Page:Carroll - Euclid and His Modern Rivals.djvu/249

§ 1.] that they have different directions: and then we bring in the Axiom that Lines having different directions will meet if produced.'

Euc. All that you have satisfactorily disposed of in your review of Mr. Wilson's Manual.

Min. The only other substitute, that I know of, belongs to the 'equidistant' theory, which replaces your Axiom by three or four new Axioms and six new Theorems. That substitute, also, I have seen reason to reject.

My general conclusion is that your method of treatment of all these subjects is the best that has yet been suggested.

Euc. Any noticeable innovations in the treatment of Right Lines and Angles?

Min. Those subjects I should be glad to talk over with you.

Euc. With all my heart. And now how do you propose to conduct this our final interview?

Min. I should wish, in the first place, to lay before you the general charges which have been brought against you: then to discuss your treatment of Lines and Angles, as contrasted with that of your 'Rivals'; and lastly the omissions, alterations, and additions proposed by them.

Euc. Good. Let us begin.

Min. I will take the general charges under three headings:—Construction, Demonstration, and Style. And first as to Construction:—