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