14 Maryland Road, Harrow Road, W.—Discusses Cotterill’s paper on polygons.
(A summary of Cotterill’s paper, which was read to the London Mathematical Society on 8 Feb. 1872, was printed with the proceedings of the Society's meeting on 13 June, with a note referring to Clifford’s interest in it.)
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Transcript
14 Maryland Road
Harrow Road W.
Dear Mr Cotterill
I have read your paper on polygons {1} with great interest, but I was disappointed that the analysis was not in the part already sent in to the Society. I hope we shall have another part containing an account of the beautiful metrical theorem on which (from your verbal account of the paper) it all depends; about the area of a polygon being zero when the class. curve cn–3 {2} touching all its diagonals touches also the line infinity. I think if you were to draw a pentagon with its diagonals touching a parabola it would make a pretty illustration. It would be interesting to experiment with a planimeter or such a figure carefully drawn, and find that the area really was about zero. But I wish you would investigate the analogous theory in solid geometry. Taking a n-acron to be a polyhedron with 2(n–2) triangular faces, the diagonal planes will all touch a surface of order n–4—this is a theorem, there being more diagonal planes than are sufficient to determine it—and when this surface touches also the plane infinity the volume of the solid is zero. From the want of symmetry of this figure, however, the volume-surface contains in general one or more diagonal lines—besides
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Letter-head of Trinity College, Cambridge. The printed address has been struck through. This draft, which was presumably written between 8 February and 13 June 1872 (see below), was not completed.
{1} ‘On an Algebraical Form, and the Geometry of its Dual Connexion with a Polygon, plane or spherical’, read before the London Mathematical Society on 8 February 1872 (Proceedings, iv. 139–43). It was printed among the proceedings of the Society’s meeting of 13 June with these prefatory words: ‘The following is a resumé of some of the results and the method employed in a paper on this subject, read before the Society, Feb. 8, 1872. A portion of the MS. having been mislaid, and some improvements having suggested themselves, the whole paper will be given hereafter.’ It concludes: ‘The Author is happy to add that the subject in space has attracted the attention of Prof. Clifford, who has laid the results of his investigations before the Society.’
{2} 'n–3' is a subscript.