240 Chapter 15
For the first time in his writings on the problem of space, Helmholtz described in detail
non-Euclidean geometries and the pseudosphere of Beltrami, whose criticisms had first
moved Helmholtz to address this issue directly. Helmholtz also brought forward a striking
device for comparing and contrasting these different geometries: “ Think of the image of
the world in a convex mirror. ” In the mirror-world, the theorems of Euclidean geometry
would instantly be translated into non-Euclidean image-theorems, at least as seen from
our side of the mirror. “ In short I do not see how men in the mirror are to discover that
their bodies are not rigid solids and their experiences good examples of the correctness of
Euclid ’ s axioms. But if they could look out upon our world as we can look into theirs,
without overstepping the boundary, they must declare it to be a picture in a spherical
mirror, and would speak of us just as we speak of them; ... neither, so far as I can see,
would be able to convince the other that he had the true, the other the distorted relations. ”
As further evidence, Helmholtz also adduces the eye ’ s ability to accommodate seeing
through “ convex spectacles, ” of the sort he had experimented with in the course of his
visual studies: “ After going about a little the illusion would vanish. ... We have every
reason to suppose that what happens in a few hours to anyone beginning to wear spectacles
would soon enough be experienced in pseudospherical space. In short, pseudospherical
space would not seem to us very strange, comparatively speaking, ” once we had gotten
used to it, just as our eyes would quickly get used to those “ distorting ” spectacles.^44 Helm-
holtz ’ s penetrating insight into the relative consistency of these seemingly antithetical
geometries, Euclidean and non-Euclidean, was directly indebted to his studies of visual
physiology.
Looking back at his recent work, he remarks that “ while Riemann entered upon this
new field from the side of the most general and fundamental questions of analytical geom-
etry, I myself arrived at similar conclusions, partly from seeking to represent in space the
system of colors, involving the comparison of one threefold extended manifold with
another, and partly from inquiries on the origin of our ocular measure for distances in the
field of vision. ”^45 As in his 1868 paper, Helmholtz locates his own “ facts ” as confirming
Riemann ’ s “ hypotheses. ”
In his 1870 exposition, besides adducing the three-dimensional manifolds of “ the space
in which we live ” and “ the system of colors, ” Helmholtz adds that “ time also is a manifold
of one dimension. ”^46 Here, for the first time, time enters the discussion as a manifold,
albeit one-dimensional.^47 Nor did Riemann include time explicitly in his geometrical
(hence implicitly spatial) manifolds. Immediately after his mention of time, Helmholtz
goes on to include the manifold of musical tones, whose time-dependence he had studied
so closely:In the same way we may consider the system of simple tones as a manifold of two dimensions, if
we distinguish only pitch and intensity and leave out of account differences of timbre. This gener-
alization of the idea is well-suited to bring out the distinction between space of three dimensions
and other manifolds. We can, as we know from daily experience, compare the vertical distance of