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, 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, 8 : 27 is the triplicate ratio of 2 : 3.
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.