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.1 Hyp. 2 1 Let A B the 1st = mC the 2nd, & D E the 3rd m F the 4th; & BG the 5th = n C the 2nd, and EH the 6th =n F the 4th ; then AGAB + BC=(m + n) C; & DH = DE + EH AB m C, and DE =mF; 2 Def. 2, V.obs... magns. in AB (each (each F); So magns. in BG (each (each = F); (m + n) F. C)= magns. in EH A G same mult. of C that D H is of F; i. e. AG (the 1st + 5th) same mult. of C the 2nd, that DH (the 3rd + 6th) is of F the 4th. If therefore the first be the same, &c., Q. E.D. COR.-Hence, if any no. of magns. AB, BG, GH, be mults. of C, and as many, DE, EK, KL, the same ms of F, each of each; then, A H i.e. (AB+ BG + GH) the same m of C that DL, i. e. (DE+ EK + KL) is of F. E F Alg. & Arith. Hyp.-Take 6 quantities A 24, B8, C21, D = 7, = 16, F = 14, m= 3, and n = 2. = Alg. Arith. E the 5th: =n B, and F the 6th nD; +equals, then A+ E = (m + n) B and C+ F= (m+ n) D. .. Def. 2, V. A+E is the same mult. of B that C + F is of D. 24 = 3 x 8, and 21 equals, 24+ 16 =(3+2) 8, and 21 + 14 (3 + 2) 7. But 40 contains 8, five times, and 35 contains 7, five times. .. Def. 2, V. 40 the same mult. of 8 that 35 is of 7. SCHOL.-Allied to this Proposition is the Theorem": If the first of three magnitudes contain the second as often as there are units in a certain number;— and if the second contain the third also as often as there are units in a certain number, the first will contain the third as often as there are units in the product of these two numbers." 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. CON. Pst. 2, V.--DEM. Def. 2, V.-Cor. 2, V. E. 1 Hyp. 1. Let A the 1st = m B the 3 Conc. 2. D. 1 Hyp. 2. 2 Def. 2, V. 2nd; and C the 3rd = Also, let E F = n A, and then EF the same m of B as G H is of D. ... EF same m of A as G H .. as many magns. in E F, C. 1 Pst. 2, V. Divide EF into EK, KF, each= A, and 3 Conc. D. 3 Hyp. 1. 4 C. 1. 5 Def. 2, V. 6 Sim. 1. 7 2. GH into GL, LH, 30. 15. 5. 24. 12. 4. each, C; the no. of Ms in EK, KF no. of Ms in GL, L H. A same m of B, that C is of D; and EK = A, and GL .. EK same m of B, that GL is of D; and .. KF same m of B, that LH is of D. and so, if more Ms in EF & GH each = A, C. 8 Cor, 2, V. Hence the 1st EK same m of the 2nd B, as 9 102, V. 11 Conc. 3rd GL, of 4th D; and the 5th KF same m of the 2nd B, as 6th LH of 4th D; .. EF (1st + 5th) same m of the 2nd B, as GH (3rd+6th) of 4th D. If, therefore the first be the same, &c. Q. E. D. COR. If A, A' be equimults. of B, B' and also of C, C'; and if B be a m of C, the other B' shall be the same m of C'. 5; m3 & n = 2. Alg. & Arith. Hyp.-Take a — 12, b — 4, c — Alg.-Let a i. e., the equims. na & ne of the 1st and 3rd, are mults, of the 2nd and 4th. Arith. * 12 = 3 × 4, and 15 = 3 × 5, .°. 2 × 12 = 6 × 4, & 2 × 15 = = 6 X 5; i. e. the equims. 24 & 30, of the 1st 12, and 3rd 15, are equims. of the 2nd 4, and of the 4th 5. SCH.-"If any equimultiples m A, m C, be taken of the antecedents of an analogy, A: B:: C:D, and any equimultiples, n B, n D, of the consequents, these multiples, taken in the order of the terms, are proportional,” i. e. mA n B:: mCnD. Of m A, m C take equims. p times, and of n B, n D, equims. q times; thenm A, m C, contain A and C, p m units of times; .*. equims. m A, m C by p are equims. of A and C, and equal pm A, pm C. So n BX q and n D x q = qn B and qn D. : Since A B C : D, and equims. of A and C are pm A, pm C, and equims. of B, D are q n B, q n D, .. if pm A > q n B, p m C > qn D; if, =, and if and <,<. But pm A, pm C are also equims. of m A and m C, If the first of four magnituties has the same ratio to the second which the third has to the fourth; then any equal multiples whatever of the first and third shall have the same ratio to any equimultiples of the second and fourth; viz., "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. Of E and F take any equims. K, L, and of G and H and K, L are equims. of A and C; .. K, >, = or < M, so L >, = or < N. and M, N equims. of G and H; COR. 1.-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 second and fourth ; and in like manner, the first and third shall have the same ratio to any equimultiples whatever of the second and fourth." Or, "If 4 Ms be proportional, then 1o, any equims. being taken of the 1st and 3rd, the m of the 1st 2ndm of 3rd: 4th; and II, any equims. being taken of 2nd and 4th, the 1st: m of 2nd 3rd m of the 4th." : E. 1 Hyp. 1. Let A B = : C: D and let E, F be any equims. of A & C; take of E, F any equims. K, L, and of |