PRO P. XI. THEOR E M. Ratios which are the same to the same ratio, are the same to each other. Let A be to B as c is to D, and c to D as E is to F; then will a be to B as E is to F. For, of A, C and E take any equimultiples whatever G, H and K; and of B, D and F any equimultiples whatever L, M and N: Then, since A is to B as c is to D (by Hyp.), and G and H are equimultiples of A and c, and L and m of B and D, if G be greater than 1, H will be greater than M; and if equal, equal ; and if less, less (V. Def. 5.) And, because c is to D as E is to F (by Hyp.), and H and K are equimultiples of c and E, and m and x of p and F; if be greater than M, K will be greater than N; and if equal, equal ; and if less, less (V. Def. 5.) But if G be greater than 1, it has been shewn that h will also be greater than m; and if equal, equal; and if less, less; whence, if G be greater than L, K will also be greater than n; and if equal, equal; and if lefs, less. And since g and K are any equimultiples whatever of A and E, and 1 and n are'any equimultiples whatever of B and F, A will have to B the same ratio that E has t0 F (V. Def. 5.) Q. E. D. H If any number of magnitudes be proportional, either of the antecedents will be to its consequent, as the sum of all the antecedents is to the sum of all the consequents. Let A be to e as c is to D, and as E is to F; then will A be to B, as A, c and E together, are to B, D and Fitogether. For, of A, c and E take any equimultiples whatever G, H and K; and of B, D and F any equimultiples whatever i, M and N: Then, since a is to e, as c is to.D (by Hyp.), and G, are equimultiples of A, C, and L, M of B, D, if G be greater than 1, h will be greater than M, and if equal, equal ; and if less, less (V. Def. 5.) And because a is also to B as E is to F (by Hyp.), and C, K are equimultiples of A, E, and 1, N of B, F, if G be greater than L, K will be greater than N; and if equal, equal; and if lefs, less (V. Def. 5.) From hence it follows, that if G be greater than 1, G, H and K together, will be greater than 1,- M and n together; and if equal, equal ; and if less, lefs. But G, and G, H, K together, are any equimultiples whatever of A, and A, C, E together (by Const.); and L, and L, M, N together, are any equimultiples whatever of B, and B, D, F together ; whence, as A is to B, fo is A, C and e together, to B, D and F together (V. Def. 5:) Q. E. D. PRO P. XIII. THEOREM. Equimultiples of any two magnitudes have the same ratio as the magnitudes themfelves, Let cd be the same multiple of A that er is of B; then will cd have the same ratio to ef that A has to B. For, since cd is the fame multiple of A that er is of B3 there are as many magnitudes in CD equal to A, as there are in EF equal to B. Let cd be divided into the magnitudes CG, GH, HD each equal to A (I. 25.); and EF into the magnitudes EK, KL, LF, each equal to B. Then, the number of magnitudes cG, GH, Hd in the one, will be equal to the number of magnitudes pk, KL, LF in the other. And, because DH, HG, Gc are all equal to cach other, as are also FL, IK, KE, DH will be to FL xś HG to LK, and as GC to KE (V.9) L 3 Anda And, fince any antecedent is to its consequent, as all the antecedents are to all the consequents (V. 12.), FL will be to DH, as fe is to DC. But di is equal to A (by Const.), and Fl is equal to B; therefore B will be to A, as FE to DC; and, inversely, DC to FE as A to B, & E. D. PROP. xly. THEOREM. If four magnitudes of the same kind be proportional, and the first be greater than the third, the second will also be greater than the fourth; and if equal, equal; and if less, less. Let A be to B as c is to d; then if á be greater than C, B will also be greater than D; and if equal, equal ; and if less, less. First, let a be greater than c; then B will be greater tian D. ior, of A, c take the equimultiples E, G, and of the multole F such, that E may be greater than F, but G not greate (V.4.); and make h the same multiple of p that F is of 2 : Then, because A is to B as c is to d (by Hyp.), and E, G are quimultiples of A, C, and F, H of B, D (by Conf.), e beng greater than F, G will also be greater than 1 (V. D&. 5.) And, fince, by construction, F is not less than G, and G has been proved to be greater than H, F will likewise be greater than H. But F and h are equimultiples of B and D (by Conft.); therefore, fince F is greater than H, B will also be greater than D. Secondly, let A be equal to c; then will B be equal to D. For, a is to B as c is to D (by Hyp.); or, since A is equal to c, A is to B as A is to D; therefore B is equal to D (V. 10.) Thirdly, let A be less than c; then will B be less than D. For, c is to D as a is to B, by the proposition; therefore c being greater than A, D will also be greater than B, by the first case. Q. E. D. PRO P. XV. THEOREM. If four magnitudes of the same kind be proportional, they will be proportional also when taken alternately. Let A be to B as c is to D; then, also, alternately, A will be to c as B is to D. For, of A and B take any equimultiples whatever e and and of ç and D any equimultiples whatever F and h: L 4 Then, |