| Dem.—Since | a : b :: | b′ : c′, | |||||||||
| we have | = | . | |||||||||
| In like manner, | = | . | |||||||||
| Hence, multiplying | = | . |
Therefore, if a be greater than c, a′ is greater than c′; if equal, equal; if less, less.
PROP. XXII.—Theorem.
If there be two sets of magnitudes, which, taken two by two in direct order, have equal ratios, then the first : the last of the first set :: the first : the last of the second set (“ex aequali,” or “ex aequo”).
Let a, b, c; a′, b′, c′ be the two sets of magnitudes, and if a : b :: a′ : b′, and b : c :: b′ : c′, then a : c :: a′ : c′.
| Dem.—Since | a : b :: | a′ : b′, | |||||||||
| we have | = | ||||||||||
| In like manner, | = | . | |||||||||
| Hence, multiplying, | = | . | |||||||||
| Therefore | a : c :: | a′ : c′, |
and similarly for any number of magnitudes in each set.
Cor. 1.—If the ratio b : c be equal to the ratio a : b, then a, b, c will be in continued proportion, and so will a′, b′, c′. Hence [Def. xii. Annotation 3],
| but | = . | [xxii.] | |||||||||
| Therefore | = . |