IX.2

If two numbers by multiplying one another make a square number, they are similar plane numbers.

Proof

Let A, B be two numbers, and let A by multiplying B make the square number C; I say that A, B are similar plane numbers.

For let A by multiplying itself make D; therefore D is square.

Now, since A by multiplying itself has made D, and by multiplying B has made C, therefore, as A is to B, so is D to C. VII. 17

And, since D is square, and C is so also, therefore D, C are similar plane numbers.

Therefore one mean proportional number falls between D, C. VIII. 18

And, as D is to C, so is A to B; therefore one mean proportional number falls between A, B also. VIII. 8

But, if one mean proportional number fall between two numbers, they are similar plane numbers; VIII. 20 therefore A, B are similar plane numbers. Q. E. D.