PROPOSITION 115.
From a medial straight line there arise irrational straight lines infinite in number, and none of them is the same as any of the preceding.

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](elem.10.def.4)] for that which is contained by an irrational and a rational straight line is irrational. [deduction from [X. 20](elem.10.20)]

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](elem.10.20)]

Therefore D is irrational; [[X. Def. 4](elem.10.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.