For let A by multiplying itself make D.
Therefore D is square.
Since then 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 A, B are similar plane numbers,
therefore one mean proportional number falls between A, B. [VIII. 18]
But, if numbers fall between two numbers in continued proportion,
as many as fall between them, so many also fall between those which have the same ratio; [VIII. 8]
so that one mean proportional number falls between D, C also.
And D is square;
therefore C is also square. [VIII. 22]
Q. E. D.
Complete text
Book I