Page:Carroll - Euclid and His Modern Rivals.djvu/157

. § 1.] in those days) 'A plate of, and look sharp about it! I'm in a hurry!'

Min. If the gentleman wanted to catch his train—by the way, had they trains in Egypt in ancient days?

Nie. Certainly. Read your 'Antony and Cleopatra,' Act I, Scene 1. 'Exeunt Antony and Cleopatra with their train.'

Min. In that case, wouldn't it be enough to say 'A plate of '?

Nie. Most certainly not—at least not in a fashionable restaurant. But this is a digression. I am willing to adopt the word 'sepcodal.'

Min. Now, before you read any more, let us get a clear idea of your Definition. We know of two real classes of Pairs of Lines, 'coincidental' and 'intersectional'; and to these we may (if we credit you with a Corollary to Euc. I. 27, 'It is possible for two Lines to have no common point') add a third class, which we may call 'separational.'

We also know that if a Pair of Lines has a common point, and no separate point, it belongs to the first class; if a common point, and a separate point, to the second. Hence all Pairs of Lines, having a common point, must belong to one or other of these classes. And since a Pair, which has no common point, belongs to the third class, we see that every conceivable Pair of Lines must belong to one of these three classes.

We also know that