Page:Carroll - Euclid and His Modern Rivals.djvu/216

178 read 'with the conception of straightness in a Line we naturally associate that of the utmost possible shortness of path between any two of its points; allow this to be assumed, &c.' This I consider a most objectionable Axiom, obliging us, as it does, to contemplate the lengths of curved lines. This matter I have already discussed with M. Legendre (p. 56).

Secondly, for the host of new Axioms with which we are threatened in the Preface, I have searched the book in vain: possibly I have overlooked some, as he never uses the heading 'Axiom,' but really I can only find one new one, at p. 5. 'Every angle has one, and only one bisector,' which is hardly worth stating. Perhaps the writer means that his proofs are not so full as those in Euclid, but take more for granted. I do not think this any improvement in a book meant for beginners.

Another change, claimed in the Preface as an improvement, is the more constant use of superposition. I have considered that point already (p. 47) and have come to the conclusion that Euclid's method of constructing a new figure has all the advantages, without the obscurity, of the method of superposition.

I see little to remark on in the general style of the book. At p. 21 I read 'the straight Line AI satisfies the four following conditions: it passes through the vertex A, through the middle point I of the base, is a perpendicular on that base, and is the bisector of the vertical angle. Now, two of these four conditions suffice to determine the straight Line AI,… Hence a straight Line fulfilling any two of these four conditions necessarily fulfils the other two.' All