b Take of A and of D; that is, if E be greater than G, C any equirul F is greater than H: In like manner, tiples whatever if E be equal to G, or less, F is equal E and F; and of to H, or less than it. But E, F are B & D any equis equimultiples, any whatever, of A, C, multiples what and G, H, any equimultiples whatever G and H: ever of B, D. Therefore A is to B, Then, because A B F H as C is to D. (5. Def. 5.) is the same multiple of B that C Next, let the is of D; and that first A be the E is the same same part of the multiple of A, second B, that that F is of C; the third C is of E is the same the fourth D: A multiple of B, is to B, as C is, that is of D'; (3. 5.) therefore E to D: For B is the same multiple of and F are the same multiples of B A, that D is of C: wherefore, by the and D: But G and H are equimul- preceeding case, B is to A, as D is to tiples of B and D; therefore, if E be C; and inversely (B. 5.) A is to B as a greater multiple of B than G is, F C is to D. Therefore, if the first be is a greater multiple of D than H is the same multiple, &c. Q. E. D. PROP. D. THEOR. If the first be to the second as the third to the fourth, and if the first be a multiple, or part of the second; the third is the same multiple, or the same part of the fourth, Let A be to B, as C is to D; and But A is equal to E, therefore C is first let A be a multiple of B; C is equal to F: (A. 5.) And F is the the same multiple of D. same multiple of D, that A is of B. Take E equal to A, and whatever Wherefore C is the same multiple of multiple A or E D, that A is of B. is of B, make F Next, let the first A be a part of the same mul the second B; C the third is the sarne tiple of D: Then, part of the fourth D. because A is to Because A is to B, as C is to D; B, as C is to D; D and of B the se then, inversely, B is (B. 5.) to A, as D to C: But A is a part of B, therecond, and D the E fore B is a multiple of A; and, by the fourth equimul preceding case, D is the same multiples have been tiple of C; that is, C is the same part taken E and F; of D, that A is of B: Therefore, if the A is to E, as c first, &c. Q. E. D. to F: (Cor. 4.5.) R PROP. VII. THEOR. Equal magnitudes have the same ratio to the same magnitude ; and the same has the same ratio lo equal magnitudes. Let A and B be equal magnitudes, and if equal, equal; if less, less : And and C any other. A and B have each D, E are any equimultiples of A, B, of them the same ratio to C, and C and F is any multiple of C. Therehas the same ratio to each of the mag. fore, (5. Def. 5.) as A is to C, so is B nitudes A and B. to C. Take of A and B any equimultiples Likewise C has the same ratio to whatever D and A, that it has to B: For, having made E, and of C any the same construction, D may in like multiple whai manuer be shewn equal to E: Thereever F: Then, fore, if F be greater than D, it is likebecause D is the wise greater than E; and if equal, same multiple of DÅ equal; if Jess, lesg : And F is any A, that Eis of multiple whatever of C, and D, É B, and that A is E B equal to B; D is с Е are any equimultiples whatever of A, B. Therefore C is to A, as C (i. Ax. 5.) equal is to B. (5. Def. 5.) Therefore equal to E: Therefore, magnitudes, &c. Q. E. D. if D be greater than F, E is greater than F; PROP. VIII. THEOR. Of unequal magnitudes, the greater has the greater ratio to the same than the less has ; and the same magnitude has a greater ratio to the less, than it has to the greater. Let AB, BC be unequal magni- in Fig. 2. and 3.) this magnitude can tudes, of which AB is the greater, be multiplied, so as to become greater and let D be any magnitude what than D, whether it be AC, or CB. ever: AB has a Fig. 1. Let it be multiplied, until it become greater ratio to D greater than D, and let the other be than BC to D: E multiplied as often; and let EF be And D has a the multiple thus taken of AC, and greater ratio to F FG the same multiple of CB : ThereBC than unto AB. fore EF and FG are each of them If the magni greater than D: And in every one of tude which is not GB the cases, take H the double of 'D, K the greater of the L KH its triple, and so on, till the multiple two AC, CB be of d be that which first becomes not less than D, greater than FG: Let L be that multake EF, FG, the tiple of D which is first greater than toubles of AC, PG, and K the inultiple of D which is CB, as in Fig. 1. next less than L. But if that which is not the greater Then, because L is the multiple of D, of the two AC, CB be less than D (as which is the first that becoines greater than FG, the next preceding multiple Also'D has to BC agreater ratio than K is not greater than FG; that is, FG it has to AB: For, having made the is not less than K: And since EF is the same construction, it may be shewn, same multiple of AC, that FG is of in like manner, that L is greater than ÇB; FG is the same multiple of CB FG, but that that EG is of AB; (1. 5.) wherefore it is not great E EG and FG are equimultiples of AB er than EG; and CB: And it was shewn, that FG and I is a FI А was not less than K, and, by the con- multiple of struction, EF is greater than D; D); and FG, Fc therefore the whole EG is greater EG are GB than K and D together: But K, to- quimultiples gether with D, is equal to L; there of CB, AB; | LKD fore EG is greater than L; but FG therefore D is not greater than L; and EG, FG has to CB a are eqnimultiples of AB, BC, and L greater is a multiple of D; therefore (7. Def. tio (7. Def. 5.) AB has to Da ter ratio than BC 5.) than it has to AB. Wherefore, has to D. of unequal magnitudes, &c. Q. E. D. e ra PROP. IX. THEOR. A Magnitudes which have the same ratio to the same magnitude are equal to one another; and those to which the same magnitude has the same ratio are equal to one another. Let A, B have each of them the that D is greater than F; E shall alsame ratio to C: A is equal to B: so be greater than F; (5. Def. 5.) For, if they are but E is not greater than F, which is not equal, one of