Page:Carroll - Euclid and His Modern Rivals.djvu/60

22 By 'coincidental Lines,' then, I shall mean Lines which either coincide or would do so if produced: and by 'intersectional Lines' I shall mean Lines which either intersect or would do so if produced; and, by 'separational Lines,' Lines which have no common point, however far produced.

In the same way, when I speak of 'Lines having a common point,' or of 'Lines having two common points,' I shall mean Lines which either have such points or would have them if produced.

It will also save time and trouble to agree on the use of a certain conventional phrase respecting transversals.

It admits of easy proof that, if a Pair of Lines make, with a certain transversal, either (a) a pair of alternate angles equal, or (b) an exterior angle equal to the interior opposite angle on the same side of the transversal, or (c) a pair of interior angles on the same side of the transversal supplementary; they will make, with the same transversal, (d) each pair of alternate angles equal, and (e) every exterior angle equal to the interior opposite angle on the same side of the transversal, and (f) each pair of interior angles on the same side of the transversal supplementary.

You will accept that as a simple Theorem, though with a somewhat lengthy enunciation?

Min. Certainly.

Euc. The phrase I propose is as follows. When I speak of a Pair of Lines as 'equally inclined to' a transversal, I wish it to be understood that they fulfil some one of the three conditions (a), (b), (c), and therefore all the three conditions (d), (e), (f).