Def. 9. When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second.
Def. 10. When four magnitudes are continuously proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually, whatever be the proportion.
| In the illustration A, B, and C form three terms for the proportion A:B = B:C, therefore the ratio A:C is the duplicate ratio of A:B. For a numerical example, 9:4 is the duplicate ratio of 3:2. | |
| The illustration also shows a continued proportion of four magnitudes, A, B, C, and D, since A:B = B:C = C:D. Also, A:D is the triplicate ratio of A:B. For a numerical example, 27:8 is the triplicate ratio of 3:2. | |
Book V Introduction - Definition V.7 - Definitions V.11 through V.13.