- DIFFERENTIAL GEOMETRY 379
// a curved surface is mapped on any other surface, the measure of curvature at
each point remains unchanged.
Among other consequences, this meant that surfaces that can be developed on
a plane, such as a cone or cylinder, must have Gaussian curvature 0 at each point.
With the first fundamental form Gauss was able to derive a pair of differential
equations that must be satisfied by geodesic lines, which he called shortest lines,^26
and prove that the endpoints of a geodesic circle—the set of geodesies originating at
a given point and having a given length—form a curve that intersects each geodesic
at a right angle. This result was the foundation for a generalized theory of polar
coordinates on a surface, using ñ as the distance along a geodesic from a variable
point to a pole of reference and q as the angle between that geodesic and a fixed
geodesic through the pole. This topic very naturally led to the subject of geodesic
triangles, formed by joining three points to one another along geodesies. Since he
had shown earlier that the element of surface area was
do = y/EG-F^2 dpdq,
and that this expression was particularly simple when one of the sets of coordinate
lines consisted of geodesies (as in the case of a sphere, where the lines of longitude
are geodesies), the total curvature of such a triangle was easily found for a geodesic
triangle and turned out to be
A + B + C-ô,
where A, B, and C are the angles of the triangle, expressed in radians. For a
plane triangle this expression is zero. For a spherical triangle it is, not surprisingly,
the area of the triangle divided by the square of the radius of the sphere. In
this way, area, curvature, and the sum of the angles of a triangle were shown
to be linked on curved surfaces. This result was the earliest theorem on global
differential geometry, since it applies to any surface that can be triangulated. In its
modern, developed version, it relates curvature to the topological property of the
surface as a whole known as the Euler characteristic. It is called the Gauss-Bonnet
theorem after Pierre Ossian Bonnet (1819-1892), who introduced the notion of the
geodesic curvature of a curve on a surface (that is, the tangential component of the
acceleration of a point moving along the curve with unit speed)^27 and generalized
the formula to include this concept.
3.6. The French and British geometers. In France differential geometry was of
interest for a number of reasons connected with physics. In particular, it seemed ap-
plicable to the problem of heat conduction, the theory of which had been pioneered
by such outstanding mathematicians as Jean-Baptiste Joseph Fourier (1768-1830),
Simeon-Denis Poisson (1781-1840), and Gabriel Lame (1795-1870), since isother-
mal surfaces and curves in a body were a topic of primary interest. It also applied
to the theory of elasticity, studied by Lame and Sophie Germain, among others.
Lame developed a theory of elastic waves that he hoped would explain light prop-
agation in an elastic medium called ether. Sophie Germain noted that the average
(^26) According to Klein (1926, Vol. 2, p. 148), the term geodesic was first used by Joseph Liouville
(1809-1882) in 1850. Klein cites an 1893 history of the term by Paul Stackel (1862-1919) as
source.
(^27) According to Struik (1933, 20, pp. 163, 165), even this concept was anticipated by Gauss in
an unpublished paper of 1825 and followed up on by Ferdinand Minding (1806-1885) in a paper
in Crelle's Journal in 1830.