222 
ON THE NON-EUCLIDIAN PLANE GEOMETRY. 
[827 
Hence also in any triangle ABC, drawing a perpendicular, say AD, from A to the 
side BC, and so dividing the triangle into two right-angled triangles, we prove that 
the sum A + B+C of the angles is = two right angles, and we further establish the 
relations 
a=b cos C + c cos B, b = c cos A + a cos C, c = a cos B + b cos A, 
which are the fundamental formulae of plane trigonometry; that is, we derive the 
metrical geometry or trigonometry of the plane from the two original equations 
a = c cos B, b = c sin B. 
5. Supposing the plane bent in any manner, that is, converted into a developable 
surface or torse, and using the term straight line to denote a geodesic of the surface, 
then the straight line of the surface is in fact the form assumed, in consequence of 
the bending, by a straight line of the plane. The sides and angles of the rectilinear 
triangle ABC on the surface are equal to those of the rectilinear triangle ABC on the 
plane, and the metrical relations hold good without variation. But it is not simpliciter 
true that the descriptive properties of the torse are identical with those of the plane. 
This will be the case if the points of the plane and torse have with each other a 
(1, 1) correspondence, but not otherwise. For instance, consider a plane curve (such 
as the parabola or one branch of the hyperbola) extending from infinity to infinity, 
and let the torse be the cylinder having this curve for a plane section; then to each 
point of the plane there corresponds a single point of the cylinder ; and conversely to 
each point of the cylinder there corresponds a single point of the plane; and the 
descriptive geometries are identical. In particular, two straight lines (geodesics) on the 
cylinder cannot inclose a space; and Euclid’s twelfth axiom holds good in regard to 
the straight lines (geodesics) of the cylinder. But take the plane curve to be a closed 
curve, or (to fix the ideas) a circle; the infinite plane is bent into a cylinder con 
sidered as composed of an infinity of convolutions; to each point of the plane there 
corresponds a single point of the cylinder, but to each point of the cylinder an infinity 
of points of the plane; and the descriptive properties are in this case altered; the 
straight lines (geodesics) of the cylinder are helices; and we can through two given 
points of the cylinder draw, not only one, but an infinity of helices; any two of 
these will inclose a space. And even if instead of the geodesics we consider only the 
shortest lines, or helices of greatest inclination; yet even here for a pair of points on 
opposite generating lines of the cylinder, there are two helices of equal inclination, 
that is, two shortest lines inclosing a space. We have, in what precedes, an illustration 
in regard to the descriptive geometry of the pseudosphere ; this is not identical with 
the Lobatschewskian geometry, but corresponds to it in a manner such as that in 
which the geometry of the surface of the circular cylinder corresponds to that of the 
plane. 
6. The surface of the sphere is, like the plane, homogeneous, isotropic, and palin- 
tropic. We may on the spherical surface construct, as above, a right-angled triangle 
ABC, wherein the side c and the angle B are arbitrary; and (corresponding to the 
before-mentioned formulas for the plane) we then have 
tan a = tan c cos B, sin b = sin c sin B,