Page:Carroll - Euclid and His Modern Rivals.djvu/107

.] new Theorems. In point of 'axiomaticity' I do not think there is much to choose between the two methods. But in point of brevity, clearness, and suitability to a beginner, I give the preference altogether to Euclid's axiom.

The next subject to consider is your practical test, if any, for two given Lines meeting when produced.

Nie. One test is that one of the Lines should meet a Line parallel to the other.

Min. Certainly: and that will suffice in such a case as Euc. I. 44 (Pr. M. p. 60, in this book) though you omit to point out why the Lines may be assumed to meet. But what if the diagram does not contain 'a Line parallel to the other'? Look at Pr. (h) p. 69, where we are told to make, at the ends of a Line, two angles which are together less than two right angles, and where it is assumed that the Lines, so drawn, will meet. That is, you assume the truth of Euclid's 12th Axiom. And you do the same thing at pp. 70, 123, 143, and 185.

Nie. Euclid's 12th Axiom is easily proved from our Theorems.

Min. No doubt: but you have not done it, and the omission makes a very serious hiatus in your argument. It is not a thing that beginners are at all likely to be able to supply for themselves.

I have no adverse criticisms to make on the general style of the book, which seems clear and well written. Nor is it necessary to discuss the claims of the book to supersede Euclid, since the writer makes no such claim, but has been careful (as he states in his preface) to avoid any arrangement incompatible with Euclid's order. The