PROP. II. THEOR. If the first magnitude be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth, then shall the first together with the fifth be the same multiple of the second, that the third together with the sixth is of the fourth. D Let AB the first be the same multiple of C the second, that DE the third is of F the fourth, and let BG the fifth the same multiple of C the second, that EH the sixth is of F the fourth then shall AG, the first together with the fifth, be the same multiple of C the second, that DH, the third together Bwith the sixth, is of F the fourth. For because AB is the same multiple of G HF C that DE is of F, there are as many magnitudes in AB equal to C, as there are in DE equal to F; and, in like manner, there are as many magnitudes in BG equal to C, as there are in EH equal to F: Therefore, there are as many magnitudes in the whole AG equal to C, as there are in the whole DH equal to F, that is, AG, the first and fifth together, is the same multiple of C, the second, that DH, the third and sixth together, is of F, the fourth. E Wherefore, If the first be the same multiple &c. Q.E.D. COR. In like manner it is plain, that, if any number of magnitudes be multiples of another, C, and as many magnitudes be the same multiples of another, F, each of each, then the whole of the first magnitudes is the same multiple of C, that the whole of the last magnitudes is of F. PROP. III. THEOR. If the first be the same multiple of the second which the third is of the fourth, and if of the first and third there be taken equimultiples, these shall be equimultiples, the one of the second, and the other of the fourth. Let A the first be the same multiple of B the second, that C the third is of D the fourth, and of A and C let the equimultiples EF and GH be taken; then EF shall be the same multiple of B that GH is of D. For because EF is the same multiple of A that GH is of C, there are as many magnitudes in EF equal to A, as there are in GH equal to C: Divide EF into magnitudes EK, KF, each equal to A, and GH into GL, LH, each equal to C: Then the number of the magnitudes EK, KF, will be the same as the number of the magnitudes GL, LH: And because A is the same multiple of B that C is of D, and that EK is equal to A, and GL to C, therefore EK is the same multiple of B that GL is of D: For the like reason, KF is the same multiple of B that LH is of D : Therefore because EK, the first, is the same multiple of B, the second, which GL, the third, is of D, the fourth, and FK, the fifth, is the same multiple of B, the second, which LH, the sixth, is of D, the fourth, therefore EF, the first together with the fifth, is the same multiple of B, the second, which GH, the third together with the sixth, is of D, the fourth (5.2): And, in like manner, if there be more parts in EF and GH equal to A and C, it may be shewn by the help of (5. 2. Cor.) that EF is the same multiple of B which GH s of D. Wherefore, If the first &c. Q. E. D. FI K H EABG C D PROP. IV. THEOR. If the first of four magnitudes has the same ratio to the second which the t'ird has to the fourth, and if of the first and third there be taken any equimultiples whatever, and also any whatever of the second and fourth, then the multiple of the first shall have the same ratio to that of the second which the multiple of the third has to that of the fourth. Let A, the first, have to B, the second, the same ratio which C, the third, has to D, the fourth, and of A and C let there be taken any equimultiples whatever E and F, and of B and D, any equimultiples whatever G and H: then E shall have the same ratio to G which F has to H. Take of E and F any equimultiples whatever K and L, and of G and H, any equimultiples whatever M and N : Then, because E is the same multiple of A that F is of C, and of E and F KEABG M 11 have been taken equimultiples K and L, therefore K is the same multiple of A that L is of C (5. 3): For the like reason, M is the same multiple of B that N is of D: And because that A is to B as C is to D, and of A and C have been taken certain equimultiples K and L, and of B and D certain equimultiples M and N, therefore (5. Def. 5) if K be greater than M, L is greater than N, and if equal, equal, and if less, less: But K and L are any equimultiples whatever of E and F, and M and N are any equimultiples whatever of G and H; therefore (5. Def. 5) E is to G as F to H. Wherefore, If the first &c. Q. E.D. COR. So also, if the first has the same ratio to the second which the third has to the fourth, then any equimultiples whatever of the first and third shall have the same ratio to the second and fourth, and, in like manner, the first and the third shall have the same ratio to any equimultiples whatever of the second and fourth. Let A the first have to B the second, the same ratio which C, the third, has to D, the fourth, and of A and C, let there be taken any equimultiples whatever, E and F: then E shall be to B as F is to D. Take of E, F, any equimultiples whatever K, L, and of B, D, any equimultiples whatever G, H: Then it may be shewn, as before, that K is the same multiple of A that L is of C: And because A is to B as C is to D, and of A and C have been taken certain equimultiples K and L, and of B and D, certain equimultiples G and H, therefore, if K be greater than G, L is greater than H, and if equal, equal, and if less, less: But K and L are any equimultiples whatever of E and F, and G and H are any equimultiples whatever of B and D; therefore E is to B as F is to D. And in the same way the other case may be demonstrated. PROP. V. THEOR. If one magnitude be the same multiple of another, which a magnitude taken from the first is of a magnitude taken from the other, the remainder shall be the same multiple of the remainder that the whole is of the whole. Let AB be the same multiple of CD, that AE, taken from the first, is of CF, taken from the other: the remainder EB shall be the same multiple of the remainder FD that the whole AB is of the whole CD. GI B Take AG the same multiple of FD that AE is of CF; therefore AE is the same multiple of CF that EG is of CD (5.1): But AE is the same multiple of CF that AB is of CD (Hyp.); therefore EG is the same multiple of CD that AB is of CD, and therefore EG is equal to AB (5. Ax. 1): From each of these take AE, which is common to both; then the remainder AG is equal to the remainder EB: Therefore, because AE is the same multiple of CF that AG is of FD, and that AG is equal to EB, therefore AE is the same multiple of CF that EB is of FD: But AE is the same multiple of CF that AB is of CD; therefore EB is the same multiple of FD that AB is of CD. Wherefore, If one magnitude &c. Q.E.D. A A E K F PROP. VI. THEOR. If two magnitudes be equimultiples of two others, and if equimultiples of these be taken from the first two, the remainders shall be either equal to these others, or equimultiples of them. C Let the two magnitudes, AB, CD, be equimultiples of the two, E, F, and let AG, CH taken from the first two be equimultiples of the same E, F; the remainders GB, HD shall be either equal to E, F, or equimultiples of them. D First, let GB be equal to E; HD shall be equal to F. Make CK equal to F: Then, because AG is the same multiple of E that CH is of F, and that GB is equal to E and CK to F, therefore AB is the same multiple c of E that KH is of F: But AB is the same multiple of E that CD is of F (Hyp.); therefore KH is the same multiple of F that CD is of BDEF |