two? "It cannot be," say Messrs. Walmsley and Co.: "these Lines must intersect, if produced far enough." What? In consequence of their relative situation? That relative situation being, for each pair of them, that they make with a certain transversal two interior angles together less than two right angles? In assuming this, I very much fear that Messrs. Walmsley and Co. are performing the not-wholly-unprecedented feat of assuming Euclid's 12th Axiom! § 6. [Pg 73] The Outlook. In conclusion let me address myself to the young and eager explorer who, Alpine-staff in hand, and duly furnished with all the necessaries for his perilous quest—pick-axe, theodolite, paper-collars, Brown's Sticking-Plaster, Jones's Cough-Pills, and Robinson's Insect-Powder, "the only known remedy for Phlebitis"—is preparing to sally forth, to do or die! [Pg 74] To him let me address myself, as being, perchance, an older and a more experienced traveller—one who has wandered much, and pondered long, and who can best describe himself in the words of a lady well-known in the literary world (I am glad to have this opportunity of recording them, as they have never been printed. They were written "for music," for which purpose, I imagine, the amount of sense required is not excessive). "I have wandered, I have pondered, I have squandered Many a boon: In the sadness, In the gladness, In the madness Of the moon. "Seek thy pillow By the billow, Where the willow Doth not weep: Few will wonder Who lies under, Hearing thunder, Fast asleep!" Poetry like this speaks for itself: vain were it to hope that any poor words of mine could serve to illuminate, or even elucidate, its almost ethereal beauty! To what point of the compass, then, should this young and eager explorer be advised to direct his steps? I think his best chance—and that only a slender one—is to find some elementary proof for my Axiom, or for one of the many Theorems which will serve the same purpose, a few of which I will enumerate. In the first place, any Polygon will do, and any ratio, between it and the out-lying Segment, so long as[Pg 75] it is a finite ratio. What I want it for is to prove that there is some isosceles Triangle, with a definite vertical angle (i. e. some named fraction of a right angle), whose base is less than one of its sides. And that, again, is wanted in order to prove it possible to draw, on a given base, an isosceles Triangle, whose base-angles shall have some nameable value. And that, again, is wanted in order to prove that there is some finite minimum value for the sum of the angles of a Triangle. And that, again, is wanted in order to prove Prop. 3, at p. 19. So, if any one of these propositions could be either assumed as an Axiom or proved as a Theorem, it would suffice for the proof of Euc. I. 32 and Co.