Book V.

a16. 5.

b 17. 5.

c. 11. 5.

PROP. XIX. THEOR.

If a whole magnitude be to a whole, as a magnitude taken from the first, is to a magnitude taken from the other; the remainder will be to the remainder as the whole to the whole.

If A: B::C: D, and if C be less than A,
A-C: B-D::A: B.

A-Č: C:: B―D: D.
A-C: B-D: C: D;
A-C: B-D :: A: B.

Because A: B:: C: D, alternately a, A: C:: B: D; and therefore by division Wherefore, again, alternately, but A: B:: C: D, therefore Therefore, &c. Q. E. D.

COR. A-C: B-D:: C: D.

17. 5.

b. A, 5.

PROP. D. THEOR.

If four magnitudes be proportionals, they are also proportionals by conversion, that is, the first is to its excess above the second, as the third to its excess above the fourth.

If A: B:: C: D, by conversion,

A: A-B:: C: C-D.

For, since A: B:: C: D, by division, A-B: B:: C-D: D, and inversely, B: A−B:: D:C—D ; 18, 5. therefore, by composition, A: A-B :: C: C—D. Therefore, &c. Q. E. D.

COR. In the same way, it may be proved that A: A+B :: C: C+ D.

Book V.

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,

If there be three magnitudes, A, B, and C, and other three D, E, and F; and if A: B :: D: E; and also B: C:: E: F, then if AC, DF; if A=C, D=F; and if A≤C, D≤F.

D,

E, F,

But A: B:: Now B: C:: and it has been

a8. 5,

b 13. 5.

c A. 5.

First, let AC; then A: BC: B. D: E, therefore also D: EC : B. E: F, and inversely, C: B:: F: E; shewn that D: E C: B, therefore D: E>F : E', and consequently DF

d 10. 5.

Next, let A=C; then A: B:: C: Be, but A: B :: e 7. 5. D: E; therefore, C: B:: D: E, but C: B:: F:E, therefore DE:: F:E, and D-F. Lastly, let f 11. 5. A C. Then CA, and because, as was already shewn, g 9. 5. C: B:: F: E, and B: A :: E: D; therefore, by the first case, if CA, FD, that is, if AC, 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, and F, such that A: B:: E: F, and B: C :: D: E; if AC, DF; if A-C, D=F, and if A≤C, D≤F.

Book V.

a 8. 5.

b 13. 5.

c 10. 5.

d 7. 5. e 11. 5.

f.9.5.

a 4. 5.

b 20. 5.

Then

First, let AC.
A:B-C: Ba, but A: B:: E: F,
therefore E: FC: Bb. Now,
B:C: D: E, and inversely,

A, B, C,

D, E, F.

C: B: E: D; therefore E: FE: D', wherefore,
D-Fc.

Next, let A=C. Then A: B:: C: B; but A: B:: E: F, therefore, C: B:: E: Fe; but B: C:: D: E, and inversely, C: B:: E: D, therefore, E: F:: E: D, and, consequently, D=F.

Lastly, let AC. Then CA, and, as was already proved, C: BE: D; and B: A:: F: E, therefore, by the first case, since CA, FD, that is, DF. Therefore, &c. Q. E. D.

PROP. XXII. THEOR.

If there be any number of magnitudes, and as many others, which, taken two and treo 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, viz. A: B:: D: E, and B: C :: E: F; then A: C:: D: F.

A, B, C,

Take of A and D any equimultiples whatever, mA, mD; and of B and E any whatever, nB, nE; and of C and F any whatever, qC, qF. Because A: B:: D: E, mA : nB ::: mD: nEa; and for the same reason, nB: qC: nE: qF. Therefore, according as mA is greater than qC, equal to it, or less, mD

D, E, F, mA, nB, qC,

mD, nE, qF.

is greater than qF, equal to it, or less; but mA, mD are any equimultiples of A and D; and qC, qF are any equie def. 5. 5. multiples of C and F; therefore, A: C:: D: F.

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

Again, let there be four magnitudes, and other four Book V. which, taken two and two in order, have the same ratio, viz. 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 threemagnitudes, 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 that same case, A: D:: E: H. In the same manner is the demonstration extended to number of magnitudes. Therefore, &c. Q. E. D.

any

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, and F, which, taken two and two, in a cross order, have the same ratio, viz. A: B:: E: F, and B: C:: D: E, then A: C:: D : F.

b 11. 5.

Take of A, B, and D, any equimultiples mA, mB, mD; and of C, E, F, any equimultiples nC, nE, nF. ⚫ Because A: B:: E: F, and because also A: B:: mA: mB, and E: F:: nE: nF; therefore, mA: mB:: a 15. 5. nE: nF. Again, because B: C:: D: E, mB: nČ:: mD: nEe; and it has been just shewn that mA: mB:: nE: nF; therefore, if mAnC, mDnFd; if mAnC, mD=nF; and if mAnC, mDnF. Now, mA

d 21, 5.

A, B, C, c 4. 5. D, E, F, mA, mB, nC, mD, nE, nF,

and mD are any equimultiples of A and D, and nC, nF,

any equimultiples of C and F; therefore, A: C:: D: Fe, e def. 5. 5.

*N. B. This proposition is usually cited by the words "ex æquali in "proportione perturbatą :" or, "ex æquo inversely."

Book V.

Next, let there be four magnitudes, A, B, C, and D, and other four, E, F, G, and H, which, taken two and two, in a cross order, have the same ratio, viz. A: B::G:H; B:C::

A,

B, C, D,

E,

F,

E, F, G, H,

F: : G, and C: D:: E: F, 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 ra-
tio, by the first case, A: C:: F: H. But C: D:: E: F,
therefore, again, by the first case, A: D:: E: H. In
the same manner, may the demonstration be extended to
any number of magnitudes. Therefore, &c.
Therefore, &c. Q. E. D.

a 22.5. b 18. 5.

PROP. XXIV. THEOR.

If the first has 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, shall have to the second, the same ratio which the third and sixth together, have to the fourth.

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

Because E: B:: F: D, by inversion, B: E::D : F. But by hypothesis, A: B:: C: D, therefore, ex æqualia, A: E:: C: F, and, by composition ", A+E: E::C+ F: F. And again, by hypothesis, E: B: F: D, therefore, ex æqualia, A+E:B::C+F: D. Therefore, &c. Q. E. D.

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