You enter a room, where there is a book-case containing (say) five shelves, and your eye wanders carelessly along a shelf, making a rough estimate of the number of books in it. Now shut your eyes, and try to guess how many books there are altogether. Your hasty reckoning of one shelf gave a total (say) of 19 or 20, you are not sure which: so you feel safe in saying "I think there are about a hundred." "Are you certain," I ask, "that there are more than fifty?" "Quite certain," you reply. "And also certain that there are less than a hundred and fifty?" "Quite certain," you repeat. Here, then, are the three districts. The numbers up to 50 are certainly too small; the numbers over 150 are certainly too great; the intermediate numbers contain some doubtful ones—the most doubtful being very near 100—and this doubt shades off into certainty as we approach either of the out-lying districts. You would not risk five shillings on the chance of the true number being under 100, or on the chance of its being over 100, but you would feel quite at your ease, if told that you would forfeit a thousand pounds, in case the number turned out to be under 50, or over 150. Another objection, that has already been raised to my Axiom, and so will probably be raised again, and which I may as well meet here by anticipation, is that, on the supposition of Euclid I. 32 not being true, it may be proved that this relationship of magnitude, between the Tetragon and the Segment, changes as the Circle increases, until, with an infinitely great Circle, the Tetragon may[Pg xxiii] actually be proved to be less than the Segment! This phenomenon, however, does not appal me so much as might be expected: for I have often observed it to occur that, when Theorem logically leads to Theorem , then, on the supposition of Theorem not being true, it may be proved that Theorem also is not true. (The second sequence is, in fact, what De Morgan calls the 'contranominal' of the first.) Hence this objection, if worth anything, proves too much: to dispute the validity of an argument, on the ground that, if it were valid, its contranominal would also be valid, is to upset the whole edifice of Logic itself: and, if you tell me, on such grounds as these, that I cannot prove what I assert, I may fairly retort upon you, that you cannot prove anything at all! You have destroyed the only machinery available for the purpose, and must henceforth dispense with all Logical methods, and console yourself with the cynical American adage "There's nothing true: and there's nothing new: and it don't signify!"