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

Set out a rational straight line *B*, and let the square on *C* equal the rectangle *B* by *A*. Then *C* is irrational, for that which is contained by an irrational and a rational straight line is irrational.

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 will produce as breadth a medial straight line.

Again, let the square on *D* equal the rectangle *B* by *C*. Then the square on *D* is irrational.

Therefore *D* is irrational, 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, will produce *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.