The construction, and vse of the line of proportion By helpe whereof the hardest questions of arithmetique & geometry, as well in broken as whole numbers, are resolved by addition and subtraction. By Edm: Wingate, gent. Wingate, Edmund, 1596-1656. 1628 Approx. 26 KB of XML-encoded text transcribed from 15 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2013-12 (EEBO-TCP Phase 2). A15585 STC 25850 ESTC S102193 99837990 99837990 2344

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 Early English books online. (EEBO-TCP ; phase 2, no. A15585) Transcribed from: (Early English Books Online ; image set 2344) Images scanned from microfilm: (Early English books, 1475-1640 ; 1053:09) The construction, and vse of the line of proportion By helpe whereof the hardest questions of arithmetique & geometry, as well in broken as whole numbers, are resolved by addition and subtraction. By Edm: Wingate, gent. Wingate, Edmund, 1596-1656. [4], 20 p. Printed by Iohn Dawson, London : 1628. Some pages marked, stained and torn. Reproduction of the original in the British Library.

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The Line of Proportion depiction of a logarithmic number line

continuation of logarithmic number line

continuation of logarithmic number line

THE CONSTRVCTION, And Vſe of the Line of PROPORTION.

By helpe whereof the hardes • Queſtions of Arithmetique & Geometry, as well in broken as whole numbers, are reſolved by Addition and Subſtraction.

BY EDM WINGATE, Gent.

Nulla dies ſine Linea.

LONDON Printed by Iohn Dawſon. 1628.

¶ The Preface.

HAVING not many moneths agoe publiſhed a diſcourſe, declaring the nature and vſe of the Logarithmeticall Tables, and obſerving the Table of Numbers there to be too ſmall for ordinary vſe, not giving indeed without much difficultie the Logarithme of any number that exceeds 1000; I haue invented this tabular Scale, or Line of Proportion, by meanes whereof (as I take it) you ſhall find that defect fully ſupplyed: this Inſtrument yeelding you the reſolution of the hardeſt queſtions of Arithmetique or Geometry, both in broken, and whole Numbers, onely by Addition and Subſtraction, when the terme required happens not to exceed 10000. although the termes propounded conſiſt of never ſo many places, as ſhall further appeare by the Treatiſe following.

CHAP. I. The Definition of the Line of Proportion. CHAP. II. The Deſcription and Vſe of the Scale of Logarithmes. CHAP. III. The Deſcription, Conſtruction, and Vſe of the Scale of Numbers. CHAP. IIII. The ioynt Vſe of the Scale of Numbers, and the Scale of Logarithmes together.
THE CONSTRVCTION, and Vſe of the Line of PROPORTION.
CHAP. I. The Definition thereof.

THE Line of Proportion is a double ſcale, broken off into tenne Fractions, vpon which the Logarithmes of numbers are found out.

To vnderſtand the nature of Logarithmes, I referre you to Maſter Brigges his learned Worke, intituled Arithmetica Logarithmica, and to the Treatiſe mentioned in the Preface.

A Fraction is a tenth part of the Line of Proportion, conſiſting of ſix Lines and fiue ſpaces; ſuch as are the parts a b c d, & c d e f.

The Lines are thoſe, by which the ſpaces are diſtinguiſhed; So a b is the firſt, g h the ſecond, and c d the laſt line of the firſt Fraction, which c d is alſo the firſt line of c d e f the Fraction following.

The ſpaces are the diſtances betwixt the lines; And they are either greater, as the firſt and laſt ſpaces of each fraction; or leſſe, ſuch as are the other three placed in the middeſt of each fraction.

Theſe fractions, together with their Lines and ſpaces, muſt be vnderſtood to ioyne reſpectiuely one to another, in ſuch ſort that the whole Line of Proportion may be conceived to be one intire and continued Line; As the left end of the firſt fraction, marked by the Letters a g c muſt be conceived to ioyne with the right end of the ſecond fraction, noted by d f, and the left end of the ſecond fraction, ſigned by c e, muſt be vnderſtood to ioyne with the right end of the third Fraction, marked by f k; and ſo of the reſt: So that the whole Line of Proportion, beginning at the right end of the firſt Fraction, marked by b h α d, and ending at the left end of the laſt Fraction, ſigned by l Ω m, muſt be conceived to be one intire Line, as is aforeſaid.

A double ſcale, is when two ſeverall ſcales meete both vpon one common Line; So the Line of Proportion being compoſed of the two ſcales, which meete vpon the fourth Line (marked at the beginning by α, and at the end by Ω) may fitly be called a double ſcale.

