A History of Mathematics From Mesopotamia to Modernity

Geometries andSpace 203

are congruent, of which more later. Staying for the moment with ‘internal’ factors, we can see an
obvious reason for the relativelylatedevelopment of the theory; that the actual function
(p)was
quite a sophisticated one, whose formulation would have been difficult for (say) Leibniz but was
fairly accessible a century later. It is in fact given by

sin(

(p))=sech(Kp)

or

tan

1

2

(

(p))=e−Kp

HereKis the constant of ‘curvature’ appropriate to the space. It is analogous to 1/Rfor a sphere
of radiusR—the biggerKis, the more curved the space. From this, a great deal of detailed
understanding of ‘non-Euclidean space’ follows; in particular the triangle formulae (Appendix B),
and Lambert’s area-defect formula of the previous section. For the fuller development of
Lobachevsky–Bolyai geometry, you are referred either to the expositions in Gray (1979), or the
actual sources, which are quite readable, in Bonola (1955). For a modern explanation of what it is,
and how it works, in what is called the ‘Poincaré model’, see Thurston (1997, Chapter 2).

Exercise 4.How would you prove that two equal line-segments determine the same angle of parallelism?

Exercise 5.Check that the two formulae given for
(p)are equivalent; and that
(p)is a decreasing
function of p, with
( 0 )=π/2,
(p)→ 0 as p→∞.

6. The ‘time-lag’ question

Gray’s question—why did it take so long?—actually divides into two parts, as the story is usually
understood. The first is the delay from Saccheri (1733) to Lambert to Lobachevsky–Bolyai (1820s);
this, it has been argued, can be accounted for. More serious is the delay from the invention to the
wider reception of the new geometries, which was in about 1866, that is, roughly 40 years. One
could ask (for example):

1. Why did it take so long for the new geometries to reach the ‘public domain’?


2. If the conditions were favourable for two (three counting Gauss) independent discoveries in the
   1820s, why were there no further such discoveries in the next 40 years?

If we say that the discoveries occurred in the 1820s because the problem, and its particular
solution were ‘in the air’, we have to explain why the solutions were neither noticed nor duplicated
in the years that followed. Conversely, if the historical conditions were not right for a solution we
have more of a problem in explaining the occurrence of three. There is no particularly easy answer
to this dilemma. The isolation of Lobachevsky and Bolyai and the caution of Gauss are usually
invoked as sufficient reasons for the neglect of their work.

The development of non-Euclidean geometry in Central and Eastern Europe was half-hidden from the public owing to
the obscurity of two of its creators and the shyness of the third. (Torelli 1978, p. 110)

However, as Torelli implicitly acknowledges, the fact is that, since geometry was now firmly believed
to be about ‘space’, or the world of physics and of everyday life, the question which non-Euclidean