For, since A is commensurable in length with B, therefore A has to B the ratio which a number has to a number. [X. 5]
Let it have to it the ratio which C has to D.
Since then, as A is to B, so is C to D,
while the ratio of the square on A to the square on B is duplicate of the ratio of A to B,
for similar figures are in the duplicate ratio of their corresponding sides; [VI. 20, Por.]
and the ratio of the square on C to the square on D is duplicate of the ratio of C to D,
for between two square numbers there is one mean proportional number,
and the square number has to the square number the ratio duplicate of that which the side has to the side; [VIII. 11]
therefore also, as the square on A is to the square on B, so is the square on C to the square on D.