"If the vertical angle of a Sector of a Circle be divided by radii into equal angles, thus forming equal Sectors; and if the chord of each such Sector be not less than the radius of the Circle: the original Sector is not less than times the Triangle cut off from it by its chord." My controversy with Nature, on this enunciation, will be best given in the form of a dialogue. (Of course I quote verbatim.) Nature. (Dec. 6, 1888.) "How are the figures to be constructed, if n be greater than 2?" [Pg xi] Author. (In the Preface to the 2nd Edition, at p. x.) "Well, suppose n were equal to 4: i.e. we have to divide the vertical angle into 24 equal parts. Bisect it: that gives halves. Bisect the halves: that gives quarters. Bisect again: that gives eighths. Bisect once more: that gives sixteenths. Voila tout!" Nature. (June 13, 1889.) "Shade of Euclid! Who knows not such things? We admitted the same, but stated that our difficulty in the construction was the condition imposed in the enunciation: viz., 'the chord of each such sector not less than the radius of the circle.' Take Mr. Dodgson's illustration of a sixteenth: this would necessitate that the original angle should be at least 960° ... we ... have further noted that no one of the chords in Mr. Dodgson's figures is even equal to the radius." Author. "What you call 'the condition imposed' is introduced with an 'if': it is merely an hypothesis: all I undertake to prove is that, if certain things were true, certain other things would be true. Surely I need not remind you that the validity of a Syllogism is quite independent of the truth of its Premisses! 'I have sent for you, my dear Ducks', said the worthy Mrs. Bond, 'to enquire with what sauce you would like to be eaten?' But we don't want to be killed!' cried the Ducks. 'You are wandering from the point' was Mrs. Bond's perfectly logical reply. So here. 'I beg you to observe, my dear Nature, that, if the chord of each Sector were not less than the radius, the logical result would be so-and-so.' 'But the chord is less than the radius!' you cry. All I need say, in reply, is 'You are wandering from the point.'" "But I will be generous, and will say more. I take exception to two assertions of yours. Remember our[Pg xii] logical stand-point. We may use Euclid's Axioms, all but the last; and his Propositions as far as I. 28. Now be good enough to prove to me, with this machinery, first, that my hypothesis 'necessitates that the original angle should be at least 960°'; secondly, that 'no one of the chords' in my Figure 'is even equal to the radius.' Your logical position is, I fear, this. You dispute the validity of a certain argument, on the ground that its premisses are false. My reply is, first, that you cannot prove them false; and secondly, that, even if you could, it wouldn't affect the question!"