and no whole numbers .

Example .

If you will adde 33 ∶ 1 with 13 ∶ 4 , multiply the numerator 33 of your whole number , by the Denominator of your fraction 4 , makes 132 ∶ 4 , which adde vnto 13 ∶ 4 , makes the totall 145 ∶ 4

2. Example .

If you will subtract 13 ∶ 4 from 33 ∶ 1 , reduce them , and subtract 13 from 132 , rest 119 ∶ 4

Example .

If you will multiply 33 ∶ 1 by 13 ∶ 4 ; multiply the numerators , 33 by 13 , makes 429 ; to the which subscribe the Denominator 4 , makes 429 ∶ 4

4. Example .

If you will diuide 33 ∶ 1 by 13 ∶ 4 , multiply crosse 33 by 4 , makes 132 , to be set aboue ; then 13 by 1 makes 13 for denominator .

8. Rule . How to worke whole numbers and fractions with fractions .

Reduce your whole numbers into fractions in multiplying your whole number by the denominator of your fraction ; and vnto that product adde the numerator of your fraction , and subscribe the old denominator .

1. Example .

If you will multiply 28 3 ∶ 4 by 3 ∶ 5 reduce 28 3 ∶ 4 into fourths in multiplying by the fractions denominator 4 , saying , 28 by 4 makes 112 , to the which adde the numerator of your fraction 3 , makes 115 ; which multiplied by 3 ∶ 5 , makes 345 ∶ 20

If you will diuide 28 3 ∶ 4 by 3 ∶ 5 ; reduce them as before , and then multiply them crosse , makes 115 ∶ 4 by 3 ∶ 5 , is 575 ∶ 12

Example .

9. Rule . How to Abreuiate a fraction .

Take one halfe of the numerator , and 1 ∶ 2 of the denominator , as oft as you may vntill the lowest numbers in valew of your fractions comes to be primes together , which are such numbers , as cannot be abreuiated no lower .

Example .

In the first example of fractions of fractions , the fraction was 72 ∶ 504 , which was abreuiated vnto 1 ∶ 7 of a pound : first , take halfe the numerator 72 , which is 36 , then halfe the Denominator 504 , which is 252 ; then 1 ∶ 2 of 36 , is 18 ; and 1 ∶ 2 of 252 is 126. Againe , 1 ∶ 2 of 18 is 9 , and 1 ∶ 2 of 126 is 63 ; then I see I cannot take 1 ∶ 2 of the remainer , wherefore I see I may abreuiate them by 3 still , saying , the third part of 9 is 3 , and 1 ∶ 3 of 63 is 21 : lastly , 1 ∶ 3 of 3 is 1 , and 1 ∶ 3 of 21 is 7 , which place thus , 1 ∶ 7 ▪ so that I find by abreuiation that 72 ∶ 504 of a pound , is one seuenth part of a pound .

Example .

72 ∶ 504 36 ∶ 252 18 ∶ 126 9 ∶ 63 3 ∶ 21 1 ∶ 7

If you cannot take halfe the numbers , then mark whether they wil abreuiate by 3 4 , or 5 , or any other number vnder 9 ; as for example , I would abreuiate 92 ∶ 144 , I see I may abreuiate both by 4 ; then taking 92 , diuide by 4 , makes 23 , and 144 by 4 makes 36 , totall 23 ∶ 36 &c.

If you will abreuiate , 375 ∶ 625 of a pound , you may easily see , they wil be both abreuiated by 5 : wherefore diuide the numerator and denominator both by 5 , as o●t as you can , vntill they become primes together , and you shall find the value of that fraction to be 3 ∶ 5 of one pound , or 12 shillings .

Example .

10. Rule . How to find the value of any Fraction .

Multiply the numerator of your fraction by the parts contained in the whole , and diuide that product by the old Basse , and the quotient will bee the value of that fraction in the knowne parts of Coyne .

Example .

If you would know what 24 ∶ 32 parts of a pound is in Coyne : multiply your numerator 24 by 240 , the pence in one pound , makes 5760 ; which diuided by 32 , the denominator , makes 180 pence , or 15 shillings , the true value of that fraction .

Example .

What is 343 ∶ 522 parts of a yard , multiply 343 by 16 , the number of nayles in one yard makes 5448 ; which diuide by 522 , makes 10 nayles , and 268 ∶ 522 parts of a nayle .

Example .

11. Rule . How to change the Sirname of a Fraction .

Multiply the numerator of your fraction by the parts , or new Sirname of that you would change your fraction into , and diuide by your denominator , and the quotient will be your desire .

1. Example .

I haue 324 ∶ 1620 parts of a yeare , which I would conuert into dayes ; I multiply 324 by 365 , the number of dayes in one yeare , makes 118260 ; which diuided by 1620 , makes 73 dayes , the value of that fraction .

Example .

I would change 256 ∶ 5292 parts of a pound into pence ; multiply the numerator 756 by 240 pence , makes 181440 , which diuide by the denominator 5292 , and the quotient is 34 pence 1512 ∶ 5292

Example .

12. Rule . Questions of Fractions .

What number is that to the which if you doe adde 3 ∶ 4 , the totall will bee 5 ∶ 6 of a pound . Answere ; reduce them to one denomination , and they are for 3 ∶ 4 of a pound 18 ∶ 24 , and the 5 ∶ 6 are 20 ∶ 24 , from which subtract 18 , rest 2 ∶ 24 of a pound , or 20 pence : the proofe , take 3 ▪ 4 of a pound , which is 15 shillings , and adde 20 pence to it , and the totall is 16 shillings , 8 pence ; which is 5 ∶ 6 of a pound .

Example .

2. Example .

What number is that , from which if you doe subtract 8 ∶ 12 , the remayner will bee 6 ∶ 10. Answere , reduce them , and adde them both into one totall , makes 152 ∶ 120 of a pound for the number you doe seeke .

The proofe in coyne ; 152 ∶ 120 of a pound is 304 pence , and 8 ∶ 12 of a pound is 160 pence , which taken from 304 , leaues 144 pence remaining , which is 6 ∶ 10 of a pound , or 12 shillings , as appeareth by the worke .

What number is that , which being multiplied by 3 ∶ 5 , the product wil be 9 ∶ 20. Answere diuide 9 ∶ 20 by 3 ∶ 5 , and the quotient is 45 ∶ 60 , or 3 ∶ 4. For the proofe , multiply 108 pence , which is 9 ∶ 10 of a pound , by 240 , the product is 25920 ; which diuide by 144 , or 3 ∶ 5 , which is 12 shillings , makes 180 pence , or 3 ∶ 4 of a pound .

3. Example .

Example .

What number is that , which being diuided by 7 ∶ 8 , the quotient will be 4 ▪ 5. Answere , multiply 7 ∶ 8 by 4 ∶ 5 , the product is 28 ∶ 40 , or 7 ∶ 10 , which makes 14 shillings .

The proofe in Coyne ; 7 ∶ 8 , which is 210 pence , by 4 ∶ 5 , which is 192 pence , and the product is 40320 ; which diuide by 240 , makes 168 pence , or 14 shillings : behold the example following .

Example .

Rules of Practice .
Rules of Practice by the first Table .

TO worke by the Aliquot parts of a pound , search in the first Table for your giuen price , and by that number found , diuide your number giuen , and the quotient is your answere in pounds , and the remayner is the fraction of one pound .

But if the giuen price be not found exactly at the first entrance , then find 2 , or more numbers , to make the giuen price , and then worke as followeth .

Example .

If one yard cost 3 shillings 4 pence , what will 7859 yards cost at that rate : I enter the Table , and against 3 shillings 4 pence , I find 1 ∶ 6 of a pound ; wherefore I diuide 7859 by 6 , makes 1309 pound , 5 ∶ 6 of one pound , or 16 shillings 8 pence .

The first Table . The second Table . The Aliquot parts of a pound . Shillings . s. d. part . s. d. par. s. par. s. part . 1 240 1. 4 15 1 ½ 11 5. ½ 2 120 1. 8 12 2 ●/10 12 6. 3 80 1. 0 10 3 1. ½ 13 6. 1 / 2 4 60 2. 6 8 4 2. 14 7. 5 48 3. 4 6 5 2. ½ 15 7. ½ 6 40 4. 0 5 6 3. 16 8. 8 30 5. 0 4 7 3. ½ 17 8. ½ 10 24 6. 8 3 8 4. 18 9. 1. 0 20 10. 0 2 9 4. ½ 19 9. ½ 1. 3 16 20. 0 1 10 5. 20 10. Diuisors . Multipliers .

At 16 pence an ell , what will 8976 elles cost , I find for 16 pence my diuisor , to bee 15 , and so diuiding 8976 by 15 , the quotient is 598 pound , 6 : 15 , or 2 : 5 , which is 8 shillings .

Example .

Adde a Cypher to your number giuen , and the last figure of your quotient will bee primes , euery one in value 2 shillings , and the remainer is the fraction of a prime , alwaies lesse then 2 shillings . In the first example , the remainer was 5 : 6 of one pound , but if you adde a cypher , the quotient will be 1309 pound , 8 primes or 16 shillings , and the remayner is 2 : 6 of one prime , or 1 : 3 , which is 8 pence .

At 2 shillings 6 pence a pound pepper , what will 2436 pound cost : find 2 shillings , 6 pence 1 : 8 of a pound , wherefore adde a cypher , and diuide , by 8 , makes 304 pound ; 10 shillings .

At 8 pence a pound Ginger , will 77856 pound cost ; diuide by 30 , adding a cypher , makes 2595 pound , 2 primes , or 4 shillings .

At 17 pence a pound Sugar ; what shall 23459 pounds cost : for 12 pence , diuide by 20 , makes 1172 pound , 9 primes , 1 : 2 , or 19 shillings : then for the rest of your giuen price , which is 5 pence ; take 48 , and diuide , and the quotient is 488 pound , 7 primes , which added together into one sum , makes the totall 1661 pound , 13 shillings , 7 pence .

Example .

At 6 shillings 8 pence a pound Cloues , what will 3769 pound waite cost : diuide by 3 , makes 1256 pound , 3 primes , 1 : 3 , or 6 shillings 8 pence .

At 22 pence an elle of Holland , what 3768 elles cost : for 20 pence diuide by 12 , makes 314 pound , and for 2 pence by 120 , makes 31 pound , 4 primes , or 8 shillings ; the totall is 345 pound , 8 shillings .

If one elle of Holland cost 20 pence , how many elles shall I buy for 345 pound : multiply 345 by the price , which is ● : 12 , or by 12 , makes 4140 elles , the summe desired .

If one elle of Ozenbrigs cost 8 pence , what summe of elles will 78 pound buy me : multiply by 30 , makes 2340 elles .

At 15 pence an elle of Canuas , how many elles will 100 pound buy : multiply by 16 , makes 1600 elles .

If one elle of parchment lace cost 1 penny , how many ells shall I haue for 73 pound : multiply by 240 , makes 17520 elles .

Example .

If one Acre of land bee 5 shillings , how many Acres may I hire for 132 pound : multiply by 4 ; makes 528 Acres .

Rules of Practice by the second Table .

If the price giuen be any number of shillings , search in the second Table for the price giuen , and by the number there found : multiply your number of yards , ells , pounds or pieces , and cut off the last figure with a dash of the penne for primes , euery one in value 2 shillings , and the product is the sum of pounds and shillings that your giuen number will cost .

Example .

At 2 shillings an elle of Holland , what will 956 elles cost : in the second table ● find the tenth of the number giuen , so that if you take the tenth of 956 , it is 95 pound , 12 shillings , onely by cutting off the last figure by a dash of the penne .

956 elles at 2s . an ell , makes 956 , or 12s .

At 7 shillings an ell of Cambricke , what will 789 elles cost : multiply by 3 1 ∶ 2 , or take halfe of the giuen number , and multiply the whole number giuen , by 3 , makes in one summe , cutting off the prime line , 276 pound , 3 shillings .

Example .

At 25 shillings a piece Raysons , what will 356 pieces cost : take alwaies half the number of shillings of your giuen price for your multiplier , and worke as before , and the product is 456 poued , 0 prime .

Example .

Also 75032 pieces at 26 shilling a piece .

If one barrell of Sope cost 47 shillings , what will 3584 barrels cost : multiply by 23 1 ∶ 2 , makes 8422 pound , 8 shillings .

Example .

At 3 pound 6 shillings a Barrell , what will 124 cost .

If one Acre of land cost 6 pound 8 shillings , what will 758 Acres cost : multiply by 64 shillings , which is halfe the price , the product is 4851 pound , 4 shillings , or two primes .

How to proue the last question , or any other of like kind . If one Acre of land cost 6 pound 8 shillings , how many Acres shall be bought for 4851 pound , 4 shillings : diuide your number of pounds and shillings by one halfe of the number of shillings in the price giuen , adding a cypher to your number of pounds , and the quotient is the number of Acres of land the said summe wil buy at that rate .

Example .

The giuen sum is 4851 pound , 2 primes , or 4 shillings ; which diuided by halfe the giuen price , which is 64 shillings , brings into the quotient 758 Acres : and so of any other summe .

A Merchant bought Cambricks , cost him 855 pound , 18 shillings ; the question is , how many pieces hee had , paying for euery piece 27 shillings . Answere , ad●e a cypher to your number giuen , which 855 pound , 9 primes , makes 85590 ; which diuide by half the price giuen , which is 13 1 ∶ 2 ; or diuide by 135 the quotient will bee 634 pieces : now the reason wherefore a cypher is added to the number giuen , hauing 9 primes in it is , because I diuide by 13 1 ∶ 2 , which hath one fraction ; and this rule is generall .

Example .

What cost 634 pieces , at 27 shillings .

How to prooue one question in the Rules of Practice , by working of another .

If you wil proue any question in the rules of Practice , by a second example marke the complement , or want of your giuen price from one pound , and worke the same number at that price which doth want , and the totall of those two summes added together , makes the iust number of pounds of the giuen summe .

Example .

At 16 shillings a piece of Fustian , what will 320 pieces cost . Answere ; multiply by 8 , makes 256 pound , 0 prime .

Againe , 16 shillings your giuen price wanted 4 shillings of one pound , wherefore worke 320 at 4 shillings , which is multiplied by 2 primes , makes 64 pound , 0 prime , the totall is 320 pound , which prooues the former worke .

Example .

At 13 shillings a piece of Lawne , what will 752 pieces cost : 752 by 6 1 ∶ 2 , makes 488 pound , 8 primes .

At 7 shillings a piece , what 752 pieces : 752 by 3 1 ∶ 2 , makes 263 pound , 2 primes , totall is 752 pound .

Example .

Rules of Practice by the third Table , the most excellent of all the other .

The third Table . The Aliquot parts of 24. d. part . d. part . 1 24. 13 2. 24 2 12. 14 3. 4 3 8. 15 2. 8 4 6. 16 3. 3 5 12. 8 17 3.4 . 8 6 4. 18 2. 4 7 8. 6 19 2.8 . 6 8 3. 20 2. 3 9 4. 8 21 2.4 . 8 10 4. 6 21 2.4 . 6 11 3. 8 23 2.3 . 8 12 2. 24 ●/10

Diuisors . The parts of a Shilling . d. par. d. part . ¼ 48 7 2. 12 ½ 24 8 1. ½ ⅓ 18 9 2. 4 ¾ 16 10 2. 3 11 2.3 . 6 d. 12 Idem . 1 12 Diuisors . 2 6 3 4 4 3 6 2

Diuide the number of ells , yards , pounds , or pieces giuen by the number , or numbers found in the third Table , alwaies cutting the last figure for primes ; if that any remaine after Diuision , it is alwaies lesse then one prime , or 2 shillings .

Example .

At 3 pence a pound Licoras , what will 123728 pound cost . Answere ; for 3 pence in the third Table , I find my Diuisor to bee 8 , by which I diuide my giuen number , makes 1546 pound , 6 primes , or 12 shillings .

At 9 pence the pound Ginger , what will 8768 pound cost : for 6 pence diuide by 4 , makes 219 li. 2 primes ; then for 3 pence the residue of the price , diuide by 8 , makes 109 pound , 6 primes , totall is 328 pound , 16 shillings . Or otherwise , diuide by 4 for 6 d. and then take half that product for 3 pence , and adde them into one summe , as before .

Example .

At 11 pence the yard Canuas , what will 2356 cost : for 8 pence diuide by 3 , makes 78 pound , 5 primes , 1 ∶ 3 or 8 pence ; and for 3 pence , diuide by 8 , makes 29 pound , 4 primes , 1 ∶ 2 , or 12 pence ; the totall is 107 pound , 19 shillings , 8 pence .

A second example , the proofe of the last .

At 13 pence a pound fine Sugar , what will 2356 pound cost : for 12 pence diuide by 2 , makes 117 pound , 8 primes , or 16 shillings ; then for 1 penny , diuide by 24 , makes 9 pound , 8 primes , 4 pence , the totall is 127 pound , 12 shillings , 4 pence ; which added to the former summe in the last example , makes 235 pound , 12 shillings ; and so much will 2356 pound cost at 2 shillings a pound , because the two giuen prices make one prime , or 2 shillings .

Example .

At 16 pence a pound Sugar , what will 78432 pound cost : worke for 8 pence , and double the summe , makes 5228 pound , 8 primes , or 16 shillings .

At 8 pence a pound Almonds , what will 78432 pound cost : diuide by 3 , makes 2614 pound , 4 primes , or 8 shillings : which added with the former example , makes 7843 pound , 2 primes , which is the price that 78432 pound will cost at 2 shillings a pound , and prooues both examples true .

Example .

At 18 pence a pound Comfits , what will 78432 pound cost : for 12 pence take halfe the giuen number , and for 6 pence take half of that summe , which added into one totall , makes 5882 pound , 4 primes .

At 6 pence a pound small Ginger , what will 78432 li. cost : diuide by 4 , makes 1960 li. 8 primes , or 16 shillings ; which added to 5882 pound , 8 shillings , makes 7843 pound 2 primes , the price at two shillings .

These tables may serue also , if the price bee aboue 2 shillings , or one prime : as if you shall say at 3 shillings , 6 pence an ell , what 782 elles : here I see the giuen price is compounded of 7 times 6 pence ; wherefore I worke first for 6 pence in diuiding by 4 , makes 19 pound , 11 pence ; which multiply by 7 , makes 136 pound , 17 shillings for the price of 782 elles at 3 shillings , 6 pence the elle .

At 6 pence an elle , what 782 elles : find for 6 pence , 19 pound 11 shillings , which added to the former summe in the last example , makes 156 pound , 8 shillings , which is the summe that 782 elles will cost at 4 shillings the elle .

Example .

At 4 shillings 8 pence the elle Holland , what will 2148 elles cost . I find 4 shillings 8 pence to be 14 Groats , so diuiding by 6 for one Groat , makes 35 pound , 8 primes ; which multiply by 14 , makes 501 pound , 4 shillings .

At 15 pence a groce of points , what will 2256 grosse cost . 15 Pence is 5 times 3 d. and so I diuide 2256 by 8 , makes 28 pound , 2 primes ; which multiply by 5 , makes 141 pound .

Rules of Pr●●●ice by the fourth Table .

If the number of the price giuen bee any Aliquot part of a shilling : enter the fourth Table , and there you shall find a Diuisor , by the which if you diuide your number giuen , the Quotient will be shillings , and the remainer parts of one shilling . Then to conuert your shillings into pounds , take one halfe of the Quotient , cutting off the lower number for shillings , and the rest is pounds .

Example .

At 3 far things a pound prunes , what will 756 pound wayte cost . Search in the fourth Table , and you shall find 16 for your Diuisor : by the which if you diuide 756 , the Quotient is 47 shillings , 1 ∶ 4 , or 3 pence .

At 1 halfe penny a pound Coporas , what will 8756 pound cost . Diuide by 24 makes 364 shillings , of which the one half cutting of the 4 shillings , is 18 pound 4 shillings ; and 20 halfe pence remaining , totall is 18 pound , 4 shillings , 10 pence .

At 4 pence a pound Licoras , what will 789 pound cost . Diuide by 3 , makes 13 pound , 3 shillings .

Againe , at 6 pence a pound , what will 8579 pound cost . Diuide by 2 , makes 214 pounds , 9 shillings , 6 pence .

1. Example .

2. Example .

Generall Rules of Practice without Tables .

Multiply your number giuen by the sum of pence , that one yard , piece , pound , or elle doth cost , and the product will be the summe of pence , the whole number giuen will cost ; and then diuide that summe of pence by 4 , ma ▪ es the Quotient Groats , and if any reremaine , they are pence , alwaies lesse then 4 pence , or one Groat : and secondly againe , diuide that Quotient will bee pounds and primes , euery prime in value 2 shillings , and the remaines is Groats , alwaies lesse then 6 Groats , or one prime , which is value 2 shillings .

At 17 pence an ell Canuas , what will 3245 elles cost : Multiply by 17 , makes 55165 pence , which diuided by 4 , makes 13791 Groats , and there will remaine one penny . Secondly , diuide that Quotient againe by 6 , makes 229 pound , 8 primes , and the remayner is 3 Groats , or one shilling ; and so the totall is 229 pound , 17 shillings , 1 penny .

Example .

At 3 shillings , 7 pence a yard Holland , what will 752 elles cost : multiply 752 by 43 pence , the price of one ell , makes 32336 ; which diuided , as is before taught , makes 134 pound , 14 shillings , 8 pence .

Example .

At 7 shillings , 11 pence the elle Cambricks , what will 85● ells cost . Reduce 7 shillings , 11 pence into pence , makes 95 pence ; by which multiply 856 , makes 8●320 ; which diuided as before , makes 338 pound , 16 shillings , 8 pence .

Example .

At 2 shillings , 11 pence an elle Holland , what will 7856 elles cost : multiply , and diuide as is before taught , makes 1145 li. 13 shillings , 4 pence .

Example .

At 17 shillings , 7 pence a yard Broad Cloath , what will 7856 yards cost : multiply by 211 ▪ the price of one yard , and diuide as before , makes 6906 pound , 7 primes .

