PROP. VI. THEOR. If from a multiple of a magnitude by anynumber 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; mA-nA contains A as oft as m-n contains unity, or mA-nA=(m-n) 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 mA-nA 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: С. 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 mA 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, nDmC. 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 numbern; and of A and B equimultiples by any number p; these will be nmA (3. 5.), рА, 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 nimA=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: C:: B: D, and inversely (A. 5.) C: A :: D: B. There. fore, &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 A=mB, 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 C, and therefore C is the same part of D that A is of B. Therefore, &c. Q. E. D. 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 Cany 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 mAZnC, mBnC. 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. PROP. VIII. THEOR. 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, 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; mA-nA contains A as oft as m-n contains unity, or mA-nA=(m-n) 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 mA-nA 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: С. 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 mA 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, nDmC. 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 numbern; and of A and B equimultiples by any number p; these will be nmA (3. 5.), рА, 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 pВ. In the same manner, if nimA=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: C:: B: 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 A=mB, 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 C, 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: С. 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, mBnC. 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 AB, 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. PROP. VIII. THEOR. 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 Ca 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 mA+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. 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 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 C: A:: C: B, AB. For by inversion (A. 5.) A: C:: B:C; 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:BZA. Fortwo 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, nB∠nA, and therefore BLA. Therefore, &c. Q. E. D. 1 |