On May 27, 1884, Professor Cayley submitted to the Royal Society of London a paper "On the Non-Euclidean Plane Geometry". In this article he considered hyperbolic or Lobachevskian geometry. By taking the imaginary spherical surface x(exp2) + y(exp2) + z(exp2) = -1 and 'bending' it, that is, without extension or contraction, one can generate the real surface of revolution (considered by Beltrami) defined by the equations: x = log cot(1/2t) - cos(t), (y(exp2) + z(exp2))(exp1/2) = sin(t). Cayley called this surface the Pseudosphere from this point on, taken from Beltrami's work on surfaces of constant negative curvature in which he used the term 'pseudosferiche'. Cayley expressed the notion that because we now have on the imaginary spherical surface imaginary points corresponding to real points on the pseudosphere, imaginary lines (arcs of great circles) corres- ponding to real lines (geodesics) of the pseudosphere, and preservation of distance and angles, the geometry of the pseudosphere is the Lobachevsky geometry, using the expression straight line to denote a geodesic on the surface. He remarked that this realization supports the opinion that Euclid's twelfth axiom is undemonstrable.

Here Cayley drew an interesting parallel to Abbott's "Flatland". He used a notion of a curved 'flat'land to further introduce his discussion. Here is an exerpt:

"We may imagine rational beings living in a two-dimensional space, and conceiving of space accordingly, that is having no conception of a third dimension of space; this two-dimensional space need not however be a plane, and taking it to be a pseudospherical surface, the geometry of this surface, that is, the Lobatschewskian geometry. With regard to our own two-dimensional space, the plane, I have, in my Presidential Address (B.A., Southport, 1883) expressed the opinion that Euclid's twelfth axiom in Playfair's form of it does not need demonstration, but is part of our notion of space, of the physical space of our experience, but which is the representation lying at the foundation of all physical experience." This is an interesting coincidence--apparently Abbott was not alone in some of his imagery, although Cayley abandons the 'beings' at this point and launches into the purely mathematical. Cayley does show, however, that even as the flatlanders conceive of their space as two-dimensional based on physical experience, they cannot really even tell what sort of two dimensional world they inhabit. Perhaps Flatland is actually the surface of a constantly curved surface, in which case it would not be truly flat at all. The remainder of Cayley's paper is a development of the geometry of the pseudosphere.