Page:Carroll - Euclid and His Modern Rivals.djvu/59

. II. § 4.] such a Pair may be defined, and what other properties it possesses.

After that we will take a Pair of Lines which have a common point and a separate point ('a separate point' being one that lies on one of the Lines but not on the other), and which therefore have no other common point, and treat it in the same way.

And in the third place we will take a Pair of Lines which have no common point.

And let us understand, by 'the distance between two points,' the length of the right Line joining them; and, by 'the distance of a point from a Line,' the length of the perpendicular drawn, from the point, to the Line.

Now the properties of a Pair of Lines may be ranged under four headings:—
 * (1) as to common or separate points;
 * (2) as to the angles made with transversals;
 * (3) as to the equidistance, or otherwise, of points on the one from the other;
 * (4) as to direction.

We might distinguish the first two classes, which I have mentioned, as 'coincident' and 'intersecting': and these names would serve very well if we were going to consider only infinite Lines; but, as all the relations of infinite Lines, with regard to angles made with transversals, equidistance of points, and direction, are equally true of finite portions of them, it will be well to use names which will include them also. And the names I would suggest are 'coincidental,' 'intersectional,' and 'separational.'