The Line of Proportion depiction of a logarithmic number line continuation of logarithmic number line continuation of logarithmic number line THE CONSTRUCTION, And Use of the Line of PROPORTION. By help whereof the hardest Questions of Arithmetic & 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 john Dawson. 1628. ¶ The Preface. HAVING not many months ago published a discourse, declaring the nature and use of the Logarithmeticall Tables, and observing the Table of Numbers there to be too small for ordinary use, not giving indeed without much difficulty the Logarithme of any number that exceeds 1000; I have invented this tabular Scale, or Line of Proportion, by means whereof (as I take it) you shall find that defect fully supplied: this Instrument yielding you the resolution of the hardest questions of Arithmetic or Geometry, both in broken, and whole Numbers, only by Addition and Substraction, when the term required happens not to exceed 10000 although the terms propounded consist of never so many places, as shall further appear by the Treatise following. CHAP. I. The Definition of the Line of Proportion. CHAP. II. The Description and Use of the Scale of Logarithmes. CHAP. III. The Description, Construction, and Use of the Scale of Numbers. CHAP. FOUR The joint Use of the Scale of Numbers, and the Scale of Logarithmes together. THE CONSTRUCTION, and Use of the Line of PROPORTION. CHAP. I. The Definition thereof. THE Line of Proportion is a double scale, broken off into ten Fractions, upon which the Logarithmes of numbers are found out. To understand the nature of Logarithmes, I refer you to Master Brigges his learned Work, entitled 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 five 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 less, such as are the other three placed in the midst of each fraction. These fractions, together with their Lines and spaces, must be understood to join respectively one to another, in such sort that the whole Line of Proportion may be conceived to be one entire and continued Line; As the left end of the first fraction, marked by the Letters a g c must be conceived to join 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 understood to join 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 entire Line, as is aforesaid. A double scale, is when two several scales meet both upon one common Line; So the Line of Proportion being composed of the two scales, which meet upon 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 Use 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 under the common Line α Ω; viz. in the two last spaces of the Line of Proportion, which are first divided into ten equal parts by the fractions themselves (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 use of this scale for the finding of any number upon it, are called thousands: Again, the same spaces are divided upon each fraction (by cross lines struck through them) into ten other equal 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 equal parts, which hereafter are termed tenths: Lastly, each of those tenths is again supposed to be divided into ten parts, which are called vnits. The use of this scale follows in the resolution of the proposition following. A number being given that exceeds not 10000, to find it upon the scale of Logarithmes. BEfore we come to the resolution of this proposition, it must be observed that the numbers propounded to be found upon this scale, must always consist of four places, being either significant figures of cyphers, such as are 2372. 2370. 2300. 2080. 2008. 2000 0264. 0064. 0008. 0004. etc. This being premised, you may find any such number upon 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 upon the fraction, unto which that first figure directs you, search the second figure of the number given; again, for the third figure count so many tenths, as that figure hath unities; 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 upon 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 upon that fraction by the figure 3; For 7. the third figure I count seven tenths of that hundred, viz. to the point p; and for 2 the last figure I count two units of that tenth: which done, I find the number given to be represented upon the third fraction, at the point n. So 2370. is represented at the point p; 2300. upon 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 upon the same fraction at the point q, the cipher in the second place showing that no hundreds, and the other in the last place, that no units are to be taken in finding out that number upon the scale: In like manner 2008, is represented upon the same fraction at the point r: And 0264. 0064. 0008. & 0004. upon 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 Use 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 proportional 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 upon the fourth fraction, marked by the figure 2 on the right hand, and by a little cipher on the left: The second beginning there, and ending at such another cross Line upon the fift fraction, signed by the figure 3: The third reaching from thence to another cross line upon the seventh fraction, noted by the figure 4. In like manner, shall you find the fift part to begin near the left end of the seaventh fraction, the sixth upon the eight, the seaventh upon the ninth, and the eight and nine parts upon 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 again 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 unto which they belong annexed respectively unto them) which parts are hereafter called seconds, which seconds are each of them again subdivided into ten other parts by cross Lines struck through the second and third spaces, and are hereafter termed thirds; which thirds are each of them again 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 only divided into two parts, and therefore each of those parts are conceived to have the value of five, which ten parts of the thirds are hereafter called fourth's: Lastly, each fourth in the first second & third Primes is conceived to be again divided into ten parts, which are hereafter termed fifts. Now the construction of this scale is in this manner: Repair unto Mr. Brigges his Tables of Logarithmes, and supposing 1000 to be represented at the beginning of the Line of Proportion, find in those Tables the Logarithme of 1001. which is 3,00042, 40774, 7932, whereof take only 3,0004, the first five figures; then casting away 3 the Characteristique (his office being only to show of how many places the number, unto which the Logarithme belongs, consists, as I have formerly showed in the Treatise abovementioned) find by the proposition of the last chapter upon the scale of Logarithmes 0004, the figures that remain, which are represented upon the first Fraction at the point x; this done, just against that point in the scale of Numbers mark the point z, which represents the number 1001 upon that scale, then taking the Logarithme of 1002. do in like manner, and so proceed till you have described all the divisions of the scale of Numbers upon the Line. The use of this scale appears in the resolution of the propositions following. PROPOSITION I. A whole number being given to find it upon the scale of Numbers. Find 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 fourth's 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 upon the common Line α Ω, is the point that represents the number given. Example, 17268. being given, I demand the point upon 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 upon 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 fourth's 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 upon 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 etc. at the beginning of the Line; And 2. 20. 200. 