Proposition 22
If three numbers are in continued proportion, and the first is square, then the third is also square.
Let A, B, and C be three numbers in continued proportion, and let A the first be square.
I say that C the third is also square.
Since between A and C there is one mean proportional number, B, therefore A and C are similar plane numbers. But A is square, therefore C is also square.
Therefore, if three numbers are in continued proportion, and the first is square, then the third is also square.
Q.E.D.