Page:Carroll - Euclid and His Modern Rivals.djvu/180

142 the limit of size is, in my opinion, objectionable. In other respects your language, though hazy, agrees on the whole with Euclid.

(3) As to 'Parallels,' there is a good deal to be said, and that not very flattering, I fear.

In Ax. 6, you assert the reality of different Lines having the same direction—a property you can neither define, nor construct, nor test.

You also assert (by implication) the reality of separational Lines, which Euclid proves.

You also assert the reality of Lines, not known to have a common point, but having different directions—a property you can neither define, nor construct, nor test.

In Ax. 8, you assert that the undefined Lines last mentioned would meet if produced.

These Axioms, therefore, are not axiomatic.

In proving result (2), you are guilty of the fallacy 'Petitio Principii.'

In Ax. 9 and Th. 4 taken together, if the word 'angle' in Ax. 9 means 'variable angle,' you are guilty of the fallacy A dicto secundum Quid ad dictum Simpliciter; if 'constant angle,' of the fallacy 'Petitio Principii.’ In Ax. 9, you assert that Lines possessing a certain real geometrical property, viz. making equal angles with a certain transversal, possess also the before-mentioned undefined property. This is not axiomatic.

In Ax. 9 combined with Ax. 6, you assert the reality of Lines which make equal angles with all transversals. This is not more axiomatic than Euc. Ax. 12.