Example .

If your giuen price haue any farthings in it , then reduce your price into farthings , and multiply your giuen number by those farthings , and the product will be the number of farthings , which your summe will cost : then diuide that product by 16 , makes the quotient Groats , and the remainer will be farthings , alwaies lesse then 16 , or one Groate . Secondly , diuide that quotient of Groats by 6 , makes pounds and primes , as before .

Examples .

At 5 shillings , 1 penny , one halfe-penny an ounce Plate , what will 356 ounces cost . Reduce 5 shillings , 1 penny , half-penny into farthings , makes 246 fa●things : by which multiply 356 , makes 87567 farthings ; which diuided by 16 , makes 5473 Groats , and 8 farthings will remaine ; which diuide againe by 6 , makes 91 pound , 2 primes , and one Groat will remaine , totall is 91 pound , 4 shillings , 6 pence .

At 6 shillings , 9 pence , farthing an ounce of gilt plate , what will 3542 ounces cost : multiply your shillings by 48 , the farthings which are in one shilling , makes 288 ; to the which adde 37 farthings , which are in 9 pence , farthing , makes 325 farthings ; and then worke as before is taught , and you shal find 1199 pound , 2 shillings ; 3 pence , halfe-penny .

Example .

Another way to worke Practice .

Diuide your number of yards , elles , or pieces by 240 , adding a Cypher to your number giuen , and then multiply the Quotient by your price , and the Product is the summe of pounds , and shillings , that the giuen number will cost .

At 17 pence the elle Canuas , what will 7848 elles cost : adde a Cypher , and diuide 78480 by 240 , and the Quotient will bee 32 pound , 7 primes ; which multiply by 17 pence , the price , makes 555 pound , 9 primes or 18 shillings .

Example .

At 3 shillings , 5 pence an ell of Holland , what will 702 elles cost : diuide 7020 by 240 , makes 2 pound , 9 primes , and there will remaine 6 ; which multiply by 41 pence , the price of ●e ell , makes 118 pound , 9 primes , or 18 shillings , and then the 6 elles , makes 1 pound , 6 pence , the totall is 119 pound , 18 shillings , 6 pence .

Example .

At 19 pence the elle of Holland , what will 32544 elles cost : diuide 325440 by 240 , makes 1356 ; which multiply by 19 pence , the price of one elle , makes 2576 pound , 8 shillings .

The Golden Rule . Of single proportion Direct , or the Rule of three , called The Goulden Rule .

IN this Rule of 3 Direct , there is alwaies three termes giuen , and a fourth required , and it is called the Goulden Rule , in regard of the excellency of this Rule aboue all others . The difficulty of this rule consisteth in the right placing of the three numbers giuen , set the terme next your right hand , whhereupon the question is moued , and a terme of the same nature towards the left hand , & the third terme in the middle . Then multiply the second number by the third , and diuide the product by the first , and the Quotient is the fourth proportional number sought or desired to be found out ; whose denomination is euer like vnto the middle number .

1. Example .

If 90 yards of Cloath cost 23 pound , what cost 346 yards .

If 124 pound gaine 37 pound , 12 shillings , what will 758 pound gaine .

How to worke this last example , and all other , after a more briefe and exact manner .

Diuide the third number by the first , and by the Quotient multiply the second , and the product is the answere .

Example .

If 356 elles cost 137 pound , 12 shillings , 9 pence , what cost 2848 elles .

First , diuiding 2848 by 356 , the Quotient is 8 ; by which I multiply 137 pound , 12 shillings , 9 pence , the products are 1096 pound , 96 shillings , 72 pence ; then diuide 72 by 12 , is 6 shillings ; which added to 96 shillings , makes 102 shillings , or 5 pound , 2 shillings ; the totall is 1101 pound , 2 shiliings , as before .

2. Example .

If 124 yards cost 17 pound , 10 shillings , 1 penny , what cost 744 yards .

If 32 pieces of Raysons cost 19 pound , 2 shillings , 2 pence , what will 112 pieces cost at that rate .

3. Example .

If 356 pieces cost 137 pound , 12 shillings , 9 pence ; what will 2848 pieces cost at that rate .

Example .

Example .

How to know whether any question giuen be to be answered by the Rule Direct , or Conuersed .

By these notes following , you shall find , whether any question propounded be to be answered by the Rule of 3 Direct , or conuersed ; for alwaies the third number is the number whereon the question dependeth , and is distinguished from the other two , by some one of these notes following .

And the answer is alwaies , more or lesse , so that if it bee more then the lesser of your two extreame numbers is the diuisor : if lesse , then the greater of your two extremes is your diuisor . If the number whereon the question bee depending , be your Diuisor , them the answer is , by the conuerse Rule , and you must multiply your two former numbers for Diuidend . If the first number be the Diuisor , then the question is answerable by the Direct Rule , and the product of the two latter numbers is your Diuidend .

Example .

If 13 Cannons spend 358 pound of powder , what will 5 Cannons spend , now here the question is , what 5 Cannons will spend . I answere , lesse then 13 Cannons ; wherefore by this rule , the greater of the two extreames , 13 is the diuisor : wherefore I multiply 358 by 5 , and diuide by 13 , makes 137 pound , 6 ∶ 13 that 5 Cannons will spend .

2. Example .

If 13 Cannons spend 358 powder , what will 5 Cannons spend .

2. Example .

I lent my friend 115 pound for 7 moneths , and when I came to him to require the like kindnesse he could lend me by 54 pound , the question is , how long hee should forbeare that 54 pound to make requitall , or to equal my time , and kindnesse .

If 115 pound require 7 monthes , what will 54 pound require : here the answere in reason is , that 54 pound must bee longer time forborne then 115 pound , and so the answere is more times then 115 pound ; so that I find the lesser of my exteames 54 , is my Diuisor , and the question answerable by the Rule conuersed , so that I multiply 115 by 7 , makes 805 ; which diuided by 54 , makes 14 moneths , 49 ∶ 54 of a moneth , or 14 moneths , 25 dayes , 23 ∶ 25

Example .

4. Example .

A Captaine of a Band of men is besieged in a Citie , hauing with him 7200 men , and his victuals will serue the whole Company but 7 moneths , but there is no hope left to haue any fresh victuals vntill 16 moneths ; the question is , how many men he shall send away to make the victuals serue for 16 moneths . Answere , lesse then 7200 men .

If 7 moneths require 7200 men , how many will 16 moneths aske .

When Wheate was sold at 3 shillings , 8 pence the bushell , the penny loafe of bread waighed 6 ounces , what shall the same loafe of bread waigh , when Wheat is sould for 2 shillings the bushell : I answere more then a 11 ounces .

If 44 pence giue 6 ounces , what will 24 pence giue .

If 356 men digge a trench in 24 dayes , in how many dayes will 200 men make the same ? Answere , in more dayes ; 42 dayes , 17 houres , 7 ∶ 25.

If 356 men require 24 dayes , how many will 200 men require .

Or thus ; Considering the numbers , 200 may be had in 156 once , therefore for 200 take 24 dayes ; then for 150 take 18 dayes , totall 42 dayes ; then there will remaine 6 to bee multiplied by 24 , makes 144 ∶ 200 parts of a day , as before .

If 112 pound cost 3 pound , 5 shillings , 5 pence , what will 3136 pound cost ? diuide 3136 by 112 , makes 28 ; which multiply by 3 pound , 5 shillings , 5 pence , makes 91 pound , 11 shillings , 8 pence .

If 100 pound gaine 7 pound , what summe of money will gaine 85 at that rate ? Answere .

If 7 pound require 100 pound , what will 85 pound , require .

Or otherwise , diuide 85 by 7 , makes 12 1 ∶ 7 ; by which multiply 100 , makes 1214 pound 2 ∶ 7 of a pound .

Or otherwise , diuide 100 by 7 , makes 14 2 ∶ 7 ; by which multiply 85 , makes 1214 pound , 2 ∶ 7

Example .

Carseys at 54 shillings the piece , are put in Barter , at 3 pound the piece , how shall Wooll worth 24 shillings the Tod , be set in Barter , to make the bargaine equall ?

If 54 shillings be 60 shillings , what shall 24 shillings make .

Answere : for more then 24 shillings , and lesse then 54 , so that 54 is the diuisor , and multiplying 24 by 60 , makes 1440 ; which diuided by 54 , makes 26 shillings , 2 ∶ 3 , or 8 pence .

If 54 shillings be 60 shillings , what will 24 shillings make .

If 6 sheepe cost 58 shillings , how many shall I buy for 124 pound ? multiply 124 by 58 , makes 7192 ; which diuide by 6 , makes 1198 sheepe 2 ∶ 3.

Or otherwise , diuide 58 by 6 , makes 9 2 ∶ 3 , by which multiply 124 , makes 1198 2 ∶ 3 , as before .

Example .

A Merchant at Siuill deliuereth 1500 Rialls , to receiue for euery 11 , being a ducat in London 5 shillings , 10 pence sterling money , how much must he receiue ?

If 11 Rialls be 70 pence , what are 1500 Rials ?

At 13 pound in the 100 pound profit , of what stocke came 3274 pound ? Answer : diuide 3274 pound by 113 pound , makes 2897 pound , 39 ∶ 113 of a pound , adde two cyphers to the giuen number .

A Merchant receiued for principall and gaine 328 wherein he found he had gained cleare 56 pound , what did he gaine vpon the 100 pound . Answere , multiply 100 by 56 , the gaines makes 5600 ; which diuide by 328 , and the Quotient is 17 pound , 3 ∶ 41 in smallest termes .

If 112 pound cost 7 pound , 6 shillings , how may I sell to gine 10 pound vpon the 100 pound . Answere : Take the tenth part of 7 pound , 6 shillings , or of 146 shillings , which is 14 shillings , 3 ∶ 5 of a shilling ; which added to the price , makes 8 pound , 7 pence , 1 ∶ 5 of a penny .

If 100 pound exchange be 7 pound 2 shillings , what is one pound . Answere , 71 ∶ 100 parts of a pound : wherefore multiply 71 by 240 , and diuide by 100 , makes 17 pence ▪ 12 ∶ 5 of a penny .

If 107 ells of cloth cost 17 pound 12 shillings , what will 321 ells cost at that rate ? Here if you consider the proportion betweene the first number , and the third , you shall find the third number doth containe the first exactly three times ; wherefore you need not to multiply the second by the third , and diuide by the first number , but only take the second number , and multiply by 3 , makes 52 pound , 16 shillings for the price that 321 ells will cost : behold the worke at large .

If 107 ells of cloth cost 17 pound , 12 shillings , what will 321 ells ?

How to find whither that your numbers giuen be proportionall , or not .

Diuide your third number by the first , and if the quotient be an euen number , and nothing remaine of your diuidend , then the first and third numbers are euen proportionall in whole numbers , as in the last example , the first number was 107 , and the third number 321 , so that in deuiding the third number by the first , the quotient is 3 & 0 remaines : wherefore I conclude , that the first and third numbers are proportionals in whole numbers , and that the third doth containe the first iust three times , and so often must the fourth number sought for , containe the second ; and I conclude , that three times 17 pound 12 shillings , which is 52 pound 16 shillings , is the fourth proportionall number sought , as appeareth by the ordinary forme of worke in the last example .

If 36 elles of cloth cost 13 pound , 4 shillings , 1 penny , what will 432 elles cost at that rate : diuide 432 by 36 , makes 12 ; by which multiply your second number 13 pound , 4 shillings , 1 penny , makes 158 pound , 9 shillings .

A. doth lend vnto B. 600 pound for 8 moneths , the question is , how much B shall lend vnto A. for 12 moneths to recompence him , not reckoning compound , interest . Answere . If 8 moneths require 600 pound , what will 12 moneths require ▪ the reason is lesse then 600 pound ; wherefore diuide 600 pound by 12 , makes 50 ; which multiply by 8 , makes 400 pound .

Or otherwise by proportion , as 8 is to 12 so must 600 bee to 400 pound , 2 ∶ 3 parts of 600 pound .

If the number bee not exactly proportionall , yet there is a great abreuiation to bee made of the worke of Reduction , Multiplication , and Diuision , in the working of most examples in the Golden Rule ; as for example .

If 19 Barrels of Figgs cost 16 pound 12 shillings , what shall 58 barrels cost , here diuiding 58 by 19 , the Quotient is 3 , and 1 will remaine ; wherefore I take 3 times 16 pound , 12 shillings for 57 barrels , and I haue to worke but for the one remaining ; which is but to diuide 16 pound , 12 shillings , by 19 , makes 17 shillings , 9 ∶ 19 of one shilling , the totall is 50 pound , 13 shillings , 9 ∶ 19 shillings .

If 356 elles of Holland cost 124 pound , 2 shillings , 3 pence , what will 7259 elles cost at that rate . Reduce 124 pound , 2 shillings 3 pence , into pence , makes 29787 pence ; which multiply by 7259 , makes 216223833 pence , which diuide by 356 , make 607370 ▪ which diuided by 240 pence , makes 253 pound , 170 pence , or 14 shillings 2 pence .

Example .

A second way more briefly to worke this question , or any other of like nature , is this : multiply the third number by the pounds and primes , or shillings and pence , and diuide the product by the first number , and the quotient will be the fourth number sought . In the last example , 7259 elles was the third number , which multiply by 124l . 1 prime , or 2s . makes 900841 l. 9 primes : then also 7259 by 3 pence , makes 21777 pence ; which diuided by 240 , makes 90 pound , 14 shillings , 9 pence : then adde those two sums into one totall , makes 9009326 primes , 9 pence ; leaue out ▪ 9 , and then diuide the residue by 356 , makes 2530 pound , 7 primes , and 54 ∶ 356 ; which with the 9 d. brings out the two pence , as in the last example .

Example .

If 24 pieces of Raysons cost 25 pound , 8 shillings , what will 324 pieces cost : multiply 324 by 25 pound , 4 primes , makes 8229 , 6 primes : which if you diuide by 24 , the Quotient will be ●42 pound , 9 primes , or 18 shillings without Reduction , as in the example following .

Example .

If 25 pound gaine 1 pound , 8 shillings , what will 725 pound gaine at that rate ? Multiply 725 by 1 pound 4 primes , makes 10150 ; which diuided by 25 , makes 40 pound , 6 primes , or 12 shillings .

And in this sort may diuers other questions bee wrought in pounds and shillings without Reduction , which I thought good to giue a taste of , but I will proceed here no further , because I purpose in the second part of this Booke to speake of them at large in the Treatise of Decimal Arithmatick , whereby all manner of questions are to bee wrought of Multiplication and Diuision in pounds , shillings and pence , without Reduction , as shall appeare in their seuerall places following . And now I will proceede to speake something of the Rule of Three Direct and Conuerst in Fractionall operations , wherein I will be as briefe as I may , not intending to increase this little Treatise intended for a pocket booke , into ouer large a volume .

The Rule of 3 in Fraction .

If your three numbers giuen be all fractions , multiply the third by the second , and diuide the product by the first , and the quotient will be the fourth proportionall number sought for .

Example .

If 3 ∶ 4 of a yard of Holland cost 4 ∶ 5 of a pound , what shall 5 ∶ 6 of one yard cost at that rate ? Multiply 5 ∶ 6 by 4 ∶ 5 , makes 20 ∶ 30 or 2 ∶ 3 , which diuide by 3 ∶ 4 , makes 8 ∶ 9 of one pound , or 17 shillings , 7 ∶ 9 of one shilling .

If 7 ∶ 8 of one ell of cloth cost 9 ∶ 12 of a pound , what will 17 elles cost ? Make 17 fraction wise , and multiply 17 ∶ 1 by 9 ∶ 12 , makes 153 ∶ 12 , which wil be both abreuiated by 3 , makes 51 ∶ 4 , which diuided by 7 ∶ 8 makes 408 ∶ 28 parts of a pound , or in smaltermes 102 ∶ 7 ; then diuide 102 by 7 , makes 14 pound , 4 ∶ 7 of one pound for the price .

2. Rule .

If all your three numbers giuen be fractions , multiply the Numerator of the first fraction by the Denominator of the other two fractions , for to make your Diuisor . Then multiply the Denominator of your first fraction by the Numerators of your other two fractions , to make your Diuidend and then diuide by your Diuisor , and the Quotient is the answere sought : but if your Diuisor bee greater then your Diuidend , then the Quotient is a fraction , lesse then a vnite .

Example .

If 3 ∶ 4 of a yard cost 4 ∶ 5 of a pound , what cost 5 ∶ 6 of a yard ? Multiply 3 , the Numerator of the first fraction by 5 and 6 , the denominators of the other two fractions , makes 90 for your diuisor ; then multiply 4 , the denominator of your first fraction by 4 and 5 , the numerators of your other two fractions , makes 80 for your diuidend : now because your diuisor is greater then your diuidend , place them fraction wise thus , 80 ∶ 90 of 1 li. or in least termes , 8 ∶ 9 of a li.

Example . 90 If 3 ∶ 4 of a yard cost 4 ∶ 5 of a pound makes 8 ∶ 9 l. what 5 ∶ 6 of a yard 80

Againe , if 7 ∶ 8 of an ell cost 2 ∶ 3 of a shilling , what will 34 ells cost .

If 18 Pioners in 3 ∶ 5 of a day doe make 22 Rodds of Barricadoe , what will they make in 7 dayes .

3 If 3 ∶ 5 of a Day make 22 ∶ 1 of a Wall , what 7 ∶ 1 of a Day ? 770

If 12 hundred 3 ∶ 7 of Allum cost 15 pound 1 ∶ 3 li. what will 324 , 1 ∶ 8 of a hundred cost ? Reduce the whole and broken numbers into broken , and worke as is before taught .

If 7 ∶ 9 of an ell cost 8 ∶ 11 of a pound , what will 15 ∶ 13 of an ell cost ? 1 pound , 1 shilling , 6 pence , 3 ∶ 4 , fere .

Example .

If 3 ∶ 4 of a yard of Veluet cost 7 ∶ 8 of a pound , what wil 28 yards cost . 32. l. 13 s. 4 d.

Example .

If 3 ells 1 ∶ 8 cost 5 ∶ 7 of a pound , what wil the whole piece cost , containing 28 ells 1 ∶ 2 at that rate ? Answere .

If 12 pound , 4 ounces of Quichanella cost 4 pound , 3 shillings , 4 pence , how much will 100 pound buy me at that rate ? If 49 ∶ 4 of a pound cost 25 ∶ 6 of a pound sterling , what will 600 ∶ 6 parts of a pound buy . Answere .

The proofe of this last example . If 100 pound starling buy me 294 pound of quichanella , how much shall 4 pound , 3 shillings , 4 pence buy mee ? to find the valew of the hundred , the rate of one pound being giuen ; abate 2 places from 294 , and it will bee 2 pound 94 ∶ 100 parts of one pound : which multiply by 4 pound , 1 ∶ 6 , makes 12 pound 25 ∶ 100 parts , or one fourth for the proofe .

If 30 men cast a Trench in 3 dayes 2 ∶ 3 ; how many men would cast it in 5 ∶ 6 of a day ? here by comparing these proportions together , I find that 5 ∶ 6 the third number , will desire a greater quantitie of men to performe the worke , then 11 ∶ 3 of a day will require ; wherefore this proportion is reciprocall or backward ; wherefore I multiply the two former numbers together , makes 333 ∶ 3 , or in smaller termes , 110 ∶ 1 ; which diuided by 5 ∶ 6 , makes 660 ∶ 5 ; which diuide by the denominator 5 , makes 132 men .

In the Backward Rule , or Conuerst in fractions , multiply the Denominator of your third number , by the numerators of both your other numbers for diuidend , then multiply the numerator of your third number , by the denominators of your other two numbers for diuisor , and then worke as before .

Example .

If when the bushell of Wheate was sold for 4 shillings , the penny loafe wayghed 6 ounces 1 ∶ 2 , what shall the same loafe weigh when Wheat is sold for 2 shilling , 8 pence p●nce the bushell ? Multiply 48 by 13 , makes your diuidend 624 : then 2 by 32 , makes 64 for your diuisor , and then diuide 624 by 64 , makes 9 ounces , 48 ∶ 64 , or 3 ∶ 4 of an ounce .

If when one ounce of sterling siluer was worth 1 ∶ 4 of a pound the penny of siluer weighed 30 graines , what shall the same penny weigh , when the ounce shal be worth 1 ∶ 3 of a pound .

Example .

If when a loade of Hay was sold for 24 shillings , 8 pence , the penny bottle weighed 3 pound , 1 ∶ 4 , what shall it weigh , now the load is sold 37 shillings . Answere , 2 pound , 71 ∶ 76 of a pound .

If 3 yards 1 ∶ 8 cost 9 shillings , 9 pence , what will 380 yards cost at that rate ? Reduce 3 yards 1 ∶ 8 into eights , makes 25 ∶ 8 ; then reduce 380 yards into eights , makes 3040 ∶ 8 parts : then 9 s. 9 d. into pence , makes 117 pence ; by which multiply 3040 , makes 355680 , which diuided by 25 , makes 14227 pence , 5 ∶ 25 of one penny in the whole 59 pound , 5 shillings , 7 pence , 5 ∶ 25 or 1 ∶ 5 of a penny . Behold the worke .

The proofe of the former worke . If 380 yards cost 59 pound , 5 shillings , 7 pence 5 ∶ 25 of one penny , what will 3 yards 1 ∶ 8 cost at that rate ? reduce your coyne into 25 , makes 355680 ; then reduce your 380 yards into 8 , makes 3040 ; by which diuide 144227 pence , makes 9 shillings , 9 pence , as before .

If 34 ship Carpenters build a ship in 8 moneths 3 ∶ 5 , in how long time will 120 Carpenters build the same ? Reduce 8 moneths 3 ∶ 5 into fifthes , makes 43 ∶ 5 ; then multiply 34 by 43 , makes 1462. Also put your diuisor 120 into fifthes , makes 600 ∶ 5 ; then diuiding of 1462 by 600 , the Quotient will bee 2 moneths 262 ∶ 600 parts of one month , or in smallest termes 131 ∶ 300 parts . And this Rule generall if one of your numbers bee a fraction , put alwaies your diuisor into the same fraction of your diuidend , and the quotient will bee of the same denomination of your diuidend , and so the answere was moneths , and parts of a moneth .

