[TURN OVER. A NEW THEORY OF PARALLELS In every Circle, the inscribed equilateral Tetragon is greater than any one of the Segments which lie outside it. Curiosa Mathematica PART I A NEW THEORY OF PARALLELS BY CHARLES L. DODGSON, M.A. Student and late Mathematical Lecturer of Christ Church Oxford FOURTH EDITION PRICE TWO SHILLINGS London MACMILLAN AND CO. 1895 [All rights reserved] Oxford HORACE HART, PRINTER TO THE UNIVERSITY [Pg ix] PREFACE TO THIRD EDITION. The chief novelty, in the First Edition of this treatise, was the Axiom, by means of which I proved Euc. I. 32 without making use of his 12th Axiom. And the chief novelty, in this Third Edition, is the change I have made in that Axiom, by substituting 'Tetragon' for 'Hexagon'. The new Figure is more simple, and more easily constructed, than its predecessor: while the Axiom is, I hope, as obviously true as ever. The proof of my "New Theory of Parallels" is, I think, greatly simplified and improved in this new Edition—the Propositions, which do not require any disputable Axiom, being placed by themselves in 'Book I,' while those, which require the new Axiom for their proof, are placed in 'Book II.' At the end of Book II will be found a proof (so far as finite magnitudes are concerned) for Euclid's celebrated 12th Axiom, preceded by, and dependent on, the Axiom tacitly assumed by him in his Book X, Prop. 1, and also assumed, I believe, by every subsequent writer who has attempted to prove his 12th Axiom. My proof is borrowed, with some slight alterations, from Cuthbertson's 'Euclidean Geometry.' One advantage, in thus separating the Propositions into two classes, is that it calls attention to the very remarkable[Pg x] and interesting fact that the Theorem "There is a Triangle whose angles are together not-greater than two right angles" is actually provable without any disputable Axiom whatever. If only it could be proved, with equal ease, that "there is a Triangle whose angles are together not-less than two right angles"! But alas, that is an ignis fatuus that has never yet been caught! The man, who first proves that Theorem, without using Euclid's 12th Axiom or any substitute for it, will certainly deserve a place among the world's great discoverers. I take this opportunity of replying to one or two criticisms, which have been published, on the Second Edition—earnestly assuring the writers of those criticisms that, in treating the questions at issue between us from a not-wholly-solemn point of view, I have been actuated by no feeling of disrespect towards them, but simply from the wish to lighten a subject, naturally somewhat too heavy and sombre, and thus to make it a little more palatable to the general Reader. At p. 12 of the 2nd Edition, the enunciation of Prop. VI (which re-appears, in a modified form, at p. 34 of the 3rd Edition) stood thus:—