2. To prove that the ratio c : b is greater than the ratio c : a.
Dem.—Since b is less than a, the quotient which is the result of dividing any magnitude by b is greater than the quotient which is got by dividing the same magnitude by a;
| therefore | is greater than . |
Hence the ratio c : b is greater than the ratio c : a.
PROP. IX.—Theorem.
Magnitudes which have equal ratios to the same magnitude are equal to one another; 2. magnitudes to which the same magnitude has equal ratios are equal to one another.
1. If a : c :: b : c, to prove a = b.
| Dem.—Since | a : c | :: b : c, | |||||||||
| = . |
Hence, multiplying each by c, we get a = b.
2. If c : a :: c : b, to prove a = b.