PROP. XX. THEOR.

IF there be three magnitudes, and other three, which, taken two and two, have the same ratio; if the first be greater than the third, the fourth is greater than the sixth; if equal, equal; and if less, less.

A, B, C,

D, E, F.

Book V.

If there be three magnitudes, A, B, C, and other three, D, E, F; and if A: B:: D: E, and B:CE: F; then if A>C, D>F; if A=C, D=F; and if A<C, D<F. First, let A>C; then A: B>C: Ba. But A: B :: D: E; a 8. 5. therefore also D: E>C: B. Now B: C:: E: F, and in- b 13. 5. versely, C: B:: FE; and it has been shown that D: E >C: B, therefore D : E>F : Eb, and consequently D>Fd. Next, let A=C; then A: B:: C: Be. But A:B::D:E; therefore C:B:: D: Ef. But C: B::F:E; therefore D: E:: F: Ef, therefore D=Fɛ.

::

c A. 5.

d 10. 5.

e 7. 5.

f11. 5.

Lastly, let A<C, then C>A; and because C: B: 89. 5. F: Ec, and BA :: E: D, by the first case, if C>A, F>D, that is, if A<C, D<F. Therefore, &c. Q. E. D.

PROP. XXI. THEOR.

IF there be three magnitudes, and other three, which have the same ratio taken two and two, but in a cross order; if the first magnitude be greater than the third, the fourth is greater than the sixth; if equal, equal; and if less, less.

If there be three magnitudes, A, B, C, and other three, D, E, F, such that A: B:: E: F, and B: C:: D: E; if A>C, D>F; if A=C, D=F; and if A<C, D<F.

Next, let A=C; then A : B :: C: Bd. But A: B:: E: F; therefore C: B:: E: Fe. But BC: D: E, and inversely, C B E D, therefore E: F:: E: De, and, consequently, D=Ff.

:

Lastly, let A<C, then C>A; and because C: B::E: Do, and B: A: F: E, by the first case, since C>A, F>D, that is, D<F. Therefore, &c. Q. E. D.

a 4. 5.

PROP. XXII. THEOR.

IF there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio; the first will have to the last of the first magnitudes the same ratio which the first of the others has to the last*.

First, let there be three magnitudes, A, B, C, and other three, D, E, F, which, taken two and two in order, have the same ratio, A: B:: D: E, and B: C:: E: F, then A: C :: D: F.

whatever, mA, mD, and of C and F any

Take of A and D any equimultiples and of B and E any whatever, nB, nE, whatever, qC, qF. Because A: B:: D: E, mA: nB:: mD: nEa; and for the same reason, nB: qCnE : qF. Therefore, according as mA is greater than qC, equal to it, or less, mD is greater than 9F, equal to it, or lessb. But mA, mD are any equimultiples of A, D; and 9C, qF are c Def. 5. 5 any equimultiples of C, F; therefore A: C:: D: Fc.

b 20. 5.

A, B, C, D, E, F, mA, nB, qC, mD, nE, qF.

* N. B. This proposition is usually cited by the words "ex æquali," or " ex æquo."

Again, let there be four magnitudes, and other four, which, Book V. taken two and two in order, have the same ratio, A : B ::

E: F, B: C:: F: G, C: D::G: H; then A: D:: E: H.

A, B, C, D,

E, F, G, H.

For, since A, B, C are three magnitudes, and E, F, G other three, which taken two and two, have the same ratio, by the foregoing case, A: C:: E:G; and because also C: D:: G: H, by the foregoing case, A: DE: H. In the same manner is the demonstration extended to any number of magnitudes. Therefore, &c. Q. E. D.

PROP. XXIII. THEOR.

IF there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio, the first will have to the last of the first magnitudes the same ratio which the first of the others has to the last*.

First, let there be three magnitudes, A, B, C, and other three, D, E, F, which, taken two and two, in a cross order, have the same ratio, A: B:: E: F, and B: C::D: E; then A: C:: D: F.

Take of A, B, D any equimultiples mA, mB, mD; and of C, E, F any equimultiples nC, nE, nF.

c 4. 5.

d 21. 5.

Because A: B::E: F, A: B:: mA: mBa, and E: F a 15. 5. :: nE nF; therefore mA : mB: : : nE: nFb. Because b 11. 5. B:C:: D: E, mB: nC:: mD: nEc. But mA mB:: nE: nF; therefore, if mAnC, mD>nFd; if mA=nC, mD=nF; and if mA<nC, mD<nF. Now mA, mD are any equimultiples of A, D, and nC, nF any equimultiples of C, F; therefore A: C:: D: Fe,

A, B, C, D, E, F, mA, mB, nC, mD, nE, nF.

*N. B. This proposition is usually cited by the words "ex æquali in pro"portione perturbata;" or "ex æquo inversely."

e 5. Def. 5.

Book V.

A, B, C, D,
E, F, G, H.

Next, let there be four magnitudes, A, B, C, D, and other four, E, F, G, H, which, taken two and two, in a cross order, have the same ratio, A: B:: G: H, B: C:: F: G, and C: DEF; then A: D:: E: H. For, since A, B, C are three magnitudes, and F, G, H, other three, which, taken two and two, in a cross order, have the same ratio, by the first case, A: C:: F: H. But C: D:: E: F, therefore, by the first case, A:D :: E: H. In the same manner may the demonstration be extended to any number of magnitudes. Therefore, &c. Q. E. D.

a A. 5.

b 22. 5.

c 18. 5.

PROP. XXIV. THEOR.

IF the first have to the second the same ratio which the third has to the fourth, and the fifth to the second the same ratio which the sixth has to the fourth; the first and fifth together will have to the second the same ratio which the third and sixth together have to the fourth.

Let A B C : D, and E: B:: F: D; then
A+EB::C+F: D.

Because E: B:: F: D, B: E:: D: Fa. But A: B::
C: D; therefore, ex æqualib, A: E::C: F, therefore A+E
E: C+F: Fc. But E: B:: F: D; therefore, ex æqualib,
A+E BC+F: D. Therefore, &c. Q. E. D.

Book V.

PROP. E. THEOR.

IF four magnitudes be proportionals, the sum of the first two is to their difference as the sum of the other two to their difference.

Let A B C : D,

then A+B A-B:: C+D: C-D, if A>B;

:

and A+B : B—A : : C+D : D—C, if A<B.
If A>B, then, because A : B :: C: D,
A-B: B:: C-D: D, by division",

therefore B: A-B:: D: CD, by inversion.
But A+B B:: C+D: D, by composition;

:

therefore, ex æqualia, A+B: A-B:: C+D: C-D.

If B>A, it may be proved in the same manner that A+B : B-A :: C+D: D-C. Therefore, &c. Q. E. D.

a 17. 5.

b A. 5,

c 18. 5.

d 22.5

PROP. F. THEOR.

RATIOS which are compounded of equal ratios are equal to one another.

Let the ratios of A to B, and of B to C, which compound the ratio of A to C, be equal, each to each, to the ratios of D to E, and E to F, which compound the ratio of D to F; A: CD: F.

First, if the ratio of A to B be equal to that of D to E, and the ratio of B to C equal to that of E to F; ex æqualia, A: C:: D: F.

A, B, C,

D, E, F.

a 22. 5.

Next, if the ratio of A to B be equal to that of E to F, and the ratio of B to C equal to that of D to E; ex æquali inversely, A ; C:: D: F. In the same manner may the b 23. 5proposition be demonstrated, whatever be the number of ratios. Therefore, &c. Q. E. D.