× 6 = 2 × 3, which gives 1: 2 ∷ 3: 6, 3: 1 ∷ 6: 2, &c.
- Hence, if four numbers be proportional, they are also proportional in any other order, provided it be such that similar terms still remain similar. For since, when
| a | = | c | , |
|---|---|---|---|
| b | d |
it follows (181) that ad = bc, all the proportions which follow from ad = bc, by the last article, follow also from
| a | = | c | . |
|---|---|---|---|
| b | d |
- From (114) it follows that
| 1 + | a | = | b + a | , |
|---|---|---|---|---|
| b | b |
| and if | a | be | less than | 1 , |
|---|---|---|---|---|
| b |
| 1 - | a | = | b - a | , |
|---|---|---|---|---|
| b | b |
| while if | a | be | greater than | 1 , |
|---|---|---|---|---|
| b |
| a | - 1 | = | a - b | . |
|---|---|---|---|---|
| b | b |
| Also (122), if | a + b | be | divided by | a - b |
|---|---|---|---|---|
| b | b | |||
| the result is | a + b | |||
| a - b |
Hence, a , b ,