Geometries andSpace 193We have given the traditional outline of the story, and it is easy to criticize. An up-to-date,
serious history of mathematics such as this one claims to be obviously ought to be cautious of a
narrative which (a) supposes that a single project has occupied researchers for over 2000 years
(from Euclid’s time to the nineteenth century) and (b) points to a single discovery, at the end of this
time-span, as a founding event or revolution. The problem of the story of non-Euclidean geometry
is the problem of stories in general in history. How far has a generally confused situation been
simplified to produce a neat narrative? Has the meaning of the terms changed over the period? What
other issues, of philosophy, or the varying meaning of the word geometry need to be taken into
account?
In presenting the traditional history first, the intention is not to expose it to ridicule, but to raise
some genuine problems. In a thoughtful discussion (cited in Fauvel and Gray 16.C.5), Gray raises
the main problems of what he terms the ‘standard narrative’ for the major revolutionary period,
that is, roughly from 1730 to 1860; but before dealing with these, a similar assessment needs to
be attempted for the much longer earlier period. There seems to be an essential continuity in the
history from 300bceto the mid-eighteenth century, and a discontinuity for some time after that,
whether it is termed a ‘revolution’ or not. Is the continuity genuine, where does the discontinuity
come from, and what do either of them have to do with wider questions about how we conceive of
space and the world? The fact that Kant, whose famously influential ideas on space were founded
on Euclidean geometry, wrote just before Lobachevsky and Bolyai is often remarked on; but is it just
chance?
Instead of the usual lengthy discussion of source-material, it is easy to give a relatively short
reading-list here; and at the head of it will naturally stand Jeremy Gray’s excellent study (1979).
This book is not only about the history of parallels, and it is the better for that; and it is useful in
covering both mathematical and philosophical questions, with a natural bias to the mathematical.
The lengthy history of attempts to prove the postulate (particularly in the Islamic period) is dealt
with rather briefly, but this can be justified by the greater interest of the eighteenth and nineteenth
centuries. And Gray consistently pays attention to the context—what other kinds of geometry were
of importance, and receiving attention—so that non-Euclidean geometry is given its proper place
as one contender in an often quite diverse field.
As Gray remarks, the older work of Bonola (1955) is still the final authority; it is all the more
important since it includes the main founding works of Lobachevsky and Bolyai as appendices.
Probably because of Gray’s particular interest, the source material reproduced in the chapter on
the subject in Fauvel and Gray (chapter 16) is generous, with extracts from Greek and Islamic
writers in earlier chapters in particular; as regards the standard narrative, it is a very useful
complement to Gray’s book.
Dating from the same period as Gray is Torrelli (1978), which is important in covering, one
would think, much the same questions (how did geometry change in the nineteenth century, and
why?), but with relatively little common ground. Much more attention is paid to rival methods of
axiomatics, and to ideas of what the subject matter should be. As a result, the cast of characters is
richer, including not only mathematicians but physicists and philosophers as well as those who, like
Helmholtz, Mach, and Poincaré, tried to combine the various disciplines. Finally, Joan Richards’
(1988) provides an enlightening antidote to a narrative centred on research mathematicians,
showing the reception of the new ideas on geometry in the rather special case of England, where
the teaching curriculum and humanistic values played a central part in what one might have
expected to be purely mathematical debates.