Page:Carroll - Euclid and His Modern Rivals.djvu/64

26

By examining the second figure (in which, as you see, there are three points with double names) we find that the alternate angles AGF, EHD, have become vertical angles; that the exterior and interior opposite angles EGB, EHD, have become the same angle; and that the two interior angles BGF, DHE, have become adjacent angles.

Min. That is quite clear.

Euc. Let us do on to the second class of Pairs of Lines.

If we are told that a certain Pair of Lines fulfil some one of the following conditions:—
 * (1) they have a common point and a separate point;
 * or (2) they have a common point, and are unequally inclined to a certain transversal;
 * or (3) they have a common point, and one of them has two points not-equidistant from the other;
 * or (4) they have a common point and different directions;

we may conclude that they fulfil all the following conditions:—
 * (1) they are separational;
 * (2) they are unequally inclined to any transversal;
 * (3) any two points on one, which are on the same side of the other, are not equidistant from it;