Page:Carroll - Euclid and His Modern Rivals.djvu/158

120 Nie. (sighing deeply) You are heaving the lead again!

Min. I am: but we shall be in calmer water soon.

We also know that the 'Coincidental' class possesses two properties—they are coincidental and have identical directions; and that the 'Intersectional' class also possesses two properties—they are intersectional and have different directions.

Now if you choose to frame a Definition by denying one property of each of these two classes, any Pair of Lines, so defined, is excluded from both of these classes, and must, if it exist at all, belong to the 'Separational' class. Remember, however, that you may have so framed your Definition as to exclude your Pair of Lines from existence. For instance, if you choose to combine two contradictory conditions of direction, and to say that Lines, which have identical and intersectional directions, are to be called so-and-so, you are simply describing a nonentity.

Nie. That is all quite clear.

Min. Your Definition, then, amounts to this:—Lines, which are not coincidental, but which have identical directions, are said to be 'sepcodal.'

Nie. It does.

Min. Well, here is another Definition for Parallels, which will answer your purpose just as well:—'Lines, which are not intersectional, but which have different directions.'

Nie. But I think I can prove to you that you have now done the very thing you cautioned me against: you have annihilated your Pair of Lines.

Min. That is a matter which we need not consider at present. Proceed.