Whence (186), bbb(2xxx + 3xx): bbb(1 + x) ∷ ddd(2xxx + 3xx): ddd(1 + x), which, when instead of these expressions their equals just found are substituted, becomes 2aaa + 3aab: bbb + abb ∷ 2ccc + 3ccd: ddd + cdd.
The same reasoning may be applied to any other case, and the pupil may in this way prove the following theorems:
If
a : b ∷ c : d
2a + 3b : b ∷ 2c + 3d : d
aa + bb : aa - bb ∷ cc + dd : cc - dd
mab : 2aa + bb ∷ mcd : 2cc + dd
191. If the two means of a proportion be the same, that is, if a : b ∷ b: c, the three numbers, a, b, and c, are said to be in continued proportion, or in geometrical progression. The same terms are applied to a series of numbers, of which any three that follow one another are in continued proportion, such as
| 1 | 2 | 4 | 8 | 16 | 32 | 64 | &c. |
| 2 | 2 | 2 | 2 | 2 | 2 | 2 | &c. |
| 3 | 9 | 27 | 81 | 243 | 729 | ||
Which are in continued proportion, since