378 12. MODERN GEOMETRIESAdpdq, where Ä = sjEG - F^2. Gauss denoted the analogous expression for the
mapping (p,q) ^ (X(p,q),Y(p,q), Z(p,q)), by
(3) D dp^2 + 2D'dpdq + D" dq^2.
It turns out that D is just Ä times the cosine of the angle between the normal line
to the surface and the line through the origin passing through the point
id^2 ÷ d^2 x d^2 x\
V dp^2 ' dpdq' dq^2 ) '
and similarly for D' and D" with ÷ replaced by y and æ respectively. This coinci-
dence is particular to three-dimensional space, since there just happen to be three
different second-order partial derivatives.
The expression in formula (3) is now divided by Ä and the quotient, called the
second fundamental form, is written edp^2 + 2/dpdq + gdq^2. The element of area
on the sphere is (DD" - [D')^2 )dpdq. Hence the measure of curvature—what is
now called the Gaussian curvature and denoted k—is
DD" - (D')^2
(EG - F^2 )^2 'or as it is now written,
eg-Ñ
EG-F^2 '
Gauss found another expression for k involving only the quantities E, F, and
G and their first and second partial derivatives with respect to the parameters ñ
and q. The expression was complicated, but it was needed for theoretical purposes,
not computation.
In a very prescient remark that was later to be developed by Riemann, Gauss
noted that "for finding the measure of curvature, there is no need of finite formula?,
which express the coordinates x, y, æ as functions of the indeterminates p, q; but
that the general expression for the magnitude of any linear element is sufficient."
The idea is that the geometry of a surface is to be built up from the infinitesimal
level using the parameters, not derived from the metric imposed on it by its position
in Euclidean space. That is the essential idea of what is now called a differentiable
manifold.
It is also clear from Gauss' correspondence (Klein, 1926, p. 16) that Gauss
already realized that non-Euclidean geometry was consistent. In fact, the question
of consistency did not trouble him; he was more interested in measuring large
triangles to see if the sum of their angles could be demonstrably less than two right
angles. If so, what we now call hyperbolic geometry would be more convenient for
physics than Euclidean geometry.
Gauss considered the possibility of developing one surface on another, that is,
mapping it in such a way that lengths are preserved on the infinitesimal level. If the
mapping is (x,y,z) -* (u,v,w), then by composition, u, v, and w are all functions
of the same parameters that determine x, y, and z, and they generate functions
E', F', and G' for the second surface that must be equal to E, F, and G at the
corresponding points, since that is what is meant by developing one surface on
another. But since he had just derived an expression for the measure of curvature
that depended only on E, F, G and their partial derivatives, he was able to state
the profound result that has come to be called his theorema egregium (outstanding
theorem):