to overcome this difficulty in a better way than Euclid has done. We shall not give an account of these attempts. The student who wishes to examine them may consult Camerer's Euclid, Gergonne's Annales de Mathématiques, Volumes sx and xvi, the work by Colonel Perronet Thompson entitled Geometry without Axioms, the article Parallels in the English Cyclopædia, a memoir by Professor Baden Powell in the second volume of the Memoirs of the Ashmolean Society, an article by M. Bouniakofsky in the Bulletin de l' Académie Impériale, Volume v, St Petersbourg, 1863, articles in the volumes of the Philosophical Magazine for 1856 and 1857, and a dissertation entitled Sur un point de Vhisioire de la Geometrie chez les Grecs par A. J. H. Vincent. Paris, 1857.
Speaking generally it may be said that the methods which differ substantially from Euclid's involve, in the first place an axiom as difficult as his, and then an intricate series of propositions; while in Euclid's method after the axiom is once admitted the remaining process is simple and clear.
One modification of Euclid's axiom has been proposed, which appears to diminish the difficulty of the subject. This consists in assuming instead of Euclid's axiom the following; two intersecting straight lines cannot he both parallel to a third straight line.
The propositions in the Elements are then demonstrated as in Euclid up to I. 28, inclusive. Then, in I. 29, we proceed with Euclid up to the words, "therefore the angles BGH, GHD are less than two right angles." We then infer that BGH and GHD must meet: because if a straight line be drawn through G so as to make the interior angles together equal to two right angles this straight line will be parallel to CB, by I. 28; and, by our axiom, there cannot be two parallels to CB, both passing through G.
This form of making the necessary assumption has been recommended by various eminent mathematicians, among whom may be mentioned Playfair and De Morgan. By postponing the consideration of the axiom until it is wanted, that is, until after I. 28, and then presenting it in the form here given, the theory of parallel straight lines appears to be treated in the easiest manner that has hitherto been proposed.
I. 30. Here we may in the same way shew that if AB and EF are each of them parallel to CD, they are parallel to each other. It has been said that the case considered in the text is so obvious as to need no demonstration; for if AB and CD can
