Riemann and the Sound of Space 237
For Helmholtz, questions about “ the origins of spatial intuition in general ” emerge from
studies of the visual system and lead directly to considerations about the nature of geom-
etry. As a result, he breaches the customary barrier between the propositions of geometry
and physical reality, previously considered separate from one another.
Though during this period he had been mainly working on experimental physiology,
Helmholtz reveals that he had gone remarkably far in his own self-directed reconsideration
of the mathematical and philosophical problems concerning the nature of space: “ The
author had already begun such an investigation and had completed it in the main when
Riemann ’ s habilitation lecture ‘ On the Hypotheses That Lie at the Foundations of Geom-
etry ’ was made public, in which an identical investigation is carried out, having only a
slightly different formulation of the question. On this occasion, we learned that Gauss had
also worked on the same subject matter, of which his famous essay on the curvature of
surfaces is the only published part of that investigation. ”^30 Riemann ’ s argument assumed
a generalized quadratic line element but did not prove its necessity. Helmholtz asked
whether there is some fundamental reason that would necessarily mandate this assumption,
rather than other, more general possibilities.
Helmholtz ’ s 1868 paper summarized his response to this problem.^31 Though he shared
with Riemann the fundamental idea that geometry ultimately rested on physics rather than
on transcendental ideas, Helmholtz replaced Riemann ’ s “ hypotheses ” with “ facts. ” Steeped
in Goethe, like his educated contemporaries, Helmholtz knew by heart Faust ’ s emendation
of the Gospel of St. John ’ s opening line from “ In the beginning was the Word ” to “ In the
beginning was the Deed ” ( “ Im Anfang war die That ” ); like Faust, Helmholtz moved from
the Word (or Riemann ’ s “ hypotheses ” ) to the Deed, understood as the Fact.^32
Helmholtz argued that fundamental physical facts necessitate the quadratic form of the
line element. Specifically, he assumes “ (1) continuity and dimensions ” (each point in space
is determined by n continuous, independent variables), “ (2) the existence of moving and
rigid bodies , ” “ (3) free mobility ” ( “ each point can pass over into any other along a continu-
ous path ” ), and finally “ (4) the invariance of the form of rigid bodies under rotation. ”
From these premises, he deduced that “ if we desire to find the degree of rigidity and
mobility of natural bodies attributable to our space in a space of otherwise unknown
properties, the square of the line element ds would have to be a homogenous second-degree
function of infinitely small increments of the arbitrarily chosen coordinates u , v , w. This
proposition ... [is] the most general form of the Pythagorean Theorem. The proof of this
proposition vindicates the assumption of Riemann ’ s investigations into space. ”^33 In his
original draft, Helmholtz thought this meant that quadratic form had to correspond to
Euclidean geometry, but Eugenio Beltrami and Sophus Lie soon objected to this erroneous
overspecialization of a more generalized result that Helmholtz should have found: in fact,
the quadratic form was, in general, non-Euclidean, as Helmholtz acknowledged in a note
appended to his 1868 paper.^34 In the subsequent literature, this issue became known as the
Helmholtz – Lie Raumproblem , the so-called problem of space; not a merely technical