Page:Carroll - Euclid and His Modern Rivals.djvu/65

. II. § 4.] :(4) a point may be found on each, whose distance from the other shall exceed any assigned length;
 * (5) they have different directions.

And thirdly, if we are told that a certain Pair of Lines fulfil some one of the following conditions:—
 * (1) they have a separate point, and are equally inclined to a certain transversal;
 * or (2) they have a separate point, and one of them has two points on the same side of, and equidistant from, the other;

we may conclude that they are separational.

Min. Why not use your own word 'parallel'?

Euc. Because that word is not uniformly employed, by modern writers, in one and the same sense. I would advise you, in discussing the works of my Modern Rivals, to disallow the use of the word 'parallel' altogether, and to oblige each writer to adopt a word which shall express his own definition.

Min. When you speak of two points on one Line, which are on the same side of the other, being 'equidistant from it,' do you include the case of their lying on the other Line?

Euc. Certainly. You may take them as lying on either side you like, and at zero-distances. The only case excluded is, where both points are outside the other Line, and on opposite sides of it.

Min. I understand you.