Dem.—By hypotheses,
Since these fractions are all equal, let their common value be r; then we have
| = r, | = r, = r; | ||||||||||
| therefore | a | = br, | |||||||||
| c | = dr, | ||||||||||
| e | = fr; | ||||||||||
| therefore | a + c + e | = (b + d + f)r. |
| Hence | = r; | ||||||||||
| therefore | = , | ||||||||||
| and | a : b :: | a + c + e : b + d + f. |
Cor.—With the same hypotheses, if l, m, n be any three multipliers, a : b :: la + mc + ne : lb + md + nf.
PROP. XIII.—Theorem.
If two ratios are equal, and if one of them be greater than any third ratio, then the other is also greater than that third ratio.
If a : b :: c : d, but the ratio of c : d greater than the ratio of e : f; then the ratio of a : b is greater than the ratio of e : f.
Dem.—Since the ratio of c : d is greater than the ratio of e : f,