If 34 Carpenters aske 43 ∶ 5 moneths , what 600 ∶ 5 moneth .

If 100 pound in 12 moneths gaine 10 pound , what will 336 pound gaine in 8 moneths ? Take the tenth part of 336 , which is ●● li. 6 primes , or 12 s. makes 369 li. 12 s.

Secondly , if 12 moneths gaine 33 pound 6 primes , what will 8 moneths gaine ? I answere , lesse then 33 pound 6 primes ; wherfore multiply by 8 , and diuide by the greater extreme , 12 , makes 22 pound , 4 primes , or 8 shillings , the answere .

If 120 Pioners in 6 dayes cast 300 rods of Trench , how many shall 600 men cast vp in 4 dayes . If 120 giue 300 , what will 600 giue ? Answere , 1500 Rods.

Secondly , if 6 dayes giue 1500 rods , how many will 4 dayes giue ? I answere , lesse ; multiply by 4 , and diuide by 6 , makes 1000 Rods.

If 112 pound in 12 months gaine 100 li. what wil 340 li. gaine in 7 months ? Answer : 303 li. 4 ∶ 7.

Secondly , if 12 moneths gaine 303 li. 4 ∶ 7 what will 7 moneths gaine .

Example .

A generall Rule ▪

Put alwaies your diuisor into the same fraction of your diuidend , and your quotient will be of the same denomination , that your diuidend was : as in the last example , 12 moneths was turned into seuenths , and also 303 pound 4 ∶ 7 was turned into seauenths of pounds , and so the quotient of that diuision was pounds , and the fraction of a pound remaining .

If 7 pound in 13 months gaine 3 pound , in how long time will 340 pound gaine 60 pound . First , if 7 pound Gaine 3 pound , what will 340 pound gaine , makes 145 pound , 5 ∶ 7 of a pound . Secondly , if 145 pound , 5 ∶ 7 or 1020 ∶ 7 ask 13 moneths , what will 60 pound , or 420 ∶ 7 gaine . Multiply by 13 , and diuide by 1020 , makes 5 months 6 ∶ 17 of a month .

If 600 great Horses in 5 dayes doe spend 1125 Bushels of oats , how many bushels wil serue ? 1400 Horses for 22 Dayes . First , say , if 600 giue 1125 , what 1400 , makes 2625 bushels . Secondly , if 5 spend 2625 bushels , what will 22 dayes spend ? Multiply by 22 , and diuide by 3 , makes 11550 bushels .

How to worke the double Rule at one operation .

This last question , or any other of like nature which is wrought by the double Rule at two seuerall operations may be answered at one in this manner : multiply the three latter numbers , to make your diuidend one into the other ; then multiply the two former numbers for to make your diuisor , and then diuide the diuidend by the diuisor , and the quotient will be the same , as in the last example , 1125 being multiplied by 1400 , makes 1575000 ; which againe increased by 22 , makes your diuidend 34650000. Then multiply your two former numbers 600 by 5 , makes 3000 for the Diuisor ; and then diuiding your diuidend by your diuisor 3000 , the quotient will bee 11550 bushels , as before at two operations .

Example .

If 35 s. in 7 months gaine 6 s. in how long time will 340 l. gaine 100 l. First , if 35 s. gaine 6 s. what will 340 l. require ? Reduce 340 l. into pence , and multiply by 6 , makes 40800 ; which diuided by 35 , makes 1165 s. 5 ∶ 7 s. Secondly , if 1165 ● . 5 ∶ 7 require 7 moneths , what will 100 l , require ? Makes 12 moneths , 8 ∶ 816 parts of a moneth .

Fellowship without Time .

This Rule differeth very little from the Rule of three ; for in this Rule the summe of all the moneys disbursed , is the first number in the Golden Rule . Then the gaines or losse is the second number : the third number is each seuerall partners money disbursed so that the Rule must bee seuerally wrought for each seuerall Partners portion .

Example .

Foure Merchants made a company together ; the first , viz. A. put in stock 74 pound , B. put in 90 pound , C. put in 100 pound , and D. put in 120 pound , and they found that they had gained 84 pound ; now the question is , what each man must haue of the gaines , according to the proportion of his money disbursed . First , adde all the moneys disbursed into one totall summe , viz. 74 , 90 , 100 , 120 , totall is 384 for the first number in the Golden Rule . Then the second number is 84 pound , the gaines ; and the third number is each particular mans stock ; then worke as followeth .

If 384 pound gaine 84 pound , what will A. B. C. D. summs gaine to them .

The like reason is in losse , as is in gaines . Example : A certaine ship being in a tempest on the sea was forced to cast ouer board so much of her lading , as amounted vnto the summe of 642 pound , then there is great reason that all the ventures should beare part of that losse , according to the proportion of his stocke which hee ventured . As suppose : A. ventured 700 pound , B. 530 pound , C. 640 pound , D. 800 pound ; totall is 2670. Then say ; If 2670 pound loose 642 pound , what will each of A. B. C. D. loose ? as in the example following .

Example .

If 2670 pound loose 642 pound , what will A. B. C. D. summes loose to them .

fioure Merchants bought a ship , which cost them 3600 pound , whereof A ▪ must pay one third part of the money , B. one fourth , C. one fifth , D. one sixth ; the question is , what each man must pay of the saidsumme . Answere .

Seeke a number wherein the like parts may bee had , which is 60 , and take the like parts of that number for the numbers that you seeke , for to find each mans portion of the money ▪ which he should pay . First , 1 ∶ 3 of 60 is 20 , the 1 ∶ 4 is 15 , the 1 ▪ 5 is 12 , the 1 ∶ 6 is 10 ; which adde into one totall , makes 57 for the first number in the Golden Rule .

Example .

If 57 be 3600 , what will bee the summes of A. B. C. D.

The said ship made a Voyage to Sea , and hath gotten all charges , deducted out 240 pound , the question is , what each man must haue of the gaines . Answere .

If 57 gaine 240 , what will A. B. C. D. summes gaine to them .

Foure Merchants made a Company ; A. put in 320 pound , 13 shillings , 3 pence ; B. put in 840 pound , 16 shillings , 6 pence ; C. put in 560 pound , 18 shillings , 9 pence ; D. 1000 pound ; and in one yeare they found they had gained 400 pound , 18 shillings , 6 pence : the question is , what each man must haue of the gaines . First , the totall summe of all their moneys makes 2721 pound , 8 shillings , 6 pence , or 653142 pence , for the first number . Then reduce each seuerall mans money disbursed into pence for the third number , the second is the gaines also reduced into pence , and then worke according to the Rule .

Example .

If 2721 pound , 8 shillings , 6 pence gaine 400 pound , 18 shillings , 6 pence , what will A. B. C. D. summes gaine to them .

Rules of Fellowship , with diuersitie of Time .

Multiply each mans money disbursed by the time that it continued in stock , and gather the totals , as in the last Rule , to make the first terme in the Golden Rule , and the gaines or losse is the second , and then each mans product of money and time for the third terme in the Golden Rule , and worke as followeth .

Example .

Three men made a stock , A. B. and C. and in long continuance of time by dangerous aduentures they gained , and got by prizes taken at Sea 2345 pound ; A. put in stocke 40 pound , 14 moneths ; B. put in 50 pound , 8 moneth ; C. put in 85 pound 6 moneths , what shall each man haue of this gaines .

Example .

If 1470 pound gaine 2345 pound , what will A. B. C. summes gaine them .

The second question with more diuersity of time , foure Merchants made a Company ; A. put in 340 li. 19 s. 2 d. for 10 moneths ; B. put in 930 li. for 9 moneths ; C. put in 760 li for 12 moneths ; D. put in 583 li. 13 s. 4 d. for 5 moneths , wherewith they gained 740 li. now the question is , to know what each man must haue of this gaines .

Cut off two Cyphers from each number , and then worke as followeth .

If 57163 pence gaine 1776 pence , what what will A. B. C. D. summes gaine them .

Example .

There is a Bootie or Spoyle taken by 3 men worth 7851 pound , and they agree to diuide it in this sort ; A ▪ is to haue one half , B. one third , C. one fourth , what is each mans share .

To worke this question , and all other of like nature , seeke a number which may bee diuided by all the denominators of your three fractions in whole numbers , and the smaller such a number bee that you choose , the more easie will your worke be ; which for to find , multiply your denominators of your fractions one into another ; that is to say , 2 by 3 makes 6 ; and 6 by 4 , makes 24 ; so 12 , one halfe of 24 will bee euenly diuided by all the three denominators , 2 , 3 and 4. Wherefore I take 1 ∶ 2 of 12 is 6 , and 1 ∶ 3 of 12 is 4 , and 1 ∶ 4 of 12 is ● ; which added into one summe , makes 13 for first number in the Golden Rule ; the second is 7851 pound , and the third numbers are each seuerall mans portion imagined to be , viz 6 , 4 , 3 , and then worke as before .

If 13 giue 7851 pound , what will A. B. C. summes giue .

4. Example .

Foure Merchants bought a house together , which cost 3000 pound ; A. was to pay 1 ∶ 2 and 6 pound ouerplus ; B. 1 ∶ 3 and 12 pound more ; C. 8 pound lesse then 2 ∶ 3 ; D 1 ∶ 4 with 20 pound ouerplus . Now the question is , what each Merchant must pay of this sum . Answer : First , the pounds ouerplus must be subtracted from the summe giuen ; and the pounds wanting must bee added to the summe giuen ; as for A. 6 pound , B. 12 pound , for D 20 pound , totall is 38 pound , to bee subtracted then ; for C. adde 8 pound , therefore subtract 30 pound from 3000 pound , there wil remaine 2970 pound ; then worke by the Rule of Fellowship , taking 12 for a number , which will bee diuided by all the denominators , 2 , 3 and 4 , viz. take for A. 6 , for B 4 , for C. 8 , for D. 3 ; totall is 21 for diuisor , the second number is 2970 pound , the third , each mans part imagined .

Example .

If 21 giue 2970 pound , what will A. B. C. D. summes giue .

The numbers found to A. are 848 pound 4 ∶ 7 , to which if you adde 6 pound , makes 854 pound , 4 ∶ 7.

To B 565 pound , to which 12 pound added , makes 577 pound , 5 ∶ 7

To C. 1131 pound , 3 ∶ 7 , from which subtract 8 , leaues 1123 pound , 3 ∶ 7

To D. 424 pound , 2 ∶ 7 , to which adde 20 pound , makes 444 pound , 2 ∶ 7 ; the which added into one total , makes 3000 pound , the proofe .

And in this manner may infinite varietie of questions bee propounded , and their doubts easily resolued ; and here will I ende concerning this Rule , and goe in hand with some pleasant questions to bee wrought by position , which is the most excellent Rule of all others in Arithmatick , as shal appeare in the second part of this Booke in Decimall Arithmatick .

Position .
The Rule of Position requiring one number to bee imagined , before the principall proportion can be found .

TO worke by this Rule ; Take any number at pleasure , which you shall imagine to be the true number sought , and proceed with it , as if it were the true number , wherein if you haue failed , by doubling or tripling according to the nature of the question , you shall then attaine vnto the true number desired , by aide of the Golden Rule , in manner following : for looke what proportion is betweene the false conclusion , and the false position , such proportion hath the giuen number , to the number sought .

Example .

A. B. and C. consent to buy a ship , which will cost them 2700 pound , so that B. must pay twice so much as A , and C. must pay 4 times so much as B : the question is , what each man must pay of this summe ? I suppose A must pay 8 pound , then B must pay twice as much as A , which is 16 pound ; then C must pay 64 pound , which is 4 times as much as B : but yet 8 pound , 16 pound , and 64 pound , is but 88 pound , and it should be 2700 pound , so that now I resort to the Golden Rule , and worke as followeth . If 88 pound come of my Position 8 pound , of what comes 2700 ? Multiply 2700 by 8 , and then diuide by 88 , makes 245 pound 40 ∶ 88 , or 5 ∶ 11 of a pound for the part that A must pay ; then B must pay 490 pound , 10 ∶ 11 of a pound , which is twice as much as A ; and C must pay 1960 pound , 40 ∶ 11 of a pound , which is 4 times as much as B. The totall summe is 2700 pound . Behold worke as followeth .

If 88 pound come of 8 pound , of what comes 2700.

2. Example .

A Captaine of a Band of Men being asked , what number of Souldiers were in his Band , answered , I doe not readily know ; yet ( quoth he ) of this I am certaine , that the 1 ∶ 2 and 2 ∶ 3 , and 4 ∶ 5 , and 1 ∶ 6 of their number added together into one sum , are 384 men : now the question is , what summe of men he had in his Band. I suppose he had 60 men , or 30 men in his Band , but the least number is best , viz. 30 , whereof 1 ∶ 2 is 15 , and 2 ∶ 3 is 20 , and 4 ∶ 5 is 24 , also 1 ∶ 6 is 5 , their totall is but 64 men , but that should be 384 men . Then say by the Golden Rule , as followeth .

If 64 come of 30 , of what number comes 384.

Answere : hee had 180 men in his Band , whereof

The solution of this Question another way more briefe .

Diuide 384 by 64 , makes 6 ; which multiply by 30 , makes 180 men , as before .

3. Example .

A certaine man hauing spent 120 pound , had yet remaining 1 ∶ 2 and 1 ∶ 3 of his whole substance ; the question is , what his substance was . Answere : First , 1 ∶ 2 and 1 ∶ 3 is 5 ∶ 6 , which being taken from 6 ∶ 6 , the whole substance leaues remaining 1 ∶ 6 ; therefore if 1 ∶ 6 be 40 pound , what is 6 ∶ 6 ? makes 240 pound .

4. Example .

A Merchant bought 384 yards of broad Cloth of three seuerall prices , of each a like quantitie , and he was to pay halfe as much more for the second sort , as he payed for the first , and twice as much for the third sort as he payed for the second : now the question is , what each sort cost him , and at what price euery yard was rated vnto him ? I suppose the first sort cost him 4 pound , then the second sort must cost him 6 pound , which is halfe as much more as the first ; and then the third sort cost him 12 pound , which is twice as much as the second ; the totall is but 22 pound , but it should be 248 pound : wherefore if 22 pound come of 4 pound , of what number comes 248 pound ?

Example .

The first cost him 45 pound , 1 ∶ 11 of a pound ; then the second sort cost 67 pound , 7 ∶ 11 of a pound ; the third sort cost 135 pound , 3 ∶ 11 of a pound , total is 248 pound : then diuide 384 by 3 , and you shall find hee had 128 yards of each sort , and by Practise , you shall find the first sort cost 7 shillings , 1 ∶ 2 d. a yard ; the second sort cost 10 shillings 7 pence a yard almost , the third sort cost 21 shillings , 1 penny , 1 ∶ 2 d.

Double Position .
The Rule of double Position .

SVppose a number at pleasure , as in the last Rule of single Position , and proceed as if you had found the right number , and if by working you find the true number , then your Position was the right number , which doth seldome happen . First , if by your working there commeth out more then the true number ; then note it thus + with a crosse ; if lesse , then thus − with a long line , which doth signifie lesse .

Secondly , suppose another number , greater or smaller , and worke as before , vntill you doe find the true number sought ; which if you doe not find , see the difference also from the true number sought , and note it with the signe + or − as it shall bee found .

Then thirdly , set your suppositions with their errours , more or lesse , as in the examples following .

Fourthly , multiply crosse the first position by the seconds errour , and the second position by the errour of the first , and then if the signes be both alike + or − , abate the lesser from the greater , and the remaines shall be the diuidend . Also the lesser error abated from the greater , leaues the diuisor ▪ but if the signes be contrary one + , the other lesse , add both together to make the diuidend , and adde the two errors to make the diuisor : and lastly diuide the diuidend by the diuisor , and the quotient is the true number desired .

1. Example .

A certaine man seeing a purse in his friends hand , saith vnto him : It seemeth vnto me , that there is 100 Crownes in your purse ▪ To whom the other answered : Nay ( quoth hee ) there are not 100 Crownes , but ( saith he ) if they were increased 1 ∶ 2 and 1 ∶ 3 ▪ and 1 ∶ 4 , and lastly , one Crowne ouerplus , then would they be iust 100 crownes .

I suppose there were 12 Crownes in his purse , to which if I adde one half ; of 12 , which is 6 ; and one third of 12 , which is 4 ; and one fourth of 12 , which is 3 ; and lastly , one Crowne more , the totall will be but 26 Crownes , but they should be 100 Crownes , so that this errour is two little by 74 Crownes , which I note thus :

74 − 12

Secondly , I suppose he had 24 Crownes , to which I adde 1 ∶ 2 of 24 , which is 12 and 1 ∶ 3 , which is 8 and 1 ∶ 4 , which is 6 : and lastly , one Crowne ouerplus , the totall is 51 , but it should bee 100 Crownes , so that this is an errour of 49 , too little , which I also note thus : 49 − 24

The answere is ; that hee had 47 pound 13 ∶ 25 parts of a pound in his purse . The proofe followeth .

2. Example .

Twenty yards of Sattin , and 12 shillings is equall vnto 12 yards of veluet lesse , 10 shillings ; the price of either sort is required .

To answere this , or any other like question , take any number for the price of a yard of the lesser number , which here is veluet , which at 20 shillings a yard , lesse 10 shillings , amounteth vnto 230 shillings . Now admit a yard of Sattin at 14 shillings , so 20 yards and 12 shillings amounteth vnto 292 shillings ; from which subtract 230 shillings , rests 62 s. more then the truth . Againe , rate a yard at 12 shillings , so the 20 yards and 12 shillings makes 252 shillings ; from which take 230 shillings , rests 22 shillings more then the truth also . Now multiplying 22 by 14 , and 62 by 12 , the productes are 308 , and 744 , and the difference of those numbers is 436 ; then take 22 from 62 , rests 40 for diuisor , by which diuide the difference , makes 10 shillings , 9 ∶ 10 shillings for the price of a yard of Sattin .

Example .

3. Example .

Otherwaies if 40 , the difference of errors gaine a , the difference of positions , then 62 the first error yeelds 3 and 1 ∶ 10 ▪

Or if 40 yeeld 2 , what 22 ? makes 1 and 1 ∶ 10 ; this taken from 12 , or 3 , 1 ∶ 10 from 14 , leaues 10 , 9 ∶ 10 for the price , as before .

4. Example .

A Carpenter was hired to work 20 daies at 12 pence a day , but euery day that hee was idle , hee was to abate 18 pence of his wages , and in the end he receiued but 8 shillings : now the question is , how many daies he wrought .

First , suppose he wrought 12 daies , which commeth to 12 shillings , then must the 8 dayes that hee played , come to 12 shillings at 18 pence a day also : but this question saith , there came due to him 8 shillings ? Behold an error of 8 shillings too little .

Againe , I say that he wrought 14 dayes , amounting to 14 shillings ▪ then 6 dayes that he played at 18 pence a day , commeth to 9 shillings ; this taken from 14 shillings , leaues 5 shillings , and it should bee 8 shillings , which is an errour of 3 shillings too little . Now multiplying 12 by 3 , and 14 by 8 , the products are 36. and 112 , and the excesse is 76 ; which being diuided by 5 , the difference of the errours , quoteth out 15 , 1 ∶ 5 for the number of working dayes , and 4 dayes 4 ∶ 5 for the number of playing dayes .

12 − 8 5 14 − 3

Otherwayes .

If 5 ▪ the difference of errours , yeeld 2 , the difference of positions , what 8 the first errour ? makes 3 , 1 ∶ 5 to be added to 12.

Or if 5 be 2 , what is 3 ▪ makes 1 , 1 ∶ 5 to be added to the second position 14 , whereby all three wayes the numbers of the Dayes he wrought are found out .

Barter or Exchange .

TWo men barter , one hath Ginger of 10 pence a pound ready money , & in barter hee will sell it for 12 pence a pound . The other hath sugar of 12 pence a pound ready money , but in barter hee will sell it for 14 pence a pound ; the Question is , how much Sugar will pay for 756 pound of Ginger ? First , put your price of your Ginger into pence , makes 9072 pence ; which diuide by 14 pence , makes 648 pound of Sugar , which must be giuen for 756 pound of Ginger , at 12 pence the pound .

2. Example .

Two Merchants will barter , one hath Raysons of 34 shillings the hundred readie money , and in barter hee will sell them for 40 shillings : the other hath Nut megs of 4 shillings the pound ready money , how shall he set his Nut megs to make the like profit . Put your coyne into pence , and say ; If 408 d. be 480 d. what is 48 d. Multiply 480 by 48 , and diuide by 408 , makes 56 d. 2●●5● of one penny for the price of the Nutmegs ; vid. 4 s. 8 d. 1 ∶ 2 of a pound .

3. Example .

Two Merchants wil barter , one hath Holland of 2 shillings , 7 pence the ell ready money , which he will sell in barter for a shillings , 10 pence the ell , and yet he wil gaine priuately 10 pound in 100 pound ouer that gaine ▪ at what price must hee then set his Holland ? Answere : Set downe 2 shillings 10 pence in pence , makes 34 pence ; of which take the tenth part , which is 3 pence , 4 ∶ 10 , or 2 ∶ 5 , and adde to 34 pence , makes 37 pence 2 ∶ 5 of a penny for the price , to sell one ell to make that gaines . Now the other Merchant hath wooll at 7 shillings a Todde ready money , how shall he set his wooll to make like profit that he be not deceiued in the bargaine . If 31 pence be 37 pence , 2 ∶ 5 , what is 84 pence ? Multiply 374 primes by 84 , makes 31416 ; which diuide by 31 , makes 101 pence , 3 ∶ 10 penny , or 8 shillings , 5 pence , 3 ∶ 10 of one penny , which is the price for him to sell his wooll to make like profit .

Example .

