In Ax. 9 (α) combined with Ax. 9 (β), you assert that Lines, which make equal angles with a certain transversal, do so with all transversals. This I believe to be the most unaxiomatic Axiom ever yet proposed.
(4) You furnish no practical test for the meeting of finite Lines, and consequently you never prove (however necessary for the matter in hand) that any particular Lines will meet. And when we come to examine what practical test can possibly be extracted from your Axioms, the only result is an imperfect edition of Euclid's 12th Axiom!
The sum total of the chief defects which I have noticed is as follows:—
- fourteen of Euclid's Theorems in Book I. omitted;
- seven unaxiomatic Axioms;
- six instances of 'Petitio Principii.'
The abundant specimens of logical inaccuracy, and of loose writing generally, which I have here collected would, I feel sure, in a mere popular treatise be discreditable—in a scientific treatise, however modestly put forth, deplorable—but in a treatise avowedly put forth as a model of logical precision, and intended to supersede Euclid, they are simply monstrous.
My ultimate conclusion on your Manual is that it has no claim whatever to be adopted as the Manual for purposes of teaching and examination.
