Page:Carroll - Euclid and His Modern Rivals.djvu/80

42 have the advantage. In fact, I could not form any mental picture at all of the relative position of two finite Lines, if all I knew about them was their never meeting however far produced: and it would be equally impossible to form any mental picture of the position which a Line, crossing one of them, would have relatively to the other. But, though your picture may be more easy to conceive, I doubt if it is enough so to constitute an axiom.

Euc. Taken by itself, it may be, as you say, not entirely axiomatic. But I think I can put before you a few considerations which will make it more acceptable.

Min. They will be well worth having. An absolute proof of it, from first principles, would be received, I can assure you, with absolute rapture, being an ignis fatuus that mathematicians have been chasing from your age down to our own.

Euc. I know it. But I cannot help you. Some mysterious flaw lies at the root of the subject. Probabilities are all I have to offer you.

Now suppose you were assured, with regard to two finite Lines placed before you, that, when produced in a certain direction, one of them approached the other, that is, contained two points, of which the second was nearer, to the other Line, than the first, would you not think it probable—if not absolutely certain—that they would meet at last?

Min. Utilising—as I suppose you will allow me to do—my knowledge of the properties of asymptotes, I should say 'No. The mere fact of approach, granted as to two Lines, does not secure a future meeting.'