| Hence, if | a | : b :: a′ : b′, | |||||||||
| a2 | : b2 :: a′2 : b′2 |
Or if four magnitudes be proportional, their squares are proportional.
Cor. 2.—If four magnitudes be proportional, their cubes are proportional.
PROP. XXIII.—Theorem.
If there be two sets of magnitudes, which, taken two by two in transverse order, have equal ratios; then the first : the last of the first set :: the first : the last of the second set (“ex aequo perturbato”).
Let a, b, c; a′, b′, c′ be the two sets of magnitudes, and let the ratio a : b = b′ : c′, and b : c = a′ : b′; then a : c :: a′ : c′.
| Dem.—Since | a : b :: | b′ : c′, | |||||||||
| we have | = | . | |||||||||
| In like manner, | = | . | |||||||||
| Hence, multiplying, | = | ; | |||||||||
| therefore | a : c :: | a′ : c′, |
and similarly for any number of magnitudes in each set.
This Proposition and the preceding one may be included in one enunciation, thus: “Ratios compounded of equal ratios are equal.”
PROP. XXIV.—Theorem.