PROP. VI. THEOR. If from a multiple of a magnitude by any number a multiple of the same magnitude by a less number be taken away, the remainder will be the same multiple of that magnitude that the difference of the numbers is of unity. Let mA and nA be multiples of the magnitude A, by the numbers m and n, and let m be greater than n; mAnA contains A as oft as m-n contains unity, or mA-nA=(mn) A. Let m―n=q; then m=n+q. Therefore (2. 5.) m A=nA+qA; take nA from both, and mA―nA=qA. Therefore mAnA contains A as oft as there are units in q, that is, in m—n, or mA—nA= (m- n) A. Therefore, &c. Q. E. D. COR. When the difference of the two numbers is equal to unity, or m--n=1, then mA—nA=A. PROP. A. THEOR. If four magnitudes be proportionals, they are proportionals also when taken inversely. If A: B:: C: D, then also B: A:: D: C. Let mA and mC be any equimultiples of A and C; nB and nD any equimultiples of B and D. Then, because A: B::C: D, if mĂ be less than nB, mC will be less than nD (def, 5. 5.), that is, if nB be greater than mA, nD will be greater than mC. For the same reason, if nB=mA, nD=mC, and if nBmA, nD4mC. But nB, nD are any equimultiples of Band D, and mA, mC any equimultiples of A and C, therefore (def. 5. 5.), B: A::D: C. Therefore, &c. Q. E. D. PROP. B. THEOR. If the first be the same multiple of the second, or the same part of it, that the third is of the fourth; the first is to the second 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 n; and of A and B equimultiples by any number p; these will be nmA (3. 5.), pA, nmB (3. 5.), pB. Now, if nmA be greater than pA, nm is also 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 same manner, if ninA=pA, nmB=pB, and if nmApA, nmB ZpB Now, nmA, nmB are any equimultiples of mA and mB; and pA, pB are any equimultiples of A and B, therefore mA : A :: mB : B (def. 5. 5.). Next, Let C be the same part of A that D is of B; then A is the same multiple of C that B is of D, and therefore, as has been demonstrated, A CB: D, and inversely (A. 5.) C: A :: D: B. Therefore, &c. Q. E. D. PROP. C. THEOR. If the first be to the second as the third to the fourth; and if the first be a multiple or a part of the second, the third is the same multiple or the same part of the fourth. Let A B C : D, and first let A be a multiple of B, C is the same 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 2m, viz. 2mB, 2mD (3. 5.); then, because AmB, 2A=2mB; and since A: B :: C: D, and since 2A=2mB, therefore 2C=2mD (def. 5. 5.), and C=mD, that is, C contains Dm times, or as often as A contains B. : Next, Let A be a part of B, C is the same part of D. For, since A B C D, inversely (A. 5.), B: A :: D: C. But A being a part of B, B is a multiple of A, and therefore, as is shewn above, Dis the same multiple of Ĉ, and therefore C is the same part of D that A is of B. Therefore, &c. Q. E. D. PROP. VII. THEOR. Equal magnitudes have the same ratio to the same magnitude; and the same has the same ratio to equal magnitudes. Let A and B be equal magnitudes, and C any other; A : C :: B :C. Let mA, mB, be any equimultiples of A and B ; and nC any multiple of C. Because A=B, mA=mB (Ax. 1. 5.), wherefore, if mA be greater than nC, mB is greater than nC; and if mA=nC, mB=nC; or, if mA ▲nC, mB≤nč. But mA and mB are any equimultiples of A and B, and nC is any multiple of C, therefore (def. 5. 5.) A: C: B: C. Again, if A=B, C: A :: C: B; for, as has been proved, A: C:: B: C, and inversely (A. 5.), C: A:: C: B. Therefore, &c. Q. E. D. Of unequal magnitudes, the greater has a greater ratio to the same than the less has; and the same magnitude has a greater ratio to the less than it has to the greater. Let A+B be a magnitude greater than 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. Let m be such a number that mA and mB are each of them greater than C ; and let nC be the least multiple of C that exceeds mA+mB; then nC-C, that is, (n-1)C (1. 5.) will be less than mA+mB, or mA+mB, that is, m(A+B) is greater than (n-1)C. But because nC is greater than mÀ+mB, and C less than mB, nC-C is greater than mA, or mA 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; therefore A+B has a greater ratio to C than A has to C (def. 7. 5.). 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+B (def. 7. 5.). Therefore, &c. Q. E. D. PROP. IX. THEOR. Magnitudes which have the same ratio to the same magnitude are equal to one another; and those to which the same magnitude has the same ratio are equal to one another. 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 proposition, such that mA shall exceed nC, while mB does not exceed nC. But because A: C :: B: C; if mA exceed nC, mB must also exceed nC (def. 5. 