Page:Carroll - Euclid and His Modern Rivals.djvu/116

78 Before going any further let us have them all clearly stated.

We assume, in all three figures, that the ray-heads, as drawn, do not intersect the given line; but that either of them would, if it began to revolve towards the given Line, instantly intersect it. In other words we assume that any half-ray, drawn from P in the dotted angular space, would intersect the given Line: but that any half-ray, drawn from P in the undotted angular space, as well as the two ray-heads which limit that angular space, would not intersect it. And as to the ray-tails, it is obvious that in fig. 1 they do intersect the given Line, but in figs. 2, 3, they do not do so.

Nie. That is all clear enough.

Min. Then these are the five cases:—

Figure 1. The head of each ray must revolve downwards through a finite angle before it can coincide with the tail of the other ray.

Same figure. The head of each ray, on beginning to revolve downwards, instantly coincides with the tail of the other ray.

Figure 2. The head of each ray must revolve upwards through a finite angle before it can coincide with the tail of the other ray.

Same figure. The head of each ray, on beginning to revolve upwards, instantly coincides with the tail of the other ray.

Figure 3.

These five cases suggest a few observations.

In case a number of Lines may be drawn through