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](elem.7.17)]

And, since A, B are similar plane numbers, therefore one mean proportional number falls between A, B. [[VIII. 18](elem.8.18)]

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](elem.8.8)] so that one mean proportional number falls between D, C also.

And D is square; therefore C is also square. [[VIII. 22](elem.8.22)] Q. E. D.