- DIFFERENTIAL GEOMETRY 377
in geometric astronomy. This unit sphere is used in mapping a curved surface by
taking the normal line at each point of the surface and mapping it to a point on
the sphere, as described, so that the sphere and the surface have parallel normal
lines at corresponding points. Obviously a plane maps to a single point under this
procedure, since all of its normal lines are parallel to one another. Gauss proposed
to use the area of the portion of the sphere covered by this map as a measure of
curvature of the surface in question. He called this area the total curvature of the
surface. He refined this total curvature by specifying that it was to be positive if the
surface was convex in both of two mutually perpendicular directions and negative if
it was convex in one direction and concave in the other (like a saddle). Gauss gave
an informal discussion of this question in terms of the side of the surface on which
the normals were to be erected. When the quality of convexity varied in different
parts of a surface, Gauss said, a still more refined definition was necessary, which
he found it necessary to omit. Along with the total curvature he defined what we
would call its density function and he called the measure of curvature, namely the
ratio of the total curvature of an element of surface to the area of the same element
of surface, which he denoted k. The simplest example is provided by a sphere of
radius R, any region of which projects to the similar region on the unit sphere.
The ratio of the areas is k = l/R^2 , which is therefore the measure of curvature of
a sphere at every point.
Gauss used two mappings from the parameter space (p, q) into three-dimensional
space. The first was the mapping onto the surface itself:
(p, q) i-+ (x(p, q), y(p, q), z(p, q)).The second was the mapping(p, q) H-> (X{p, q), Y{p, q), Z{p, <?))
to the unit sphere, which takes (p, q) to the three direction cosines of the normal
to the surface at the point (x(p, q), y(p, q), z(p, q)).
From these preliminaries, Gauss was able to derive very simply what he him-
self described as "almost everything that the illustrious Euler was the first to prove
about the curvature of curved surfaces." In particular, he showed that his mea-
sure of curvature k was the reciprocal of the product of the two principal radii of
curvature that Euler called / and g. He then went on to consider more general
parameterized surfaces. Here he introduced the now-standard quantities E, F, and
G, given byE- (%)'+<&)' + $)'•
ñ _ dx dx dy dy dz dz
dp dq dp dq dp dq '
c- (!)'+(£)'•(£)'·
and what is now called the first fundamental form for the square of the element of
arc length:
ds^2 = Edp^2 + 2Fdpdq + G dq^2.
It is easy to compute that the element of area—the area of an infinitesimal par-
allelogram whose sides are (^j dp, |* dp, || dp) and (|| dq, |" dq, |^ dq)—is just