[AGNITUDES are faid to have a ratio to one another; when the less can be multiplied so 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 A, and taken four times exceeds it. The Rgles M&N bave also a ratio to one another, because the Rgle N taken three times and a half, is = to Rgle M, and repeated oftner exceeds it. But the line B, and the Rgle M bave no ratio to one another, because 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 same kind can have a ratio to one another, as numbers to numbers, lines to lines, surfaces 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 supposed of the same kind. For a magnitude conceived infinite, is conceived without bounds, confequently a finite magnitude repeated ever so 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 incommensurable when the terms of the ratio are incommenfurable. § 4. The antecedent of the ratio of M to N, is the first 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 second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth. DEFINITION S. If the multiple of the first, be less than that of the second, the multiple of the third is also less than that of the fourth; or if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth, or if the multiple of the first be greater than that of the second, 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 same as that of the number 8 to the number 24, for if the two antecedents 258 be multiplied by the same number M, and the two confequents 6 & 24 by another number N; the multiple 2 M of the first antecedent cannot be = or > or < the multiple 6 N of its confequent, unless the multiple of the second 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 beto 6 N, 2M+2M+2M+2 M is allo=6N+6N+6N+6N, that is, 8 M= 24 N. Likewise, If 2 M be > 6 N, then 2M+2M+2M + 2 M is also >6N+6N+6N+6N, that is, 8 M > 24 N. And in fine, If 2 M be <6N, then 2M+2M+2M + 2 Mis also <6N+6N+6N+6N, that is, 8 M < 24 N. §. 2. On the contrary, the numbers 2, 3 & 7, 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 equal to the multiple 6 of its confequent, whilst 21 multiple of the II. antecedent is greater than 16 multiple of its confequent. §. 3. Incommensurable magnitudes can never have their equimultiples equal, otherwise they would be commensurable to one another, wherefore incommenfurables are shewn to be proportional only from the joint excess or defect of their equimultiples; whereas commenfurable magnitudes being capable of a joint equality, and inequality of their equimultiples, are shewn to be proportional from the joint equality or excess of their equimultiples, bence 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 respect to that of fubmultiples. But it is probable that Euclid preferred the use of multiples to that of fubmultiples, because be could not prescribe to take fubmultiples without first spewing how to divide magnitude into equal parts, whilst the formation of multiples required no such principle. This Geometer bad aright to assume for granted, that the double triple, or any multiple of a magnitude could be taken, but was under the necessity of shewing by the Resolution of a problem, how to take away an aliquot part from a given line, and the resolution of this problem supposing the doctrine of fimilitude, could not be given but in the IX. Propofition of the VI. Book. VI. Magnitudes which have the fame ratio, are called proportionals. When four magnitudes A,B,C,D are proportional, it is usually expreft thus, A: BC: Dand in words, the first is to the second as the third to the fourth. VII. When of the equimultiples of four magnitudes (taken as in the 5th definition) the multiple of the first is greater than that of the second, but the multiple of the third is not greater than the multiple of the fourth; then the first is faid to have to the second a greater ratio than the third magnitude has to the fourth, and on the contrary, the third is faid to have to the fourth a less ratio than the first has to the fecond. §. 1. Such are the ratios 3 : 2 & 11 : 9 for if the antecedents be multiplied by 9, and the confequents by 13, there will refult 27:26;99:117. 3:2; II:9 27:26;99:117 Where the correspondence of the multiples does not bold, the first antecedent 27 being greater than its confequent 26 whilst the second antecedent 99 is less than its confequent I17. §. 2. To discover by inspection the inequality of two ratios A: B & C : D by this character of the non correspondence of multiples, it fuffices to chuse for multiples, the two terms of one of the two ratios, for Ex. C: D, and to multiply the antecedents A & C by the confequent D of this ratio; and the twe confequents B & D by the antecedent C of the fame ratio, in this manner. Which being done, the two products C.D & D.C will be found equal, whilst the two others A.D & B.C are unequal, and in particular, if the multiple of one of the antecedents be greater than that of its confequent, whilst the multiple of the other is equal to its, then the terms of the leffer ratio bave been chofen for multipliers. On the contrary, if the multiple of one of the antecedents be less than that of its confequent, whilst the multiple of the other is equal to its, then the terms of the greater ratio have been chosen for multipliers. VIII. Analogy or proportion, is the fimilitude of ratios. As a fign and character of proportionals has been already given (in Def. 5.) this is a fuperfluous definition, a remark of some scholiaft soufled into the text which interrupts the coherence of Euclid's genuine definitions. IX. Proportion consists in three terms at least. : §. 1. Proportion confifting in the equality of two ratios, and each ratio baving two terms, in a proportion there are four terms, of which the first and fourth are called the extreames, and the second and third the means, those jour terms are confidered as only three, when the consequent of the first ratio at the fame time bolds the place of the antecedent of the second ratio: it is for this reason, that proportions are diftinguished into discrete, and continued. A proportion is discrete when the two means are unequal, and it is called continued when these Same terms are equal, thus this proportion 2 : 4 = 5 : 10 is discrete because the two mean terms 4 & 5 are unequal, on the contrary, the proportion 2:4 = 4:8 is a continued proportion on account of the equality of the mean terms 4 & 4. §. 2. A feries of magnitudes incontinued proportion, forms a geometrical progreffion, fuch are the numbers 1, 2, 4, 8, 16, 32, 64, &c. Χ. When three magnitudes are proportional the first is said to have to the third the duplicate ratio of that which it has to the second. ΧΙ. When four magnitudes are continual proportionals, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and fo on quadruplicate, &c. increasing the denomination still by unity in any number of proportionals. XII. In proportionals, the antecedent terms are called Homologous to one another, as also the consequents to one another. XIII. Proportion is faid to be alternate when the antecedent of the first ratio is compared with the antecedent of the second, and the consequent of the first ratio with the consequent of the second. If A : B = C:D then by alternation. {A:CB:D 4:5 = 16:20 4:16 5:20 When the proportion is disposed after this manner, it is faid to be done by permutation or alternately, permutando or alternando. XIV. But when the consequents are changed into antecedents, and the antecedents into consequents in the same order, it is said that the comparison of the terms is made by inversion or invertendo. A: B = C:D } therefore invertendo. { XV. B: A=D:C 3 But the comparison is made by composition or componendo, when the fum of the confequents and antecedents is compared with their respective consequents. A:B C: DS therefore A + B : B = C + D :D 3:9 = 4:12{componendo 3 + 9: 9 = 4 + 12:12 XVI. The comparison is made by division of ratio, or dividendo when the excess of the antecedent above its consequent, is compared with its consequent. The comparison is made by the conversion of ratio, or convertendo, when the antecedent is compared to the excess of the antecedent above its consequent. If A: B = C: D{ therefore A:A-B = C: 9:3=12: 4 C-D convertendo.59 : 9 -3 = 12:12-4 XVIII. A conclusion is drawn from equality of ratio or ex equo, when comparing two series of magnitudes of the same number, such that the ratios of the first be equal to the ratios of the second, each to each, (whether the comparison be made in the fame order or in an inverted one), it is concluded that the extreames of the two series are in proportion. The sense of this definition is as follows, if A, B, C, D be a feries of four magnitudes, and a, b, c, d a series of four other magnitudes, such that |