4. Example .

Two Merchants will barter , one hath Sugar of 6 pound , 4 shillings ready money , and he will sell it for 7 pound the hundred . The other hath Ginger of 4 pound , 6 shillings the hundred , and in barter he will sell it for 5 pound the hundred ; now the question is , at what rate each of them doth gaine per cent ' and which hath the aduantage of the other . First , if 6 pound , 2 primes gaine 8 primes , what will 100 pound gaine ? Multiply 8 primes by 100 , makes 800 primes ; then ad 2. or 3 cyphers more to it , which diuide by 6 ∶ 2 primes , makes 12 l. 9 primes , 10 ∶ 31 of a prime , or neare 12 l. 18 shilling , 8 pence , which the first man doth gaine per cent● .

Secondly , if 4 pound , 3 primes gaine 7 primes , what will 100 pound gaine ? Multiply 7 primes by 100 , and adde 2 cyphers more , makes 70000 ; which diuide by 4 ∶ 3 primes , makes 16 pound , 2 primes , 34 ∶ 43 of a prime ; from which subtract 12 pound , 18 shillings , 8 pence , rests 3 pound , 6 shillings , 2 pence , which the second man hath gained more then the first gained .

6. Example .

Two Merchants barter , one hath a certaine number of pieces of Sakkins at 18 shillings a piece , for the which the other doth giue him 1806 ells of linnen Cloth , at 16 pence the ell , and yet 30 pound in readie money ; the Question is , how many pieces of Sakkin he had . First , find what 1806 ells of linnen Cloth cost by Practice ? makes 120 pound , 8 shillings : to the which adde 30 pound , makes 150 pound , 8 shillings : then diuide 150 pound , 4 primes , by 18 shillings , or 9 primes , makes 167 pieces of Sakkin , and 1 ∶ 9 of a piece .

Example .

6. Example .

Two men will barter , one hath Pepper of 22 pence the pound ready mony , but in barter hee will sell it for 27 pence the pound : the other hath Sinamond of 3 shilling , 6 pence the pound readie monie , and in barter hee will sell it for 4 shilling the pound ; the question is , how much sinamond wil pay for 384 pound of Pepper at that rate ? First , 384 pound of Pepper at 27 pence the pound is 43 pound , 4 shillings ; which diuide 43 ∶ 2 primes , makes 216 pound Sinamond , which he must giue .

7. Example .

If 4 English ells make 5 yards , and 13 yards , makes 50 Pawnes at Geanes , how many Pawnes is in 100 ells English . If 5 be 4 , what is 13 , makes 10 2 ∶ 5. Secondly , if 10 2 ∶ 5 be 50 , what is 100 , 480 10 ∶ 13

8. Examples .

Euery 4 ells at Antwerpe maketh 5 at Frankford , and 25 there makes 24 Braces at Luques , the question is , how many braces is 100 in Antwerpe . If 25 bee 24 , what is 5 , makes 4 4 ∶ 5. Secondly , if 4 bee 4 4 ∶ 5 , what are 100 , makes 120.

9. Example .

If 3 yards at London be 4 ells at Antwerpe , how many yards at London make 84 ells at Antwerpe . If 4 be 3 , what 84 ? makes 63 ells .

10. Example .

At Roan 112 ells make but 98 , and 100 ells at Roan is 112 at Siuil , how many of ours in 100 ells of Si●ull . If 98 Roan be 112 ells , what 100 Roan , makes 114 ells , 1 ∶ 7 of an ell . Secondly , if 112 ells be 114 , 1 ∶ 7 , what is 100 Siuill , makes 102 , 19 ∶ 25.

11. Example .

If 67 yards at London bee 100 in Venice ; how many are 7894 ? multiply by 67 , makes 5288 yards , 98 ∶ 100 parts .

12. Example .

A Merchant doth deliuer 400 pound sterling in London by exchange for Antwerpe , at 23 shillings , 5 pence the pound sterling , the question is , how much Flemish money , hee shall receiue at Antwerpe : put your 23 s. 5 d. into pence , makes 281 pence ; which multiply by 400 , makes 112400 pence ; which diuide by 240 , makes 468 pound , 6 shillings , 8 pence , which he must receiue at Antwerpe .

Example .

13. Example .

If 100 pound starling be 134 pound , 6 shillings , 4 pence Flemmish , what is one pound starling worth ? Reduce your coine 134 l. 6 s. 4 pence , into pence makes 32236 ; which diuided by 100 , makes 322 pence , 9 ∶ 25 pence , or 26 shillings , 10 pence , 9 ∶ 25 of one penny , for one pound sterling .

If one pound sterling be 1 pound , 14 shilling , 7 pence , ob . Flemish , how much sterling money is in 100 li. Flemmish ? Reduce 100 pound into pence , makes 24000 pence ; then put it into halfe pence , makes 48000 halfe pence ; then put 1 pound , 14 shillings , 7 pence , ob . into half pence , makes 831 ; by which diuide 48000 , makes 57 pound , 15 shillings , 1 penny almost , and so much sterling money is in 100 pound of Flemish money at that rate .

Of Gaine and Losse .

IF 13 pieces of Canuas cost 17 pound , 12 shillings , how may I sell them to gaine 8 pound in the hundred ? Multiply 176600 by 8 , makes 19 pound , 19008 , or two pence almost , and so much must he sell them for to gaine 8 pound in the hundred .

If 17 pound , 12 shillings gaine 1 pound , 8 shillings , 2 pence , what will 100 pound gaine ? Multiply 1 pound , 8 shillings , 2 pence in Decimalls by 100 , and diuide by 17 pound , 6 primes , makes 8 pound in the 100 , the proofe .

Example .

A Merchant hath lent 630 pound at interest for 10 pound in the 100 for 3 yeeres interest vpon interest , the Question is , vnto what summe it will amount vnto at the end of the terme ? Answere : Take the tenth part , and adde it into one totall 3 seuerall times , makes 838 pound , 10 shillings , 7 pence , 1 ∶ 5 of a penny for principall and interest , at the rate giuen , to bee paid at the end of three yeares .

Example .

2. Example .

A Merchant receiueth for principall and interest 838 pound , 10 shillings , 7 pence , 1 ∶ 5 of a penny at 10 pound in the hundred compound interest , which was for money deliuered out for 3 yeares ; now the Question is , what was the summe of money that was lent ? To doe this , or any other the like question , diuide the summe of mony receiued by 110 three seuerall times , and the three quotients will shew the yearely increase of the money lent , and the last quotient will be the answere to the question , or the money disbursed , as in the example following , which is the proofe of the former question .

Example .

3. Example .

A Merchant lent 100 pound for 7 yeares at 10 pound in the hundred Compound Interest , the Question is , what he shall receiue at the end of the terme .

Example .

Makes at 7 yeares end 194 li. 17 s. 5 d.

How to worke Compound interest at any rate per cent .

What is the principall and interest of 352 pound , put out at 8 pound in the hundred compound Interest , to be paid at the end of two yeares ? Adde two cyphers to 352 pound , makes 35200 ; then place your Interest 8 vnder the lowest cypher next the right hand , and multiply 352 by 8 , placing the product vnder the line , and that will be the Interest ; which added into the summe lent , makes the totall of the principall and interest ▪ and so worke for the second , third , and fourth yeare , as in the Example .

First I multiply 35200 by 8 , makes 2816 , which I adde vnto 35200 , makes 38016 ; then I multiply 3801600 by 8 , makes 4105728 , or 11 shillings , 5 pence , abating 4 figures for the 4 cyphers , which I added to the summe for to find out the prime line , as appeareth in the example ; and so of any other summe or rate in the hundred .

At 17 pound the hundred per annum compound interest , what wil 879 pound amount vnto to bee all forborne vnto the end of 5 yeares ? Adde 2 cyphers to your summe giuen , and multiply by your Interest 17 , and adde into the principall , and so worke 5 yeares , and the last product will bee the summe of money to bee receiued , viz. 1927 pound , 3 shillings , 5 pence .

Example .

If a Merchant buy a parcell of Holland , at 3 pound , 6 shillings the piece ; and another parcel at 4 pound , 2 shillings the piece ; the third sort at 4 pound 10 shillings the piece , the fourth sort at 5 pound the piece ; how may he sell 40 pieces , of each sort 10 pieces to gaine 18 pound in the hundred , and giue 9 moneths time for the payment ; as in the Example following .

Example . 10 Pieces at 3. 6. a piece , 33. 0. 10 Pieces at 4. 2. a piece , 41. 0. 10 Pieces at 4. 10. a piece , 45. 0. 10 Pieces at 5. 0. a piece . 50. 0. The summe is 169. 0.

Take the 3 ∶ 4 of the interest , makes 191 pound , 16 shillings , 3 pence , 3 ∶ 5 of one penny , to sell to gaine 18 pound in the hundred , for to giue 9 moneths time .

A Merchant sold 300 quarters of wheat , cost him 352 pound ready money , and lost 7 pound in the hundred , what did one quarter cost him , and at what rate did he sell a quarter , to loose 7 pound in the hundred ? Take the interest at 7 pound in the hundred , which is 24 pound , 12 shillings , 9 pence , 3 ∶ 5 , which subtract from 352 li. makes 327 pound , 7 shillings , 2 pence , 2 ∶ 5 of a penny , and diuide the remainer by 300 , makes 1 pound , 1 shilling , 10 pence for the price sold : secondly , diuide 352 pound by 300 , makes 1 pound , 3 shillings ▪ 5 pence ob . for the price which it cost him

Rie sold for 3 shillings a bushell looseth 20 pound in the hundred , what will then be lost , if it bee sold for 3 shillings 6 pence a bushell ? If 3 shillings be 80 pound , what is 3 shillings 6 pence ? Multiply 80 pound by 3 1 ∶ 2 , or by 3 shillings , 6 pence , makes 2800 ; which diuide by 3 , makes 93 li. 1 ∶ 3 Or otherwise , if 36 pence bee 80 pound , what is 42 pence ? Multiply 80 by 42 , and diuide by 36 , makes 93 pound 1 ∶ 3 of a pound as before .

If in one ell of Cloth sold for 3 shillings , 2 pence there were gained after the rate of 10 pound in the hundred , what did that ell of cloth cost ? diuide 385 , or 38 penny 1 ∶ 2 by 110 , makes 35 pence that the ell cost .

If one yard of Holland cloth cost 2 shillings , 11 pence , how many yards shall I buy for 34 pound , 6 shillings , put it into pence , makes 8232 pence ; which diuide by 35 pence , makes 235 yards , 1 ∶ 5 yard .

How to gaine any rate in the Hundred you desire .

Put your price that one yard , ell , pound or piece doth cost you into pence ; and then for 10 pound in the hundred , take the tenth part of that summe , which is the same number , placed one place nearer to the right hand , and that is the profit or Interest ; which added vp into the price giuen , makes the price to sel one yard , pound , ell , or piece , to gaine 10 pound in the hundred ready money .

Example .

If one ell of Holland cloth cost 3 shillings , 9 pence , how may I sell to gaine 10 pound per cent ' ready money ? Put 3 shillings 9 pence into pence , makes 45 pence : then take the tenth part of 45 pence , which is 4 pence 5 ∶ 10 , or one half , makes 49 d. 1 ∶ 2 for the price to sell an ell to gaine 10 li. per cent .

Example .

If your price you would gaine , bee not 10 pound in hundred , then adde 2 Cyphers to your number of pence giuen and multiply that number by your Interest , omitting to multiply by the cyphers , and the product vnder the line is your Interest or gaine , which added vp into one summe ; makes the price to sell one yard , ell , pound , or piece , to gain according to the rate desired example .

If one pound of Cloues cost 4 shillings , 10 pence , how may I sell to gaine 9 pound per cent ready money ? Put 4 s. 10 d. into pence , makes 58 d. then ad 2 cyphers , makes 5800 ; which multiply by 9 , makes 5 ∶ 22 or 5 pence , 22 ∶ 100 parts of one penny ; which added vp to the vpper numbers , is 63 pence , 22 ∶ 100 parts of one penny , or 5 shillings , 3 pence , 1 ∶ 5 of a penny for the price to sell one , to gaine 9 pound in the hundred .

If one piece of Raysons cost 18 shillings , 9 pence , how may I sell to gaine 18 pound in the hundred ready money ? put your money into pence , makes 225 pence , to which adde 2 cyphers , makes 22500 ; which multiply by 18 , makes 40 ∶ 50 , or 40 pence , ob . which added into the price , makes 265 pence , ob . for the price to sell one piece to gaine 18 pound in the hundred .

Example .

A Merchant lent wares for 10 pound in the hundred profit for 12 moneths , and at the end of 6 moneths he receiued principall and interest 356 ls. the question is , what was the summe lent ? Answere : adde 2 cyphers to 356 pound , and diuide by 105 pound , which is 6 moneths interest and principall , makes 339 pound 1 ∶ 21 parts of a pound for the sum lent .

Example .

Equation of Payment . The Rule of payment is to bring diuers paymen●s due at seuerall dayes to be payed at one intire payment .

AMerchant is to pay at diuers payments 600 pound : viz. 200 pound present , 200 pound at 8 moneths , 140 pound at 6 moneths , and 60 pound at 2 moneths : now hee is willing to pay all at one payment , what time must be giuen ? The ready mony being omitted , set the rest as numerators thus , 200 ∶ 600 140 ∶ 600 60 ∶ 600 partes , which in their least termes ▪ abreuiated , makes 1 ∶ 3 ▪ 7 ∶ 30 and 1 ∶ 10. Now multiply 1 ∶ 3 by 8 , makes 2 , and 2 ∶ 3 ; secondly , 7 ∶ 30 by 6 , makes 1 and 2 ∶ 5 ; thirdly , 1 ∶ 10 by 2 makes 1 ∶ 5 ; totall is 4 moneths , and 4 ∶ 15 of a month for the time sought .

Examples .

A Merchant hath owing him 752 pound , to be payd 200 pound content ; 200 pound at 3 moneths , 130 pound at 5 moneths , and the rest at 12 moneths ; now at what time ought this money to be payd all at one payment ?

Example .

A Merchant hath owing vnto him 782 pound , 12 shillings , to bee paid 1 ∶ 3 at 4 moneths 1 ∶ 2 at 7 moneths , the rest at 12 moneths , what time must it bee all at one payment .

Makes 6 moneths , 5 ∶ 6 of a moneth .

Wines worth 14 pound ready money are sold for 16 pound , to pay 1 ∶ 3 at 3 moneths , 1 ∶ 2 at 4 moneths , and the rest which is 1 ∶ 6 at 12 moneths , the question is , what is gained in 100 pound in 12 moneths .

Makes at 5 pound in the hundred .

Sugars worth 21 pound ready money are sold for 25 pound , to pay 1 ∶ 5 ready money , 1 ∶ 8 at 4 moneths , 3 ∶ 10 at 7 moneths , 3 ∶ 8 at 15 moneths ; the question is , at what rate per cent ▪ per annum they were sold .

Makes 8 pound , 9 ∶ 40 per cent ▪ .

Alligation Mediall .

ALlegation is an Artem teaching to combine or knit together diuers things vnequally prised , and thereby to find an equall price of any part of the said mixture , Alligation Mediall , is that which by the augmenting the quantitie of euery seueral portion to be mixed by his owne price , and diuiding the summe of all the products by the totall of the seuerall portions to bee mixed , findeth the thing sought .

Example .

Three seueral sorts of Barly are to be mixed ; viz. 34 bushels at 18 pence , and 76 at 20 pence , and 100 at 22 pence ; the Question is , what one bushell of that mixture will be worth ? First , multiply each number by his price , viz. 34 by 18 , 76 by 20 , and 100 by 22 , makes 612 , 1520 , and 1200 , the totall is 4332 : then adde the number of bushells into one summe , makes 210 ; by which diuide 4332 d. makes 20 pence , 132 ∶ 210 of one penny for the price of one bushell so mixed .

2. Example .

If you will mixe 30 gallons of Sacke at 4 shillings a gallon , with 150 gallons of White Wine at 2 shillings the gallon , what will a gallon of that mixture bee worth ? Multiply 30 by 4 , makes 120 shillings ; also 150 by 2 shillings , makes 300 shillings , totall is 420 shillings : then adde 30 and 150 , makes 180 gallons ; by which diuide 420 shillings , makes 2 shillings , 1 ∶ 3 of a shilling , or 2 shillings , 4 pence , for the price of one gallon so mixed .

3. Example .

Admit there were 6 portion of Siluer of 7 ounces fine , 12 of 8 ounces fine , and 25 of 10 ounces fine , which are to bee mingled with 10 pound of Copper , what is a pound of that mixture worth ? For answer : multiply 6 by 7 , makes 42 ; also 12 by 8 makes 96 , and 25 by 10 , makes 250 , the totall is 388 , which being diuided by 53 , the totall of 6 , 12 , 25 and 10 makes 7 ounces , 17 ∶ 53 of an ounce ; and so much fine is a pound of that mixture .

4. Example .

A Merchant hath 6 seuerall sorts of Spices , of which he will sell , of each an equall quantitie of seuerall prices for the summe of 323 pound , 8 shillings : viz. Sinamond large at 4 shillings , 6 pence a pound ; Nutmegs Case at 3 shillings , 8 pence a pound ; Large Maces at 8 shillings a pound ; and Pepper Case at 2 shillings 2 pence a pound , Pepper Callico at 22 pence the pound , and Ginger large at 10 pence a pound ; the Question is , how many pound he must haue of each to make the iust summe of 323 pound , 8 shillings ? Answer : first , put your money into shillings , makes 6468 shillings ; secondly , put all your prices of the Spice into one summe , and by that summe , which is 21 shillings diuide 6468 , makes 308 pound which he must sell of each .

Example .

Alligation Alternat .

ALligation Alternat is that , which altereth the places of such excesse as commonly fall betweene the meane price , and the extremes ; in which counter-change , if the extremes be equall , then the difference betweene the meane price , & lesser extreme is to be set against the greater extreme , and of the contrary if otherwise .

1. Example .

White Wine of 20 pence the gallon is to be mixed with Sacke of 3 shillings a gallon , so that there must be mixed 300 gallons to make the price to bee but 2 shillings , 4 pence the gallon , the question is , how much of each sort must bee taken . The numbers set downe , as in this example thus , the difference of 20 the lesser extreme from 28 , is 8 ; also the difference of 36 the greater extreme is also 8 , so that I find you must take as many of one sort , as of the other to make this mixture : viz. 150 gallons of each sort .

2. Example .

White Wine of 16 pence a gallon is to be mixed with Sack of 40 pence the gallon , how many gallons must bee taken of either sort , so that 120 gallons may be of 30 pence the gallon .

The numbers being set downe , as in this example , the difference of 16 the lesser extreme from 30 the meane price , there wil remaine 14 , which I plate against 40 ; then take the difference of 40 , the greater extreme , from 30 the meane price , there will rest 10 to be linked with the lesser extreme ; whereby I find , that so often as I take 14 gallons of Sacke I must take 10 gallons of White Wine to make the mixture :

3. Example .

A certaine Clothier is desirous to mingle 144 pound of wooll of 4 sorts : viz. blew wooll of 10 shillings the stone , red wooll of 11 shillings the stone , greene wooll of 12 shillings , white wooll of 9 shillings the stone , how many stones of each shal he take , that one stone of the mixture may be worth 14 shillings .

The counter-change being made , according to the Rule , as is in the Margent it is plaine , that so often as you take 5 of Blew , you must take 3 of Greene , and 2 of Red , and 2 of White . Therefore if 12 bee 144 , what

The end of the first Booke .

THE SECOND BOOKE .

Containing a Treatise of Decimall Arithmatick :

Wherein is taught how to work all manner of operations in Decimall Arithmatick , more speedy and easie , then by vulgar Arithmatick ; and first of the Decimall Table .

LONDON , Printed by Augustine Matthewes dwelling in the Parsonage-house in Saint Brides lane , neere Fleetstreet , 1623.

THE VSE OF THE Decimall Table .

THe Decimall Table following doth begin from one Farthing vnto a Prime , or two Shillings ; so that if you haue a Decimall Fraction giuen , which doth containe 90625 sixths : search it in the Decimall Table , and you shall find it ouer against 21 pence , three farthings and that is the value of that fraction giuen .

Or if you would know how to set out 16 pence halfe-penny in Decimalls ; search in the Table against 16 d. 2 g. and you shall find 6875 fifthes for the decimall sought .

But if you would set out any number of shillings from one shilling vnto one pound , or ●o shillings ; search in this little Table following , and you shall find your desire . As if you would set out 15 shillings in Decimalls , you shall find 7 primes , 5 seconds for 15 shillings , and so of any other summe , as in the example following .

Example . ●●ill . 1. 2. 1 05 2 10 3 15 4 20 5 25 6 30 7 35 8 40 9 45 10 50 11 55 12 60 13 65 14 70 15 75 16 80 17 85 18 90 19 95 20 1 li.
q 1. 2. 3. 4. 5. 6. 7 q. 1. 2. 3. 4. 5. 6. 7 1 0010416 0 025 2 0020833 1 0260146 3 003125 2 0270833 3 028125 1 0041666 7 0291666 1 0052083 1 0302083 2 00625 2 03125 3 0072916 3 0322916 2 0083333 8 0333333 1 009375 1 034375 2 0104166 2 0354166 3 0114583 3 0364583 3 0125 9 0375 1 0135416 1 0385416 2 0145833 2 0395833 3 015625 3 040625 4 0166666 10 0416666 1 0177082 1 0427082 2 01875 2 04375 3 0197916 3 0447916 5 0208333 11 0458333 1 0218746 1 046875 2 0229166 2 0479166 3 0239582 3 0489584 6 0●5 12 05

q. 1. 2. 3 4. 5. 6. 7 q. 1. 2. 3. 4. 5. 6. 7 12 05 18 075 1 0510416 1 0760146 2 0520833 2 0770833 3 053125 3 078125 13 0541666 19 0791666 1 0552083 1 0802083 2 05625 2 08125 3 0572916 3 0822916 14 0583333 20 0833333 1 059375 1 084375 2 0604166 2 0854166 3 0614583 3 0864583 15 0625 21 0875 1 0635416 1 0885416 2 0645833 2 0895833 3 065625 3 090625 16 0666666 22 0916666 1 0677082 1 0927082 2 06875 2 09375 3 0697916 3 0947916 17 0708333 23 0958333 1 0718746 1 096875 2 0729166 2 0979166 3 0739582 3 0989584 18 075 24 1000000

THE SECOND BOOKE , CONTAINING A TREATISE of Decimall Arithmatick .
The declaration of the parts of the Decimall Table .

