Book V. } a 3. 5. IF PROP. B. THEOR. F the first be the fame multiple of the second, or the fame part of it, that the third is of the fourth the firft is to the fecond as the third to the ; fourth. First, if mA, mB be equimultiples of the magnitudes A and B, mA: A:: mB: B. Take of mA and mB equimultiples by any number ; and of A and B equimultiples by any number p; thefe will be nmА a, þA, nmВa, pB. Now, if nmA be greater than pA, nm is alfo greater than p; and if nm is greater than p, nmB is greater than pB; therefore, when nmA is greater than pA, nmB is greater than pB. In the fame manner, if nmA=pA, nmB=pB, and if nmA <pA, nmB <pB. Now, nmA, nmB are any equimultiples of mA and mB; and pA, pB are b def. 5. 5.any equimultiples of A and B, therefore mA: A:: mB: Bb. c A. 5. Next, Let C be the fame part of A that D is of B; then A is the fame multiple of C that B is of D; and therefore, as has been demonftrated, A: C:: B: D, and inversely c C ; A::D: B. Therefore, &c. Q.E. D. PROP. VI. THEOR. F the first be to the fecond as the third to the fourth; and if the firft be a multiple or a part of the fecond, the third is the fame multiple or the fame part of the fourth. Let A: B:: C: D, and firft let A be a multiple of B, C is the fame multiple of D; that is, if A= mB, C≤mD. Take of A and C equimultiples by any number as 2, viz. 2A and 2C; and of B and D, take equimultiples by the number number 2m, viz. 2mB, 2mD a; then, because AmB, 2A Book V. 2mB; and fince A: B::C: D, and fince 2A=2mB, therefore 2C2mDb, and CmD, that is, C contains D m times, b def. 5.5. or as often as A contains B. a 3. 5. Next, Let A be a part of B, C is the fame part of D. For, fince A: B::C:D, inversely c, B; A::D:C. But A CA. 5. . being a part of B, B is a multiple of A, and therefore, as is fhewn above, D is the fame multiple of C, and therefore C is the fame part of D that A is of B. Therefore, &c. Q.E. D. E PROP. VII. THE OR. QUAL magnitudes have the fame ratio to the fame magnitude; and the fame has the fame ratio to equal magnitudes. Let A and B be equal magnitudes, and C any other; A: C::B: C. b def. 5. 5. Let mA, mB be any equimultiples of A and B; and C any multiple of C. Because AB, mAmBa; wherefore, if mA be greater a A. 1. 5. than nC, mB is greater than #Cb; and if mAnC, mB=nC; or, if mA <nC, mB <nC. But mA and mB are any equimultiples of A and B, and C is any multiple of C, thereforeb A: C:: B: C. Again, if AB, C: A:: C: B; for, as has been proved, A:C::B:C, and inversely c, C: A::C: B. Therefore, c A. 5. &c. Q. E. D. OF PROP. VIII. PRO B. Funequal magnitudes, the greater has a great、 er ratio to the fame than the less has; and the fame magnitude has a greater ratio to the lels than it has to the greater. Let Book V. Let A+B be a magnitude greater han A and C a third magnitude, A+B has to C a greater ratio than A has to C; and C has a greater ratio to A than it has to A+B. a 1. 5. Let m be fuch a number that mA and mB are each of them greater than C; and let nC be the leaft multiple of C that exceeds mA+mB; then nC-C, that is, n-1. Ca will be lefs than mA+B, or mA+B, that is m. A+B is greater than 7. C. But because nC is greater than mA+mB, and C lefs than mB, nC-C is greater than A, or "A is less than nC-C, that is, than n-1. C. Therefore the multiple of ', A+B by m exceeds the multiple of C by n-1, but the multiple of A by m does not exceed the multiple of C by n−1, 7. def. 5. therefore A+B has a greater ratio to C than A has to C b. Again, because the multiple of C by n-1, exceeds the multiple of A by m, but does not exceed the multiple of A+B by m, C has a greater ratio to A than it has to A+Bb. Therefore, &c. Q. E. D. MA PROP. IX. THE OR. AGNITUDES which have the fame ratio to the fame magnitude are equal to one another; and thofe to which the fame magnitude has the fame ratio are equal to one another. If A: C: B: C, A= B. For if not, let A be greater than B; then, because A is greater than B, two numbers, m and n, may be found as in the last propofition, fuch that mA fhall exceed nC, while mB does not exceed nC. But because A: C:: B: C; if mA exa 5. def. 5. ceed nC, mB must also exceed nCa; and it is alfo fhewn that mB does not exceed nC, which is impoffible. Therefore A is not greater than B; and in the fame way it is demonstrated that B is not greater than A; therefore A is equal to B. Next, let C: A:: C: B, A =B. For by inverfion b, A:C: B: C; and therefore by the first case, A =B. b A. 5. PROP. Book V. PROP. X. THEO R. THAT magnitude which has a greater ratio than another has unto the fame magnitude is the greater of the two: And that magnitude to which the same has a greater ratio than it has unto another magnitude is the leffer of the two. If the ratio of A to C be greater than that of B to C, A is greater than B. Because A: C> B: C, two numbers m and n may be found, fuch that mA > nC, and mB <Ca. Therefore alfo a def. y. 5. mA> mB, and A> B b. Again, let C: B>C: A; B<A. For two numbers, m and n may be found, fuch that mC > mB, and mC <mA a. Therefore, fince mB is lefs, and mA greater than the fame magnitude, mC, mB <mA, and therefore B < A. b 4. Ax. 5. R PRO P. XI. THE OR. ATIOS that are equal to the fame ratio are If A: B::C: D; and alfo C:D:: E:F; then A: B:: E: F. Take mA, mC, mE any equimultiples of A, C, and E; and nB, nD, nF any equimultiples of B, D, and F. Because A: B:: CD, if mA>nB, mC > nD a; but if mC>nD, a def. 5. 5. mEnFa, because C: D:: E: F; therefore if mA>nB, mE>nF. In the fame manner, if mAnB, mE =nF; and if mA <nB, mE <nF. Now, mA, mE are any equimultiples whatever of A and E; and nВ, nF any whatever of B and F; therefore A: B::E: Fa. Therefore, &c. Q. E. D. PROP. Book V. PROP. XII. THEOR. F any number of magnitudes be proportionals, as one of the antecedents is to its confequent, fo fhall all the antecedents taken together be to all the confequents. If A: B:: C: D, and C: D::E:F; A:B:: A+C+E: B+D+F. Take mA, mC, mE any equimultiples of A, C, and E; and nB, D, F, any equimultiples of B, D, and F. Then, bea 5. def. 5. cause A: B:: C: D, if mA> nB, mC > nDa; and when mC > nD, mE> nF, because C: D:: E: F. Therefore, if mA>nB, mA+mC+mE>nB÷nD+nF: In the fame manner, if mAn3, mA+mC+mEnB+nD+nF; and if mA< nB, mA+mC+mE <nB+nD+nF. Now mA+mC+mE= b 1. 5. m.A+C+Eb, fo that mA and mA+mC+mE are any equi multiples of A and of A+C+E. And for the fame reason nB, and B÷nD+nF are any equimultiples of B, and of B+D+F; therefore a A: B:: A+C+E:B+D+F. There fore, &c. Q.E. D. PROP. XIII. PROB. F the firft has to the fecond the fame ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the fixth; the firft fhall alfo have to the second a greater ratio than the fifth has to the fixth. If A: B::C: D; but C: D>E: F; then alfo, A: B> Because C: D> E: F, there are fome two numbers m and ", a 7. def. 5. fuch that mC > nD, but mE <nFa. PROP |