I. .

A Lefs magnitude is faid to be a part of a greater magnitude, when the less

measures the greater.

§. 1. By the expreffion of measuring a magnitude Euclid means to be contained in it a certain number of times without a remainder, that is a lefs magnitude N (fig. 1.) measures a greater M, when the magnitude N is contained in M without a remainder twice, thrice, four times, and in general, any number of times whatsoever, or which comes to the fame, when the lefs magnitude N repeated twice, thrice four times, and in general any number of times produces a whole, equal to the greater M.

§. 2. Those parts which measure a whole without a remainder, are called aliquot parts, and fuch as are not contained in a whole exactly, but are meafured by fome other determined quantity which measures alfo the whole, are called aliquant parts.

Thus the numbers 2, 3, 4, 6 are so many aliquot parts of the number 12 confidered as a whole; as each of the numbers 2, 3, 4, 6 is found repeated in 12 a certain number of times without a remainder. But the numbers 5, 7, 9 &c. are aliquant parts of the fame whole 12; as they do not measure 12 but with a remainder: although they are all measured by unity as well as 12; which often bappens in other numbers different from unity, as in the number 9 which is commenfurable to 12 by the number 3, as alfo by unity.

Likewife the magnitude N (fig. 2.) is an aliquant part of the magnitude M(=N+N+N+R&c), if N meafures M leaving a remainder R, and this remainder R be fuch, that it measures Nor at leaft that one of its determined parts asr measures this remainder R, as alfo the magnitude N& confequently the whole

M.

§. 3.IN general numbers are faid to be commenfurable to each other which may

refult from unity or one of its aliquot parts repeated a determined number of times or what amounts to the fame that which is measured by unity or one of its aliquot parts.

Thus the numbers 6, 9, 17, and the fractions, are commenfurable numbers; because the firft may be conceived to refult from the determined and fucceffive addition of unity; and the last from that of the fractions & aliquot parts of unity.

§. 4. According to this definition, a commenfurable quantity, is that which refults from the determined repetition of any determined quantity. A quantity is therefore commenfurable, when it contains one of its parts, as often as a determined number contains unity.

§. 5. Commenfurability is therefore fomething relative. The magnitudes M and N are commensurable, as having a common and determined measure t which can be taken for unity, and measure them both exactly; or, as those two magnitudes may arife from the determined repetition of the fame quantity R, be

it what it will.

§. 6. It follows from this notion of commenfurable numbers, that they are all multiples of each other, or aliquot parts, or aliquant parts. For if the quantities M and N, are commenfurable, N meafures M, or M measures N, or jome other determined number 1 measures them both. In the firft cafe, the number M, is a multiple of N, in the fecond cafe M, is an aliquot part of N, and in the third, the leffer of the two is an aliquant part of the leaft. The fame is true with respect to rational magnitudes in general.

§. 7. The number which cannot refult from a determined repetition of unity or of one of its aliquot parts is called, irrational or incommenfurable, with refpect to unity. And in general, magnitudes which cannot refult from the determined repetition of the fame determined quantity confidered as unity, are, incommenfurable to one another, or irrational.

THUS

DEFINITIONS.

HUS the fide (AD or DC) of the square (ABCD) is incommenfurable to its diagonal (AC), or bow much one contains of the other is inaffignable (Fig. 1). §. 8. From whence it follows, that if two magnitudes M and N, are incommenfurable to each other, M cannot be a multiple of N; nor an aliquot part, nor in fine an aliquant part of this fame N, for if it was, the magnitudes M and N could be measured by the fame determined magnitude, which is repugnant to the notion of incommenfurability (Fig. 2)

II.

A greater magnitude is faid to be a multiple of a lefs, when the greater is measured by the less.

Thus, the number 12 is faid to be a multiple, of the number 4, becaufe 4 meafures 12 without a remainder.

To the term of multiple corresponds that of submultiple, which fignifies, that a lefs magnitude is an aliquot part of a greater; thus 4 is a fubmultiple of 12, as 12 is a multiple of 4.

III.

Ratio, is a mutual relation of two magnitudes of the fame kind to one another in respect of quantity.

