Before discussing the first of these two Axioms, permit me to remind the Reader that the question before us is not "is it true?" but "is it axiomatic?", that is, 'does the average human intellect accept it as true, without proof?' It is a question in Mental Physiology rather than in Geometry. The 47th Proposition of Euclid is quite as true as the Axiom 'things that are equal to the same are equal to one another'; but the average human intellect, while accepting the latter without proof, does most certainly demand a good deal of proof before it will accept the former. Intellectual beings may conceivably exist, to whom Euc. I. 47 is axiomatic; but our books are not written for them. Now there is one preliminary step, that is absolutely indispensable before the human intellect can accept any Axiom[Pg 68] whatever: and that is, it must attach some meaning to it. We cannot, rationally, either assent to, or deny, any Proposition the words of which convey to us no idea. We have, then, two questions to answer: first, "what idea is conveyed to the average human intellect by the phrase 'in the same direction,' when applied to non-coincident Lines?", secondly, "in accepting the Axiom, that 'Lines, which have the same direction, make equal angles with all transversals,' to what other assertions are we committing ourselves?" If we contemplate a fixed Point in a Plane, and imagine one or more Lines passing through it, it is not difficult to grasp the following ideas—that the 'direction' of any such Line is that property of it which determines its position, now that one Point in it is already fixed—that any two such Lines form 4 angles, whose common vertex is the fixed Point—that, if one of those 4 angles be zero, the 2 Lines coincide; if not, they intersect—that, in the first case, they have the same direction, in the second, different directions—and that the difference of the directions of such Lines is measured by the angle between them. But these ideas are of little use to us, when confronted with a given Line and a given Point outside it, and when told to imagine a new Line drawn, through the given Point, and 'in the same direction' as the given Line, the 2 Lines having no common Point. For the directions of the Lines are no longer directly comparable. There is no use in asking "do they contain a zero-angle?" when they contain no angle whatever. An angle cannot exist without a vertex: and where is the vertex?