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.
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, [Page 2]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 [Page 3]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 [Page 4]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 [Page 5]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 [Page 6]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.) [Page 7]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 [Page 8]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 M r. 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 [Page 9]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 [Page 12]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 [Page 13]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 [Page 14]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 [Page 15]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, [Page 16]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, [Page 17]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 [Page 18]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 [Page 19]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.
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