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](elem.7.17)]

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](elem.8.18)]

And, as D is to C, so is A to B; therefore one mean proportional number falls between A, B also. [[VIII. 8](elem.8.18)]

But, if one mean proportional number fall between two numbers, they are similar plane numbers; [[VIII. 20](elem.8.20)] therefore A, B are similar plane numbers. Q. E. D.