Riemann and the Sound of Space 241
two points with the horizontal distance of two others, because we can apply a measure first to the
one pair and then to the other. But we cannot compare the difference between two tones of equal
intensity and different pitch. Riemann showed by considerations of this kind that the essential foun-
dation of any system of geometry is the expression that it gives for the distance between two points
lying in any direction from one another.^48Helmholtz ’ s concept of manifold includes music and sound in the same arena as space,
time, color, and vision. Though simple tones may be described as a manifold of two dimen-
sions, Helmholtz had investigated the parameters of timbre that distinguish complex
musical sonorities from simple tones. At this point, the question of dimensionality seems
open: going beyond the two dimensions of simple tones, how many dimensions are needed
to describe the full character of musical “ space ”? And what then of the dimensional rela-
tions between space and time?^49 Though he does not go further with these questions,
Helmholtz leaves the Raumproblem as the shared heritage of the manifolds of music,
vision, space, and time.
The conclusion of Helmholtz ’ s 1876 revised version of his essay “ On the Origin and
Meaning of Geometrical Axioms ” clarified his current understanding that(1) The axioms of geometry, taken by themselves out of all connection with mechanical propositions,
represent no relations of real things. ... They constitute a form into which any empirical content
whatever will fit and which therefore does not in any way limit or determine beforehand the nature
of the content. This is true, however, not only of Euclid ’ s axioms, but also of the axioms of spherical
and pseudo-spherical geometry.
(2) As soon as certain principles of mechanics are conjoined with the axioms of geometry we
obtain a system of propositions which has real import, and which can be verified or overturned by
empirical observations.^50Helmholtz stepped decisively beyond Kant by including Euclidean and non-Euclidean
geometries on the same footing, each “ a form into which any empirical content whatever
will fit. ”^51 Hence, the axioms of geometry must meet “ certain principles of mechanics ” in
ways that finally rest on empirical observations. Helmholtz ’ s view of this empirical con-
frontation was informed both by optics (and visual physiology) and mechanics (and its
connection to acoustics and music).
In this revised, 1876 version, Helmholtz also added a mathematical appendix on “ the
elements of the geometry of spherical space, ” the same four-dimensional manifold he had
mentioned to Schering in his 1868 letter, cited above, described by the expression x 2 +
y 2 + z 2 + t 2 = R 2. Though there is no hint that t is not a fourth spatial coordinate, its common
identification as time pervaded contemporary mathematical physics; Helmholtz also
allowed t to become an imaginary quantity, further increasing the similarity with the
pseudo-Euclidean space-time later used by Einstein and Hermann Minkowski.^52
Such beguiling speculations aside, it would go too far to conclude that Helmholtz had
(even unknowingly) written down an expression from relativistic physics, fifty years in
advance. His appendix, however, does illustrate his ability to invoke a four-dimensional