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, and which I will call the Pseudosphere, viz., this is the surface of revolution defined by the equations
x
= log cot ½θ—cosθ, √y
2
+ z
2
= sinθ. 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, 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; the space, that is, which we become acquainted with by experience, but which is the representation lying at the foundation of all physical experience.
Metrical Geometry Classification of Conic Section in Hyperbolic Space Form