CHAP. II. The Deſcription and Vſe of the ſcale of Logarithmes.

THe ſcales, whereof the Line of Proportion conſiſts, are 1. the ſcale of Logarithmes, 2. the ſcale of Numbers.

The ſcale of Logarithmes, is that deſcribed vnder the common Line α Ω; viz. in the two laſt ſpaces of the Line of Proportion, which are firſt divided into ten equall parts by the fractions themſelues (each fraction being the tenth part of the whole Line;) and theſe parts are ſigned at the right end of the fractions by the figures 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. and in the vſe of this ſcale for the finding of any number vpon it, are called thouſands: Againe, the ſame ſpaces are divided vpon each fraction (by croſſe lines ſtrucke through them) into ten other equall parts, which are likewiſe noted in the laſt ſpace of each fraction, at the beginning of each part by the figures 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. and are hereafter called hundreds; then each of theſe hundreds is ſubdivided in the fourth ſpace of each fraction into ten other equall parts, which hereafter are termed tenths: Laſtly, each of thoſe tenths is againe ſuppoſed to be divided into ten parts, which are called vnits.

The vſe of this ſcale followes in the reſolution of the propoſition following.

A number being given that exceedes not 10000, to finde it vpon the ſcale of Logarithmes.

BEfore we come to the reſolution of this propoſition, it muſt be obſerved that the numbers propounded to be found vpon this ſcale, muſt alwayes conſiſt of foure places, being either ſignificant figures of ciphers, ſuch as are 2372. 2370. 2300. 2080. 2008. 2000. 0264. 0064. 0008. 0004. &c. This being premiſed, you may finde any ſuch number vpon that ſcale by this direction following.

Find the firſt figure of the number given amongst the thouſands, viz. the figures placed at the right end of the fractions; thou amongst the hundreds deſcribed vpon the fraction, vnto which that firſt figure directs you, ſearch the ſecond figure of the number given; againe, for the third figure count ſo many tenths, as that figure hath vnities; And for the laſt figure count ſo many vnits: This done the point of the common Line α Ω, where the laſt figure happens to fall, is the point that repreſents the number given.

Example, 2 3 7 2 being given, I demand the point vpon the common Line, that repreſents the ſame number; 2 the firſt figure directs me to the third fraction, ſigned by the figure 2; 3 the ſecond figure leads me to the hundred, marked vpon that fraction by the figure 3; For 7. the third figure I count ſeaven tenths of that hundred, viz. to the point p; and for 2 the laſt figure I count two vnits of that tenth: which done, I find the number given to be repreſented vpon the third fraction, at the point n. So 2370. is repreſented at the point p; 2300. vpon the ſame fraction at the beginning of the hundred, ſigned by the figure 3; and 2000. at the beginning of the ſame fraction, the three cyphers following 2. ſignifying that no hundreds, tenths, or vnits, are to be taken in finding the point, which repreſents that number: So likewiſe 2080. is found vpon the ſame fraction at the point q, the cypher in the ſecond place ſhewing that no hundreds, and the other in the laſt place, that no vnits are to be taken in finding out that number vpon the ſcale: In like manner 2008, is repreſented vpon the ſame fraction at the point r: And 0264. 0064. 0008. & 0004. vpon the firſt fraction at the points s, t, u, x.

Contrariwiſe, by inverting the rules of this propoſition, any point of the common Line being given, you may find the number repreſented by it: So the points p n q r being given, the numbers repreſented by them are 2370. 2372. 2080. and 2008.

CHAP. III. The Deſcription, Conſtruction, and Vſe of the Scale of Numbers.

THe ſcale of Numbers, is that deſcribed abouc the common Line α Ω, viz. in the three firſt ſpaces of the Line of Proportion, which are firſt divided into nine proportionall parts (diſtinguiſhed by the great figures 0. 1. 2. 3. 4. 5. 6. 7. 8. 9.) the firſt beginning at the beginning of the Line noted by 1. and ending at the line that croſſeth thoſe three ſpaces vpon the fourth fraction, marked by the figure 2 on the right hand, and by a little cypher on the left: The ſecond beginning there, and ending at ſuch another croſſe Line vpon the fift fraction, ſigned by the figure 3: The third reaching from thence to another croſſe line vpon the ſeventh fractiō, noted by the figure 4. In like manner, ſhall you finde the fift part to begin neere the left end of the ſeaventh fraction, the ſixt vpon the eight, the ſeaventh vpon the ninth, and the eight and nine parts vpon the laſt fraction, all ſigned by their proper figures 5. 6. 7. 8. 9. Which parts are hereafter called Primes, and are each of them againe divided into ten other parts, according to the ſame proportion, noted in the firſt ſpace of the Line by the little figures 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. (each of them having the Primefigure vnto which they belong annext reſpectiuely vnto them) which parts are hereafter called ſeconds, which ſeconds are each of them againe ſubdivided into ten other parts by croſſe Lines ſtrucke through the ſecond and third ſpaces, and are hereafter termed thirds; which thirds are each of them againe divided, or at leaſt ſuppoſed to be divided into ten parts, viz. the thirds contained in the firſt, ſecond, and third Primes, are really divided into ten parts; but thoſe betwixt the beginning of the fourth Prime, and the end of the Line, are onely divided into two parts, and therefore each of thoſe parts are conceived to haue the value of fiue, which ten parts of the thirds are hereafter called fourths: Laſtly, each fourth in the firſt ſecond & third Primes is conceived to be againe divided into ten parts, which are hereafter termed fifts. Now the conſtruction of this ſcale is in this manner:

