Page:Carroll - Euclid and His Modern Rivals.djvu/184

146 properties 'different' and 'having the same direction.' Bear in mind that you have yet to prove the reality of such Lines. And may I request you in future to call such Lines 'sepcodal'? But if you wish to assert any thing of them which is also true of coincidental Lines, you had better drop the 'sep-' and simply call them 'Lines which have the same direction,' so as to include both classes.

Nie. Very well.

P. 9. § 28. Th. 'Sepcodal Lines cannot meet, however far they are produced.'

Min. Or rather 'could not meet, if they existed.' Proceed.

P. 9. § 29. Th. 'Two angles are equal, when their sides have the same direction.'

Min. How do you define 'same direction' for different Lines?

Nie. We cannot define it.

Min. Then I cannot admit that such Lines exist. But even if I did admit their reality, why should the angles be equal?

Nie. Because 'the difference of direction' is the same in each case.

Min. But how would that prove the angles equal? Do you define 'angle' as the 'difference of direction' of two lines?

Nie. Not exactly. We have stated (p. 6, § 19) 'The