FIrst , the Decimall Table in the left Margent containes certaine numbers in great and small letters ; first , from 1 farthing vnto one prime , or tenth of a pound , or two shillings . Then from one prime for euery shilling vnto one pound starling , or 20 shillings .

First , beginning in the left margent is set downe one farthing in the vttermost paralell to the left hand , in the first paralell of the Table , and so continuing from one farthing to one prime , or 2 shillings ; and ouer against euery number in the left side in a right line towards the right hand is contained the numbers in decimals , answering vnto euery farthing from one farthing to one prime , or 2 shillings ; and in the vpper margent in the head of the Table is contained , the true denominations of the said are all numbers in primes , seconds , thirds , fourths , fifths , sixths , and seuenths , which are small enough to worke any question exact to a small fraction of one penny in a summe of great value , as shall appeare by examples following . But here you shall note , that all the numbers in the said Table cannot be exact and perfit .

To find the value of a Decimall fraction in the parts of Coyne .

Suppose the number giuen to bee 2 seconds , 4 thirds , 5 fourths , and 7 fifthes , and you desire to know the true value thereof in coyne ; set downe your numbers , as in the example following , and marke your prime line , and then multiplie the fraction by 240 , the pence in one pound , and the numbers that arise by multiplication ouer the prime line are the summe of pence , the value of that fraction giuen , and the remainer on the right hand of the prime line is the fraction of one penny .

Example .

Here by multiplication of 2457 fifthes by 240 pence , I find 5 pence is gone ouer the prime line , and there remaines 82080 : 100000 parts of one penny . Now to know the value of that fraction in farthings , multiply the same by 4 , and so many as goe ouer the prime line , are farthings , the rest is the fraction of a farthing .

Example .

Numeration in Decimals .

If you haue a number to be expressed in Decimals of money , or Coyne sterling , learne first by the Decimall Table how to expresse your Coyne , from one penny vnto one pound sterling , or from one farthing to one pound sterling , for which the Table going before was calculated . If you would know the manner how to calculate the said Table ; diuide 1 pound , adding 7 cyphers vnto it , by your part you would know how to set forth in Decimals : as if you would know how a farthing will stand in Decimals ; diuide 1 pound with cyphers by 960 , the number of farthings in one pound sterling , and the quotient will be the numbers in Decimals , signifying one farthing .

Example :

So that I find , that diuiding of 1 pound by 960 farthings , the Quotient is 1 third , 0 fourth , 4 fifths , 1 sixth , and 6 seuenths : for if you should haue proceeded , adding more Cyphers , the Quotient would haue been alwaies 6 , because I see the number remaining to be the same it was at the last , that is 64. And although a farthing cannot bee set out exact in Decimals , yet it will serue in Multiplication and Diuision : for in 10000 yards or ells , it wil not differ 1 penny , as shal appeare afterwards by examples in their places .

How to set out a penny in Decimalls .

Diuide 1 penny with Cyphers by 240 , the number of pence in one pound sterling , and the quotient wil be a penny in decimals .

2. Example .

Here seeing that after I find the first quotient 6 , and the remainer 16 , as before I cease Diuision , as needlesse any further , knowing it will produce 6 in the quotient infinitely , and therfore I put as many times 6 in the quotient , as I find expedient and needfull , and 1 penny stands thus :

And these and diuers other numbers will not be set exact in Decimals , but yet they will serue to great purpose and exactnes in a multitude of questions , in sauing an infinite labour in Reduction , and Multiplication and Diuision .

How to breake a pound into his exact parts .

Set downe 1 pound thus , 10 ; then take the tenth , which is one prime , or 2 shillings , which I note thus ,

Then take halfe of that prime or 2 shillings , saying , the one halfe of 10 is 5 , or the one halfe of one prime is 5 seconds , or one shilling ; then the one halfe of 5 seconds is 2 seconds , and 5 thirds , saying , the one halfe of 5 seconds , is 2 seconds , and 5 thirds , which is 6 pence : then halfe of 2 seconds , 5 seconds , is 1 second , 2 thirds , 5 fourths , which doth represent 3 pence in Decimals . Againe , one halfe of 1 second , 2 thirds , 5 fourths , is 6 thirds , 2 fourths , 5 fifths , representing 1 penny , half-penny , or three halfe pence . Againe , halfe of that number is 3125 , or 3 thirds , 1 fourth , 2 fifths , 5 sixths ; signifying three farthings in Decimalls ; behold the worke .

Example .

It is also very necessary to vnderstand the proportionall parts of a pound , for by them are many questions speedily wrought in Decimals , as shall appeare in the examples of Multiplication and Diuision afterwards .

How to expresse the value of any number in Decimals .

Admit for example this number following , is to bee expressed according to the computation of Decimall Arithmatick , viz. 3785725 thirds : then for the expressing the signification of that number in the knowne parts of Coyne , first , marke out your prime line , to distinguish the whole numbers from the fractions with a right downe stroke with the penne , and then you shall find the numbers to stand thus 3785 pound , 7 primes , 2 seconds , and 5 thirds ; which search in your Decimall Table , and it doth signifie 14 shillings , 6 pence ; so that the whole number is 3785 pound , 14 shillings , 6 pence , and so of all numbers ; for you shall vnderstand , that euery prime doth signifie in value 2 shillings , euery second 2 pence and 2 ∶ 5 parts of 1 penny , and euery 5 thirds 1 penny , and 1 ∶ 5 of 1 penny : or ells euery prime is 1 ∶ 10 of one pound ; euery second 1 ∶ 100 part of one pound , and euery third 1 ∶ 1000 part of one pound , &c. infinitely .

How to remoue a Decimall number from one place to another .

If you haue a Decimall number giuen : as for example , ●3 pence , which doth thus stand in Decimalls , 1 second , 2 thirds , 5 fourths ; then you desire to know how it will stand in the place of primes , pounds , or in the place of 10 l. or hundreds or thousands . remoue it one place towards the left hand , and it is 1 prime , 2 seconds , 5 thirds , or in knowne parts of coine 2 shillings , 6 pence . Againe , remooue them one place more towards the left hand , and it will be 1 pound , 2 primes , 5 seconds , or 1 pound , 5 shillings . Againe , remoue one place more : and it is 12 pound , 19 shillings : Againe , remoue it one place more , and all your fractions are in whole numbers , and will signifie 125 pound , &c.

And this Rule is very necessary to bee well and perfectly vnderstood , for by it any price be giuen of a vnite in decimals . you may speedily know what 100 , or 1000 , or 10000 will cost at that rate , onely by adding of one , two , or more Cyphers .

As for example , if one ell cost 6 shillings 3 pence , what will 100 ells cost at that rate ? first , s●t out your price in decimals thus , 3 primes , 1 second , 2 thirds , 5 fourths , and adding of two Cyphers , because 100 hath 2 Cyphers , the summ will be 312500 : and because your fractions were fourths , cut off 4 figures and Cyphers towards the right hand , or marke your prime line , and you shall find , that 100 ells will cost 31 pound , 5 shillings at that rate .

1. Example .

If the numbers of the price giuen will not be exactly set downe in Decimals : as for example , at 7 pence , 3 farthings a yard , what will 100 yards cost ? Set downe your price as neere as may be , by your Decimall Table , which is 322916 seuenths , adde vnto it two cyphers , makes 32291600 ; and because your fractions are seuenths , cut off 7 figures , and there will bee 3 pound , 4 shillings , 7 pence .

2. Example .

And thus much shall suffice for Numeration in Decimalls , and I will now proceede vnto the second Rule of Arithmatick , viz. Addition in Decimals .

CHAP. II. Addition in Decimals of Coyne .

I If you haue diuers seuerall numbers giuen in Decimalls to bee added together into one summe , place them in order euery one right vnder his like denomination , or kind , Integers vnder Integers , Primes vnder Primes , Seconds vnder seconds , &c. Then begin your Addition at the right hand at the least Denomination first , and adde them all according to the Rule of Addition , as if they were all whole numbers , alwaies hauing a care to marke out your prime line , and the totall of your Addition will shew you the iust value of those whole numbers and fractions .

1. Example .

CHAP. III. Subtraction in Decimalls .

IF you haue two numbers in Decimals , the one to be subtracted from the other , place them aboue one the other , as in Addition , the greater numbers in the vpper part , and the smaller numbers right vnderneath , and then subtract them as if they were whole numbers , and note downe the remayners each in their proper places , as in this example .

1. Example .

2. Example .

CHAP. IV. Multiplication in Decimalls .

IF you haue any two numbers giuen to be multiplied in decimals , place your multiplicand vppermost , and your multiplier right vnder-neath , as if the same were absolute whole numbers , and no fractions at all ; and when your numbers are placed , marke how many fractions your two numbers doth contain , and note that number downe , and multiply according to any of my former instructions in the first booke ; and when the product is gathered , cut off your prime line , iust so many figures and ciphers , as your multiplicand and multiplier had fractions betweene them , and the worke is ended .

Example .

If you will multiply 758325 thirds , by 3857 primes , I place first my numbers , and then I find my multiplicand to haue 3 fractions , to wit , primes , seconds & thirds , and I find my multiplier to haue one fraction , onely primes , which makes 4 fractions , and so many figures I cut off from the product .

Example .

2. Example .

If you will multiply 34 pound , 5 shillings 3 pence , by 16 pound , 6 shillings , 6 pence , set them in Decimals , 342625 fourths , by 16325 thirds , and multiply them together , and cut from the product 7 figures to the right hand , and the product will be 559 pound , 6 shillings , 8 pence ob . almost .

Example .

3. Example .

If you will multiply 758 Integers by 3 primes , 7 seconds , 5 thirds , which is by 7 shillings , 6 pence ; place them as in the last example , and from the product cut off the 3 figures for the 3 fractions , and the totall is 284 pound , 5 shillings , the sum that 758 ells will cost at 7 shillings , 6 pence an ell , &c.

Example .

If you will multiply fractions by fractions in decimals ; as to multiply 5 primes , 2 seconds , 6 thirds , 3 fourths , by 7 primes , 2 seconds , 5 thirds ; set them as before , and cut off 7 figures .

4. Examples .

Makes 7 s. 7 d. ob .

If you will multiply in Decimals by 10 , or by 100 , or by 1000 , &c. set downe your numbers , and marke how many fractions there bee in your multiplicand , and then ad so many cyphers as your multiplier hath to the right hand , and cut off your prime line , and the worke is ended , as in this example .

Example .

How to change any fraction giuen into Decimalls .

Admit there be a quotient of a diuision , which is 358 pound , 126 ∶ 255 of one pound , which fraction you would turne into Demalls ; adde a cypher to your numerator of your fraction , makes 1260 : but because your number will not be euenly diuided by your denominator 255 , therefore adde more cyphers , and then diuide the number by 255 makes 49411 fifths in Decimals to be ioyned with the whole numbers 35849411 fifthes , and are now fit for multiplication and diuision in Decimals .

5. Example .

Admit there be a fraction to be set out in Decimals thus , it is required to know what 156 yards of cloth will cost at 196 : 784 of a pound one yard ? Adde to 156 , 2 , 3 , or more cyphers , and diuide by the denominator 784 , makes 25 seconds , by which multiply 156 yards , makes 39 pound .

6. Example .

7. Example .

For the proofe of this worke , multiply 156 by 196 , makes 30576 ; which diuided by 784 , makes 39 pound , as before .

CHAP. V. Diuision in Decimalls .

IF you will diuide any number in Decimals , either whole numbers by fractions , or fractions by whole numbers , or whole numbers and fractions by whole numbers and fractions ; set them downe according to the Rules in Decimalls in the operations before going . As for example , a certaine Merchant bought as much cloath as cost him 284 pound , 5 shillings , at 7 shillings , 6 pence an ell , the question is , how many elles he had for his money ? To doe this , or any other the like question ; diuide your summe of money 284 pound , 5 shillings by 7 shillings , 6 pence , and the quotient will shew you , what number of ells , and parts of an ell , if any bee , were bought for that money .

1. Example .

How to Diuide the smaller number by the greater .

If you will diuide 34 pound , 6 shillings amongst 36 men : place your numbers , adding , 3 , or 4 , or 5 cyphers ; and then diuide by 36 , makes 95271 fifthes ; or in Coyne 19 shillings , 0 pence , ob . for euery mans portion .

2. Example .

What is the quotient of 724 pound ? Diuided by 3 ∶ 4 of a vnit , or 15 shillings ? Answer : diuide 724 by 75 seconds , makes 965 1 ∶ 3 ; for triall whereof multiply 965 1 ∶ 3 by 15 shillings , or 75 seconds , makes 724 , as in the Example .

2. Example .

This last question is in effect no other but as the former : for if I shall say , a merchant buyes Broad Cloth , costs him 724 pound at 15 shillings , or 3 ∶ 4 of a pound one yard , the question is , what number he had for his money , and by Diuision I find he had 965 yards , and one third part of a yard , as is proued in the example ; and so diuiding 724 by 3 ∶ 4 , the quotient is 965 , 1 ∶ 3

3. Example .

If you will diuide the product of the second example in multiplication , which was 559●●53125 seuenths by 16325 for the proofe of that worke , which ought to bring out the multiplicand 34●2625 ; or rather if you will diuide 559 pound , 6 shillings , 8 pence , ob . almost , by 16 pound , 6 shillings , 6 pence , the quotient will be 34 pound , 5 shillings , 3 pence .

Example .

How to find the Prime line in any Diuision decimall , or to find the true denomination of of the Quotient .

In any diuision decimall , alwaies marke out your prime line in your diuidend with a streight do vne line with the pen , then set your Decimall fractions in primes , seconds , thirds , fourths , &c. beyond the line ; also do the like in your diuisor , and then mark how often you may remoue your diuisor , that the whole numbers of your diuisor may stand vnder the whole numbers of your diuidend , and so many figures shall your quotiont haue in whole numbers , the rest are to bee marked with prickes in the quotient for primes , seconds thirds , &c.

If you will diuide 93861375 fifthes by 34 pound 35 seconds , then place them with pricks as in the example following . I find hauing placed my diuisor vnderneath my diuidend , that I may remoue my diuisor twice vnder the whole numbers of my diuidend , and therefore I conclude , the first two numbers of my quotient wil be whole numbers , which I marke from the rest of the numbers in the quotient with a line , and then diuiding according to the former instruction , you shall find the quotient will bee 27 pound , 3 primes , 2 seconds , and 5 thirds .

Example .

2. Example .

If you would diuide 15554 pound , 2 primes , 5 seconds , or 5 shillings , by 45 pound ? Place them as in the Example following , and you shall find , that there will be in the quotient 3 figures in whole numbers , and the rest will be primes and seconds , so that diuiding of 15554 pound , 5 primes by 45 pound , the quotient is 345 pound , 13 shillings .

Example .

3. Example .

If the greatest number of your Diuisor be primes , then the figures of your whole numbers in the quotient will be , once greater in value ; then the times you can remoue your Diuisor , as if you would diuide 241 pound , 5 primes , by 7 primes : then whereas you can remoue your diuisor by two times vnder the whole numbers 241 , yet you shall haue 3 numbers in the quotient in whole numbers , because your first figure of your diuisor is primes ; so that in diuiding 241 pound , 5 primes by 7 primes , I find the quotient will be 345 pound , or integers ; and so many yards , at 14 shillings a yard , which is 7 primes , wil 241 pound , 10 shillings buy .

Example .

4. Example .

If you will diuide 16 pound , 875 thirds , which is 16 pound , 17 shillings , 6 pence by 375 thirds , which is 7 shillings , 6 pence , or which is all one , imagine there is as much cloth of 7 shillings , 6 pence a yard , as cost 16 pound , 17 shillings , 6 pence ; the question is , how many yards was bought for that money ? placing your numbers as in the example following , I find 45 yards is the answere to the question .

Example .

5. Example .

If you will diuide whole numbers and fractions by whole numbers , place the whole numbers and fractions vppermost , and marke out your prime line , and then set your diuisor vnder-neath , and the lowest figure in valew of your diuisor , will shew you what is the denomination of the first figure of your quotient . As if you will diuide 13 pound 95 seconds by 45 ; or which is all one if you shall say ; if 45 pieces of figgs cost me 16 pound , 19 shillings , what did one piece cost ? Diuide 1395 seconds by 45 , makes 31 seconds , or 6 shillings , 2 pence , 2 ∶ 5 of a penny for the price of one piece . And in this sort the price of any number of yards , ells , or pounds being giuen in diuiding it by the number of yards , elles , or pounds , the quotient will bee the price of one ; and by this Rule you saue a labour of Reduction , alwaies diuiding the price by the number giuen , the greater by the lesser , or the lesser by the greater .

Example .

6. Example .

If 456 ells of cloth cost 575 pound , 7 primes , what will one ell cost ? Diuide 575 pound , 7 primes by 456 ells , makes 1 pound 2625 fourths , or in Coyne , 1 pound , 5 shillings , 3 pence for the price of one ell .

Reduction in Decimals .

If you will reduce 75 pound , 12 shillings , 9 pence into Decimals , enter your Decimal Table , and for 12 shillings find 6 primes ; then looke for 9 pence , and you shall find 375 fourths ; so the totall is 75 pound , 6375 fourths and are now fit and apt for any Decimall operation .

If you multiply or diuide 84 pound , 13 shillings , 6 pence , by 17 pound , 3 shillings , reduce them into Decimals by the Table , makes for 84 pound , 13 shillings , 6 pence 84 ∶ 675 , and for 17 pound , 3 shillings , 17 ∶ 15 , and are now fit to be multiplied or diuided one by the other .

If you will reduce 189 ∶ 756 parts of one pound into Decimals : diuide 189 , adding 3 cyphers to it by 756 makes 25 seconds for that fraction in Decimalls : and now for example , If 158 ells of cloth & 189 ∶ 756 parts of an ell cost 79 pound , 2 shillings , 6 pence , what will 640 ells cost at that rate ? Now according to vulgar Arithmatick , either I must reduce 158 ells 189 ∶ 756 parts of an ell into 756 parts , or otherwise I must Reduce the fraction into his least termes , makes 1 ∶ 4 ; then I multiply or reduce 158 ells into fourths , makes 633 fourths for the first number in the Golden Rule . Secondly , reduce 79 pound , 2 shilling , 6 pence into pence , makes 18990 pence for the second number ; then put 640 ells into fourths , makes 2560 fourths ; then multiply ●8990 by 2560 , makes 48614400 ; which diuide by 633 , makes 320 pound .

Example .

The same example wrought by Decimalls .

If 158 ells 1 ∶ 4 ell cost 79 pound , 2 shilling 6 pence , what will 640 ells cost at that rate ? Place them in Decimals thus : If 15825 seconds cost 79125 thirds , what 640 ells ? Multiply 79125 thirds by 640 , makes 50640000 ; which diuide by 15825 , makes 320 pound the quotient .

Example .

Or otherwise .

Diuide 15825 by 79125 , adding one cypher , makes 2 primes for the Quotient ; wherefore I conclude , that one halfe of 640 pound , which is 320 pound , is the answere to the question demanded . Also diuide 7912● by 15825 , the quotient is 5 primes ; by which multiply 640 pound , makes 320 pound for the answere to the question as before .

If a Phillips Dollar be worth 4 shillings , 8 pence , what are 465342 Dollars worth in sterling money ? Answer multiply 465342 by primes , which is 4 shillings , and take the sixth part of that product , and adde into it , makes 1085798 primes for the answer .

Or otherwise , multiply by 2 primes , and 1 ∶ 3 of a prime , because 8 pence is 1 ∶ 3 of a prime , and both wayes will produce the same answere .

Example .

If a common Dollar be worth 4 shillings , and a Princes Dollar bee worth 4 shillings , 6 pence , how many Princes Dollars will pay for 7584 common Dollars ? Multiply 7584 by 4 shillings , and diuide by 4 shillings , 6 pence , makes 6741 Dollars , and 7 seconds , and 5 thirds will remaine , which is 18 pence ; so that I conclude , 6741 Princes Dollars at 4 shillings , 6 pence a piece will pay for 7584 common Dollars , and there will remaine 18 pence .

Example .

In 654 pound , how many Dollars of 3 shillings a piece ? Adde two Cyphers to 654 , makes 65400 , because 3 shillings hath 2 fractions in Decimals , viz. primes and seconds , which is 1 prime and 5 seconds , by which diuide 65400 , makes 4360 Dollars at 3 shillings a piece .

Example .

In 756 pound how many Dollars of 3 shillings , 9 pence a piece ? Adde 4 Cyphers to 756 , makes 7560000 ; which diuide by 1875 , which is 3 shillings , 9 pence in Decimals , makes 4032 Dollars . Behold the example following ▪

Example .

If I doe sell 346 yards of Veluet for 298 pound , 8 shillings , 6 pence , how doe I sell one yard ? Answere : diuide the price by the quantitie of yards in decimals , makes 8625 fourths , or in Coyne 17 shillings , 3 pence for the price of one yard .

Example .

Makes 17 s. 3 d. a yard .

A Merchant would buy seuerall sorts of Spices of seuerall prices , to wit , of 3 shillings a pound of 2 shillings , of 2 shillings 3 pence , of 1 shillings 7 pence , and of 2 shillings , 2 pence a pound , and would haue of each a like quantitie ; for 324 pound , the question is , how many pound hee must haue of each ? First , adde all the prices into one summe , makes 11 shillings , by which diuide 324 pound , makes 584 pound , 1 ∶ 11 of a pound ; and so many pound must he haue of each sort .

