| × = | ×, | ||||||||||
| or | = | ; | |||||||||
| therefore | a : c :: | b : d. |
Hence, Proposition [A], if a be greater than c, b is greater than d; if equal, equal; and if less, less.
PROP. XV.—Theorem.
Magnitudes have the same ratio which all equimultiples of them have.
Let a, b be two magnitudes, then the ratio a : b is equal to the ratio ma : mb.
Dem.—The ratio a : b =
, and the ratio of ma : mb =
; but since the value of a fraction is not altered by multiplying its numerator and denominator by the same number,
| = ; | |||||||||||
| therefore | a : b | :: ma : mb. |