-ROPOSITION I. THEOREM.-If any number of magnitudes be equimultiples of as many others, each of each; what multiple soever any one of the first is of its part, the same multiple shall all the first magnitudes taken together be of all the others taken together. Let any number of magnitudes AB, CD, be equimultiples of as many others E, F, each of each; whatsoever multiple AB is of E, the same multiple shall AB and CD together be of E and F together. DEMONSTRATION. Divide AB into magnitudes equal to E, viz. AG, GB; and CD into CH, HD, equal each of them to F; the number, therefore, of the magnitudes CH, HD is equal to the number of the others AG, GB (a). And because AG is equal to E, and CH to F, therefore AG and CH together are equal to E and F together (6). For the same reason, GB and HD together are equal to E and F together; wherefore, as many magnitudes as 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 are AB and CD together, of E and F together; and the same demonstration would hold if the number of magnitudes were greater than two. Therefore, if any number of magnitudes, &c. A G B H D E F (a) Hypoth. (b) I. Ax. 2. SCHOLIUM. In order to the elucidation of Euclid's demonstrations we shall append to each proposition an algebraical investigation and proof, preserving his train of reasoning unaltered. THEOREM. If A, B, C, &c., be equimultiples of a, b, c, fc., then whatsoever multiple A is of a, the same multiple is A+B+C+ &c., of a+b+c+ &c. Let A contain n parts each equal to a, then A = na and because B, C, &c., are the same multiples of b, c, &c., that A is of a, therefore therefore A+B+C+ &c. = n a + n b + nc+ &c. = = a + a + a + &c. to n terms +6 +6 + b + &c. to n terms (a+b+c+ &c.) + (a + b + c + &c.) to n terms = n (a + b + c + &c.) PROPOSITION II. THEOREM.-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. Let AB the first be the same multiple of C the second, that DE the third is of F the fourth, and 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 with the sixth, is of F the fourth. D E DEMONSTRATION. Because AB is the same multiple of C that DE is of F, there are as many magnitudes in AB equal to C as there are in DE equal to F; in like manner, as A many as there are in BG equal to C, so many are there in EH equal to F; therefore, as many 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, that is, AG, the first and fifth together, is the same multiple of the second C, that DH, the third and sixth together, is of the fourth E. Therefore, if the first magnitude, &c. B G d H COROLLARY. From this it is evident 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; the whole of the first, viz. AH, is the same multiple of C, that the whole of the last, viz, DL, is of F. SCHOLIA. 1. This proposition, algebraically expressed, as follows: THEOREM. If A, α, B, b, A,,, B be six magnitudes such that A, B are equimultiples of a and b, and A, B1, are also equimultiples of a and b, then A+A, B+ B, shall be equimultiples of a and b. A+ A1 = m. a + n. a = (m + n). a, 1 (m + n). b; d F that is, A+ A, and B + B, are equimultiples of a and b. A + A1 + A2 + A3 + &c. = m. a + n. a + p.a+q.a + &c. = (m + n + p + q + &c.) a. and B+ B+ B2 + B ̧ + &c. = m. b + n. b + p. b + q.b+ &c. 2 = (m + n + p + q + &c.) b. 1 Therefore, A+ A1 + A2 + &c., B+ B1 + B2 + &c. are equimultiples of a and b. PROPOSITION III. THEOREM.-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 is the same multiple of B that GH is of D. F E H L A B G (a) V. 2. DEMONSTRATION. 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; let EF be divided into Kthe magnitudes EK, KF, each equal to A, and GH into GL, LÍ, each equal to C, therefore the number of the magnitudes EK, KF, shall be equal to the number of the others 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 equal to C, 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, and the same holds if there be more parts in EF, GH, equal to A, C; therefore, because the first EK is the same multiple of the second B which the third GL is of the fourth D, and that the fifth KF is the same multiple of the second B which the sixth LII is of the fourth D; EF, the first together with the fifth, is the same multiple of the second B which GH, the third together with the sixth, is of the fourth D (a). Therefore, if the first be the same multiple, &c. SCHOLIUM. The foregoing proposition, algebraically expressed, is as follows: THEOREM. If of four magnitudes the first A is the same multiple of the second a, which the third B is of the fourth b, and if of A and B equimultiples be taken, these shall also be equimultiples of a and b. and if the equimultiples of A and B be taken such that they shall contain A and B, n times, they shall be respectively Now because A and B contain a and b, m times, therefore n. A and ». B contain a and b, m n times, and n. A = n. m.a and n. Bn.m. b that is, n. A, n. B are equimultiples of a and b. PROPOSITION IV. THEOREM.-If the first of four magnitudes 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 any equimultiples of the second and fourth, i. e. "the equimultiple of the first shall have the same ratio to that of the second which the equimultiple of the third has to that of the fourth." Let A the first have to B the second the same 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 shall E have the same ratio to G that has to H. DEMONSTRATION. Take of E and F any equimultiples whatever K, L ; and of G, H, any equimultiples whatever M, N; then because E is the same multiple of A that F is of C; and of E and F the equimultiples K, L, have been taken; therefore K is the same multiple of A that L is of C (a); for the same reason, M is the same multiple of B that N is of D. And because, as A is to B, so is C to D (b), and of A and C have been taken certain equimultiples K, L, and of B and D have been taken certain equimultiples M, N; therefore if K be greater than M, L is greater than N; and if equal, equal; if less, less (c); but K, L are any equimultiples whatever of E, F, and M, N any whatever of G, H; therefore as E is to G so is F to Í (c). COROLLARY. Likewise, if the first has the same ratio to the second which the third has to the fourth, then also any equimultiples whatever of the first and third shall have the same ratio to the |