Again, let c have a greater ratio to B, than it has to A : I say, B is less than a. For if it be not lels, it is either equal to it or greater, But B is not equal to A, for then c (by 7. 5.) would have the same ratio to A as to B. But it has not : Therefore A is not equal to B, nor is B greater than A; for then [by 8. 5.] c would have a less ratio to B than to A. But it has not : Therefore B is not greater than a : But it has been proved not to be equal to it: Consequently B will be less than A: Therefore that magnitude of those magnitudes which have a ratio to the fame magnitude, is the greater which has the greater ratio, and that magnitude to which the fame magnitude has the greater ratio, is the lesser. PROP. XI. THEOR. Those ratios that are the same to the same ratio, are the same between themselves. For let a be to B, as c is to D, and as c is to D, fo is E to F: I say, as a is to B, so is E to F. For take G, H, K equimultiples of A, C, E, and any other equimultiples L, M, N of B, D, F. Then because it is as A to B, fo is c to D, and G, H, are equimultiples of A,C, and any L, M other equimultiplesof B, D: If G exceeds L, [by 5: def. 5.) H will exceed M; if, the one be equal, the ther will be equal ; if less, less. A gain, because it is as c to D, so is E to F, and GABL HCDM KÉÊN H, K are taken equimultiples of c, E ; also M, N any other equimultiples of D, F. If u exceeds m, K will exceed N ; If equal, equal ; but if less, less. But if H.ex ceeds ceeds m, Ġ will exceed 1: If it be equal, equal; if less, less. Wherefore if G exceeds L, K will exceed N; if equal, equal; if less, less. And 6, K are equimultiples of A, E, and 1, N any others of B; F: Therefore [by šé def. 5.] as A is to B; so will E be to F. Wherefore those ratios that are the fame to the same ratio, are the same between themselves. Which was to be demonstrated. PROP. XII. THEOR. If any magnitudes foever be proportional ; as one of the antecedents is to its consequent, so will all the antecedents together be to all the consequents together y. Let any magnitudes sơever Å, B, C, D, E, F be proportional; and let a be to B, as c is to D, and as E is to š: I say, as A is to B, fo is A, C, E, to B, D, F. For take the equimultiples G, H, K of A, C, E, and any other equimultiples L, M, N of B, D, F. Then because as A is to B; so is c to D, and E to F: And there are taken the equimultiples G, H,K of A, C, E, and any other equimultiples L, M, N of B, D, F: If G exceeds 1 [by 5. def. 5.] # will ex ceed Mỹ and K will exceed n: If the one GH KACE BDFLMN be equal, the other will be equal; if lefs, less. Wherefore also if g exceeds I; G, H, K will exceed L, M, N; if equal; equal; and if less, less. And G, and G, H, K are equimultiples of A and A, C, E: Because [by 1. 5.) if there be any number of magnitudes soever equimultiples of an equal number of magnitudes, each the fame multiple of each, the fame muluple that one of the magnitudes is of one of the other, will all the magnitudes be of all the others. By the fame reafon L and L, M, N are equimultiples of B, and B, D, F: B, D, F,: Therefore [by si def. 5.) as a is to Ž, so is A, Wherefore if any magnitudes foever be proportional : as one of the Antecedents is to its confequent, so will all the antecedents together be to all the consequents together. Which was to be demonstrated. y The twelfth propofition of the seventh book of Euclid answers to this in numbers. PROP. XÍII. THEOR. the same ratio to the second, as the third to the For let the first of fix magnitude A, have the same ratio to the second Bị that the third c, has to the fourth d; and let the third c have a greater ratio to the fourth D, than the fifth E has to the fixth Fi I say the first A will have a greater ratio to the second B, than the fifth É has to the Then because a is to B, as c is to D, and there are taken the equimultiples M, G of A, C, and other equimultiples N, K of B, D: [by s. def. 5.] If m exceeds n, Ġ will exceed k: if m be equal to N, G will be equal to K: if less, less: But G exceeds K; therefore M will exceed n: But h does not exceed L ; and m, h are equimultiples of Ä, E, and N, L some other equimultiples of B, F: Therefore [by 7 def. 5] A will have a greater ratio to B than E has to F. Sixth F. tio of E to F, For because the ratio of c to D is greater than the ra there are [by 7. def. 5.) some equimultiples of C, E, and others of D, F; where the multiple of c exceeds the multiple of D, but the mul. tiple of e does not exceed the multi-MÁBN GCDK HEFL ple of F. Take these equimultiples. And let G, H be equimultiples of C, E, and K, L other equimultiples of D, F, such that G exceeds K, but H does not exceed L: and the same multiple that G is of c, the same let M be of A; and the same that K is of D, let n be of B. ni If therefore the first of six magnitudes has the same ratio to the second, as the third to the fourth; and the third to the fourth has a greater ratio than the fifth to the sixth : then the first to the second will have a greater ratio than the fifth has to the fixth. Which was to be demonstrated. PRO P. XIV. THEOR. If the first of four magnitudes has the same ratio to the second, as the third has to the fourth; and the first be greater than the third : then will the second be greater than the fourth. If the first be equal to the ibird, the second will be equal to the fourtb. If the first be less then the third, the second will : be less than the fourth. For let the first magnitude a have the same ratio to the fecond.B, as the third Ċ has to the fourth D, and let a be greater than c: I say B is also greater than D. For because a is greater than c, and B is any other magnitude ; [by 8. 5.1 A will have a greater ratio to B, than c has to B.' But as A is to B, fo is c to D: Therefore [by 13. 5.) the ratio of c to D, will be greater than the ratio of c to B. But [by 10. 5) that magnitude to which the same magnitude has the greater ratio, is the lesser magnitude : wherefore d is less than B: and accordingly B will be greater than D. After the like manner we demonstrate if a be equal to c; that B is equal to D: and if A be less than C, B is less А в Note, C D than Di Note, If Å be equal to C, B will be equal to D. Fol. lows [by 9. 5.) for here A is to B, as C, that is A to D; B Thall be equal to D. If A be less than c, B shall be less than D, for c is greater than A ; and here c is to D, as a to B; D will be greater than B by the first case, wherefore B is less than D. Therefore if the first magnitude has the same ratio to the second, as the third has to the fourth, and the first be greater than the third : then will the second be greater than the fourth ; if the first be equal to the third, the second will be equal to the fourth : and if the first be less than the third, the second will be less than the fourth. Which was to be demonstrated. PRO P. XV. THEOR. Magnitudes have the same ratio to one another which their equimultiples have s. For let A B be the same multiple of c, that D e is off I fay c is to F, as A B is to D E. For because A B is the fame multiple of c, that D E is of F : there will be as many magnitudes in a s equal to cg as there are in D Е, equal to F. vide A B into magnitudes equal to c, which let be AG, GH, HB; and divide De into the magnitudes DK, KL, L E B each equal to F. Then the multitude G of a G, GH, HB will be equal to the multiude of D K, KL, LE. And because A G, GH, H B are equal to one another, as also D K, KL, LE equal to K one another : [by 7. 5.) AG will be to Ht DK, as GH is to KL, and as KB is to U LE: But [by 12. 5.] as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents. Therefore as is to DK, BO E F as A B is to D E. But AG is equal to c, and DK equal to F. Therefore as c is to F, so will AB be to DE. Wherefore magnitudes have the same ratio to one an. other which their equimultiples have. Which was to be demonstrated. The seventeenth proposition of the fertuff book answera to this proposition. Q2 PROP. Di- AT Z |