A proportion in three terms is the least possible.

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.

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, 4:9 is the duplicate ratio of 2:3.
The illustration also shows a continued proportion of four magnitudes, |

See, for example, VIII.1. These are commonly called *geometric progressions* or *geometric sequences*. In a geometric progression, the ratio of each term to the next term is the same. Euclid finds the sum of numerical geometric progression in IX.35.