| 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.
182. 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; that is, if ab = pq, we have the following proportions:—
- a : p ∷ q : b
- a : q ∷ p : b
- b : p ∷ q : a
- b : q ∷ p : a
- p : a ∷ b : q
- p : b ∷ a : q
- q : a ∷ b : p
- q : b ∷ a : p
To prove any one of these, divide both ab and pq by the product of its second and fourth terms; for example, to shew the truth of a: q ∷ p: b, divide both ab and pq by bq. Then,
| ab | = | a | , and | pq | = | p | ; hence (180), |
| bq | q | bq | b | ||||
| a | = | p | , or a : q ∷ p : b . | ||||
| q | b | ||||||
The pupil should not fail to prove every one of the eight cases, and to verify them by some simple examples, such as 1 × 6 = 2 × 3, which gives 1: 2 ∷ 3: 6, 3: 1 ∷ 6: 2, &c.