Mr. Dodgson on Parallels

Abstract

MR. DODGSON has written to me thanking me heartily for my “interesting and helpful review” of his “New Theory of Parallels.” He admits his slip in the corollary on p. 11, and supposes, as I had myself thought, he took ADC to be the triangle required instead of ABF. “But there is one criticism of yours which, if true, would vitiate the whole treatise. May I ask you to reconsider the point, and, should you see reason so to do, to notify to the readers of NATURE that you withdraw it? You say that, in Props, viii., xi., I tacitly assume that the ‘amounts’ of triangles are either all greater than two right angles, or else all less... Such an assumption would indeed be monstrous.” I willingly accede to Mr. Dodgson's request, as the following form of his argument, supplied in his letter to me, does away with, my difficulty. “Either (α) there is a triangle whose ‘amount’ = two right angles, or (β) there is none. If (β) be true, then either (βI) all triangles have greater ‘amounts’, or (2) all have less amounts, or (3) some have greater amounts and all others less. Now (β1) is proved impossible, in Prop. viii.; (β2) is proved impossible in Prop. xi.; (β3) may easily be proved impossible, by means of Prop. vii. Hence (β) is impossible. Hence (α) is true.” It will be well, if, in a future edition, the missing link of (β3) be supplied. One other point puzzles Mr. Dodgson. It is my remark on Prop. vi.: “How are the figures to be constructed if n>2?” Mr. Dodgson says: “It surely does not need pointing out that the operation of bisecting an angle may be repeated ad libitum.” Certainly not. But what I meant was the effect of the e bisections upon the resultant chords. The figures to the proposition are incorrectly drawn: in the one figure: BD, DC, and in the other BE, ED, DF, FC are not drawn greater than the radius, and my point was not the bisections but the enlargement of the figure: thus if n = 3, we should have eight triangles, vertices at the centre A, with the sum of their angles greater than 480°. My apology for thus trespassing upon valuable space is my desire to meet Mr. Dodgson's natural wish, and by pointing out what I thought were faults in his “interesting” brochure, to enable him to make it more perfect in after editions.