2. To prove the theorem contained in Def. xii.
Let the magnitudes be a, b, c, d. Then the ratio of
| 1st : 2nd | = , | ||
| 2nd : 3rd | = , | ||
| 3rd : 4th | = . |
Hence the ratio compounded of the ratio of 1st : 2nd, of 2nd : 3rd, of 3rd : 4th
3. If three magnitudes be proportional, the ratio of the 1st : 3rd is equal to the square of the ratio of the 1st : 2nd. For the ratio of the 1st : 3rd is compounded of the ratio of the 1st : 2nd, and of the ratio of the 2nd : 3rd; and since these ratios are equal, the ratio compounded of them will be equal to the square of one of them.
Or thus: Let the proportionals be a, b, c, that is, let a : b :: b : c; hence we have
And multiplying each by
