On an upright Infinite-Plane let us suppose 2 horizontal Infinite-Lines, having a common perpendicular, and therefore of course never intersecting. Let us call them 'No. 1' and 'No. 2,' No. 1 being above No. 2: and let us suppose the common perpendicular to be an inch long, so that the 2 Lines are the edges of an Infinite-Strip, whose uniform width is an inch. Now let us suppose Line No. 1 to begin to revolve about the upper end of the common perpendicular. The believer in the absolute truth of Euclid's Axiom is bound to believe that, in the very act of beginning to revolve, it also begins to intersect No. 2: he cannot allow the one process any such start of the other as might enable him to say "No. 1 has begun to revolve, but has not yet begun to intersect No. 2": such a state of things must be, in his view, absolutely impossible. 'And what's impossible ca'n't be, And never never comes to pass!' Very good. The actual process of beginning to intersect No. 2 is a deep mystery, no doubt: but that the thing happens is quite undeniable. We are able to say "Now it isn't intersecting No. 2—and now it is!" So, though we cannot conceive how it managed to begin, it has certainly done it. Now let us suppose another horizontal Line, 'No. 3,' lying an inch below No. 2. The believer in Euclid's Axiom is bound to assert, as to No. 3, exactly what he asserted as to No. 2. Has he, then, any logical escape from the conclusion that the revolving Line begins to intersect Nos. 2 and 3 together? I see none, myself. And yet how can it get at No. 3, without first going through No. 2? Any point on No. 1 (I am careful not to say 'every': I know how gleefully the logical Reader would swoop down upon me with the crushing sarcasm "What does 'every' mean, when there is no limit to the number of points?": but 'any' is a safe epithet) is amenable to the simple rule of[Pg 57] "First come, first served. Cross No. 2 first: and afterwards (not by any means simultaneously) you have our gracious permission to cross No. 3." Now, what is true of any point on the Infinite-Line No. 1 is surely true of that Infinite-Line itself? To say "No. 1 intersects No. 3" is tantamount to saying "A certain point of No. 1 has reached No. 3." And how did that Point get below No. 2? Of two things, one. Either some point of No. 1 has crossed Nos. 2 and 3 at the same moment: or else no point of No. 1 has crossed No. 3 until after it had crossed No. 2. That is a logical Dilemma. Which of its two horns does the Reader prefer? The choice of the first horn involves the Reader's acceptance of ubiquitous points! In the event of his choosing the second horn, he seems logically bound to admit it as a possible state of things, that No. 1 should have begun to revolve, and yet should not have begun to intersect No. 3—which is a surrender of his belief in the universal truth of Euclid's Axiom. My final answer, then, to the question "Is Euclid's Axiom true?" is as follows:—