A Goldsmith sent his seruant to the Tower of London , to fetch him 415 pound , 18 shillings , 9 pence in pieces of 6 pence , of 4 pence , of 3 pence , of 2 pence , of 1 penny , and of one halfe penny , and bad him bring of each sort a like quantitie : First , adde all your Coyne , makes 16 pence halfe penny , which in Decimals is 6875 fifths by which diuide 4157375 fourths , makes 6050 pieces of each sort .

Example .

Rules of Practice in Decimalls .

Set your price giuen in the Decimall Table of a vnite , be it yard , ell , piece , or pound , and by the price giuen , multiply the number of yards ells , pieces , or pounds , and the product will bee the summe that you seeke , if you doe but marke out the prime line , as shall appeare by examples following .

1. Example .

If one pound weight of small Ginger cost 7 pence half-penny , what will 112 pound waight cost ? Find for 7 pence half-penny 3125 fifths , which multiply by 112 pound , makes 350000 ; from which cut off fiue figures to the right hand by the prime line , and the summe is 3 pound , 5 primes , or 3 pound , 10 shillings , because your multiplicand hath 5 fractions .

Example .

How to find the price of any vnite in any place of 10 , or 100 , or 1000 , the price of one being giuen .

If the price of a vnite bee giuen at any rate , and from thence you desire to know , what 10 , or 100 , or 1000 , or 10000 will cost at that rate : or otherwise , if you desire to know , if you doe gaine any rate desired by the pound , and would know at what rate it will be in the 100 pound , or vpon exchange from place to place , the exchange of one pound being giuen , you desire to know , what 100 pound will amount vnto ? Place your rate or gaines giuen in Decimalis by helpe of the Table , and then adding of one , two , three , or more Cyphers , cutting off your prime line , you shal know your desire , marking the denominations of your fractions , if the least to the left hand be primes , seconds , thirds , fourths , fifthes , cutting off your prime line so many figures from the right hand .

2. Example .

If one pound sterling be 1 pound , 14 shillings , 3 pence Flemish , what is 100 pound sterling worth ? Place 1 pound , 14 shillings , 3 pence in decimals , makes 17125 fourths : then because 100 pound hath 2 Cyphers , makes 1712500 : then cutting off 4 figures to the right hand , you shall find 171 pound , 5 shillings for 100 pound sterling , to make as appeareth before .

If one ell of Cambrick cost 7 shillings , 6 pence , ● farthings , what will 100 ells cost at that rate ? Place 7 shillings , 6 pence , 3 farthings in Decimals , makes 378125 fixths , and adding two Cyphers for 100 , makes 37812500 : from which cut off 6 figures to the right hand , makes 37 pound , 16 shillings 3 pence for the summe that 100 elles will cost .

Makes 37 l. 16 s. 3 d.

If one pound or piece cost 1 pound , 2 shillings , 3 pence , what will 1000 pieces cost ? Set 1 d. 2 s. three pence , in Decimalls makes 11125 fourths : to the which adde 3 Cyphers , because 1000 hath 3 Cyphers , and from the totall cut off 4 figures , makes 1112 pound , 10 shillings , as is in the 4 example aboue

If one ell of Holland cost 3 shillings , 3 pence , what will 343 ells cost ? Multiply 343 by 3 shillings , 3 pence in Decimalls , which is 1625 fourths , makes 55 pound , 14 shillings , 9 pence .

If one yard of Veluet cost 15 shillings , 6 pence , what will 972 yards cost ? Find for 15 shillings 75 seconds ; then for 6 pence find 25 thirds , total is 775 thirds ; by which multiply 972 , makes 753 pound , 6 shillings , as aboue in the sixth Example .

If one yard of Veluet cost 17 s. 7 d. 3 q. what will 857 yards cost ? First , find 17 ● . to be 85 seconds ; then 7 d. 3 q. makes 322916 , totall is 8822916 ; which multiply by 857 , makes 756 l. 2 s. 5 d. 3 q.

If one Dollar be worth 4 shillings , 9 pence what are 758 Dollars worth in sterling money ? Multiply 4 shillings , 9 pence , which is 2375 fourths by 758 , makes 180 pound , 6 pence , as in the eighth example aboue .

The price of any number of yards , ells , pieces , or pounds giuen to find the price of a vnite .

If the price of any number of yards , ells , pieces , or pounds be giuen , set them downe in Decimals , adding one , two , or more Cyphers , if neede require , and diuide that sum , or price by the number of the yards , elles , pounds , or pieces , and the quotient is the price of a vnite in whole numbers , primes , seconds , and thirds , without reduction , as shall appeare by examples following : and in this manner you may know what summe of money was lent , if the principall and interest be giuen at any rate in the hundred ; or you may know if the rate of one pound exchange be giuen for any place , you may know the value of 100 of that Coyne in that money giuen ; and by this Rule is to bee abreuiated almost al operations of Arithmatick , by finding the value of a vnite in any place desired .

If ●42 ells of cloth cost 22 pound 4 pence half-penny , what cost one ell at that rate ? Diuide 2201875 fifthes by 542 , makes 40625 sixths , or in Coyne 9 pence 3 farthings for the price one ell cost .

1. Example .

If 345 pound gaine 76 pound , 12 shillings , what doth one pound gaine ? Diuide 76600000 by 345 pound , makes 222028 sixth , or in Coine , makes 4 shillings , 5 pence half penny almost , that 1 pound doth gaine as in the example following .

2. Example .

If 756 pound , 3 quarters , 24 pound of sugar cost 4421 pound 12 shillings , what did one pound waight cost , accounting 112 pound to the hundred ? Reduce 756 pound 3 quarters , 24 pound into pounds suttle , accounting 112 pound to the hundred , makes 84780 pound● then diuide 4421 pound , 12 shillings by 84780 , makes 5215 fifths , or 12 pence , half-penny one pound .

3. Example .

If I sell 1000 pieces of Cambricke for 700 pound , how doe I sell one piece ? Diuide 1000 by 100 , makes 1 pound , 42857 fifthes , 1 pound , 8 shillings , 6 pence , 3 farthings , as in the Example following .

4. Example .

If one pound starling be 1 pound , 14 shillings , 3 pence Flemish , what is one pound Flemish worth : Diuide one pound with Cyphers by 17125 , makes 11 shillings , 8 pence , 1 farthing almost .

5. Example .

If 1 l. sterling be 1 l. 14 s. 7 d. ob . Flemish , what is 100 l. Flemish worth in sterling money ? Diuide 100 by 173125 fifths , which is 1 l. 14 s. 7 d. ob . in Decimals , makes 57 l. 15 s. 3 d.

6. Example .

The Golden Rule in Decimalls .

If the number giuen be pounds , shillings and pence , set them out in Decimals , and also your number of yards , ells , pieces , pounds or any other numbers , set them out also in Decimals , and then without reduction multiply the third number by the second , and diuide by the first , according to the instructions of multiplication and Diuision in the former part of this booke , and the uotient will be the third number sought .

1. Example .

If 34 ells of Canuas cost 1 pound , 4 shillings , what will 756 ells cost at that rate ? Multiply 756 by 1 pound , 2 primes , makes 9072 primes ; which diuided by 34 , adding Ciphers , makes 266823 fourth , or in Coine 26 pound , 13 shillings , 8 pence .

Example .

If 112 pound of Indico cost 34 pound , 17 shillings , what cost 789 pound , subtill accounting 100 pound to the hundred ? Multiply 3485 seconds by 789 , makes 27496 pound , 65 seconds ; which diuided by 112 pound , makes 245 pound , 5058 fourths , or 10 shillings , 1 penny farthing .

Example .

If 981 ells of Cloath cost 94 pound , 13 shillings , 6 pence , what cost 2943 ells at that rate ? Diuide the third number by the first , and by the quotient multiply the second , and the product will be the answere sought .

If 112 pound of Sugar cost 5 pound , 3 shillings , 9 pence , how many pounds will 124 pound buy at that rate ? Diuide 51875 fourths by 112 pound , to find the price of 1 pound , makes 46316 , sixths , or in Coyne 1● d. 1 ∶ 10 of a penny almost for the price that one pound cost Secondly , diuide 124 pound by the price of one pound , viz. by by 46316 sixths , makes 26773 primes , and so many pound he shall haue for 124 pound .

If one yard Broad Cloath cost 16 shillings , 9 pence , how many yards shall 56 pound buy at that rate ? Diuide 56 pound by 16 shillings , 9 pence , the price of one yard , makes 66 yards , 9 ∶ 10 almost .

Example .

If 7 yards 1 ∶ 2 of cloth cost 9 shillings , what will 8 yards 1 ∶ 3 of a yard cost ? Multiply 9 shillings , or 45 seconds by 8 1 ∶ 3 , makes 375 ; which diuide by 7 yards 1 : ● , or by 75 primes , makes 5 primes , or 10 shillings .

Example .

If 5 yards 1 ∶ 2 cost 4 shillings , 8 pence , 1 ∶ 4 of a penny , or 56 , 1 ∶ 4 , what will 30 yardes cost at that rate ? set your 56 pence 1 ∶ 4 in Decimals , makes 5625 seconds , which multiply by 30 , makes 168750 seconds ; which diuided by 5 yards on halfe , or 55 primes , makes 306 pence 8 ∶ 10 of one penny for the price of 30 yards , as in the example following .

Example .

If 34 ells 3 ∶ 4 of Holland cost 3 pound , 6 shillings , 1 penny , half penny , what will 956 ells 1 ∶ 2 cost at that rate ? Multiply 3 pound , 6 shillings , 1 penny , half penny , which is 33625 fourths by 7565 prime● , makes 254373125 ; which diuided by 34 ells , 3 ∶ 4 , or by 3475 , makes 73200 thirds or 73 pound , 4 shillings .

Example .

If 346 pound , 10 shillings gaine 32 pound 8 shillings , what will 75 pound gaine at that rate ? First , multiply 324 primes by 75 makes 24300 prime ; which diuided by 3465 primes , makes 70129 fourths , or 7 pound , 3 pence for the answere .

Example .

The same Question wrought a second way .

Diuide 324 primes , by 3465 primes , adding 5 cyphers , and the quotient wil be 935 fourths ; which multiply by 75 , makes 7 l. 0125 fourths , which doth not want one farthing of the former summe .

The same Question wrought another way .

Diuide 75 pound , adding 5 Cyphers by 346 pound , 5 primes , and the quotient will bee 21645 fifths ; which multiply by 324 primes , makes 7012980 ; from which abate 6 figures to the right hand , because of your 6 f●●ctions and the remainer wil be 7 pound 01●9 fourths , &c. as before . And in this manner you may worke any question in the Rule of Three , three seuerall manner of wayes , and proue the worke one by the other .

If 12 shillings doe buy 74 pound of Ginger , how much shall I haue for 100 pound ? Diuide 7400 , which is the product of 74 by 100 , by 12 shillings , or 6 primes , and the quotient will be 12333 pound , 1 ∶ 3 , and so much Ginger shall I haue for 100 pound at that rate . Or otherwise , diuide 100 pound by 6 primes , makes 166 2 ∶ 3 , which multiply by 74 , makes 12333 pound , 1 ∶ 3 , as before .

Briefe Rules how to abreuiate your worke in the Golden Rule , by marking the proportions bebetweene the numbers giuen .

When as any question is propounded in the Golden Rule , marke what proportion is betweene the first and second numbers , or betweene the first and third numbers , or betweene the third and second ; for if you espie them in any proportion , the question demanded is very speedily answered , vpon the first sight ; or yet if you see them not exactly to be euen proportionals , yet you may subtract the first from the third , once twice or three times , or more and so often take the middle number towards the answer to the question , and then you neede not to multiply by your whole third number , as you shall see by examples following .

1. Example .

If 34 ells cost 2 pound , 4 shillings , 1 penny , what will 340 elles cost ? Heere comparing the first & third numbers , one with another , I find the third doth containe the first 10 times , wherefore I multiply 2 pound 4 shillings , 1 penny by 10 , and the totall is 22 pound , 10 pence , the Answere .

2. Example .

If 82 ells of Cloth cost 4 pound , 2 shillings , what will 324 ells cost at that rate ? Here I find 4 pound , 2 shilling in Decimals to be one halfe of 82 , but it standeth one roome lesse in value then 82 doth , so I conclude , that halfe of 324 one roome lesse is 16 pound , 2 primes , or 4 shillings , the Answere .

3. Example .

If 345 ells of Holland cost 34 pound , 10 shillings , what will 789 ells cost at that rate ? Set downe 34 pound , 10 shillings in Decimalls , makes 34 pound , 5 primes , which is the first number placed but one roome lower ; therefore I say , if 345 ells cost 34 pound , 5 primes one roome more to the right hand , then the third number also will cost 78 pound , 9 primes one roome more to the right hand , which is 78 pound , 18 shillings .

4. Example .

If 12 ells of Cloath cost 2 shillings , foure pence , 4 ∶ 5 of one penny , what will 356 ells cost ? place 2 shillings , 4 pence , 4 ∶ 5 in Decimals , makes 1 prime , 2 seconds , or 12 seconds , which is the same number : but it stands two roomes lower ; therefore I conclude , that 356 ells cost the same numbers two rooms lower , which is 3 pound , 11 shillings , 2 pence , 2 ∶ 5 of one penny .

5. Example .

If 130 ells of cloth cost 26 pound , what will 3759 ells cost at that rate ? I find the second number to bee twice the first , but it stands one place nearer the right hand ; therfore I conclude , that the third number will cost twice asmuch in his lower roome , which is 751 pound , 16 shillings .

If 130 cost 26 pound , what cost 3759.

6. Example .

If 75 ells one halfe co●● 7 pound , 11 shillings , what will 32812 seconds cost ? Set them downe in Decimalls , and you shall find them to stand thus , 755 primes for the first number , and 755 seconds for the second number , which is the same one roome nearer the right hand : so I conclude , that the third number wil cost 3285 seconds , which is 32 pound 17 shillings .

Example .

1. Example .

If 356 ells of Canuas cost 38 pound , 12 shillings , 1 penny , what will 740 ells cost at that rate ? First , diuide 740 by 356 , the quotient will be 2 and therefore I take twice the price giuen for that quotient , and then whereas before I should haue multiplied 38 pound , 12 shillings , 1 penny by 740 , I shall neede to multiply it but by 28 the remaynor , and diuide it by 356 , makes 30368 fourths , to bee a●ded to the former summe , and the totall will be as in the example following .

Example .

Here in this last example , I multiply 38 pound , 6 primes by 28 , omitting the penny , not setting it out in decimals , and the product is 10809 primes : then multiply 1 penny by 28 , makes 28 pence , which is one prime , 166 fourths , and the totall was 1080 pound , 9116 fourths , as in the example : and in this manner you may saue a great labour in multiplying your number of pounds and shillings first , and then multiply your pence by themselues , and adde into the rest in primes , seconds , &c.

2. Example .

If 17 ells of Holland Cloth cost 3 pound 2 shillings , 5 pence , what will 515 ells cost at that rate ? Diuide 515 by 17 , makes 30 , by which multiply 3 pound , 2 shillings , 5 pence , makes 93 pound , 12 shillings , 6 pence , then the remayner of your diuision will be 5 ells , by which 5 multiply 3 pound , 2 shillings , 5 pence , makes 15 l. 10 shillings , 1 penny , or in Decimals 1550416 fifthes ; which diuided by 17 , makes 912 thirds , or 18 shillings , 3 pence almost ; which added to 93 pound , 12 shillings , 6 pence , makes the answere to bee 94 pound , 10 shillings , 9 pence : and so here in stead of multiplying 3120833 sixths by 515 , and diuiding by 17 I haue saued more then halfe the worke .

Example .

3. Example .

If 7 pound buy 100 pound waight of Sugar , how many pound waight will 156 buy me at that rate ? Diuide 156 by 7 , makes 22 , 2 ∶ 7 ; by which multiply 100 , makes 2228 pound , 4 ∶ 7

4. Example .

If 356 pieces of Callicoes cost 300 pound , 15 shillings , how much will 917 pieces cost at that rate ? Diuide 917 by 356 , makes in the quotient 2 ; therefore take the price giuen twice , and there will remaine after your diuision 205 ; by which multiply 30075 seconds , makes 6165375 seconds ; which diuided by 356 , makes 173 pound , 18 seconds , or 173 pound , 3 shillings 8 pence , to bee added to the former summe 601 pound , 10 shillings , makes 774 pound 13 shillings , 8 pence , for the Question .

The same question wrought without Reduction in Decimals .

If 356 cost 30075 seconds , what 917 ? Multiply 30075 second by 917 , makes 27578775 seconds ; which diuide by 356 , makes 77468 seconds , or 774 pound , 13 shillings , 8 pence , as before the proofe .

Example .

5. Example .

If 179 pound of Indico cost 60 pound 13 shillings , 5 pence , what will 716 pound cost at the same rate ? diuide 716 by 179 , makes 4 in the quotient , and nothing wil remaine : wherefore I conclude , that 4 times 60l . 13 ● . 5 d. which is 242l . 13 s. 8 d. and is the answere to the question demanded .

6. Example .

If 36 pound of Cloues cost 11 pound , 6 shillings , how many pound shall I haue for 354l . Diuide 113 primes by 36 , makes 31388 fifths ; which multiply by 354 , cutting of figures for the 5 fractions , makes 111 pound , 11352 fifthes , or 3 pound , 2 shillings 2 pence , 3 farthings for the answere .

Fellowship in Decimals .

To worke the Rule of Fellowship in Decimals , gather the whole number of all the moneys disbursed into one summe , and then diuide the money gained or lost by that summe , and multiply that quotient so found by each seuerall partners stocke disbursed , and the products will be each seuerall mans gaine or losse .

1. Example .

Foure Merchants made a company : A. put in 60 pound , B. 80 pound , C. 120 pound , D. 140 pound , and they gained 72 pound ; the Question is , what part each Merchant must haue of the gaines ? First the totall summe of all their moneys disbursed was 400 pound , wherefore according to the rule I diuide 72 pound , adding Cyphers vnto it by 400 , and the quotient is 1 prime , 8 seconds ; by which I multiply each seuerall mans Stock disbursed , and I find , A. shall haue 10 pound , 16 shillings ; B. 14 pound 8 shillings ; C. 21 pound 12 shillings , and D. 25 pound , 4 shillings ; totall is 72 pound , as in the example .

Example .

2. Example .

Foure Merchants made a company , and set forth a ship to sea , which cost them 3616 pound , 13 shillings ; A. must pay 1 ∶ 3 of the money ; B. 1 ∶ 4 , C. 1 ∶ 5 , D. 1 ∶ 6 , the question is , what each man must pay of the said summe ? Take a a number wherein the like parts may be had which in the former book of vulgar Arithmatick , I find to bee 60 , whereof 1 ∶ 3 is 20 and 1 ∶ 4 is 15 , and 1 ∶ 5 is 12 ▪ and 1 ∶ 6 is 10 , the totall is but 57 : wherefore I deuide 361665 by 57 , and the quotient is 63-45 seconds ; which I multiply by 20 , and I find A shall pay 1269 pound ; then I multiply by 15 , and B. shall pay 95175 second ; and by 12 , and C. shall pay 7614 primes ; and by 10 , and D. shall pay 6345 primes , the totall is 361665 seconds , the proofe of the worke .

Example .

3. Example .

● Three Merchants made a Company : A. put in 566 primes ; B put in 398 primes ; C. put in 1204 primes , and they gained 58 pound , 16 shillings , or 58 pound , 8 primes , what must each man haue of the gaines ; first , the totall disbursed is 216 pound , 4 primes ; by the which I diuide 58 pound , 8 primes , & the quotient is 27197 fifthes for one pound gaines ; which I multiply by each seuerall mans money disbursed , and I find A. shall haue 15 pound , 7 shillings ▪ 10 pence half penny ; B. 10 pound , 14 shillings , 3 pence , 3 farthings ; C. shall haue 32 pound , 13 shillings , 9 pence , 3 farthings , the totall is 58 pound , 16 shillings , the proofe .

Example .

4. Example .

Three Captaines agree together to deuide a spoyle or bootie , which they had taken , containing 7851 li: in this sort , A. is to haue 1 ∶ 2 ; B. 1 ∶ 3 ; C. 1 ∶ 4 ; the question is , what each mans share shall be ? Find a number which hath such parts in it , viz. 12 , whereof 1 ∶ 2 is 6 , 1 ∶ 3 is 4 , and 1 ∶ 4 is 3 , which in one summe makes 13 ; therefore diuide 7851 , adding cyphers to it by 13 , and the quotient will be 603 pound , 92307 fifthes ; which multiply by 6 , 4 , and 3 , and you shall find , A. shall haue 3623 pound , 5384 ▪ fifths ; B. shall haue 2415 pound , 69228 fifths ; C. shall haue 1811 pound , 76921 fifths ; the Totall is 7850 pound , 99991 fifths , which doth want but 1 fourth of 7851 pound , which in value is but 3 : 125 parts of 1 penny , and this example is to bee wrought without the Golden Rule . Behold the proofe of the worke .

Example .

The same example wrought another way .

After you haue diuided 7851 pound by 13 , find in your Decimall Table what the quotient is in Coyne , makes 603 pound , 18 shillings , 5 pence , ob . which multiply by 6 4 , and 3 , and their totall in one summe is the answere , as before .

These three seuerall products added into one sum , makes 7850 l. 19 s. 11 d. wanting but one penny in the whole sum , which is the defect of the Decimals , which cannot be exactly set out in coyne , but it wil serue to answere a question of one million with one penny errour at the most .

5. Example .

Three men made a stocke together , and they gained 244 pound , 8 shillings : A. put in 315 pound 7 moneths , B. 408 pound 10 moneths , C. 500 pound 3 moneths ; now the question is , what each man must haue of the gaines ? First , multiply each mans stocke by his time , and gather all the totals into one summe , and they make 7785 ; by which diuide your gaines , 244 pound , 4 primes , and the quotient will bee 31393 sixths ; which multiply by the seuerall products of each mans money and time , and the totall of each seuerall product is the summe desired for each mans part of the gaine .

