In proving the latter class, no substitute for Euclid's Axiom has yet been suggested, that I know of, which does not suffer from the same defect as Euclid's Axiom—the being only partially, and not universally, true—and which does not, if we attempt to modify the language so as to remedy this defect, in some way lead us into the bewildering region of Infinities and Infinitesimals. But the former class can, as I believe, be more easily proved. This is what I attempt in the following treatise—which owes its inspiration to a sudden thought (it occurred to me some two months ago) that it might be possible to prove Euc. I. 32 without getting mixed up with those spectral Infinities. Moreover, it is quite possible to bring into this class all that is valuable in Euc. I. 29. Regarding the 'separateness' of the Lines merely as a link between Props. 27, 28, and 29, we may combine the three into one grand Theorem, thus:—"Two Lines, which are equally inclined to a certain transversal, are so to every transversal." This Theorem, as well as Euc. I. 32, I prove in the following[Pg xxi] treatise. But the feat of proving them, without assuming any new Axiom at all, is at present beyond my grasp. Like the goblin 'Puck,' it has led me "up and down, up and down," through many a wakeful night: but always, just as I thought I had it, some unforeseen fallacy was sure to trip me up, and the tricksy sprite would "leap out, laughing ho, ho, ho!" And now, to come to the real gist of this over-long Preface—however, nobody ever reads a Preface, so really it does not matter—am I not right in thinking that, on mere inspection of this diagram, any sane intellect will be ready to grant that "in any Circle, the inscribed Tetragon is greater than any one of the Segments that lie outside it"? I shall be told, no doubt, that this is too bizarre and unprecedented an Axiom—that it is an appeal to the eye, and not to the reason. That it is somewhat bizarre I am willing to admit—and am by no means sure that this is not rather a merit than a defect. But, as to its being an appeal to the eye, what is "two straight Lines cannot enclose a space" but an appeal to the eye? What is "all right angles are equal" but an appeal to the eye? In all Axioms, where an appeal is made to the eye on a question of magnitude, we shall find, I think, that the whole region of certainty and probability may be roughly mapped out into three districts—an out-lying district of certainty in one direction, a similar one of certainty in the opposite direction, and a middle district of probability—the boundaries being shadowy and liable to be shifted hither[Pg xxii] and thither according to the fancies or prejudices of each individual mind. Permit me to illustrate this by an example taken from ordinary life.