Book V.

a 1. 5.

b 7.

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 fuch 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. Ca will be lefs than mA+mB, or mA+mB, that is m.A+B is greater than 7. 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; 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+Bы. Therefore, &c. Q. E. D.

AGNITUDES which have the fame ratio to the

M 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 laft propofition, such that mA fhall exceed nC, while mB does not exceed C. But because A: C:: B: C; if mA exa 5. def. 5. ceed #C, mB must also exceed nC a; and it is alfo fhewn that mB does not exceed #C, which is impoffible. Therefore A is not greater than B; and in the fame way it is demonftrated that B is not greater than A; therefore A is equal to B.

b A. 5.

Next, let C: A:: C: B, AB. For by inverfion b, A:CBC; and therefore by the firft cafe, AB.

PROP.

Book V.

TH

PROP. X. THEOR.

HAT magnitude, which has a greater ratio than another has to the fame magnitude, is the greatest of the two: And that magnitude, to which the fame has a greater ratio than it has to another magnitude, is the leaft 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 > C, and mB <nCa. Therefore alfo mA > mB, and A > B b.

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

a def. 7. 5. b 4. Ax. 5.

PROP. XI. THE OR,

ATIOS that are equal to the fame ratio are equal
to one another.

RA

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 nВ, nD, nF any equimultiples of B, D, and F. Because AB:: C: D, if mA > nB, mC > nDa; but if mC > nD, a def. 5. 5. mEnF, because C: D:: E:F; therefore if mA> nB, mEnF. In the fame manner, if mAnB, mEnF; 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.

PROP.

Book V.

F

IF

PROP. XII. THEOR.

any number of magnitudes be proportionals, as one of the antecedents is to its confequent, so are all the antecedents, taken together, to all the confequents.

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

Take mA, mC, mE any equimultiples of A, C, and E; and nВ, nD, nF, 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 man.

if mAnB, mA+mC+mE➡nB+nD+nF; and if mA< nB, mA+mC+mE < nB+nD+nF. Now, mA+mC+mE= b Cor. 1:5. m.A+C+E b, fo that mA and mA+mC+mE are any equimultiples of A, and of A+C+E. And for the fame reafon nB, and "B+nD+nF are any equimultiples of B, and of B+D+F; therefore A: B:: A+C+E: B+D+F. Therefore, &c. Q.E. D.

IF

F the firft have 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 has alfo to the fecond a greater ratio than the fifth has to the fixth.

If A: B:: C: D; but C: D> E: F; then alfo, A: B> E: F.

Because C: D> E: F, there are two numbers m and n, ́a 7. def. 5. fuch that mC >nD, but mE<nFa. Now, if mC> nD, mAnB, because A: B:: C: D. Therefore mA > nB, and mEnF, wherefore, A : B>E: F4. Therefore, &c. Q.

E. D..

PROP.

Book V.

IF

PROP. XIV. THEOR.

F the firft have to the fecond the fame ratio which the third has to the fourth, and, if the first be greater than the third, the second fhall be greater than the fourth; if equal, equal; and if lefs, less.

If A: B:: C: D; then, if A> C, B > D; if A=C, BID; and if A <C, B < D.

First, let A> C; then A: B>C: Ba, but A: B:: C: D, therefore C: D>C: Bb, and therefore B> D c. In the fame manner, it is proved, that if A=C, BD; and if A <C, B<D. Therefore, &c. Q. E. D.

M

AGNITUDES have the fame 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: Ba; A: B::A+A: B+Bb, or a 7. 5. A:B:: 2A 2B. And in the fame manner, fince A:B::

2A 2B, A: B:: A+2A:: B+2Bb, or A: B:: 3A: 3B; b 12. 5. and fo on, for all the equimultiples of A and B. Therefore, &c. Q. E. D.

F four magnitudes of the fame kind be proportionals, they will also be proportionals when taken alternately.

Book V.

If A: B:: C:D, then alternately, A: C::B: D.

Take mA, mB any equimultiples of A and B, and nC, nD a 15. 5 any equimultiples of C and D. Then a A: B:: mA: mB; 11. 5 now A: B::C: D, therefore b C: D:: mA : mB. But C:

D::nCnDa; therefore mA: mB:: nC: nDb; wherefore c 14.5 if mA > nC, mB> nDc; if mAnC, mB=nD, or if mA< d def. 5. 5. nC, mB <nD; therefore, A: C:: B : D. Therefore, &c. Q: E. D.

IF magnitudes, taken jointly, be proportionals, they will alfo be proportionals when taken separately; that is, if the first, together with the fecond, have to the fecond the fame ratio which the third, together with the fourth, has to the fourth, the first will have to the fecond the fame ratio which the third has to the fourth.

If A+B: B:: C+D: D, then by divifion A: B:: C: D. Take mA and B any multiples of A and B, by the numbers m and n; and first let mA> mB: to each of them a 1. Cor. 5. add mB, then mA+mB> mB+nB. But mA+mBm.A+Ba and mB+nB+.Bb, therefore m.A+B>m+n.B..

b2.Cor.2.5.

And becaufe A+B : B ::C+D:D, if_m.A+B> m+n.B, m.C+D>m+n.D, or mC+mD>mD+nD, that is, taking mD from both, mC> nD. Therefore, when mA is greater than nB, mC is greater than nD. In like manner, it is demonftrated, that if mAnB, mCnD, and if mA <nB, that c. 5. def. 5. mC <nD; therefore A:B::C: Dc. Therefore, &c. E. D.

C.

PROP.