As a rather interesting example of the ease with which an unwary explorer may tumble into a pit-fall, I may refer to a "Note on Euclid's 12th Axiom" by a Mr. W. Hanna, which will be found at p. 27 of Vol. XIII of "Mathematical Questions" reprinted from the "Educational Times." Mr. Hanna takes two Lines, situated as in Euclid's Axiom, and drops a perpendicular, from a point on one, upon the other: from the foot of this he drops a second perpendicular back upon the first line: and so on, backwards and forwards, till the diagram slightly resembles the side of a laced-up boot. He then easily proves that these perpendiculars continually decrease in length. From this he infers that "the perpendicular will ultimately become less than any assignable line"! The fallacy is really too obvious to be worth pointing out. Another writer in the "Educational Times" (Mr. J. Walmsley, B.A.: his article will be found at p. 103 of Vol. XVII of the Reprint) has fallen into a rather less obvious trap. He endeavours to prove that "any straight line, perpendicular to one of two parallel straight lines, will meet the other." (This, if it could be proved without assuming any disputable Axiom, would indeed be a splendid success! Mr. J. Walmsley has hardly realised, as yet, the fearful difficulty of persuading two Lines, under any conceivable circumstances, to do such a thing as "meet." Whether, at the outset of geometrical discovery, Lines were not properly introduced to each other—or whether some mischief-making Point has been insinuating that one of them went and intersected another when it was looking the other way—certain it is, that Lines will do almost anything you like to propose, rather than "meet" one another!) Mr. Walmsley assumes, as axiomatic, that when one Line lies between two others (whatever "between" may mean), those two others lie on opposite sides of it. Now let Mr. Walmsley (or any other champion of his theory) draw three Lines diverging from a Point at equal angles (of 120° each), and thus dividing the infinite Plane into three equal Sectors. In each of these Sectors let him draw a branch of a Hyperbola, having the sides of the Sector as its Asymptotes. Now, each of these Hyperbolæ lies (in a way) "between" the other two: and yet no two can be said to lie on opposite sides of the other one! "But," says Mr. Walmsley (or the champion aforesaid) "these are Curves, not straight Lines!" Most true: but how are you to know that straight Lines will not behave just like Hyperbolæ, if only they are put far enough apart? Produce those three radiating Lines, that we began with, until each is a million miles long (paper, pen, and ink, provided regardless of expense): then, across their extremities, draw three Lines perpendicular to them. Why shouldn't these three Lines (of course shunning a "meeting," as all Lines do) perpetually face inwards, so that no one of them ever commits the discourtesy of turning its back upon either of the other