Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

But CE, ED are also commensurable with AF, FB and in the same ratio; therefore, as AF is to FB, so is KM to ML.

Therefore, alternately, as AF is to KM, so is BF to LM; therefore also the remainder AB is to the remainder KL as AF is to KM. [V. 19]

But AF is commensurable with KM; [X. 12] therefore AB is also commensurable with KL. [X. 11]

And, as AB is to KL, so is the rectangle CD, AB to the rectangle CD, KL; [VI. 1] therefore the rectangle CD, AB is also commensurable with the rectangle CD, KL. [X. 11]

But the rectangle CD, KL is equal to the square on H; therefore the rectangle CD, AB is commensurable with the square on H.

But the square on G is equal to the rectangle CD, AB; therefore the square on G is commensurable with the square on H.

But the square on H is rational; therefore the square on G is also rational; therefore G is rational.

And it is the “side” of the rectangle CD, AB.

Therefore etc.

PORISM.

PROPOSITION 115.

Let A be a medial straight line; I say that from A there arise irrational straight lines infinite in number, and none of them is the same as any of the preceding.

Let a rational straight line B be set out, and let the square on C be equal to the rectangle B, A; therefore C is irrational; [X. Def. 4] for that which is contained by an irrational and a rational straight line is irrational. [deduction from X. 20]

And it is not the same with any of the preceding; for the square on none of the preceding, if applied to a rational straight line produces as breadth a medial straight line.

Again, let the square on D be equal to the rectangle B, C; therefore the square on D is irrational. [deduction from X. 20]

Therefore D is irrational; [X. Def. 4] and it is not the same with any of the preceding, for the square on none of the preceding, if applied to a rational straight line, produces C as breadth.

Similarly, if this arrangement proceeds ad infinitum, it is manifest that from the medial straight line there arise irrational straight lines infinite in number, and none is the same with any of the preceding. Q. E. D.