Repaire vnto Mr. Brigges his Tables of Logarithmes, and ſuppoſing 1000. to be repreſented at the beginning of the Line of Proportion, finde in thoſe Tables the Logarithme of 1001. which is 3,00042, 40774, 7932, whereof take onely 3,0004, the firſt fiue figures; then caſting away 3 the Characteriſtique (his office being onely to ſhew of how many places the number, vnto which the Logarithme belongs, conſiſts, as I haue formerly ſhewed in the Treatiſe aboue-mentioned) find by the propoſition of the laſt chapter vpon the ſcale of Logarithmes 0004, the figures that remaine, which are repreſented vpon the firſt Fraction at the point x; this done, iuſt againſt that point in the ſcale of Numbers marke the point z, which repreſents the number 1001 vpon that ſcale, then taking the Logarithme of 1002. doe in like manner, and ſo proceede till you haue deſcribed all the diviſions of the ſcale of Numbers vpon the Line.

The vſe of this ſcale appeares in the reſolution of the propoſitions following.

PROPOSITION I. A whole number being given to finde it vpon the ſcale of Numbers.

FInde the firſt figure of the number given amongſt the Primes of that ſcale; then find the ſecond figure amongſt the ſeconds of that Prime; 3. for the third figure count ſo many thirds of that ſecond; 4. for the fourth count ſo many fourths of that third; and laſtly, if the number fall in the firſt, ſecond, or third Prime, for the fift figure count ſo many fifts of the laſt fourth: this done, the point, where the laſt figure falls vpon the common Line α Ω, is the point that repreſents the number given.

Example, 17268. being given, I demand the point vpon the common Line, where it is repreſented: 1. the firſt figure directs me to the firſt Prime, and 7. to the ſeaventh ſecond of that Prime, placed vpon the third fraction at the little figures 71. then for 2 I count two thirds of that ſecond, viz. to the point μ, and for 6 I count ſix fourths of that third, that is, to the point ν; And laſt of all for 8 the laſt figure, I take eight fifts of that fourth, ſo that I find 17268. the number given to be repreſented at the point ε vpon the third fraction; So 1726. or 17260. is found at the point ν; 172, 1720, or 17200. at the point μ; 17, 170, 1700, or 17000 at the ſeaventh ſecond of the firſt Prime, 1. 10. 100. 1000. &c. at the beginning of the Line; And 2. 20. 200. 2000. &c. at the beginning of the ſecond Prime: In like manner 2040. is repreſented at the point φ vpon the fourth fraction, the cypher in the ſecond place ſignifying that no ſeconds, and the other in the fourth or laſt place, ſhewing that no fourths are to be taken in finding out that number vpon the ſcale: So likewiſe 2008, is repreſented vpon the ſame fraction at the point ψ, the cyphers in the ſecond and third places ſhewing that no ſeconds or thirds are to be taken in the diſcovery of that number.

Contrariwiſe, by changing the rules of this propoſition, any point of the common Line being given, you may find the number repreſented by it, ſo the points ε vpon the third fraction, and ψ vpon the fourth, repreſent the numbers 17268. & 2008.

From the premiſſes ariſetheſe corrollaries.

1. A number that conſiſts of more figures then fiue, and falls in the firſt, ſecond, or third Prime, is repreſented at the point where the fift figure falls: So 17268347. is repreſented vpon the third fraction at the points, and 20080372. vpon the fourth at the p •• nt ψ.

2. A number that conſiſts of more figures then foure, and falls betweene the beginning of the fourth Prime, and the end of the Line, is repreſented at the point, where the fourth figure falls: So 4236, and 4236873. are both repreſented vpon the ſeaventh fraction at the point θ.