Example .

Position in Decimals .

The Merchants bought a parcell of Linnen Cloth cost them 757 pound , 17 shillings whereof A. must pay 1 ∶ 4 ; B. 1 ∶ 5 ; C. 1 ∶ 8 ; what must each man pay of this sum ? I take 20 for a number , wherein I can haue those parts , viz. 1 ∶ 4 of 20 is 5 , and 1 ∶ 5 of 20 is 4 , and 1 ∶ 8 of 20 is 2 pound 5 primes , or 2 one halfe , their totall is 11 pound , 5 primes , or 11 1 ∶ 2 ; by which I diuide 757 pound , 85 seconds , and the quotient is 65 l. 9 primes , which I multiply by 5 for A. makes 329 pound , 10 shillings ; B. 263 pound , 12 shillings ; C. 164 pound , 15 shillings : the totall is 757 pound , 85 seconds .

1. Example .

2. Example .

A Ship-carpenter bought 300 timber trees of a Gentleman , and was to pay for the first 100 a summe of money vnknowne , for the second twice asmuch as for the first 100 , and for the third 100 of trees hee was to pay thrice asmuch as he paid for the first , and the whole ●00 of trees cost him 7●4 pound , 12 shillings , the question is , what each hundred cost him seuerally ? To work this question , or any other of like nature , suppose a vnite , or one pound for the first 100 , then he must pay 2 pound for the second 100 , which is twice as much , and then also he must pay 3 pound for the third hundred , which is three times as much as the first : but yet 1 pound , 2 pound , and 3 pound makes but 6 pound , and it should be 724 pound , 12 shillings ; so that now whereas in the former Booke I taught you to resort to the Golden Rule for the answere , saying ; If 6 pound cóme of my position 1 pound , of what comes 724 pound , 12 shillings . Now alwaies supposing a vnite ▪ for your first number , you shall saue multiplication ; and so diuiding of 724 pound , 6 primes by 6 , I find the first 100 of Trees cost him 120 pound , 15 shillings , 4 pence ; and the second 100 cost him 241 pound , 10 shillings 8 pence ; and the third 100 cost him 362 pound , 5 shillings ; the total makes 724 pound , 12 shillings , behold the worke .

Example .

3. Example .

Foure Merchants consent to build a ship , cost them 541 pound , 16 shillings , whereof A. must pay a certaine summe of money vnknowne ; B. must pay twice as much as A ; C. must pay twice as much as B ; and D. must pay as much as all the other three , viz. as A. B. and C. ; now the question is , what each man must pay of this summe . I suppose A. must pay 1 pound , then B. must pay 2 pound , which is twice as much as A. doth pay ; and C. must pay 6 pound , which is thrice as much as B. doth pay ; and then D. must pay 9 pound , which is as much as all the other three doe pay : but their totall is but 18 pound , and it should be 541 pound , 16 shillings : wherefore I diuide 541 pound , 8 primes by 18 , and the quotient is 30 pound , 1 prime , or 2 shillings for the first part . Then B. must pay 60 pound , 4 shillings ? C. 180 pound , 12 shillings ; and D. 270 pound , 18 shillings , their totall makes 541 pound , 8 primes ; behold the worke .

Example .

4. Example .

A Cesterne of water containing 600 gallons is filled with water , and hath 4 seuerall Cocks to emptie the same , whereof if they be all set open at once , the Cesterne will be empty in 24 houres : now the second Cock will auoyde twice as much as the first Cock in 24 houres , and the third will auoide three times as much as the first , and the fourth Cocke 5 times as much as the first ; the question is , how many gallons each Cocke doth auoide in 24 houres of the said 600 gallons ?

I suppose the first Cock will auoyde one gallon , then the second must auoyde 2 , and the third 3 , and the fourth Cock 5 : but yet they are but a 11 gallons , and they should be 600 gallons : wherefore diuiding of 600 by 11 , the quotient is 54 gallons , and 6 ∶ 11 of a gallon for the first Cocke . Behold the worke in the example following .

Example .

Of Gaine and Losse in Decimals .

If a Broad Cloth 28 yards long bee sold for 14 shillings a yard , and the seller doth gaine 10 ▪ pound in the 100 ready money , what cost that broad Cloath ? First , by Practice find the price of the 28 yards , at 14 shillings a yard , makes 19 pound , 6 primes , or 19 pound , 12 shillings ; diuide 19 pound 6 primes by 110 pound , makes 17 pound , 81818 fifthes , or in Coyne , 17 pound , 16 shillings , 4 pence , 3 farthings .

1. Example .

Secondly , if 28 yards cost 17 pound , 81818 fifthes , what did one yard cost at that rate ? Diuide 17 pound , 81818 fifthes by 28 yards , and the quotient will be 63636 , or in Coyne , 12 shillings , 8 pence , 3 farthings for the price that one yard cost .

Example .

Thirdly , for the proofe of this worke , say , If one yard cost 63636 fifths , how may I sell it to gaine 10 pound in the hundred ready money ? Take the tenth part of 63636 fifths , makes 63636 sixths ; which added into one Totall , makes 69999 fifthes , which doth want but one fifth of 7 prime● , or 14 shillings , which proues all the former works to be true .

Example .

2 Example .

A Merchant doth deliuer money at interest for 9 months after the rate of 12 pound in the hundred for 12 moneths simple interest , and at the end of 9 moneths doth receiue for interest 87 pound ; the question is , what was the summe lent ? Answere : because the interest of 9 moneths at 12 pound in the hundred is 9 pound , deuide 8700000 by 9 pound , and the quotient is 966 pound , 6666 fourths , or 966 pound , 13 shillings , 4 pence , the summe lent .

Example .

3. Example .

If 13 pieces of Canuas cost 17 pound , 12 shillings , how may I sell them to gaine 8 pound in the hundred ? Multiply 17 pound 6 primes by 8 , adding two cyphers , makes for 19 pound , 8 thirds , or 19 pound , 2 pence almost .

The proofe of the former example , if 17 pound , 12 shillings , gaine 1 pound , 8 shillings , 2d . what will 100 pound gaine at that rate ? Multiply 1 pound , 8 shillings , 2 pence ; or in Decimals , 1 pound , 408 thirds by 100 , makes 140 pound , 800 thirds ; which diuide by 17 pound , 6 primes , makes 8 li. for the rate that 100 pound will gaine , which shewes the former worke to bee truely wrought .

Example .

4. Example .

If in one ell of cloath sold for 3 shillings , there bee gained after the rate of 12 pound in the hundred for 12 moneths , how should that ell haue been sold to gaine 17 pound in the hundred for 12 moneths ? Multiply 17 pound by 3 shillings , which is 1 prime , 5 seconds , and diuide the product by 12 , makes 2125 fourths , or in coyne 4 shillings 3 pence , and so much must it haue been sold for to gaine 17 pound in the hundred .

Example .

Secondly , if 3 shillings giue 12 pound , what will 4 shillings , 3 pence giue ? Multiply 2125 fourths by 12 , and diuide by 15 seconds , and the quotient is 17 pound , the proofe of the last example .

Example .

5. Example .

A Merchant sold 24 Clothes , which cost him 342 pound , wherein hee lost after the rate of 10 pound in the hundred , and tooke in exchange 560 pieces of Raysons at 24 shillings the piece , wherein hee gained 10 pound in the hundred ready money ; now the question is , what his gaine or losse was , and what summe of money hee was to pay for the Raysons ? First , 560 pieces of Raysons at 24 shillings a piece , is 672 pound ; from which subtract 342 pound , lea●es 330 pound to pay for the Raysons . Secondly , 672 pound , at 10 pound in the hundred , is 67 pound , 4 shillings for his gaines by the Raysons . Thirdly , 342 pound lesse , 10 in the hundred , is 34 pound , 4 shillings , to be deducted from 342 pound ; and then take 34 pound , 4 shillings , from 67 pound 4 shillings , leaues his gaines more then his losse to be 33 pound .

Example .

6. Example .

A Merchant receiueth for principall and interest 352 pound , wherein he gained 9 pound in the hundred for one yeare ; now the question is , what was the summe of money lent ? Diuide 35200000 by 109 pound , makes 322 pound , 9357 fourths , o● 322 pound , 18 shillings , 8 pence , half-peny for the summe le●t .

6. Example .

7. Example .

A Merchant hath owing vnto him , 540 pound , to be paid at the end of three yeares , now his debtor will pay him ready money , if he will abate him 9 pound in the hundred . Diuide 540 pound with Cyphers by 109 three times one after the other , and the third quotient will be the summe that hee shall pay in ready money ▪ abating 9 pound in the hundred interest vpon interest . Behold the worke following .

7 Example .

The proofe is made by multiplying the last quotient by 9 , and that product againe by 9 , and thirdly againe by 9 , makes 540 pound , wanting but one fifth , which is but 3 ∶ 1750 parts of 1 penny , or 6 ∶ 875 parts of one farthing .

8. Example .

A Merchant hath owing vnto him 632 pound , to be paid at the end of 12 monthes , now his debter will pay him ready money , if he will abate him 12 pound in the hundred per annum ? Diuide 632 by 112 pound ▪ and the quotient will be the summe of money that will discharge the debt , abating 12 pound in the hundred .

Example .

9. Example .

324 pound was receiued for interest money lent a Merchant Aduenturer at 17 pound in the hundred one yeare ▪ what was the summe lent ? Answere : diuide 32400 by 17 , makes 1900 pound , and 1 ∶ 17 of a pound .

10. Example .

If 358 ells of Holland cast 124 pound , 16 shillings , how shal it be sould an ell to gaine 12 pound in the hundred ready money ? First multiply 124 pound , 8 primes by 12 , adding two cyphers , makes 139 pound , 776 or in Coyne 139 pound , 15 shillings , 6 pence . Secondly , diuide 139 pound , 776 by 358 , makes 3905 fourths , or 7 shillings , 9 pence , 3 farthings for the price to sell one ell to gaine 12 pound in the hundred .

Example .

11. Example .

If one ell of cloth cost 18 pence , how shall I sell 358 ells to gaine 7 pound , 10 shillings by the bargaine . and at what rate in the hundred doe I gaine ? First , 358 ells at 18 pence an ell makes 26 pound , 17 shillings ; to the which adde 7 pound , 10 shillings , the gaines makes 34 pound , 7 shillings for to sell 358 ells , to gaine 7 pound , 10 shillings by the bargaine . Secondly , diuide 7 pound 500000 sixths by 26 pound , 85 seconds , and the quotient is 27 pound , 9346 fourths , or 27 pound , 18 shillings , 8 pence farthing , which is the rate gained by the 100 pound of money .

Example .

12. Example .

How much Indicoe of 6 shillings , 3 pence a pound wil pay for 73 broad clothes at 16 pound one cloth , and to pay 60 pound in present money ? First , 73 broad clothes at 16 pound a cloth makes 1168 pound , from which subtract 60 pound , there will remaine 1108 pound ; which diuide by 6 shillings , 3 pence , or 3125 fourths , and the quotient is 3545 pound , 9 ∶ 10 of one pound , and so much must he giue of Indicoe for the clothes .

Example .

13. Example .

How many pounds of Cloues at 6 shillings a pound , and small Sinamond of 3 shillings a pound must bee giuen for 36 Carseyes , at 4 pound , 3 shillings a piece , to haue of each a like number of pounds ? Answer : 36 Carseys at 4 pound , 3 shillings a piece , makes 149 pound , 8 shillings ; which diuided by the price of both , viz. 9 shillings , makes 332 pound of each sort .

The proofe : 332 pound of Cloues at 6 shillings a pound , makes 99 pound , 12 shillings ; then 332 pound of Sinamon at 3 shillings , a pound , makes 49 pound , 16 shillings , the total is 149 pound , 8 shillings , the giuen price of the 36 Carseys .

Example .

14. Example .

Of what principall came 1000 pound principall and interest , at compound interest in three yeeres at 6 pound in the hundred ? Diuide 1000 pound three seuerall times by 106 , makes 839 pound 61 seconds , or 839 pound , 12 shillings , 3 pence almost , which was the summe lent at first .

Example .

15. Example .

If 34 Tun of wine cost 544 pound , how may a man sell a Tun to gaine 12 pound vpon the hundred ready money ? First , find the price of one Tun , diuiding 544 by 34 , makes 16 pound for the price of one Tun which it cost ; then multiply 1600 by 12 pound , makes 17 pound , 92 seconds , or 17 pound , 18 shillings , 4 pence , 4 ∶ 5 of a penny , for the price to sell one Tunne of that Wine to gaine 12 pound vpon the 100 pound .

How to worke gaine and losse in pence , and parts of Pence or Farthsngs .

Set out your number of pounds , shillings , pence and farthings in pence , and in tenths of one penny ; and for one farthing , set out 2 primes , 5 seconds , which is one fourth of a penny , and for two farthings set out fiue primes , which is one halfe penny ; and for three farthings set downe 7 primes , 5 seconds , which is three quarters of one penny , and then they are apt for decimall operations both for multiplication , diuision , or any other worke of Arithmatick , without reducing them into farthings , and there wil bee a great deale of labour saued in these kinds of operations , as shall appeare afterwards by the examples following .

1. Example .

What is the interest and principall of 100 pound , put forth at 10 pound in the 100 compound interest , for the space of 7 yeares to bee all receiued at the end of the terme ? First , put your 100 pound into pence , maker 24000 pence ; then worke as in this example following , and you shal find it will amount vnto 46769 pence , and 1 ∶ 5 of one penny ; which diuided by 240 pence , makes 194 pound , 17 shillings , 5 pence , 1 ∶ 5 of a penny , which is the summe that 100 pound will amount vnto at interest vpon interest in 7 yeares at 10 pound in the hundred .

Example .

2 Example .

A Merchant deliuered 358 pound at interest for three yeares for 8 pound in the hundred compound interest ; the question is , what it wil amount vnto at the end of the terme ? Put your money into pence , makes 85920 pence ; which multiply by 8 , adding 2 Cyphers , and worke for three yeares , as in the example following .

Example .

The proofe of the former example in Decimals .

A certaine Merchant receiued for principall and interest vpon interest 450 pound 19 shillings , 6 pence , which was for money lent at 8 pound in the hundred for three yeeres ; now the Question is , what was the summe lent ? Place 450 pound , 19 shillings , 6 pence in Decimals , and you will find your third quotient will be 358 pound , wanting some few seconds , which prooues the work good .

3. Example .

A Merchant lent 112 pound for 6 months at 17 pound in the hundred , for 12 months , the question is , what he shall receiue ? Put your money into pence , makes 26880 pence ; marke out your prime line , as in the former examples , and adde two cyphers , then multiply by 17 , and take halfe that product for 6 moneths interest , and adde into the principall , and the totall is the sum of pence which hee shall receiue for principall and interest at 6 moneths end .

Example ▪

Makes 121 li. 10 s. 4 d. 4 ∶ 5 of a d.

4. Example .

If a pound of Sinamond cost 4 shillings ready money , how may it be sold to gaine 12 pound in the hundred to giue 6 moneths time ? Set your 4 shillings in pence , makes 48 pence ; then adde 2 Cyphers , and multiply by halfe the interest , and adde them into one summe , and the product will bee 50 pound , 88 seconds , or 4 shillings , 2 pence , 2 ∶ 25 of one penny for the price to sell one pound to gaine 12 pound in the hundred for 6 moneths time .

4. Example .

Makes 50 pence , 9 ∶ 10 of a penny almost .

5. Example .

If 112 pound waight of Clou●s cost 33 pound , 12 shillings , how may I sell them to gaine 14 pound in the hundred , and giue 4 moneths time ? First , set downe 33 pound , 6 primes ; then adde 2 Cyphers , and multiply by 14 , makes 4 pound , 704 thirds , of which take the third part , because 4 moneths is the third part of one yeare , which is 1 pound , 568 thirds ; which added into one totall , makes 35 pound , 3 shillings , 4 pence , halfpenny for the price to sell 112 pound to giue 4 moneths time , and to gaine 14 pound in the 100 in 12 moneths

5. Example .

6. Example .

If I gaine 8 pound , 15 shillings in 100 pieces of Linnen cloth , what doe I gaine in the 100 at that rate , when as the 100 pieces are sold for 52 pound 10 shillings ? First , subtract 8 pound , 15 shillings , from 52 l. 10 s. and there will remaine 43 l. 15 s. then say , If 43 pound , 15 shillings gaine 8 pound , 15 shillings , what will 100 pound gaine ? Diuide 8750000 by 43 pound , 15 shillings , or 43 pound , 75 seconds , and the quotient will be 17 l. 14 s. 4 d. in the 100.

7. Example .

If in 112 pound waight of Sugar , sold for 7 pound , 12 shillings ready money , there were gained 11 pound in the hundred , what did one pound cost at first penny ? First , di 7 pound , 6000000 by 111 pound , which is the principall and interest giuen , and the quotient is 6 pound , 84684 fifthes , which 112 pound cost ready money . Secondly , diuide that quotient by 112 pound , makes 61132 sixths , or 14 pence , 3 farthings for the price that one pound cost at first penny .

8. Example .

If 300 pieces of Lawne cost 321 pound , 4 shillings , how may I sell them to loose 15 pound in the hundred ? First , take the rate what one cost , by diuiding 321 pound , 2 primes by 300 , makes 1 pound , 0706666 seuenths , or 1 pound , 1 shilling , 5 pence almost for the price that one piece cost . Secondly , take the interest of 10706666 seuenths at 15 pound in the 100 , and subtract it , and then makes 91006 sixths , or 18 shillings , 2 pence , 2 ∶ 5 of a penny for the price to sell one piece to lo●osse 15 pound in the 100 ready money . Thirdly , for the proofe of this work , say ; If one piece cost 910067 sixths , what will 300 pieces cost at that rate ? Multiply 910067 sixths by 300 , and cut off 6 figures to the right hand , makes 273 pound , 5 pence almost for the sum receiued for 300 pieces to loose 15 pound in the 100. Fourthly , for a second proofe ; take the interest of 321 pound , 2 primes at 15 pound in the hundred losse , and deduct it from 321 pound , 2 primes , and there will remaine 273 pound , 5 pence almost , which doth proue all the other workes to be truely wrought .

Example .

9 Example .

If in one ell of Cloth sold for 3 shillings , 2 pence half-penny , there were gained 10 pound in the hundred ready money , what did that ell cost ? Answere : set 3 shillings 2 pence ob . in decimals , makes 38 pence , 5 primes ; then diuide 38 pence , 5000 fourths by 110 pound , makes 35 pence , the price that one ell cost .

Example .

10. Example .

If in one ell of Cloth sold for 35 pence , 19 seconds , there were gained 7 pound in the hundred ready money , what did that ell cost , when there was 6 moneths time giuen ? Diuide 35 pound , 1900 fourths by halfe the interest , adding one 100 , which is 103 pence , 5 primes , and the quotient is 34 pence , the price that the ell cost .

11. Example .

A Merchant lent money at 10 pound in the hundred for 100 pound profit for 12 moneths , and at the end of 6 moneths he receiued principall and interest 356 pound , the question is , what was the summe lent ? Diuide 356 pound , by 105 pound , which is the halfe yeares Interest and principall , and the quotient is 305 pound , 5 ∶ 105 of a pound for the summe lent .

Example .

12. Example .

If 17 pound loose 12 shillings , what will 100 pound loose ? Diuide 60000 fifthes by 17 , makes 3 pound , 529 thirds , or 3 pound 10 shillings , 7 pence in the hundred pound .

13. Example .

If 37 yards of veluet cost 32 pound , how must one yard bee sold to gaine 9 pound , 10 shillings in the hundred ? First , 32 pound the price at 9 pound , 5 primes the hundred , makes 35 pound , 4 seconds ; which diuide by 37 , makes the price of one yard to bee 94702 fifthes , or 18 shillings , 11 pence , ob . to sell one yard to gaine 9 pound , 10 shillings in the hundred .

Example .

Exchange in Decimalls .

1. Example .

IF one pound sterling be 1 pound , 14 shillings , 6 pence Flemish , what is 783 pound sterling in ●emmish money ? Set out 1 pound , 14 shillings , 6 pence in Decimalls , makes 1 pound , 725 thirds ▪ which multiply by 783 pound , makes 1350 pound , 675 thirds , or 1350 pound , 13 shillings , 6 pence .

Example .

2 Example .

If one pound exchange be 5 shillings , 6 pence what is 783 pound ? Set 5 s. 6 d. in Decimals , makes 275 thirds ; which multiply by 783 , makes 215 pound , 325 thirds , or 215 pound , 6 shillings , 6 pence ; which added to the last example , is 1566 pound , and so much is the double of the summe giuen , viz. of 78● pound , because the two prices giuen , makes iust 2 pound , and this by working a second question in exchange , the first is prooued to be truly wrought , as appeareth in the example aboue .

3. Example .

If one pound exchange be 1 pound , 17 shillings , 7 pence , half-penny , what is 1000 pound at that rate ? Set 1 pound , 17 shillings , 7 pence , half-penny in Decimalls , makes 1 pound , 88125 fifthes ; then because 1000 hath ; Cyphers , adde 3 Cyphers , and cut off 5 figures , and the answere is 1881 pound , 5 shillings .

4 Example .

A Merchant doth receiue 134 pound , 6 shillings for the exchange of one hundred pound sterling from Middleborough , what was one pound sterling in Flemmish mony ? Place 134 pound , 6 shillings in Decimalls , is 134 pound , 3 primes ; then because 100 pound hath 2 Cyphers , cut off two figures more to the left hand , and it wil be 1 pound , 343 thirds ; or in Coyne , 1 pound , 6 shillings , 11 pence , farthing for the exchange of one pound at that rate .

5. Example .

A Merchant doth receiue 645 pound , 12 shillings for exchange money , at 1 pound , 7 shillings , 6 pence for one pound sterling , the question is , how much sterling money he did deliuer ? Diuide 645 pound , 6 primes by 1 li. 375 thirds , or 1 pound , 7 shillings , 6 pence , makes 4695268 fourths , or 469 pounds , 10 shillings , 6 pence , 1 farthing for the sterling money deliuered .

