Page:Carroll - Euclid and His Modern Rivals.djvu/119

.] made is, that the third case' (i.e. case ) only happens, and this will give us a new axiom.' (p. 67.)

Min. You may assume it as an axiom, if you like. Then you will merely be in the same boat with Playfair. But if you are going to discuss the consequences of its being true, and get anything out of that, look to your feet! There are pitfalls about!

Nie. 'In the second case' (i.e. case ) 'we should have to—'

Min. Oho! Then it is case, after all, that you are provisionally assuming as true?

Nie. Apparently so.

Min. Well, go on. You are on the right track now.

Nie. In this case we should have to turn the ray 'through a finite angle' before its tail would cut the given Line: 'or there would be an indefinite number of Lines through P which do not cut' it. (p. 68.)

Min. What do you mean by 'or'? That one result would follow, or the other, but not both?

Nie. We mean that the two results are equivalent.

Min. Then you should say 'that is.' 'Or' is misleading. However, I grant you that this consequence would follow, if case were true. What then? Is there any obvious absurdity in such a consequence?

Nie. That we do not assert. We merely make the remark—and we now proceed to case.

Min. A weak and pointless remark: but let that pass. Do you omit case ?

Nie. We do. We proceed thus. 'But in the third case (i.e. in case ) there would be only one Line through