This definition is imperfect, and is commonly believed to be none of Euclid's, but the addition of fome unskilful editor; for though the idea of ratio includes a certain relation of the quantities of two bomogeneous magnitudes, yet this general character is not fufficicent; because the quantities of two magnitudes are fufceptible of feveral forts of relations different from that of ratio. Thus, when in a circle the fquare of the perpendicular let fall from the circumference on the diameter, is reprefented as conftantly equal to the difference of the jquares of the ray, and of the portion of the ray intercepted between the center and the perpendicular, without doubt, this perpendicular is confidered as bearing a certain relation to this portion of the ray, but it is manifeft that this relation is not a ratio, fince the quantities are compared only by the means of the ray which is a third bomogeneous magnitude different from the quantities compared.

MAGNITUD

DEFINITIONS.

IV.

AGNITUDES are faid to have a ratio to one another; when the lefs can be multiplied fo as to exceed the other.

§. 1. The lines A & B have a ratio to one another, because the line B, for example, taken three times and a half, is equal to the line Ă, and taken four times exceeds it. The Rgles M&N have alfa a ratio to one another, because the Rgle N taken three times and a balf, is to Rgle M, and repeated oftner exceeds it.

But the line B, and the Rgle M bave no ratio to one another, becaufe the line B repeated ever so often, can never produce a magnitude which would equal or exceed the Rgle M. Therefore, only magnitudes of the fame kind can have a ratio to one another, as numbers to numbers, lines to lines, furfaces to furfaces, and folids to folids.

§. 2. In confequence of this definition, a finite magnitude and an infinite one, bave no ratio to one another, though they be fuppofed of the fame kind. For a magnitude conceived infinite, is conceived without bounds, confequently a finite magnitude repeated ever fo often (provided the number of repetions be determined) can never become equal or exceeds an infinite magnitude.

§. 3. A ratio is commenfurable, when the terms of the ratio M & N are commenfurable to each other, & a ratio is faid to be incommenfurable when the terms of the ratio are incommensurable.

§ 4. The antecedent of the ratio of M to N, is the firft of the two terms which are compared, and the other is called its confequent.

V.

The first of four magnitudes is faid to have the fame ratio to the fecond, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatfoever of the fecond and fourth.

DEFINITIONS.

If the multiple of the first, be less than that of the fecond, the multiple of the third is alfo less than that of the fourth; or if the multiple of the first be equal to that of the fecond, the multiple of the third is alfo equal to that of the fourth, or if the multiple of the first be greater than that of the fecond, the multiple of the third is also greater than that of the fourth.

§. 1. The ratio of the number 2 to the number 6, is the fame as that of the number 8 to the number 24, for if the two antecedents 28 be multiplied by the fame number M, and the two confequents 6 & 24 by another number N the multiple 2 M of the firft antecedent cannot be or> or <the multiple 6 N of its confequent, unless the multiple of the fecond antecedent 8 M, be at the fame time = or> or < the multiple 24 N of its confequent, for it is evident that

If 2 M be to 6 N,

2M+2M+2M+ M is alfo 6N+6N+6N +6 N, that is, 8 M

If 2 M be > 6 N, then

24 N. Likewife, 2M+2M+2M+ 2 M is alfo >6N+6N+6N+6 N, that is, 8 M > 24 N. And in fine, If 2 M be 6 N, then 2M+2M+2M+ 2 M is alfo < 6N+6N+6N+6 N, that is, 8 M < 24 N. §. 2. On the contrary, the numbers 2, 37, 8 are not in the fame ratio; for if the antecedents be multiplied by 3, and the confequents by 2, there will refult the four multiples 6, 6, 21, 16, where the multiple 6 of the Ift antecedent is èqual to the multiple 6 of its confequent, whilst 21 multiple of ibe II. antècèdent is greater than 16 multiple of its consequent.

§. 3. Incommensurable magnitudes can never have their equimultiples equal, otherwife they would be commenfurable to one another, wherefore incommenfurables are fewn to be proportional only from the joint excefs or defect of their equimultiples; whereas commenfurable magnitudes being capable of a joint equality, and inequality of their equimultiples, are fewn to be proportional from the joint equality or excess of their equimultiples, hence it is that -the figns in this definition by which proportionality is discovered, are applicable to all kinds of magnitude whatsoever.

§. 4. What is true with respect to the correspondence of multiples, is also true with refpect to that of fubmultiples. But it is probable that Euclid preferred the ufe of multiples to that of fubmultiples, because be could not prefcribe to take fubmultiples without firft fhewing how to divide magnitude into equal parts, whilft the formation of multiples required no fuch principle. This Geometer bad aright to affume for granted, that the double triple, or any multiple of a magnitude could be taken, but was under the neceffity of fhewing by the