Page:Carroll - Euclid and His Modern Rivals.djvu/70

32 separate point and identical directions, have no common point, i.e. are separational.'

Euc. They are valid deductions, but in neither case do we know the 'subject' to be real.

Min. The 'contranominality'—if such a fearful word be allowable—of 17, 18, 19, seems obscure.

Euc. I will do what I can to make it less so.

Let us name the three Lines 'A, B, C.'

Then 17 may be read 'A Line (C), which has a point in common with one (A) of two coincidental Lines (A, B), has a point in common with the other (B) also.'

From this we may deduce two Contranominals.

The first is 'If A, C, have a common point; and B, C, are separational: A, B have a separate point.' That is, 'a Line (A), which has a point in common with one (C) of two separational Lines (B, C), has a point separate from the other (B)': and thus we get 18.

The other Contranominal is 'If A, C, have a common point; and A, B, have a common point; and B, C, are separational: A, B, are intersectional.' That is, 'A Line (A), which has a point in common with one (C) of two separational Lines (B, C), and also a point in common with the other (B), is intersectional with that other (B).'

But we may evidently interchange B and C without interfering with the argument, and thus prove that A is also intersectional with C. Hence A is intersectional with both: and thus we get 19.

Min. That is quite clear.

Euc. We will now go a little further into the subject of separational Lines, as to which Table I. has furnished us