Geometries andSpace 201
ADBGFig. 8Lambert’s quadrilateral.was 80◦: then one would have the absolute measure of a Paris foot onAB=AD. (Lambert, in Fauvel and Gray,
pp. 517–8)
Lastly, Lambert recognized that the area formula he had found was the ‘negative’ of the area
formula for spherical triangles; the defectπ−(A+B+C)replaces the excess as the measure of
area. As Gray remarks, he was nearly there. In another sense, he was not there at all; he could see
very clearly what a non-Euclidean geometry must be like, but he went no further in claiming its
existence than the unfortunate statement that it might hold on an imaginary sphere—which no
amount of modern reinterpretation can make sense of. This perhaps is the key point at which one
is justified in asking Gray’s question (1979, p. 155), why did the development take so long—in this
case, from the 1780s to the 1820s? Not excessively long, perhaps; and it should be remembered
that the pursuit of parallels was, as already mentioned, outside the mainstream. It was notoriously
a problem for masochists, eccentrics, or those with unrealistic ambition.
Exercise 3.(a) What does Lambert’s statement about absolute measurement mean? (b) How could it be
justified?
5. The new geometries
In fact, one sees not only that no contradiction is reached, but one soon feels oneself facing anopendeduction.
While a problem given a proof by contradiction should head fairly quickly for a conclusion where the contradiction
is clear, the deductive work of Lobachevsky’s dialectic settles itself more and more solidly in the mind of the reader.
Psychologically speaking, there is no more reason to expect a contradiction from Lobachevsky than from Euclid. This
equivalence will no doubt later be technically proved thanks to the work of Klein and Poincaré; but it is already present
at the psychological level. (Bachelard 1934, p. 30)
At this point, rather than continue with the detail of the story,^6 the reader may reasonably want
to know what is meant by saying that Lobachevsky and Bolyai ‘constructed a geometry’. What is
it to construct a geometry? This is the ‘Copernican’ aspect of the discovery—no one before had
tried to do such a thing. Following on the Euclidean model, one would reasonably ask for a set of
rules or axioms—maybe not this time self-evident—which are full enough for a substantial theory
to be deduced from them. Let us suppose, as the innovators did, that you simply deny postulate 5.
- The key stages between Lambert and Lobachevsky–Bolyai can be found in Gray (1979), chs. 6–9 or Bonola (1955, Chapter III).