5.); and it is also shewn that mB does not exceed nC, which is impossible. Therefore A is not greater than B; and in the same way it is demonstrated that B is not greater than A; therefore A is equal to B. Next, let CA: C: B, A=B. For by inversion (A. 5.) A : C: BC; and therefore by the first case, A=B. PROP. X. THEOR. That magnitude, which has a greater ratio than another has to the same magnitude, is the greatest of the two: And that magnitude, to which the same has a greater ratio than it has to another magnitude, is the least 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: C7B: C, two numbers m and n may be found, such that mA7nC, and mBnC (def. 7. 5.). Therefore also mA7mB, and A7B (Ax. 4. 5.). Again, let C: B7C: A: BLA. For two numbers, m and n may be found, such that mC7nB, and mCnA (def. 7. 5.). Therefore, since nB is less, and nA greater than the same magnitude mC, nBnA, and therefore BA. Therefore, &c. Q. E. D. PROP. XI. THEOR. Ratios that are equal to the same ratio are equal to one another. If A B C D; and also C: DE: F; theu A: B:: E: F. Take mA, mC mE, any equimultiples of A, C, and E; and nB, nD, F any equimultiples of B, D, and F. Because A: B:: C: D, if MA7nB, mC7nD (def. 5. 5.); but if mC7nD. mE7nF (def. 5. 5.), because CD: E: F; therefore if mA7nB, mE7nF. In the same manner, if mA=nB, mE=nF; and if mAnB, mEZnF. Now, mA, mE are any equimultiples whatever of A and E; and B, nF any whatever of B and F; therefore A: B: E: F (def. 5. 5.). Therefore, &c. Q. E. D. PROP. XII. THEOR. If any number of magnitudes be proportionals, as one of the antecedentsis to its consequent, so are all the antecedents, taken together, to all the consequents. If A: B:: C: D, and C: D:: E: F; then also, A : B :: A+C +E: B+D+F. Take mA, mC, mE any equimultiples of A, C, and E; and nB, nD, nF, any equimultiples of B, D, and F. Then, because A: B:: C: D, if mA7nB, mC7nD (def. 5. 5.); and when mC7nD, mE7nF, because CD:: E: F. Therefore, if mA7nB, mA+mC+mE7nB+ D+nF: In the same manner, if mAnB, mA+mC+mE÷nB+nD +nF; and if mAnB, mA+mC+mE2nB+nD+nF. Now, mA+ mC+mE=m(A+C+E) (Cor. 1. 5.), so that mA and mA+mC+mE are any equimultiples of A, and of A+C+E. And for the same reason nB, and nB+nD+nF are any equimultiples of B, and of B+D+F; therefore (def. 5. 5.) A: B:: A+C+E B+D+F. Therefore &c. Q. E. D. PROP. XIII. THEOR. If the first have to the second the same ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth; the first has also to the second a greater ratio than the fifth has to the sixth. If A B : : C D; but C: D7E: F; then also, A : B7E: F. Because C D7E: F, there are two numbers m and n, such that mC7nD, but mEnF (def. 7. 5.). Now, if mC7nD,mA7nB, because A B C D. Therefore mA7nB, and mEnF, wherefore, A: B7E: F (def. 7. 5.). Therefore, &c. Q. E. D. PROP. XIV. THEOR. If the first have to the second the same ratio which the third has to the fourth, and if the first be greater than the third, the second shall be greater than the fourth; if equal, equal; and if less, less. : If A B C D; then if A7C, B7D; if A=C, B=D; and if ALC, BLD. First, let A7C; then A: B7C: B (8. 5.), but A: B::C: D, therefore C D7C: B (13. 5.), and therefore B7D (10. 5.). In the same manner, it is proved, that if A= C, B = D; and if AZC, BLD. Therefore, &c. Q. E. D. PROP. XV. THEOR. f Magnitudes have the same ratio to one another which their equimultiples have. If A and B be two magnitudes, and m any number, A : B : : mA : mB. Because A B : A B (7. 5.); A: B :: A+A : B+B (12. 5.), or A B : 2A : 2B. And in the same manner since A: B: : 2A: 2B, A: B: A+2A:: B+2B (12. 5.), or A: B : : 3A: 3B on, for all the equimultiples of A and B. Therefore, &c. Q. E. D. PROP. XVI. THEOR. ; and so If four magnitudes of the same kind be proportionals, they will also be proportionals when taken alternately. If A B C D, then alternately, A: C:: B: D. : :: Take mA, mB any equimultiples of A and B, and nC, nD any equimultiples of C and D. Then (15. 5.) A: B :: mA: mB; now A: BC D, therefore (11. 5.) C: D:: mA: mB. But CD:: nCnD (5. 5.); therefore mA : mB:: nC : nD (11.5.) : wherefore if mA7nC, mB7nD (14. 5.); if mA=nC, mB=nD, or if mAL nC, mBnD; therefore (def. 5. 5.), A : C :: B : D. Therefore, &c. Q. E. D. PROP. XVII. THEOR. If magnitudes, taken jointly, be proportionals, they will also be proportionals when taken separately; that is, if the first, together with the second. have to the second the same ratio which the third, together with the fourth, has to the fourth, the first will have to the second the same ratio which the third has to the fourth. If A+B : B :: C+D: D, then by division A: B:: C : D. |