6 Example .

If 1 l. sterling be 1 l. 7 s. 6 d. Flemmish , what is 110 l. Flemmish in Sterling Coine ? Diuide 100 pound by 1 pound , 375 thirds , makes 72 pound , 72727 fifths ; or 72 pound 14 shillings , 6 pence , ●b . that 100 l. makes .

7. Example .

If the exchange bee from Rome to London at 69 pence sterling one Duckat , how many Duckats shall bee deliuered at Rome for to receiue 356 pound , 16 shillings sterling at London ? Answere ? Diuide 356 pound , 8 primes by 2875 fourths , which is 69 pence , and the quotient will bee 1241 Duckats , 3 pence .

8. Example .

If the exchange bee from London vnto Antwerpe at 23 shillings , 5 pence , 3 farthings Flemmish the pound sterling , how much money must be deliuered at London , to receiue 146 pound , 14 s. 10 pence , 3 q. in Flemmish money ? Answere : Diuide 146 pound , 744775 sixthes , by 1 pound , 3 shillings , 5 Pence , 3 farthings : which is 1 pound , 1739582 seuenths , and the quotient is 125 pound ; and so much must he deliuer at London to receiue 146 pound , 14 shillings , 10 pence , 3 farthings in Flemmish Coyne at that rate .

Example .

9. Example .

A Merchant doth deliuer at Antwerpe 200 pound Flemmish by exchange for London at 22 shillings , 10 pence Fleminish for one pound sterling , how much must hee receiue at London ? Answere : diuide 200 pound by 1 pound , 141666 sixths , which is 22 shillings , 10 pence ; makes 175 pound .

A generall Rule for exchange in Decimals .

If the price of a vnite be giuen , then alwaies diuide the summe of money whereon the question dependeth by that vnite in decimalls , and the quotient is the answere to the question .

1. Example .

A Merchant doth deliuer 100 pound sterling by exchange for Rome , at 72 pence sterling for one Duckat De Camera ; the question is , how many Duckets he must receiue at Rome for his 100 pound sterling ? Heere the price of one Ducket is giuen to bee 72 pence , which is 6 shillings , or 3 primes ; wherefore I diuide 100 pound by 3 primes , and the quotient is 333 pound , 1 ∶ 3 of a pound , or 6 shillings , 8 pence for answere to the question .

2. Example .

A Merchant doth deliuer 756 pound sterling at London , to receiue Duckets at 66 pence sterling , the price of one Dueket , the question is , how many Duckets he must receiue at Venice ? Diuide 756 pound by 66 pence , which is 275 thirds , and the quotient is 2748 Duckats , and 300 ∶ 2750 of one Ducket for the Answere .

3. Example .

A Merchant at Venice doth deliuer 1000 Duckats , to receiue at London 287 pound , 10 shillings sterling , what is one Ducket ? Set downe 287 pound , 5 primes , and diuide by 1000 Duckets , makes at 5 shillings , 9 pence for one Ducket .

Makes 5 s. 9. d. one Ducket●

4. Example .

A Merchant at Venice doth deliuer 800 Duckats by Exchange for London at 64 pence , b. the ducket sterling money , the question is , how much sterling he must receiue at London ? Set out 64 pence , half-penny in Decimals , makes 26875 fifthes ; which multiply by 800 , and cut off 5 figures because your fractions are 5 , and the product will be 215 pound sterling .

Makes 215 pound sterling .

5. Example .

A Merchant doth deliuer 1000 duckets by Exchange for London at 71 pence sterling for one ducket , how much must hee receiue sterling money at London ? Set out 71 pence in decimalls , makes 2958 fourths , 1 ∶ 3 , and adde 3 Cyphers for 10●0 , and cut off 4 figures , makes 295 pound , 8 primes , 1 ∶ 3 , or 295 pound , 16 shillings , 8 pence for the answere .

Makes 295 l. 8 primes , 1 ∶ 3

6. Example .

One penny Flemmish is 3 ∶ 5 of one penny sterling , and one pound Flemmish is 3 ∶ 5 of one pound sterling or ●2 shillings ; wherefore to conuert Flemmish money into sterling Coyne , multiply your Flemmish mony by 3 ∶ 5 , which in decimals is 6 ∶ 10 , or 6 , and the product will bee the value of your Flemmish money in sterling Coyne . In 345 Flemmish , how much sterling Coyne ? Multiply 345 by 6 primes , and the product is 207 pound sterling .

7. Example .

In 756 pound , 18 shillings sterling , how much Flemmish coyne , when one penny Flemmish is ● : 5 of a penny English ? Denide 756 pound , 9 primes by 6 primes , makes 1201 pound , 5 primes , or 10 shillings .

Reduction of Measures from one place to another .

IF you will reduce the measure of one Country into the measures of another As if you would reduce the measures of Antwerpe , Gaunt , Brudges , Siuill , Roauen , or of any other Countrey , into the measures at London ; learne first the order of measuring of all sorts of commodities in both places , either out of the experience of Merchants and Tradesmen in those places , or out of the best and latest approued Authors that haue written Tables to that effect and note , that 4 ells at London makes 5 yards , and 100 ells at London is at

Ells. Antwerpe 166●/● Gaunt short measure 164 Gaunt long measure 154 Brudges 164 Arras 165 Calice 157 Lisse 166 Mastrich● 173 Cullen 208 Franckfort 208 Nor●mberge 174 Da●tringe 139 Ro●●● 103 Paris 95 Licons 100 Genna 480 ●/● Palmes . Millian 214 Braces . Florence 188 Braces . Venice for Silke hath 196 Ells. Venice for Linnen hath 180 Ells. Rome 56 Cana. Lisb●●●● 100 Varras . Madera 104 Varras . Seuile 135 Varras .

These I haue taken out of Mastersons Arithmatick .

The difference of one hundred Ells , Palmes , Varras , or Braces , being found of any place from London ; if you would conuert the measures of any of those places to London measure : as for example , If you would conuert 356 ells of Brudges measure into ells at London ; you shall find in the Table , that 164 ells make 100 at London ; then by the Rule of Three say ,

1. Example .

If 164 be 100 , what are 356 ells ? Multiply 356 by 100 and diuide by 164 , makes 217 ells , 12 ∶ 164 of an ell , which 356 at Brudges will make in London . But according to the order of decimalls , if you will bring the measures of other places to those of London ? Set your number of one hundred found in the Table , to a vnite in decimalls , as in the last example 164 stands thus 164 seconds , then you neede but diuide your number 356 by 1 pound , 64 seconds , and the quotient is 217 ells , 12 164 ells , as in the last example .

Againe , if you would reduce London measure to the measures of any other place ? Find the number of 100 to that place , and set it decimalls , and multiply your number of ells at London by those numbers found , and the product will be your desire .

2. Example .

In 758 ells at London , how many ells at Dantzing , I find in the Table 139 ells there make 100 at London ; so I set 139 to a vnite , and it is 1 pound , 39 seconds ; by which I multiply 758 , makes 1053 ells , 62 ∶ 100 parts .

3. Example .

If 166 ells 7 ∶ 3 at Antwerp be 100 ells at London , how many ells at London are 1756 ells at Antwerpe ? Set 166 , 2 ∶ 3 to a vnite , makes 1 pound , 66 seconds , and 2 ∶ 3 of a second : Or otherwise ; 1 ell , and 2 ∶ 3 of one ell , by which diuide 1756 , makes 1053 , 1 ∶ 2

4. Example .

In 3258 ells at London , how many Braces at Millian ? Find 214 for 100 at London so that if you set 214 to a vnite , it will be 2 pound , 14 seconds ; by which multiply 3258 , makes 6982 Braces , and 12 ∶ 100 parts of a Brace .

And in this manner you may eass●y conuert your Measures or Waights from one place to another , either by Multiplication or Diuision , without the Golden Rule : but of this , if it please God to lend me life and health , I doe purpose to speake in a Treatise at large of Decimall Arithmatick for the good of my Country-men and others , if I find these my labours and indeauours to be acceptable and beneficiall to others , and will better informe my selfe by Merchants , who haue had experience in the Reduction of Waights and Measures from place to place ; in the meane time here is a foundation laid to worke vpon ; let the difference be what it will , and so for this time I will end this Treatise of Decimall Arithmatick , and goe in hand with some operations of Annuities , as followeth .

Of Interest and Annuities .
How to frame Tables to worke Interest and Annuities , or Purchases at any rate .

FOrasmuch as these kind of operations of interest and Annuities are ●●ry tedious and trouble some , if they be to bee wrought for many yeares , althougb I haue already in the former Booke set forth many seuerall manners of working those kind of questions after a more easie kind of method , then heretofore hath been published by any other in the like kind whatsoeuer yet here I haue thought good also in this place to shew the wayes , whereby any man that is desirous to bee satisfied in the reasons or grounds of those kind of workes , may be able to calculate for his owne vse a Table or Tables , whereby to abreuiate those kind of operations by Multiplication , or Diuision , onely without the helpe of the Golden Rule , or any tedious Reductions of Multiplications and Diuisions for many yeares to come at one onely operation , as shall appeare by the examples following .

How to calculate the Table or Breuiat of 10 pound in the hundred Compound Interest .

If you will calculate a table for 10 pound in the hundred compound Interest for 21 or 30 yeares ? Place your numbers , as in the examples following , beginning with a vnite , or ● , adding 7 Cyphers vnto it , and then take the tenth part of that , which is the same numbers one roome more to the right hand , and adde them into the first numbers , and the totall will be the summe for ▪ the first yeare , and so you must work for the second , third , fourth , &c. vntill 21 , or 30 yeares : but here you shall note , that you shall not neede to set downe in your Breuiate more then 8 , 9 , or 10 numbers at the most , for because the rest wil be superfluous , as for example .

Example . Int. 1. 2. 3. 4. 5. 6. 7. 8 Yeer . Int. 1. 2. 3. 4 5. 6. 7. 8 Yeer . 1 00000000 0 2 35794769 9 1 23579476 1 10000000 1 2 59374246 10 11 25937424 1 21000000 2 2 85311670 11 121 28531167 1 33100000 3 3 13842837 12 1331 31384283 1 46410000 4 3 45227121 13 14641 3452271 ▪ 2 1 61051000 5 3 79749833 14 161051 37974983 1 77156100 6 4 17724816 15 1771561 417●2481 1 94871710 7 4 59497298 16 19487171 45949729 2 14358881 8 5 05447028 17 21435888 50544702 2 35794769 9 5 55991731 18 55599173 6 11590904 19

Here you may sec in this Table the manner of gathering the Breuiate of 10 pound in the hundred , Compound interest , which you may extend to what number of yeares you please , only adding a vnite in the eight place , as you see the figures in the ninth place doe arise , and now I will here set downe the Breuiate from one yeare vnto 40 ready gathered .

The Breuiate of 10 pound in the hundred for 40 Yeares .

Yeeres 1. 2. 3. 4. 5. 6. 7. 8. Yeeres 1. 2. 3. 4. 5. 6. 7. 8. 9 1 11000000 21 740024990 2 12100000 22 814027490 3 13310000 23 895430240 4 14641000 24 984973260 5 16105100 25 108347059 6 17715610 26 119181765 7 19487171 27 131099941 8 21435888 28 144209936 9 23579476 29 158630929 10 25937424 30 174494022 11 28531167 31 191943424 12 31384283 32 211137766 13 34522712 33 232251543 14 37974983 34 255476697 15 41772481 35 281024367 16 45949729 36 309126803 17 50544702 37 340039484 18 55599173 38 374043432 19 61159090 39 411447775 20 67274999 40 452592553

How to calculate a Table or Breuiate at any rate vnder or aboue 10 pound in the hundred , Compound Interest .

If you would calculate a Table or Breuiat any rate vnder or aboue 10 pound in the hundred compound interest , place a vnite with seuen Cypheres to it ; then if you will calculate for 12 pound in the hundred or 16 pound ; set your 12 , or 16 vnder the 2 first Cyphers next the vnite , and multiplie your vnite , omitting the cyphers by your interest , and adde the product into one totall , and the summe is the principall and interest for the first yeare , and so worke againe for the second , third , &c. to finish your Table , as aforesaid , at 10 pound in the hundred ▪ But if your interest bee vnder 10 pound in the hundred , place your number of the interest vnder the second Cypher from your vnite , and worke as is in the example following .

Example . Int. 1. 2. 3. 4. 5 6. 7. 8 Yeeres Int. 1. 2. 3. 4. 5 6. 7 8 Yeeres 1 00000000 1 86048896 4 80 8 1 08000000 1 10883904 8 ● 864 1 46932800 5 8 1 16640000 2 1175462 8 93312 1 5868743 6 8 1 25971290 3 1 7138242 7 8 16077696 1 36048896 4

In this manner you may proceede infinitely : and thus much shall suffice for making of these Breuiats .

The Breuiat of 8 pound in the hundred per annum Compound Interest for 30 yeares .

Yeeres 1. 2. 3. 4. 5. 6. 7. 8 Yeeres 1. 2. 3. 4. 5. 6 7. 8. 9 1 10800000 16 342594260 2 11664000 17 370001800 3 12597120 18 399611940 4 13604889 19 431570100 5 14693280 20 466095710 6 15868743 21 503383370 7 17138242 22 543654040 8 18509302 23 587146360 9 19990046 24 634118070 10 21589249 25 684847510 11 23316389 26 739635320 12 25181701 27 798806140 13 77196237 28 862710630 14 29371936 29 931727480 15 31721691 30 100626506

In this sort you may gather all the Tables or Breuiats for any rate in the hundred , which I will here omit in this small vollum , intending afterwards to publish this , and diuers other operations in my second Edition of my Booke of Decimall Arithmatick shortly to come forth .

The vse of these Breuiates and Tables , and of all others of like nature in working of questions of Interest and Annusties .
Rule 1.

To find what 1 pound due at any number of yeares is worth at the end of the terme ? Enter the Table of 10 pound in the hundred , and find in the left Ma●gent the number of yeares , and from that number so found , cut off seuen figures , the answere is in pounds , primes , seconds , thirds , fourths , &c. for the answere to the question demanded .

1. Example .

What is one pound put forth at interest compound , at 10 pound in the hundred worth , to be paid at the end of 18 yeares ? Find the eighteenth number in the Breuiat , which is 55599173 ; from which cut off seuen figures to the right hand , and the answere is 5 pound , 11 shillings , 2 pence , q.

Example .

Makes 5 l. 11 s. 2 d. 1q .

2. Example .

What is 100 pound due at 7 yeares end worth to be paid at the end of the terme , at 10 in the hundred compound Interest ? Find the seuenth number in the Table of 10 l. in the hundred , makes 19487171 ; to the which adde two Cyphers , because 100 pound hath two Cyphers , and cut off seauen figures to the right hand , and the sum is 194 pound , 87171 fifthes for the Answere .

3 Example .

What will 758 pound for 6 yeare make at 10 pound in the hundred compound Interest , to bee paid ▪ at the end of the terme ? Finde the sixth number in the Table of 10 pound in the hundred , which is 17715610 ; which multiply by 758 , the money named in the question , and the product , cutting off 7 figures to the right hand , makes 1342 pound , 16 shillings , 10 pence , ob . almost .

Rule 2.

How to find what any yearely Annuitie will make to bee paid all at the end of the terme ? First , find the number of yeares of the annuitie giuen , and from the number answering , deduct a vnite in the first place to the left hand , and adde a Cypher to the last figure to the right hand , and cut off seuen figures to the right hand , and the answere is found .

1. Example .

What will 1 pound annuitie make , to be payd for at the end of the terme of 16 yeeres at 10 pound in the hundred compound interest ? Find the sixteenth number in the Table of 10 pound in the hundred , and subtract a vnite from the first figure to the left hand , adding a Cypher to the right hand , makes 359497290 ; From the which cut off 7 figures to the right hand , makes 35 pound , 18 shillings , 11 pence , 3 farthings .

2. Example .

What will 1000 pound annuitie yearely amounteth vnto to be all forborne vntill the end of the terme of 5 yeares at 10 pound in the hundred compound interest ? Find the fifth number in the Table of 10 pound in the hundred , and subtract a vnite from the first figure , adding a Cypher in the last place , makes 61051000 : then because 1000 pound hath 3 Cyphers , adde 3 Cyphers , and cut off seuen figures , makes 6105 pound , 2 shillings for the answere .

3. Example .

What will 142 pound annuitie make to be paid at the end of the terme of 10 yeares ? Find the tenth number in the Breuiat of 10 pound in the hundred , and subtract a vnite in the first place , adding a Cypher to the last , makes 159374240 ; which multiply by 142 pound , the annuitie named , and from the product cut off seuen figures to the right hand , and the answere to the question is 2263 pound , 2 shillings , 2 pence , 3 farthings .

3. Rule .

How to find what any summe of money due at the end of any number of yeares is worth in ready money , at 10 pound in the hundred compound interest . Enter the Table of 10 pound in the hundred with your number of yeares , and the numbers which doth answere in the Table is your Diuisor ; then adde seuen Cyphers to your summe of money giuen , to make your diuidend ; then diuide your diuidend by your Diuisor , and the quotient , adding more Cyphers , will be your answere in pounds , primes , seconds , thirds , &c.

1. Example .

What is 1000 pound due at 7 yeares end worth in ready money , at 10 pound in the hundred compound interest ? Find the seuenth number in the Table of 10 pound in the hundred , which is 19487171 , this is your Diuisor . Then adde seuen Cyphers to 1000 pound , makes 1000000000 ; or adde more Cyphers , marking out your prime line in your diuidend , to find out how many figures your quotient will haue in whole numbers , and the rest will bee primes , seconds and thirds ; this is your diuidend , and then diuide by your diuisor , makes 513 pound , 3 shillings , 2 pence .

Hauing found what 1000 pound due at 7 yeares end is worth in ready money , if you will find what 100 pound , or 10 pound , or 1 pound is worth in ready money ; place your quotient in decimalls , and marke out your prime lines , cutting of one figure for 100 pound , ● for 10 pound , or 3 for 1 pound , the answere is as followeth .

Example .

2. Example .

What is 750 pound due at 5 yeeres end worth in ready money , at 10 pound in the hundred compound interest ? Find the fifth number in the Table of 10 pound in the hundred , which is 16105100 for diuisor ; then place 10 Cyphers before your number giuen 750 pound , and marke out your prime line , and diuide by your Diuisor , and the quotient will be 465 pound , 13 shillings 10 pence for the answere to the question giuen .

Example .

Makes 465 pound , 13 shillings , 10 pence .

3. Example .

What is 847 pound due at 21 yeares end worth in ready money , at 10 pound in the hundred compound interest ? Find the 21 number in the Table of 10 pound in the hundred for Diuisor , which is 74002499 ; then set 10 Cyphers to your numbers giuen , makes 8470000000000 for your diuedend ; then diuide , and the quotient will be 144 l. 9 s. 1 d. 1 ∶ 5 of 1 d. the answere .

Example .

Makes 114 l. 9 s. 1 d. 1 ∶ 5 of a penny .

4 Rule .

How to find what any yearely Annuities for any number of yeares is worth in ready mony at 10 pound in the hundred compound interest . Enter the Table of 10 l. per cent ▪ with your number of yeares giuen , and from the numbers found subtract a vnite in the first place and place a Cypher in the last for your diuidend ; which diuide by the number found in the Table against your yeares giuen , and the quotient is the answere to the question .

1. Example .

What is 100 pound per annum annuitie for 21 yeares worth in ready money at 10 pound in the hundred Compound ●nterest ? Looke in the Table of 10 pound in the hundred for 2● yeares , and subtract a vnite in the first place , and adde a Cypher in the last , makes 6400 ; 4990 ; Diuide this by 74002499 , the 21 number , adding Cyphers , and marking the prime line , and the quotient is 864 pound , 17 shillings , 4 pence , ● farthings for the answere to the question demanded .

Example .

2. Example .

Hauing found what 100 pound annuitie will amount vnto , if you would know what 10 pound ▪ or 1 pound annuitie will amount vnto , or 1000 pound in 21 yeares ; place it in Decimalls , and cut off 1 , 2 , or adde 3 Cyphers to the last , or remoue 3 places , and you shall find your demand .

Example .

3. Example .

What is 546 pound yearely annuitie for 14 yeares worth in ready money , at tenne pound in the hundred compound interest ?

Find the fourteenth number in the Breuiate of 10 pound in the hundred ; from it subtract a Vnite in the first place , and adde a Cypher , makes 279749830 ; which multiply by 546 , makes 152743407180 ▪ which diuide by 37974983 , the 14 number in the Breuiate , makes 4022 pound , 4 shillings , 2 pence , 3 farthings .

Makes 4022 l. 4 s. 2 d. 3 ∶ 4

1. Example .

There is a Debt bought for 513 pound , 3 shillings , 2 pence ready money , which was due at 7 yeares end , now the question is , what the debt was at 10 pound in the hundred compound interest ? Set your money paid in Decimalls , makes 513158 ; which multiply by 19487171 , the number against 7 yeares , cutting off 10 figures , makes 999 pound , 999 thirds , wanting but one third of 1000 pound ; wherefore I conclude , the debt was 1000 pound , which was due at 7 yeares end .

2 Example .

There was a Debt bought for 600 pound , which was due at 4 yeeres end , what was that debt at 10 pound in the hundred compound interest ? Multiply 600 pound by the numbers against 4 yeares , which are 14641000 makes 878 pound , 4600000 seuenths , or in Coyne 878 pound , 9 shillings , 2 pence , 2 ∶ 5 of 1 penny for the summe of that debt .

Makes 878 l. 9 s. 2 d. 2 ∶ 5 of a penny .

I haue set no exampies of the Table of 8 pound in the hundred , nor of no other rate , bectuse I intend shortly to speake more at large of this subiect in another volume , if God please to giue mee time and health , in which I intend to speake more at large of the Grounds , Reasons and proofes of these kind of Operations , and here I will finish this small Treatice of the second Booke .

FINIS .