3. A point of the common Line in the firſt, ſecond, or third Prime, alwayes giues you a number, that conſiſts of fine places; So the points ε, ν, & μ being given, the numbers repreſented by them are 17268, 17260 & 17200.

4. A point of the common Line betweene the beginning of the fourth Prime, and the end of the Line, alwayes yeelds you a number compoſed of foure places: So θ, and χ vpon the ſeaventh fraction repreſent 4236, and 4230.

PROP: 2. A broken number being given to finde it vpon the ſcale of Numbers.

PRefixe the whole parts of the number given before the numerator of the fraction, and thereby make them as it were one intire number; then by the propoſition aforegoing finde the point which repreſents that number, which alſo will be the point, that repreſents the broken number propounded.

Example, 172 68 / 100 being given, 172 being prefixed before 68, the numerator of the fraction, conſtitutes the whole nūber 17268, which by the propoſition aforegoing is repreſented vpō the third fraction at the point • : So 17.26, that is 17 26 / 100, and 1. 726, viz. 1 726 / 1000 are both repreſented vpon the ſame fraction at the point ν; in like manner 20.40. and 20.08. are found vpon the fourth fraction at the points φ, and ψ.

But here it is to be obſerved, that the fractions of the broken numbers propounded to be found vpon this ſcale, muſt alwayes haue for their denominator a number conſiſting of an vnit in the firſt place towards the left hand, and nothing but cyphers towards the right, ſuch as are 10. 100. 1000. 10000. &c. And if the fractions of the broken numbers given be not ſuch, they ought to be reduced to fractions of that kinde.

Now other fractions are reduced to fractions of that kind for the moſt part vpon view, as if the number given were 12. foote, and 9. inches, that number being reduced is 12.75. viz. 12 75 / 100; and 12. pounds 14 ſhillings after reduction is 12.7, that is 12 1 / 10. But when you meete with a broken number, whoſe fraction is not reduccable vpon view, it may be reduced by the rule of three; for as the denominator of the fraction given is to 10. 100. or 1000. &c. ſo is the numerator of the ſame fraction to the numerator of the fraction required: So 17 98 / 305, that is, 17 yeares, and 98 dayes being given, the proportion will be;

As 365 to 1000: So 98 to 268.

So that 1000 being the denominator, and 268 the numerator of the fraction required, your number after reduction will ſtand thus 17 268 / 1000, or thus 17.268. Now to find 268. the fourth proportionall by the helpe of the Logarithmes, I referre you to the third Probleme of the fift chapter of my booke abouementioned: But in this caſe let the denominator of the fraction required alwayes exceed the denominator of the fraction given, as in the example aforegoing 1000 exceeds 365.

CHAP. IIII. The ioynt Vſe of the ſcale of Numbers, and the ſcale of Logarithmes together.
PROP. I. A whole number being given to find the Logarithme therof.

FInde vpon the ſcale of Numbers, by the firſt propoſition of the laſt chapter the point that repreſents the number given, then by the propoſition of the ſecond chapter obſerue vpon the ſcale of Logarithmes the number repreſented by that point; this done, if you prefixe before that number his proper Characteriſtique, that intire number is the Logarithme required.

Now the Characteriſtique is the firſt figure of the Logarithme, conſiſting of as many vnities within one, as the number, vnto which that Logarithme belongs, conſiſts of places: So the Characteriſtique of the numbers betwixt 1, and 10 is 0; betwixt 10, and 100 is 1; betwixt 100, and 1000 is 2; betwixt 1000, and 10000 is 3, &c.

Example, 17268 being given, I demand his Logarithme, by the firſt propoſition of the laſt Chapter I find 17268 vpon the third fraction at the point • , which giues me vpon the ſcale of Logarithmes by the propoſition of the ſecond chapter the number 2372, before which, becauſe the number given conſiſts of fiue places, I prefixe foure, ſo that the intire Logarithme of 17268 the number given is 42372: So the Logarithme of 2040 is 33096; and the Logarithme of 2008 is 33028, &c.

PROP. 2. A broken number being giuen, to find the Logarithme therof.

FInde vpon the ſcale of Numbers by the laſt propoſition of the laſt chapter the point that repreſents the number given; then by the propoſition of the ſecond chapter take vpon the ſcale of Logarithmes the number repreſented by that point; this done, if you place before that number his proper Characteriſtique, that is, a figure conſiſting of ſo many vnities, ſaue one, as the whole parts of the number given conſiſts of places, that intire number is that you looke for.

