Let there be set out two rational straight lines A, B commensurable in square only and such that the square on A, being the greater, is greater than the square on B the less by the square on a straight line commensurable in length with A. [X. 29]
And let the square on C be equal to the rectangle A, B.
Now the rectangle A, B is medial; [X. 21]
therefore the square on C is also medial;
therefore C is also medial. [X. 21]
Let the rectangle C, D be equal to the square on B.
Now the square on B is rational;
therefore the rectangle C, D is also rational.
And since, as A is to B, so is the rectangle A, B to the square on B,
while the square on C is equal to the rectangle A, B, and the rectangle C, D is equal to the square on B,
therefore, as A is to B, so is the square on C to the rectangle C, D.
But, as the square on C is to the rectangle C, D, so is C to D;
therefore also, as A is to B, so is C to D.
But A is commensurable with B in square only;
therefore C is also commensurable with D in square only. [X. 11]
And C is medial;
therefore D is also medial. [X. 23, addition]
And since, as A is to B, so is C to D, and the square on A is greater than the square on B by the square on a straight line commensurable with A,
therefore also the square on C is greater than the square on D by the square on a straight line commensurable with C. [X. 14]
Complete text
Book I