1 The Birth of Science - MSRI

11.11 Recovery and Crisis of Scientific Methodology 347
formulate the advances of modern dynamics within the lucid geometric
framework that Archimedes used for the creation of mechanics.
Another important methodological loss that took a long time to be made
good was that of the role of postulates and the criteria for their choice.
Down to the late nineteenth century there was not a single counterpart to
the many Hellenistic hypothetico-deductive theories; we have seen how
Descartes, Kepler and Newton operated in a very different mode. Indeed,
even Euclid’s Elements, having sojourned for so long in a milieu that had
no notion of a scientific theory (and hence of the relations between theory
and concrete objects), had been shoehorned into the same prescientific
conceptual mold. Its five postulates were no longer conceived as the basis
of a mathematical model for the use of ruler and compass, but as
the Truth. Over the centuries there were countless attempts to derive the
fifth postulate, an ugly duckling for not being obviously “true” like the
rest, from the first four. This vain quest, launched in the imperial era, 180
only ended in the nineteenth century, with Lobachevskii and the so-called
“non-Euclidean geometries”. As is well-known, Lobachevskii discovered page 425
that if the fifth postulate is negated one can nonetheless obtain consistent
theories. (Bolyai’s work, independent of Lobachevskii’s and in essence
equivalent to it, followed a few years later. Before both there had been
observations in the same direction by Gauss, contained in private letters
going back to 1799.)
But before this a non-Euclidean geometry had already been developed,
albeit unconsciouly, by Johann Heinrich Lambert, in a Theory of parallel
lines dating from 1766 (according to Johann Bernoulli grandson, who, two
decades later, edited Lambert’s unpublished works posthumously). This
mathematician considered whether a quadrilateral with three right angles
might have an acute or obtuse fourth angle; studying these possibilities in
order to exclude them, he actually deduced from them two non-Euclidean
geometries, without realizing he had done so. Earlier attempts (notably by
Saccheri) to demonstrate the fifth postulate by contradiction had already
led to many “false” statements characteristic of non-Euclidean geometry,
but Lambert’s work is especially interesting, not least because of its influence
on later developments: it palpably opened the way for the creation
of consciously non-Euclidean geometries a few decades later. Lambert
wrote:
180 Proclus says that Ptolemy had a “demonstration” of the fifth postulate ([Proclus/Friedlein],
362, 12 – 363, 18; 365, 5 – 367, 27). Proclus himself gave another pseudo-demonstration (op. cit.,
371, 23 – 373, 2). We know from an-Nairīzī’s commentary on the Elements that Geminus had made
a similar attempt earlier; see [Heath: HGM], vol. II, 228–230.
Revision: 1.11 Date: 2003/01/06 07:48:20