If a square measure a square, the side will also measure the side; and, if the side measure the side, the square will also measure the square.
Let A, B be square numbers, let C, D be their sides, and let A measure B;
I say that C also measures D.
For let C by multiplying D make E; therefore A, E, B are continuously proportional in the ratio of C to D. [VIII. 11]
And, since A, E, B are continuously proportional, and A measures B, therefore A also measures E. [VIII. 7]
And, as A is to E, so is C to D; therefore also C measures D. [VII. 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]
And A, E, B are continuously proportional;
therefore A also measures B.
Complete text
Book I