| Dem.—Since | c : a | :: c : b, | |||||||||
| by inversion, | a : c | :: b : c; | |||||||||
| therefore | a | = b. [1]. |
PROP. X.—Theorem.
Of two unequal magnitudes, that which has the greater ratio to any third is the greater of the two; and that to which any third has the greater ratio is the less of the two.
1. If the ratio a : c be greater than the ratio b : c, to prove a greater than b.
Dem.—Since the ratio a : c is greater than the ratio b : c,
Hence, multiplying each by c, we get a greater than b.
2. If the ratio c : b is greater than the ratio c : a, to prove b is less than a.
Dem.—Since the ratio c : b is greater than the ratio c : a,
