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[827 
827. 
ON THE NON-EUCLIDIAN PLANE GEOMETRY. 
[From the Proceedinqs of the Royal Society of London, vol. xxxvn. (1884), pp. 82—102. 
Received May 27, 1884.] 
1. I CONSIDER the hyperbolic or Lobatschewskian geometry : this is a geometry such 
as that of the imaginary spherical surface x 2 + y 2 + z 2 = — 1; and the imaginary surface 
may be bent (without extension or contraction) into the real surface considered by 
Beltrami, which I will call the Pseudosphere, viz. this is the surface of revolution 
defined by the equations x = log cot ^9 — cos 6, 'fy 2 + z 2 = sin 6. 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 
(geodesics) 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 
pseudosphere, 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, 1888), [784], expressed 
the opinion that Euclid’s twelfth axiom in Playfair’s form of it does not need demon 
stration, but is part of our notion of space, of the physical space of our experience;