The History of Mathematics: A Brief Course

382 12. MODERN GEOMETRIES

ways into another, the various determinations of it form a continuous or discrete

manifold. He noted that discrete manifolds (sets of things that can be counted, as

we would say) are very common in everyday life, but continuous manifolds are rare,

the spatial location of objects of sense and colors being almost the only examples.

The main part of his lecture was the second part, in which he investigated the

kinds of metric relations that could exist in a manifold if the length of a curve

was to be independent of its position. Assuming that the point was located by a

set of ç coordinates x\,. ..,xn (almost the only mathematical symbols that appear

in the paper), he considered the kinds of properties needed to define an infinites-

imal element of arc length ds along a curve. The simplest function that met this

requirements was

where the coefficients were continuous functions of position and the expression

under the square root is always nonnegative. The next simplest case, which he

chose not to develop, occurred when the Maclaurin series began with fourth-degree

terms. As Riemann said,

The investigation of this more general type, to be sure, would not

require any essentially different principles, but it would be rather

time-consuming and cast relatively little new light on the theory of

space; and moreover the results could not be expressed geometrically.

For the case in which coordinates could be chosen so that an = 1 and = 0 when
i Φ j, Riemann called the manifold flat.
Having listed the kinds of properties space was assumed to have, Riemann
asked to what extent these properties could be verified by experiment, especially in
the case of continuous manifolds. What he said at this point has become famous.
He made a distinction between the infinite and the unbounded, pointing out that
while space is always assumed to be unbounded, it might very well not be infinite.
Then, as he said, assuming that solid bodies exist independently of their position, it
followed that the curvature of space would have to be constant, and all astronomical
observation confirmed that it could only be zero. But, if the volume occupied by
a body varied as the body moved, no conclusion about the infinitesimal nature of
space could be drawn from observations of the metric relations that hold on the
finite level. "It is therefore quite conceivable that the metric relations of space are
not in agreement with the assumptions of geometry, and one must indeed assume
this if phenomena can be explained more simply thereby." Riemann evidently
intended to follow up on these ideas, but his mind produced ideas much faster than
his frail body would allow him to develop them. He died before his 40th birthday
with this project one of many left unfinished. He did, however, send an essay to
the Paris Academy in response to a prize question proposed (and later withdrawn):
Determine the thermal state of a body necessary in order for a system of initially
isothermal lines to remain isothermal at all times, so that its thermal state can be
expressed as a function of time and two other variables. Riemann's essay was not
awarded the prize because its results were not developed with sufficient rigor. It
was not published during his lifetime.^34

(^34) Klein (1926, Vol. 2, p. 165) notes that very valuable results were often submitted for prizes at
that time, since professors were so poorly paid.