| If there are as many numbers as we please in continued proportion, and each multiplied by itself makes some number, then the products are proportional; and, if the original numbers multiplied by the products make certain numbers, then the latter are also proportional. | |
| Let there be as many numbers as we please, A, B, and C, in continued proportion, so that A is to B as B is to C. Let A, B, and C multiplied by themselves make D, E, and F, and multiplied by D, E, and F let them make G, H, and K.
I say that D, E, and F and G, H, and K are in continued proportion. | |
| Multiply A by B to make L, and multiply the numbers A and B by L to make M and N respectively. Also multiply B by C to make O, and multiply the numbers B and C by O to make P and Q respectively.
Then, in manner similar to the foregoing, we can prove that D, L, and E and G, M, N, and H are continuously proportional in the ratio of A to B, and further E, O, and F and H, P, Q, and K are continuously proportional in the ratio of B to C. | |
| Now A is to B as B is to C, therefore D, L, and E are a]so in the same ratio with E, O, and F, and further G, M, N, and H in the same ratio with H, P, Q, and K. And the multitude of D, L, and E equals the multitude of E, O, and F and that of G, M, N, and H to that of H, P, Q, and K, therefore, ex aequali D is to E as E is to F, and G is to H as H is to K. | VII.14 |
| Therefore, if there are as many numbers as we please in continued proportion, and each multiplied by itself makes some number, then the products are proportional; and, if the original numbers multiplied by the products make certain numbers, then the latter are also proportional. | |
| Q.E.D. | |
Book VIII Introduction - Proposition VIII.12 - Proposition VIII.14.