376 12. MODERN GEOMETRIESfunctions (t,u) t-* (x(t, u),y(t, u), z(t, u)). Quantities such as curvature and area
are then expressed as functions of the parameters (t, u).
Lagrange. Another study of surfaces, actually a paper in the calculus of variations,
was Lagrange's 1762 work on extremal values of integrals.^24 The connection with
differential geometry is in the problem of minimal surfaces and isoperimetric prob-
lems, although he began with the brachystochrone problem (finding the curve of
most rapid descent for a falling body). Lagrange found a necessary condition for a
surface æ = f(x, y) to be minimal.The French geometers. After these "preliminaries" we finally arrive at the tradi-
tional beginning of differential geometry, a 1771 paper of Monge on curves in space
and his 1780 paper on curved surfaces. Monge elaborated Leibniz' idea for finding
the envelope of a family of lines, considering a family of planes parametrized by
their intersections with the z-axis, and obtained the equation of the surface that is
the envelope of the family of planes and can be locally mapped into a plane without
stretching or shrinking.3.5. Gauss. With the nineteenth century, differential geometry entered on a pe-
riod of growth and has continued to reach new heights for two full centuries. The
first mathematician to be mentioned is Gauss, who during the 1820s was involved in
mapping the region of Hannover in Lower Saxony, where Gottingen is located. This
mapping had been ordered by King George IV of England, who was also Elector
of Hannover by inheritance from his great grandfather George I. Gauss had been
interested in geodesy for many years (Reich, 1977, pp. 29-34) and had written a
paper in response to a problem posed by the Danish Academy of Sciences. This pa-
per, which was published in 1825, discussed conformal mapping, that is, mappings
that are a pure magnification at each point, so that directions are preserved and
the limiting ratio of the actual distance between two points to the map distance
between them as one of them approaches the other is the same for approach from
any direction.
Involvement with the mapping project inspired Gauss to reflect on the math-
ematical aspects of developing a curved surface on a flat page and eventually, the
more general problem of developing one curved surface on another, that is, mapping
the surfaces so that the ratio that the distance from a given point Ñ to a nearby
point Q has to the distance between their images P' and Q' tends to 1 as Q tends to
P. Gauss apparently planned a full-scale treatise on geodesy but never completed
it. Two versions of his major work Disquisitiones generates circa superficies curvas
(General Investigations oj Curved Surfaces) were written in the years 1825 and- In the preface to the latter Gauss explained the problem he had set: "to find
all representations of a given surface upon another in which the smallest elements
remain unchanged." He admitted that some of what he was doing needed to be
made more precise through a more careful statement of hypotheses, but wished to
show certain results of fundamental importance in the general problem of mapping.
A simple and fruitful technique that Gauss used was to represent any line in
space by a point on a fixed sphere of unit radius: the endpoint of the radius parallel
to the line.^25 This idea, he said, was inspired by the use of the celestial sphere
(^24) (Euvres de Lagrange, Ô. 1, pp. 335-362.
(^25) An oriented line is meant here, since there are obviously two opposite radii parallel to the line.
Gauss surely knew that the order of the parameters could be used to fix this orientation.