For let C by multiplying D make E; therefore A, E, B are continuously proportional in the ratio of C to D. [[VIII. 11](elem.8.11)]

And, since A, E, B are continuously proportional, and A measures B, therefore A also measures E. [[VIII. 7](elem.8.7)]

And, as A is to E, so is C to D; therefore also C measures D. [[VII. Def. 20](elem.7.def.20)]

Again, let C measure D; I say that A also measures B.

For, with the same construction, we can in a similar manner prove that A, E, B are continuously proportional in the ratio of C to D.

And since, as C is to D, so is A to E, and C measures D, therefore A also measures E. [[VII. Def. 20](elem.7.def.20)]

And A, E, B are continuously proportional; therefore A also measures B.

Therefore etc. Q. E. D.