204 A History ofMathematics
geometry addressed—what sort of a world do we live in?—was not considered. In this sense, the new
geometries were outside the mainstream which we have referred to above. One of Lobachevsky’s
expositions of his system, the ‘Géométrie Imaginaire’ was published in the highly respectedJournal
für die reine und angewandte Mathematikin 1837; it aroused no response. This was the year of
Chasles’s ‘History’ referred to above, in which all the important modern developments focused
on were in projective geometry, then (as we have seen) considered as an extension of Euclidean
geometry.
The question of the geometry of space, as a possible subject for doubt, was famously raised in
a new and finally extremely influential form by Gauss’s student Bernhard Riemann in his 1855
paper ‘On the Hypotheses which lie at the basis of Geometry’. This also suffered a time-lag; it was
not published until 1866, and although it is a key document of modern mathematics it is not an
easy read. (W. K. Clifford’s English translation is to be found on http://www.maths.tcd.ie/pub/HistMath/
People/Riemann /Geom/WKCGeom.html, and elsewhere.) Because it allows for a very large variab-
ility of structure, it isbotha founding text of modern physics, specifically Einstein’s General Theory
of Relativityandan opening towards the modern mathematical view which divorces the study of
geometries from any ideas of what the world may be like.It will follow from this that a multiply extended magnitude is capable of different measure-relations, and consequently
that space is only a particular case of a triply extended magnitude. But hence flows as a necessary consequence
that the propositions of geometry cannot be derived from general notions of magnitude, but that the properties
which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience.
Thus arises the problem, to determine the simplest matters of fact from which the measure-relations of space can be
determined; a problem which from the nature of the case is not fully determinate, since there may be several systems of
matters of fact which suffice to determine the measure-relations of space—the most important system for our present
purpose being that which Euclid has laid down as a foundation. (Riemann 1873, p. 14)
Riemann’s aims here deserve some closer attention. They are, in his words, to determine the
‘simplest matters of fact’ from which we can discover the geometry of space. Such matters of fact
might include (for example) the rules for congruence of triangles; the possibility of prolonging lines
indefinitely; even the axiom of parallels. Equally, they might not, in which case one would have to
include something else in their place. While not questioning that geometry was the study of space,
he wished to examine what presuppositions we bring to that study and how far—by experiment,
intuition, or whatever—we can justify them, and use them for deducing what space must be. It
had three dimensions, that much was certain; that is the meaning of the forbidding phrase ‘triply
extended magnitudes’; and one had rules for measuring the lengths of curves within it. Guided by
the analogy of Gauss’s work on surfaces, Riemann thought of ‘straight lines’ in space as curves
of shortest length and gave rules, at least in outline, for how such lines could be found. He also
considered the question of what geometry space would need in order to satisfy one reasonable
presupposition: that rigid bodies could be moved around in it without changing shape. (This is to
speak in mechanical terms. A more geometrical view is that the ordinary rules for congruence
hold.) The answer is that what Riemann called the ‘curvature’ of space would have to be constant
from point to point; and that this is satisfied in three cases:- Euclidean geometry;
- Geometry on a sphere, or something like it (which Riemann considered)—this is Saccheri’s
HOA;