Page:Carroll - Euclid and His Modern Rivals.djvu/115

.] And then we proceed to consider where their 'productions' have got to.

Min. Like 'little Bo-peep,' you are anxious about their tails in fact; taking their 'heads' to be the ends which at first intersected the given Line.

Nie. We say that there are only three conceivable cases: one, where the tails fall next to the given Line; another, where the heads fall next to it; the third, where the tail of each coincides with the head of the other.

Min. I admit all that.

Nie. The first case we say is inadmissible because, if it were true, any Line through P, lying within the angle formed by the head of one ray and the tail of the other, would cut the given Line both ways.

Min. A reductio ad absurdum, no doubt; but it only holds good on the supposition that you can draw Lines through P, so as to lie within that angle. But this supposition requires a finite angle. If we suppose that, the moment one ray begins to revolve so as to bring its head nearer to the given line, it instantly coincides with the other ray, head with tail and tail with head, it will not then be possible to draw any such Line as you suggest: and then where is your reductio ad absurdum?

Nie. We do not seem to have noticed that case.

Min. In point of fact, your three cases are really five.