Page:Carroll - Euclid and His Modern Rivals.djvu/117

.] P, in the angular space contained between the head of one ray and the tail of the other: and all such Lines will intersect the given line both ways.

In case this absurdity does not arise: all Lines, through P, intersect the given Line one way or the other: there is no instance of a Line intersecting it both ways, nor of one wholly separate from it.

In case a number of Lines may be drawn as in case : and all such Lines will be wholly separate from the given Line.

In case the two rays themselves, as drawn in the figure, are wholly separate from the given Line: but no other such Line can be drawn through P.

In case there is only one Line through P wholly separate from the given Line.

Now let us hear what you make of these five cases.

Nie. We exclude case, as I told you just now, by a reductio ad absurdum. Case we have failed to notice.

Min. True: but it can be excluded by Euc. I. 27: so that if you can manage, by pure reasoning, from ordinary Axioms, and without assuming any disputable Axiom, to exclude cases and, you will have achieved what geometricians have been vainly trying to do for the last two thousand years!

Nie. We go on thus. 'But our Axioms are not sufficient to decide which of the remaining two cases actually does occur.' (p. 67.)

Min. Or rather 'the remaining three cases.'

Nie. 'In looking at the figures the reader will at once