Example, 172. 68 being given, I demand his Logariſme, that number is found by the laſt propoſition of the laſt chapter vpon the third fraction at the point • , which yeelds you vpō the ſcale of Logarithmes by the propoſition of the ſecond chapter the number 2372; And now becauſe 172 (being the whole parts of the number given) conſiſts of three places, prefixe before 2372 the Charactcriſtique 2; which done, the intire Logarithme of 172.68 will be found 22372: ſo the Logarithme of 17.26 is 12370, and the Logarithme of 1.726 is 02370.

PROP. 3. A Logarithme being given to find the number vnto which it belongs.

NEglecting the Characteriſtique of the Logarithme given, find by the propoſition of the ſecond chapter the point where the other figures thereof are repreſented vpon the ſcale of Logarithmes, then by the firſt propoſition of the laſt chapter take off vpon the ſcale of Numbers the number repreſented by that point; this done, obſerving of how many vnities the Characteriſtique of the Logarithme given conſiſts, take one more of the firſt figures, that the number taken vpon the ſcale of nūbers hath towards the left hand, as if the Characteriſtique be 0, take one of thoſe figures, if it be 1, take two, if 2, take three, &c. which figures will be the whole parts of the number required, and if there remaine any figures towards the right hand, they are the numerator of a Fraction, whoſe denominator is a number conſiſting of an vnitie in the firſt place towards the left hand, and of ſo many cyphers towards the right, as there are figures remaining, which fraction is the broken parts of the number demanded.

Example; The Logarithme 42372 being given, I demand the number vnto which it belongs; 2372 the other figures beſides 4 the Characteriſtique I finde by the prop of the 2. chap. to be repreſented in the ſcale of Logarithmes vpon the third Fraction at the point n, at which point vpon the ſcale of Numbers I find by the 1 prop. of the laſt ch. to be repreſented the number 17268; and now becauſe the Characteriſtique of the Logarithme given is 4 the whole nūber 17268 is the number, vnto which the Logarithme given appertaines, but if the Logar. given were 22372, his number would be 172.68; the Charact. 2 ſhewing that 172 the three firſt figures of the number found ought to be taken for the whole parts, and 68 for the fraction of the number, vnto which that Logarithme belongs.

From this Propoſition ariſe theſe Corrallaries.

1. When a Logarithme, whoſe Charact. exceeds 4, falls within the firſt, ſecond, or third Prime, the firſt fiue figures of the number, vnto which it belongs, can onely be knowne; So if the Logarithine given were 72372, the fiue firſt figures of the number, vnto which it belongs are 17268.

2. When a Logarithme, whoſe Charact. exceedes 3 happens to fall betwixt the beginning of the fourth Prime and the end of the Line, the firſt foure figures of the number, vnto which it belongs, are onely diſcoverable vpon the Line: So the Logarithme 76270 being given, the foure firſt figures of the number, vnto which it belongs, are 4236, which you ſhall finde repreſented vpon the ſeaventh fraction at the point θ.

But now in taking the numbers vpon either of the ſcales obſerue this rule.

When you haue directed your eye vnto a point vpon the common Line in taking a n • mber vpon either of the ſcales, firſt take the leaſt parts repreſented by that point, and then the reſt in the ſame order.

As in the Example of the laſt propoſition, the Logarithme 42372 being propounded, your eye is directed by it vpon the ſcale of Logarithmes vnto the point n; and therefore in remooving your view for taking vpon the ſcale of Numbers the number, vnto which that Logarithme belongs, firſt take the filts, viz. 8, then 6 the fourths, and ſo the reſt in order; which done, carrying in you minde, eight, ſixe, two, ſeaven, one, and beginning with 8 firſt, ſet them downe thus, 17268, as before.

In like manner, in the example of the 1. Prop. of this ch. the number 17268, being given, your eye is directed vpon the ſcale of Numbers vnto the point • ; and therefore in remooving your view for taking vpon the ſcale of Logarithmes the Logarithme of that number, firſt obſerue the vnits, viz. 2, then 7 the tenths, and ſo the reſt in order; this done, keeping in your minde the figures ſo taken, ſet them downe as before, thus, 2372. And in obſerving this Rule, after a little practice, you ſhall finde much eaſe, and readineſſe.

Thus having ſhewed you how to find vpon the Line of Proportion the number of any Logarithme, and the Logarithme of any number propounded vnder the ſeverall limitations of the rules aforegoing; for the vſe of the Logarithmes being found, I referre you to the Treatiſes mentioned at the beginning of this Diſcourſe.

*⁎*

FINIS.