- DIFFERENTIAL GEOMETRY 383
Differential geometry and physics. The work of Grassmann and Riemann was to
have a powerful impact on the development of both geometry and physics. One
has only to read Einstein's accounts of the development of general relativity to
understand the extent to which he was imbued with Riemann's outlook. The idea
of geometrizing physics seems an attractive one. The Aristotelian idea of force,
which had continued to serve through Newton's time, began to be replaced by
subtler ideas developed by the Continental mathematical physicists of the nine-
teenth century, with the introduction of such principles as conservation of energy
and minimal action. In his 1736 treatise on mechanics, Euler had shown that a
particle constrained to move along a surface by forces normal to the surface, but
on which no forces tangential to the surface act, would move along a shortest curve
on the surface. And when he discovered the variational principles that enabled him
to solve the isoperimetric problem (see Chapter 17), he applied them to the theory
of elasticity and vibrating membranes. As he said,Since the material of the universe is the most perfect and proceeds
from a supremely wise Creator, nothing at all is found in the world
that does not illustrate some maximal or minimal principle. For
that reason, there is absolutely no doubt that everything in the
universe, being the result of an ultimate purpose, is amenable to
determination with equal success from these efficient causes using
the method of maxima and minima. [Euler, 1744, P- 245]It is known that Riemann was searching for a connection between light, elec-
tricity, magnetism, and gravitation at this time.^35 In 1846, Gauss' collaborator
Wilhelm Weber (1804-1891) had incorporated the velocity of light in a formula
for the force between two moving charged particles. According to Hermann Wey]
(Narasimhan, 1990, p. 741), Riemann did not make any connection between that
search and the content of his inaugural lecture. Laugwitz (1999, p. 222), however,
cites letters from Riemann to his brother which show that he did make precisely
that connection. In any case, four years later Riemann sent a paper^36 to the Royal
Society in Gottingen in which he made the following remarkable statement:I venture to communicate to the Royal Society a remark that brings
the theory of electricity and magnetism into a close connection
with the theory of light and heat radiation. I have found that the
electrodynamic effects of galvanic currents can be understood by
assuming that the effect of one quantity of electricity on others is
not instantaneous but propagates to them with a velocity that is
constant (equal to that of light within observational error).3.8. The Italian geometers. The unification of Italy in the mid-nineteenth cen-
tury was accompanied by a surge of mathematical activity even greater than the
sixteenth-century work in algebra (discussed in Chapter 14). Gauss had analyzed
a general surface by using two parameters and introducing six functions: the co-
efficients of the first and second fundamental forms. The question naturally arises
(^35) His lecture was given nearly a decade before Maxwell discovered his famous equations connect-
ing the speed of light with the propagation of electromagnetic waves.
(^36) This paper was later withdrawn, but was published after his death (Narasimhan, 1990, pp.
288-293).