PROPOSITION 1. THEOREM. If any number of magnitudes be equimultiples of as many, each of each; whatever multiple any one of them is of its part, the same multiple shall all the first magnitudes be of all the other. Let any number of magnitudes AB, CD be equimultiples of as many others E, F, each of each: whatever multiple AB is of E, the same multiple shall AB and CD together, be of E and F together. For, because AB is the same multiple of E, that CD is of F, as many magnitudes as there are in AB equal to E, so many are there in CD equal to F. Divide AB into the magnitudes AG, GB, A, G E B H F And, because AG is equal to E, and CH equal to F, therefore AG and CH together are equal to E and F together; and because GB is equal to E, and HD equal to F, therefore GB and HD together are equal to E and F together. [Axiom 2. Therefore as many magnitudes as there are in AB equal to E, so many are there in AB and CD together equal to E and F together. Therefore whatever multiple AB is of E, the same multiple is AB and CD together, of E and F together. Wherefore, if any number of magnitudes &c. Q.E.D. PROPOSITION 2. THEOREM. If the first 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; the first together with the fifth shall be the same multiple of the second, that the third together with the sixth is of the fourth. 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 be the same multiple of C the second, that EH the sixth is of F the fourth: AG, the first together with the fifth, shall be the same multiple of C the second, that DH, the third together with the sixth, is of F the fourth. For, because AB is the same multiple of C that DE is of F, as many magnitudes as there are in AB equal to C, so many are there in DE equal to F. For the same reason, as many magnitudes as there are in BĞ equal to C, so many are there in EH equal to F. Therefore as many magnitudes as there are in the whole AG equal to C, so many are there in the whole DH equal to F. Therefore AG is the same multiple of C that DH is of F. Wherefore, if the first be the same multiple &c. Q.E.D. COROLLARY. From this it is plain, that if any number of magnitudes AB, BG, GH be multiples of another C; and as many DE, EK, KL be the same multiples of F, each of each; then the whole of the first, namely, AH, is the same multiple of C, that the whole of the last, namely, DL, is of F. B B D E d H E K G+ L PROPOSITION 3. THEOREM. If the first be the same multiple of the second that the third is of the fourth, and if of the first and the 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: EF shall be the same multiple of B that GH is of D. For, because EF is the same multiple of A that GĦ is of C, [Hypothesis. as many magnitudes as there are in EF equal to A, so many are there in GH equal to C. Divide EF into the magnitudes EK, KF, each equal to Á; and GH into the magnitudes GL, LH, each equal to C Therefore the number of the magnitudes EK, KF, K H will be equal to the number of the magnitudes GL, LH. And because A is the same multiple of B that °C is of D, [Hypothesis. and that EK is equal to A, and GL is equal to C; [Constr. therefore EK is the same multiple of B that GL is of D. For the same 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, that GL the third is of D the fourth, and that KF the fifth is the same multiple of B the second, that LH the sixth is of D the fourth; EF the first together with the fifth, is the same multiple of B the second, that GH the third together with the sixth, is of D the fourth. [V. 2. In the same manner, if there be more parts in EF equal to A and in GH equal to C, it may be shewn that EF is the same multiple of B that GH is of D. [V. 2, Cor. Wherefore, if the first &c. Q.E.D. PROPOSITION 4. THEOREM. If the first have the same ratio to the second that the third has to the fourth, and if there be taken any equi multiples whatever of the first and the third, and also any equimultiples whatever of the second and the fourth, then the multiple of the first shall have the same ratio to the multiple of the second, that the multiple of the third has to the multiple of the fourth. Let A the first have to B the second, the same ratio that 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: E'shall have the same ratio to G that 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 Fis of C, and of E and F have been taken equimultiples K and L; therefore K is the same mul- K tiple of A that Z is of C. [V. 3. L H [V. Definition 5. But K and I are any equimultiples whatever of E and F, and M and N are any equimultiples whatever of G and H; therefore E is to G as Fis to H. [V. Definition 5. Wherefore, if the first &c. Q.E.D. COROLLARY. Also if the first have the same ratio to the second that 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 the first and third shall have the same ratio to any equimultiples whatever of the second and fourth. Let A the first have the same ratio to B the second, that C the third has to D the fourth; and of A and C let there be taken any equimultiples whatever E and F: E shall be to B as F is to D. Take of E and F any equimultiples whatever K and L, and of B and D any equimultiples whatever G and 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, [Hypothesis. and of A and C have been taken certain equimultiples K and L, and of B and D have been taken 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. [V. Definition 5. 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. [V. Definition 5. In the same way the other case may be demonstrated. PROPOSITION 5. THEOREM. If one magnitude be the same multiple of another that 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. 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. [V. 1. But AE is the same multiple of CF that AB is of CD; therefore EG is the same multiple of CD that AB is of CD; therefore EG is equal to AB. [V. Axiom 1. |