To return to our Tetragon. It really contains the area of the Segment a little over 7 times. Hence anybody, I should suppose, would be ready to say "I am certain it contains the Segment more than twice: and I am equally certain it does not contain it twelve times." In guessing the actual number, observers would greatly differ: some might guess 4, others 10: but all would agree in putting it above 2. And now see how modest is the demand of my Axiom! Merely that you will find room in the Tetragon for one single Segment! If that is not a matter of certainty, is anything certain in this world of ours? I have yet one more arrow in my quiver: let me shoot it, and have done. If the gentle reader feels any the[Pg xxiv] smallest demur to granting me that once this Tetragon is greater than the Segment lying below it, will he grant me that twice it will suffice? Or four times it? Or eight times it? He may go on doubling as long as he likes, and, so long as he keeps among finite numbers, he will have granted me all I need for a logical proof (which will be found in Appendix I) of Euc. I. 32. Surely he will not need to go into the Infinities? And may I add, in conclusion, that, if any gentle Reader be found, who thinks it just possible to squeeze 512 of these Tetragons into the Segment, but is willing to allow that no amount of skilful packing will dispose of 1024 of them—it will give me real satisfaction to be supplied with that gentle Reader's name and address? C. L. D. Ch. Ch., Oxford. July, 1888. [Pg xxv] CONTENTS. Book I. Certain universally-true Propositions, provable from genuine Axioms. PAGE Definitions, 1 to 3 1 Axioms, 1 to 4 3 Propositions:— I. Theorem. If a Pair of Lines make, with a certain transversal, either (1) a pair of alternate angles equal, or (2) an exterior angle equal to its interior opposite angle on the same side of the transversal, or (3) a pair of interior angles on the same side of the transversal supplementary: they will make, with that transversal, (4), each pair of alternate angles equal, and (5) each of the four exterior angles equal to its interior opposite angle on the same side of the transversal, and (6) each pair of interior angles on the same side of the transversal supplementary. 4 Definition 5 " II. Theorem. If two isosceles Triangles have equal bases but unequal sides: that Triangle, which has the greater sides, has the greater area. 5[Pg xxvi] III. Problem. Given a certain angle; and given that any isosceles Triangle, whose vertical angle is not-greater than the given angle, has its base not-greater than either of its sides: to describe, on a given base, an isosceles Triangle having each base-angle equal to the given angle. 6 Corollary. The isosceles Triangle, so described, has its vertical angle not-less than either of its base-angles. 7 IV. Theorem. Either all Triangles have the same 'amount'; or else, if , , be two 'possible amounts,'