impossible; À therefore and B are not them is greater unequal; that is, they are equal. than the other ; D Next, let C have the same ratio to let Abe the each of the magnitudes A and B : A greater ; then, F hy what was o is equal to B : For, if they are not, one of them is greater than the other ; shown in the let A be the greater; therefore, as was preceding propoBO shown in Prop. 8th, there is some sition, there are multiple F of C, and soine equimulsome equimul tiples E and D, of B and A such, that tiples of A and B, and some multiple F is greater than E, and not greater of C such, that the multiple of A is than D; but because C is to B, as C greater than the multiple of C, but is to A, and that F, the multiple of the multiple of B is not greater than the first, is greater than E, the mul. that of C. Let such multiples be tiple of the second; F, the multiple of taken, and let D, E, be the equimul- the third, is greater than D, the multiples of A, B, and F the multiple of tiple of the fourth : (5. Def. 5.) But C, so that D may be greater than F, Fis not greater than D, which is imand E not greater than F: But, be- possible. Therefore, A is equal to B. cause A is to C, as B is to C, and of Wherefore, magnitudes which, &c. A, B are taken equimultiples D, E, Q. E. D. end of C is taken a multiple F; and PROP. X. THEOR. That magnitude which has a greater ratio than another has unto the same magnitude, is the greater of the two : And that magnitude to which the same has a greater ratio Ihan it has unto another magnitude, is the lesser of the two. Let A have to C a greater ratio greater'than F: Therefore D is greatthan B has to C; A is greater than er than E: And, because D and E B: For, because A has a greater rac are equimultiples of A and B, and D tio to C, than B has to C, there are is greater than E; therefore A is (4. (7. Def. 5.) some Ax. 5.) greater than B. equimultiples of Next, let C have a greater ratio to A and B, and B than it has to A; B is less than A: some multiple of A! For (7. Def. 5.) there is some mulC such, that the tiple F of C, and some equimultiples multiple of A is cl E and D of B and A such, that F is greater than the old greater than E, but is not greater multiple ofC, but than D: E therefore is less than D; the multiple of B and because E and D are equimulis not greater than it : Let them be tiples of B and A, therefore B is (4. taken, and let D, E be equimultiples Ax. 5.) less than A. That magnitude, of A, B, and Fa multiple of C such, therefore, &c. Q. E. D. that D is greater than F, but E is not PROB. XI. THEOR. Ralios that are the same to the same ratio, are the same to one another. Let A be to B as C is to D; and as and if equal, equal ; and if less, less. C to D, so let E be to F; A is to B, (5. Def. 5.) Again, because C is to as E to F. D, as E is to F, and H, K are taken Take of A, C, E, any equimultiples equimultiples of C, E; and M, N, of whatever G, H, K; and of B, D, F, D, F; if H be greater than M, K is any equimultiples whatever L, M, N. greater than N; and if equal, equal; Therefore, since A is to B, as C to and if less, less: But if Ġ be greater D, and G, H are taken equimultiples than L, it has been shewn that H is greater than M: and if equal, equal; and if less, less; therefore, itG be greatG H K er than L, K is greater than N; and A E if equal, equal; and if less, less : And B D T G, K are any equimultiples whatever of A, E ; and L, N any whatever of L. MN B, F: Therefore, as A is to B, so is E to F. (5. Def. 5.) Wherefore, raof A, C, and L, M of B, D; if G be tios that, &c. Q. E. D. greater than L, H is greater than M; PROP. XII. THEOR. Y any number of magnitudes be proporlionals, as one of the antecedents is to its consequent, so shall all the antecedents taken together be to all the consequents. Let any number of magnitudes A, G be greater than L, H is greater B, C, D, E, F, be proportionals: that than M, and K greater than Ñ ; and is, as A is to B, so C to D, and E to if equal, equal ; and if less, less. (5. F: As A is to B, so shall A, C, E to Def. 5.) Wherefore, if G be greater gether be to B, D, F together. than L, then G, H, K together are Take of A, C, E any equimultiples greater than L, M, N together: and whatever G, H, K; and of B, D, F if equal, equal; and if less, less. And any equimultiples whatever L, M, N: G, and G, H, K together are any equimultiples of A, and A, C, E toge ther; because, if there be any numG H K ber of magnitudes equimultiples of as А с E many each of each, whatever mul tiple one of them is of its part, the B D. F same multiple is the whole of the whole : (1. 5.) For the same reason L M N L, and I, M, N are any equimul tiples of B, and B, D, F: As thereThen, because A is to B, as Cis to D, ther to B, D, F together. Wherefore, fore A is tó B, so are A, C, E togeind as E to F; and that G, H, I if any number, &c. Q. E. D. ire equimultiples of A, C, E, and L, I, N equiinultiples of B, D, F; if PROP. XIII. THEOR. If the first has to the second the same ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has 10 the sixth ; the first shall also have to the second a greater ratio than the fifth has to the sixth. Let A the first have the same ra. be equimultiples, and K, L equimultio to B the second, which C the third tiples of D, F, so that G be greater has to D the fourth, but C the third, to D the fourth, a greater ratio than E the fifth to F the sixth : Also the H first A shall have to the second B a C- E greater ratio than the fifth E to the sixth F. B D F. Because C has a greater ratio to D, than E to F, there are some equimul. tiples of C and E, and some of D and F such, that the multiple of C is than K, but H not greater than L; greater than the multiple of D, but and whatever multiple G is of C, take the multiple of E is not greater than M the sanie multiple of A ; and what. the multiple of F: (5. Def. 5.) Let ever multiple K is of D, take N the such bs taken, and of C, E let G, H same multiple of B: Then, because K L |