express the proportion by placing dots between the numbers, thus:
a : b ∷ c : d
- Equal numbers will still remain equal when they have been increased, diminished, multiplied, or divided, by equal quantities. This amounts to saying that if
- a = b and p = q,
- a + p = b + q,
- a - p = b - q,
- ap = bq,
- a b
- and — = —.
- p q
It is also evident, that a + p-p, a -p + p, ap/p, and a/p × p, are all equal to a.
- The product of the extremes is equal to the product of the means. Let a/b = c/d, and multiply these equal numbers by the product bd. Then,
| a | × bd = | abd | (116) = ad , |
|---|---|---|---|
| b | b |
| and | c | × bd = | cbd | = cb : |
|---|---|---|---|---|
| d | d | |||
| hence (180), ad = bc . |
Thus, 6, 8, 21, and 28, are proportional, since
| 6 | = | 3 | = | 3 × 7 | = | 21 | (180); |
|---|---|---|---|---|---|---|---|
| 8 | 4 | 4 × 7 | 28 |
and it appears that 6 × 28 = 8 × 21, since both products are 168.
- If the product of two numbers be equal to the product of two others, these numbers are proportional in any order whatever, provided the numbers in the same product are so placed as to be similar terms;