Book V.

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 often as mn contains unity, or mA-nA= m-n.A.

Let m-n=q; then m=n+q; therefore mA=nA+2A; a 2.5. take nA from both, then mA-nA=qA. Therefore mA-nA contains A as often as there are units in q, that is, in mn, 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, and 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 nDa, that is, if nB be greater than mA, nD will be greater a Def. 5. 5. than mC. For the same reason, if nB=mA, then nD=mC, and if nB<mA, then nD<mC. But nB, nD are any equimultiples of B and D, and mA, mC any equimultiples of A and C; therefore B: A:: D: C. Therefore, &c. Q. E. D.

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Book V,

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 and mB be equimultiples of the magnitudes A and B, mA: A::mB: B.

a 3. 5.

c A. 5.

Take of mA and mB equimultiples by any number n, and of A and B equimultiples by any number p; these will be nmA, pA, nmB, pB. Now, if nmA be greater than pA, nm is also greater than p; and if nm be 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 nmA=pA, then nmB=pB, and if nmA<pA, then nmB<pB. Now, nmA, nmB are any equimultiples of mA, mB; and pA, pB b Def. 5. 5. are any equimultiples of A, B; therefore mA: A::mB: Bb. 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; therefore, as has been demonstrated, A:C::B:D; therefore, inversely, 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 Band D take equimultiples by the

number 2m, viz. 2mB, 2mDa; then, because A=mB, 2A= Book V. 2mB; and since A: B :: C: D, and 2A=2mB, 2C=2mDb, and C=mD, that is, C contains Dm times, or as often as A a 3.5. contains B.

b Def. 5. 5.

Next, let A be a part of B, C is the same part of D. For, since A: B:: C: D, inversely, B: A:: D: C. But AcA.5. being a part of B, B is a multiple of A, and therefore, as is shown above, D is the same multiple of C. 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 magnitude 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=mBa; wherefore, if mA be greater a Ax. 1. 5. than nC, mB is greater than nC; and if mA=nC, mB=nC; and if mA<nC, mB<nC. But mA and me are any equimultiples of A and B, and nC is any multiple of C; thereforeb A: C:: B: C.

b Def. 5.5.

Again, if A=B, C : A :: C: B; for, as has been proved, A:C:: B: C, and inverselyc, C:A::C:B. Therefore, c A. 5. &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.

Book V.

a 1.5.

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 greater ratio to A than it has to A+B.

a

Let m be such a number that mA and mB may be each of them greater thin C; and let nC be the least multiple of C that exceeds m+mB; then nC-C, that is, n-1.Ca will be less than mA+nB, 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, or 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

b 7. Def. 5. A+B has a greater ratio to C than A has to Cb.

Again, because the multiple of C by n-1 exceeds the multiple of A bym, 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.

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 ex

a 5. Def. 5. ceed nC, mB must also exceed nCa; and it is also shown that mB does not exceed nC, which is impossible. Therefore A is not greater than B. And in the same way it can be demonstrated that B is not greater than A; therefore A is equal to B.

Next, let C: A::C: B, A=B. For, by inversion', 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 greater of the two; and that magnitude to which the same has a greater ratio than it has to another magnitude is the less of the two.

If the ratio of A to C be greater than that of B to C, A is greater than B.

Book V.

Because A: C>B: C, two numbers, m and n, may be found, such that mA>nC, and mB<nC2. Therefore also a 7. Def. 5. mA>mB; therefore A>Bb.

Again, let C: B>C:A; B<A. For two numbers, m and n, may be found, such that mC>mB, and mC<mA2. Therefore, since mB is less, and mA greater than the same magnitude mC, mB<mA, and therefore B<A. Therefore, &c. Q. E. D.

b 4.Ax. 5.

PROP. XI. THEOR.

RATIOS that are equal to the same ratio are equal to one another.

If A: B:: C: D, and also C:D::E: F; then A: B :: E: F.

Take mi, mC, mE, any equimultiples of A, C, E ; and nB, nD, nF, any equimultiples of B, D, F. Because A: B :: C: D, if mA>nB, mC>nDa; but if mC>nD, mE>nFa, a Def. 5. 5. because C: D:: E: F; therefore if mA>nB, mE>nF. In the same manner, if mA=nB, mE=nF; and if mA<nB, mE<nF. Now, mA, mE are any equimultiples whatever of A and E; and nB, nF any whatever of B and F; therefore A:B::E: Fa. Therefore, &c. Q. E. D.