Let A and B be two numbers, and let A multiplied by B make the square number C.
I say that A and B are similar plane numbers.
Multiply A by itself to make D. Then D is square.
Now, since A multiplied by itself makes D, and multiplied by B makes C, therefore A is to B as D is to C.
And, since D is square, and C is so also, therefore D and C are similar plane numbers.
Therefore one mean proportional number falls between D and C. And D is to C as A is to B, therefore one mean proportional number falls between A and B also.
But, if one mean proportional number falls between two numbers, then they are similar plane numbers, therefore A and B are similar plane numbers.
Therefore, if two numbers multiplied by one another make a square number, then they are similar plane numbers.
As an example to illustrate this proposition, take any square number, such as 202 = 400. It can be factored as a product of two numbers in several ways. One such factorization is as a = 50 times b = 8. These two numbers have a mean proportional between them, namely, 20, so by VIII.20, they are similar plane numbers. (The actual shapes given by that proposition make 8 to be 2 by 4, and 50 to be 5 by 10.)
As in the last proof, this one can be shortened. When the product ab is a square, say e2, then a mean proportional between a and b is e.