Curiosa Mathematica, Vol. 1: A New Theory of Parallels (Classic Reprint) - Softcover

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Excerpt from Curiosa Mathematica, Vol. 1: A New Theory of Parallels

If only it could be proved, with equal ease, that "there is a Triangle whose angles are together not-less than two right angles"! But alas, that is an ignis fatuus that has never yet been caught! The man, who first proves that Theorem, without using Euclid's 12th Axiom or any substitute for it, will certainly deserve a place among the world's great discoverers.

I take this opportunity of replying to one or two criticisms, which have been published, on the Second Edition - earnestly assuring the writers of those criticisms that, in treating the questions at issue between us from a not-wholly-solemn point of view, I have been actuated by no feeling of disrespect towards them, but simply from the wish to lighten a subject, naturally somewhat too heavy and sombre, and thus to make it a little more palatable to the general Reader.

At p. 12 of the and Edition, the enunciation of Prop. VI (which re-appears, in a modified form, at p. 34 of the 3rd Edition) stood thus: -

"If the vertical angle of a Sector of a Circle be divided by radii into 2n equal angles, thus forming 2n 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 a times the Triangle cut off from it by its chords." 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?"

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