Geometries andSpace 207a strong belief that the way to make mathematics ‘safe’ was through the construction of axiom
systems.
In fact, in the whole period from the ‘rediscovery’ both of Lobachevsky–Bolyai and of Riemann
in the late 1860s up to 1900, the main questions about geometry were not about ‘foundations’.
This is where Joan Richards (1988) provides a useful view of working mathematicians concerned
not only with research but with what provided the best education for young men at Cambridge,
what should be taught in schools, what was most uplifting, and many other questions which hardly
seem now to be on the agenda.^8 Her restriction to England is not a serious one; although English
mathematicians were certainly less research oriented and tended to be more conservative than
their French, German, or Italian counterparts, they were in touch with the debates which were
going on and contributed to them. Helmholtz’s propaganda for non-Euclidean geometry, as we
have seen, was published in England, and promoted by Clifford, while at the end of the century the
generation of Russell and G. H. Hardy abruptly set out to force Cambridge mathematics into the
continental mainstream.
Almost coincidentally, quite different events in physics separated geometry from empirical
investigations of the world. What geometric form the universe might have was an interesting
scientific question for Riemann, for Helmholtz, indeed until 1906. In Newton’s theory it was
a perfectly flat three-dimensional Euclidean space, in which one could (theoretically) determ-
ine the place and time of any event. The nineteenth-century revisions of geometry amounted
to questioning the nature of the space component. Much more serious problems were raised,
however, by Einstein’s special theory of relativity which—by denying the idea of simultaneity—
effectively killed off the concept of a unified three-dimensional space as a physical object of study.
The geometers had been studying something which had no physical reality. The general theory
of course reintroduced Riemannian geometry (which Einstein learned with considerable diffi-
culty), but in a way so complex that questions about the shape of the universe were turned
into questions about the nature of solutions to some difficult differential equations. Axioms,
and the shape of triangles, in the Einsteinian universe were not the guide which they always
had been.
In trying to present alternative versions of the simple story of ‘Copernican revolution’
with which this chapter opened, there is no need to belittle or downgrade the work of the
founders of non-Euclidean geometry; the record speaks for itself. Rather, we hope that the
reader may be encouraged to think about geometry itself, its changing nature at different
times, and how far the work of Lobachevsky and Bolyai may be said to have influenced those
changes.
Appendix A. Euclid’s proposition I.16
In any triangle, if one of the sides is produced, then the exterior angle is greater than either of the
interior and opposite angles. Let ABC be a triangle, and let one side of it BC be produced to D.
I say that the exterior angle ACD is greater than either of the interior and opposite angles CBA
and BAC.- There was a typically ‘Victorian’ view that the certainty of geometry supported the certainty of theological arguments for the
existence of God. It accordingly acquired a religious, even a political importance.