Kitap 10
114
And it is the side of the rectangle CD, AB.
114
Therefore etc.
114
Porism. And it is made manifest to us by this also that it is possible for a rational area to be contained by irrational straight lines. Q. E. D.
PROPOSITION 115.
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.
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.
115
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]
115
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.
115
Again, let the square on D be equal to the rectangle B, C; therefore the square on D is irrational. [deduction from X. 20]
115
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.
115
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.
