Book V. See N. a. 3.5. PROP. IV, THEOR. F the first of four magnitudes has the fame ratio to the fecond which the third has to the fourth; then any equimultiples whatever of the firft and third fhall have the same ratio to any equimultiples of the fecond and fourth, viz. the equimultiple of the firft fhall have the fame ratio to that of the fecond, which the equimultiple of the third has to that of the fourth.' Let A the first have to B the second, the fame ratio which the third C has to the fourth D; and of A and C let there be taken any equimultiples whatever E, F; and of B and D any equimultiples whatever G, H. then E has the Take of E and F any equimul- that N is of D. and because as AKE A BGM 6. Hypoth. is to B, fo is C to D, and of AL F CDHN and C have been taken certain equimultiples K, L; and of B and D have been taken certain equimultiples M, N; if therefore K be greater than M, L is greater than N; c. 5. Def. 5. and if equal, equal; if less, lefs o. And K, L are any equimultiples whatever of E, F, and M, N any whatever of G, H. as therefore E is to G, fo is F to H. Therefore if the firft, &c. Q. E. D. COR. Likewife if the first has the fame ratio to the fecond, which the third has to the fourth, then alfo any equimultiples whatever of the first and third have the fame ratio to the fecond and fourth. Book V. and in like manner the first and the third have the fame ratio to any equimultiples whatever of the fecond and fourth. Let A the first have to B the second, the fame ratio which the third C has to the fourth D, and of A and C let E and F be any equimultiples whatever; then E is to B, as F to I). Take of E, F any equimultiples whatever K, L, and of B, D any equimultiples whatever G, H; then it may be demonftrated, as before, that K is the fame multiple of A, that L is of C. and because A is to B, as C is to D, and of A and C certain equimultiples have been taken, viz. K and L; and of B and D certain equimultiples G, H; therefore if K be greater than G, L is greater than H; and if equal, equal; if lefs, less . and K, L are any equimultiples of c. 5. Def. 5. E, F, and G, H any whatever of B, D; as therefore E is to B, fo is F to D. and in the fame way the other cafe is demonstrated. PROP. V. THEOR. If one de taken from the mrft is of a mag F one magnitude be the fame multiple of another, See N. nitude taken from the other; the remainder fhall be the fame multiple of the remainder, that the whole is of the whole. G Let the magnitude AB be the fame multiple of CD, that AE taken from the first, is of CF taken from the other; the remainder EB fhall be the fame multiple of the remainder FD, that the whole AB is of the whole CD. A E C 3. I. 5. F b. I. Axe 5. Take AG the fame multiple of FD, that AE is of CF. therefore AE is a the fame multiple of CF, that EGis of CD. but AE, by the hypothefis, is the fame multiple of CF, that AB is of CD. therefore EG is the fame multiple of CD that AB is of CD; wherefore EG is equal to AB 6. take from them the common magnitude AE; the remainder AG is equal to the remainder EB. Wherefore fince AE is the fame multiple of CF, that AG is of FD, and that AG is equal to EB; therefore AE is the fame multiple of CF, that EB is of FD. but AE is the fame B D Book V. multiple of CF, that AB is of CD; therefore EB is the fame multiple of FD, that AB is of CD. Therefore if one magnitude, &c. Q. E. D. See N, PROP. IV. THEOR. F two magnitudes be equimultiples of two others, and if equimultiples of thefe be taken from the first two, the remainders are either equal to these others, or equimultiples of them. Let the two magnitudes AB, CD be equimultiples of the two E, F, and AG, CH taken from the first two be equimultiples of the fame E, F; the remainders GB, HD are either equal to E, F, or equimultiples of them. K A First, Let GB be equal to E; HD is equal to F. make CK equal to F; and because AG is the fame multiple of E, that CH is of F, and that GB is equal to E, and CK to F; therefore AB is the fame multiple of E, that KH is of F. But AB, by the hypothefis, is the fame multiple of E that CD is of F; therefore KH is the fame multiple of F, that CD is of F; wherefore G 3. 4. Ax. 5. KH is equal to CD 2. take away the common magnitude CH, then the remainder KC is equal to the remainder HD. but KC is equal b. 2.5. to F, HD therefore is equal to F. H B DEF But let GB be a multiple E; then HD is the fame multiple of F, K A Make CK the fame multiple of F, that GB H B DEF PROP. A. THEOR. Book V. F the first of four magnitudes has to the second, the See N. fame ratio which the third has to the fourth; then if the first be greater than the second, the third is also greater than the fourth; and if equal, equal; if lefs, lefs. Take any equimultiples of each of them, as the doubles of each. then by Def. 5th of this Book, if the double of the first be greater than the double of the fecond, the double of the third is greater than the double of the fourth. but if the first be greater than the fecond, the double of the first is greater than the double of the fecond, wherefore also the double of the third is greater than the double of the fourth; therefore the third is greater than the fourth. in like manner, if the firft be equal to the fecond, or lefs than it, the third can be proved to be equal to the fourth, or less than it Therefore if the firft, &c. Q. E. D. IF PROP. B. THEOR. four magnitudes are proportionals, they are pro. See N. portionals also when taken inversely. If the magnitude A be to B, as C is to D, then alfo inversely B is to A, as D to C. Take of B and D any equimultiples what ever E and F; and of A and C any equimultiples whatever G and H. First, Let E be greater than G, then G is lefs than E; and because A is to B, as C is to D, and of A and C the firft and third, G and H are equimultiples; and of B and D the fecond and fourth, E and F are equimultiples; and that G is less than E, H is also a lefs than F; that is, F is greater than H. if therefore E be greater than G, F is greater than H. in like manner, if E be equal to G, F may be fhewn to be equal to H; and if lefs, lefs. and E, F are any equimultiples whatever of B and D, and G, H GABE HCD F ויין a. 5.Def. 5. ་ Book V. any whatever of A and C. Therefore as B is to A, fo is D to C. If then four magnitudes, &c. Q. E. D. See N. a. 3. 5. IF PROP. C. THEOR. [F the first be the fame multiple of the fecond, or the fame part of it, that the third is of the fourth; the first is to the second, as the third is to the fourth. Let the firft A be the fame multiple of B the fecond, that C the third is of the fourth D. A is to B, as C is to D. Take of A and C any equimultiples whatever E and F; and of B and D any equimultiples whatever G and H. then because A is the fame multiple of B that C is of D; and that E is the fame multiple of A, that Fis of C; E is the fame multiple of B, that F is of D2; therefore E and F are the fame multiples of B and D. but G and H are equimultiples of B and D; therefore if E be a greater multiple of B, than G is; F is a greater multiple of D, than H is of D; that is, if E be greater than G, F is greater than H. in like manner, if E be equal to G, or lefs; F is equal to H, or lefs than it. But E, F are equimultiples, any whatever, of A, C, and G, H any equimultiples whatever of b. 5.Def. 5. B, D. Therefore A is to B, as C is to D. Next, Let the firft A be the fame c. B. 5. part of the fecond B, that the third C A B C D EGF H A B C D |