82 Prof. Cayley. On the [June 19, 



and descriptions of the skull in the Insectivora can be brought out, 

 the present paper, as a whole, is somewhat imperfect for want of 

 standards for comparison, such as are to be found in the more normnl 

 skulls to be seen in the animals of that Order. 



II. " On the Non-Euclidian Plane Geometry." By Professor 

 CAYLEY, F.R.S. Received May 27, 1884. 



1. I consider the hyperbolic or Lobatschewskian geometry: 

 this is a geometry such as that of the imaginary spherical surface 

 ar-f y 2 + z 2 = 1; and the imaginary surface may be bent (without 

 extension or contraction) into the real surface considered by 

 Beltrami, and which I will call the Pseudosphere, viz., this is the 

 surface of revolution defined by the equations sc= log cot \6 cos 0, 

 v y- + z 2 = sin 0. We have on the imaginary spherical surface 

 imaginary points corresponding to real points of the pseudosphere, 

 and imaginary lines (arcs of great circle) corresponding to real lines 

 (geodesies) of the pseudosphere, and, moreover, any two such 

 imaginary points or lines of the imaginary spherical surface have a 

 real distance or inclination equal to the corresponding distance or 

 inclination on the pseudosphere. Thus the geometry of the pseudo- 

 sphere, using the expression straight line to denote a geodesic of the 

 surface, is the Lobatschewskian geometry ; or rather I would say this 

 in regard to the metrical geometry, or trigonometry, of the surface ; 

 for in regard to the descriptive geometry, the statement requires (as 

 will presently appear) some qualification. 



2. I would remark that this realisation of the Lobatschewskian 

 geometry sustains the opinion that Euclid's twelfth axiom is 

 undemonstrable. 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 

 the pseudospherical surface, the geometry to which their experience 

 would lead them would be 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 docs not need demonstration, but is part of our notion of 

 space, of the physical space of our experience ; the space, that is, 

 which we become acquainted with by experience, but which is the 

 representation lying at the foundation of all physical experience. 



3. I propose in the present paper to further develope the geometry 

 of the pseudosphere. In regard to the name, and the subject 