2000 etc. at the beginning of the second Prime: In like manner 2040. is represented at the point φ upon the fourth fraction, the cipher in the second place signifying that no seconds, and the other in the fourth or last place, showing that no fourth's are to be taken in finding out that number upon the scale: So likewise 2008, is represented upon the same fraction at the point ψ, the cyphers in the second and third places showing 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 upon the third fraction, and ψ upon the fourth, represent the numbers 17268. & 2008. From the premises arisethese corrollaries. 1. A number that consists of more figures than five, and falls in the first, second, or third Prime, is represented at the point where the fift figure falls: So 17268347. is represented upon the third fraction at the points, and 20080372. upon the fourth at the p●●nt ψ. 2. A number that consists of more figures than four, and falls between 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 upon the seaventh fraction at the point θ. 3. A point of the common Line in the first, second, or third Prime, always gives 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 between the beginning of the fourth Prime, and the end of the Line, always yields you a number composed of four places: So θ, and χ upon the seaventh fraction represent 4236, and 4230. PROP: 2. A broken number being given to find it upon the scale of Numbers. Prefix the whole parts of the number given before the numerator of the fraction, and thereby make them as it were one entire number; then by the proposition aforegoing find 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 number 17268, which by the proposition aforegoing is represented upon the third fraction at the point ●: So 17.26, that is 17 26/100, and 1. 726, viz. 1 726/1000 are both represented upon the same fraction at the point ν; in like manner 20.40. and 20.08. are found upon 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 upon this scale, must always have for their denominator a number consisting of an unit in the first place towards the left hand, and nothing but cyphers towards the right, such as are 10. 100 1000 10000 etc. And if the fractions of the broken numbers given be not such, they ought to be reduced to fractions of that kind. Now other fractions are reduced to fractions of that kind for the most part upon view, as if the number given were 12. foot, 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 meet with a broken number, whose fraction is not reduccable upon view, it may be reduced by the rule of three; for as the denominator of the fraction given is to 10. 100 or 1000 etc. so is the numerator of the same fraction to the numerator of the fraction required: So 17 98/305, that is, 17 years, and 98 days 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 proportional by the help of the Logarithmes, I refer you to the third Problem of the fift chapter of my book abovementioned: But in this case let the denominator of the fraction required always exceed the denominator of the fraction given, as in the example aforegoing 1000 exceeds 365. CHAP. FOUR The joint Use of the scale of Numbers, and the scale of Logarithmes together. PROP. I. A whole number being given to find the Logarithme thereof. Find upon 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 observe upon the scale of Logarithmes the number represented by that point; this done, if you prefix before that number his proper Characteristique, that entire number is the Logarithme required. Now the Characteristique is the first figure of the Logarithme, consisting of as many unities within one, as the number, unto 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, etc. Example, 17268 being given, I demand his Logarithme, by the first proposition of the last Chapter I find 17268 upon the third fraction at the point ●, which gives me upon the scale of Logarithmes by the proposition of the second chapter the number 2372, before which, because the number given consists of five places, I prefix four, so that the entire Logarithme of 17268 the number given is 42372: So the Logarithme of 2040 is 33096; and the Logarithme of 2008 is 33028, etc. PROP. 2. A broken number being given, to find the Logarithme thereof. Find upon 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 upon 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 unities, save one, as the whole parts of the number given consists of places, that entire number is that you look for. Example, 172. 68 being given, I demand his Logarisme, that number is found by the last proposition of the last chapter upon the third fraction at the point ●, which yields you upon 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, prefix before 2372 the Charactcristique 2; which done, the entire 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 unto 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 upon the scale of Logarithmes, then by the first proposition of the last chapter take off upon the scale of Numbers the number represented by that point; this done, observing of how many unities the Characteristique of the Logarithme given consists, take one more of the first figures, that the number taken upon the scale of numbers 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, etc. which figures will be the whole parts of the number required, and if there remain any figures towards the right hand, they are the numerator of a Fraction, whose denominator is a number consisting of an unity 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 unto which it belongs; 2372 the other figures beside 4 the Characteristique I find by the prop of the 2. chap. to be represented in the scale of Logarithmes upon the third Fraction at the point n, at which point upon 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 number 17268 is the number, unto which the Logarithme given appertains, but if the Logar. given were 22372, his number would be 172.68; the Charact. 2 showing that 172 the three first figures of the number found aught to be taken for the whole parts, and 68 for the fraction of the number, unto 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 five figures of the number, unto which it belongs, can only be known; So if the Logarithine given were 72372, the five first figures of the number, unto which it belongs are 17268. 2. When a Logarithme, whose Charact. exceeds 3 happens to fall betwixt the beginning of the fourth Prime and the end of the Line, the first four figures of the number, unto which it belongs, are only discoverable upon the Line: So the Logarithme 76270 being given, the four first figures of the number, unto which it belongs, are 4236, which you shall find represented upon the seaventh fraction at the point θ. But now in taking the numbers upon either of the scales observe this rule. When you have directed your eye unto a point upon the common Line in taking a n●mber upon 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 upon the scale of Logarithmes unto the point n; and therefore in removing your view for taking upon the scale of Numbers the number, unto which that Logarithme belongs, first take the filts, viz. 8, then 6 the fourth's, and so the rest in order; which done, carrying in you mind, eight, six, two, seven, one, and beginning with 8 first, set them down 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 upon the scale of Numbers unto the point ●; and therefore in removing your view for taking upon the scale of Logarithmes the Logarithme of that number, first observe the units, viz. 2, then 7 the tenths, and so the rest in order; this done, keeping in your mind the figures so taken, set them down as before, thus, 2372. And in observing this Rule, after a little practice, you shall find much ease, and readiness. Thus having shown you how to find upon the Line of Proportion the number of any Logarithme, and the Logarithme of any number propounded under the several limitations of the rules aforegoing; for the use of the Logarithmes being found, I refer you to the Treatises mentioned at the beginning of this Discourse. * ⁎ * FINIS.