"If the defect, from the sum of two right angles, be finite, the Lines certainly meet: if it be an Infinitesimal of the first order, they may meet, or not, according to circumstances: if it be an Infinitesimal of the second, third, or any higher order, they certainly do not meet." The question, with which this Appendix is headed, was sufficiently startling: but a more startling one remains to be answered. "If Euclid's Axiom be not universally true, what becomes of all the Propositions which he has made to depend upon it, such as I. 29? Has he done no more than prove them to be partially true?" To this disheartening question I will give as re-assuring an answer as the case seems to admit of. It must be admitted that Euc. I. 29 requires—if we would make it strictly true—the same limitation as the Axiom: it should run as follows:—"Two[Pg 58] Lines, which do not meet, make, with all transversals, angles which are equal so far as finite differences are concerned" (i. e. angles so nearly equal that the difference, if any, is infinitesimal). And of course Euc. I. 32, as proved from Euc. 1. 29, would need a similar qualification. It seems to me very doubtful whether Euclid ever noticed this defect in his Axiom. If he did, it is possible that he may have thought fit to ignore it, on the ground that, when Finite Magnitudes differ only by an Infinitesimal, they are, for all practical purposes, equal. But I feel bound to admit that, for the purpose of proving Euc. I. 32 to be, as it really is, universally true, neither Euclid's Axiom, nor any other that deals with intersection of Lines, will suffice: and that some Axiom, not involving that principle, must be substituted for it. Let me say in conclusion that, though I assert the absolute truth of Euclid's Axiom—with the limiting clause I have introduced, 'the defect being a finite angle'—it still remains, in my opinion, a 'disputable' Axiom; i.e. it is not properly admissible as an Axiom, but ought to be, if possible, proved as a Theorem. [Pg 59] APPENDIX III. How should Parallels be defined? We know that, if a Pair of Lines has either of the following properties (1) they are equally inclined to a certain transversal, (2) one of them contains 2 Points on the same side of, and equidistant from, the other, it has all the following properties (3) they are equally inclined to all transversals, (4) any 2 Points, on either of them, are on the same side of, and equidistant from, the other, (5) they do not meet, however far produced. Any one of these properties may be used as a Definition. Let us take them one by one.