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 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:—

"If the vertical angle of a Sector of a Circle be divided by radii into equal angles, thus forming equal Sectors; and if the chord of each such Sector be not less than the radius of the Circle: the original Sector is not less than times the Triangle cut off from it by its chord." My controversy with Nature, on this enunciation, will be best given in the form of a dialogue. (Of course I quote verbatim.)

Nature. (Dec. 6, 1888.) "How are the figures to be constructed, if n be greater than 2?"

Author. (In the Preface to the 2nd Edition, at p. x.) "Well, suppose n were equal to 4: i.e. we have to divide the vertical angle into 2⁴ equal parts. Bisect it: that gives halves. Bisect the halves: that gives quarters. Bisect again: that gives eighths. Bisect once more: that gives sixteenths. Voila tout!"

Nature. (June 13, 1889.) "Shade of Euclid! Who knows not such things? We admitted the same, but stated that our difficulty in the construction was the condition imposed in the enunciation: viz., 'the chord of each such sector not less than the radius of the circle.' Take Mr. Dodgson's illustration of a sixteenth: this would necessitate that the original angle should be at least 960° ... we ... have further noted that no one of the chords in Mr. Dodgson's figures is even equal to the radius."

Author. "What you call 'the condition imposed' is introduced with an 'if': it is merely an hypothesis: all I undertake to prove is that, if certain things were true, certain other things would be true. Surely I need not remind you that the validity of a Syllogism is quite independent of the truth of its Premisses! 'I have sent for you, my dear Ducks', said the worthy Mrs. Bond, 'to enquire with what sauce you would like to be eaten?' But we don't want to be killed!' cried the Ducks. 'You are wandering from the point' was Mrs. Bond's perfectly logical reply. So here. 'I beg you to observe, my dear Nature, that, if the chord of each Sector were not less than the radius, the logical result would be so-and-so.' 'But the chord is less than the radius!' you cry. All I need say, in reply, is 'You are wandering from the point.'"

"But I will be generous, and will say more. I take exception to two assertions of yours. Remember our logical stand-point. We may use Euclid's Axioms, all but the last; and his Propositions as far as I. 28. Now be good enough to prove to me, with this machinery, first, that my hypothesis 'necessitates that the original angle should be at least 960°'; secondly, that 'no one of the chords' in my Figure 'is even equal to the radius.' Your logical position is, I fear, this. You dispute the validity of a certain argument, on the ground that its premisses are false. My reply is, first, that you cannot prove them false; and secondly, that, even if you could, it wouldn't affect the question!"

At p. 19 of the 2nd Edition, the new Axiom, on which my Theory rests, (which re-appears, in a modified form, at p. 14 of the 3rd Edition), stood thus:—"In every Circle, the inscribed equilateral Hexagon is greater than any one of the Segments which lie outside it." My controversy with the Athenæum, on this Axiom, shall also be given in the form of a dialogue.

Athenæum. (Oct. 27, 1888.) "... a stronger objection, in our opinion, is the implied assumption of the possibility of the inscribed equilateral Hexagon, a possibility which is not demonstrated till we reach the fifteenth Proposition of Euclid's fourth book."

Author. (In the Preface to the 2nd Edition, at p. xi). "But does it need demonstrating? May we not assume (1) that the Magnitude 'four right angles' contains 6-6ths of itself; (2) that it is theoretically possible to draw radii dividing it into these 6-6ths? Once grant me this, and I ask no more. I have then the logical right to join the ends of these radii, and to prove (by Euc. I. 4) that the chords are equal."

Athenæum. (Oct. 5, 1889.) "We objected that it was not consistent with the spirit or practice of Euclid's reasoning to assume the 'theoretical possibility' of a regular Hexagon inscribed in a Circle, without first proving that such a figure could be actually constructed from his three postulates. Euclid's restrictions may be arbitrary, unnecessary, cramping, vexatious, absurd—indeed, we think they deserve these and many other epithets—but there they are, and, if Mr. Dodgson accepts them, he is bound to keep his assumptions within the boundaries which they prescribe."

Author. "You're particular to a shade (as Scrooge said to Marley's ghost): however, I'll do what I can to oblige you. I presume you will be satisfied if I can, without using more of Euclid than his first 28 Propositions, construct an angle which shall be 1-6th of 4 right angles? Very good. First, then, with the help of his arbitrary Prop. I, I construct an equilateral Triangle. Next, by his unnecessary Prop. IX, I draw the bisectors of 2 of its angles. Next, by his cramping Post. 1, I join their point of intersection to the third vertex. Next, by his vexatious Prop. IV, I prove the 3 angles, whose common vertex is this point, to be equal. From which I draw the absurd conclusion that each of them is 1-3rd (and that therefore its half is 1-6th) of 4 right angles. How does that strike you?"

Another objection, to this same Axiom, appeared in the Academy for Feb. 9, 1889, viz. "What the Axiom practically assumes is the existence of similar figures." Permit me to reply, to this Reviewer, as follows:—"In what sense do you use the word 'similar'? In Euclid's, no doubt. That is to say, you charge me with assuming that, if the Circle and its inscribed Hexagon were supposed to expand, the magnitude of each angle, and the ratio subsisting between the sides which contain it, would remain constant? The 'ratio' part of the question we may set aside at once: there is no doubt that, since the figure continues to be equilateral, the ratio continues to be a ratio of unity: hence, if I needed this assumption (which I don't), I should have a perfect right to make it. All, that remains for discussion, is the assumption, which you say I have made, that each angle of the expanding Hexagon remains constant in magnitude. Will you, then, be kind enough to point out, first, where the need for any such assumption arises; secondly, where I have made any such assumption? For myself I cannot in the least see why, in estimating the area of the Hexagon, I should trouble myself about the size of its angles."

Let me take this opportunity of pointing out, once more, that not one Proposition in this Treatise depends, in the slightest degree, on the speculations about Infinities, &c., which occur in the Appendices.

The one merit, the one novelty, of my Theory (if it has any merit, or any novelty) is that, while every other Theory (that I have seen), which attempts to supersede Euclid's 12th Axiom, introduces the ideas of Infinities and Infinitesimals, mine dispenses wholly with their aid, and deals with nothing but what is, by universal consent, absolutely within the field of Human Reason.

C. L. D.

Ch. Ch., Oxford. August, 1890.