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[Pg xiv] 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. [Pg xv] INTRODUCTION.