Page:Carroll - Euclid and His Modern Rivals.djvu/62

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Let us now consider the properties of Pairs of Lines.

Such pairs may be arranged in three distinct classes. I will take them separately, and enumerate, for each class, first the 'subjects,' and secondly the 'predicates,' of Propositions concerning it.

Min. Let us make sure that we understand each other as to those two words. I presume that a 'subject' will include just so much 'property' as is needed to indicate the Pair of Lines referred to, i.e. to serve as a sufficient Definition for them?

Euc. Exactly so. Now, if we are told that a certain Pair of Lines fulfil some one of the following conditions:—

(1) they have two common points;

or (2) they have a common point, and are equally inclined to a certain transversal;

or (3) they have a common point, and one of them has two points on the same side of, and equidistant from, the other;

or (4) they have a common point and identical directions;

we may conclude that they fulfil all the following conditions:—

(1) they are coincidental;

(2) they are equally inclined to any transversal;

(3) they are 'equidistantial,' i. e. any two points on one are equidistant from the other;

(4) they have identical directions.