I may as well state briefly what the feat actually is, which Mathematicians have been vainly trying, since Euclid's day, to perform. In I. 27, 28, he proves (so far, without invoking the aid of any doubtful Axiom) that "two Lines, which are equally inclined to a certain transversal (whether by making a pair of alternate angles equal, or an exterior equal to its interior opposite angle, or two interior, on the same side of[Pg xix] the transversal, together equal to two right angles), will never meet." Next, in logical order, comes his 12th Axiom, viz. that "two Lines, which are unequally inclined to a certain transversal (he only names the case where they make two interior angles together less than two right angles, but he might fairly have included the others), will meet." This Axiom, as I hope to prove in Appendix III, is only partially, and not universally, true. Next, in I. 29, he proves (with the aid of this Axiom, of which it is what De Morgan calls the 'contranominal') the partially-true Theorem that "two Lines, which never meet, are equally inclined to any transversal." And from this, in I. 32, he proves that "the three angles of a Triangle are together equal to two right angles." These are only specimens of a set of Theorems which can be proved when once Axiom 12 is granted (e.g. there are several about 'equidistantial Lines,' which Euclid has altogether ignored): but they are all so connected as to follow easily from these. Now the great difficulty, which besets this subject, is that Euclid's Axiom (this, I think, is universally admitted) is not axiomatic—the intellect has not yet occurred, among that species of Vertebrates which may be defined as 'bimanous bipeds,' which accepts it as a genuine Axiom—and the great question to be answered is "can a better Axiom be found?" In Appendix IV, I will mention some of the substitutes that have been suggested, and will give some account of the 'outlook' in the direction of the new Axiom I have chanced on. In this place it will suffice, first, to explain what the task is that the long-desiderated Axiom has to[Pg xx] perform, and secondly, to state the grounds on which I claim acceptance for my Axiom. First, then, what is 'the coming Axiom' expected to do for us? It will be convenient to divide the whole class, of Theorems needing proof, into two sub-classes—one including those which are universally true: the other those which are only partially true—the error, if any, being infinitesimal when compared with the Magnitudes with which the Theorem is concerned. Euc. I. 32 is a specimen of the one kind, and Euc. I. 29 of the other.