Geometries andSpace 191
Right angle Obtuse angle Acute angle
Fig. 2Saccheri’s three hypotheses.HRA corresponds to Euclid’s geometry with postulate 5 included; it is what we normally take to be
true. His idea was to get a contradiction by carefully spelling out the consequences of the HOA and
HAA, so that the HRA would be left as the only true geometry. The proof which he thought he had
was in fact wrong, but the idea of the three hypotheses was to be very useful; and in developing his
‘proof ’ he deduced a great many consequences which must follow if we assume the HAA; this, as
we will see, is the difficult case, which amounts to denying Postulate 5.
Saccheri proved that in fact these are three mutually exclusive choices: if, say, the HAA is true for
one quadrilateral then it is true for all. There are various other ways of looking at this distinction.
For example, with the HOA there are no parallels (we shall consider how this can happen later),
while with the HAA there are an infinite number of lines through a point P which do not meet a
given linel. Again, with the HOA (the HAA), all triangles have angle-sum greater than (less than)
two right angles.
- After Saccheri, attempts at proof continued, but gradually new elements involving explicit
measurement (such as trigonometry) were brought in—and at the same time we see an increasing
tendency to doubt the possibility of effectively proving the postulate. Gauss^1 in particular became
convinced (some time around 1800) that a consistent geometry in which the postulate was untrue
could be constructed; but he confined his thoughts to private correspondence. - In the 1820s, two independent researchers, N. I. Lobachevsky and Janos Bolyai, both of whom
had been trying to prove the postulate, chose a different aim: to construct a consistent geometry
based on the ‘acute angle’ hypothesis. Note the similarity, though, to Saccheri’s programme. In
each case the idea was to assume such a geometry possible, but Saccheri hoped to deduce a
contradiction, while Lobachevsky and Bolyai did not. Both published their results in obscure places
(in Russia and Hungary) in the 1820s^2 ; and both works were more or less forgotten. However,
each of them proved some important and unexpected properties of the alternative ‘non-Euclidean’
geometry, and went a long way towards making it an interesting study in its own right. This is the
‘Copernican revolution’ referred to in our opening quote. - Although Lobachevsky and Bolyai hadconstructedtheir non-Euclidean geometries, they had
not proved them consistent. This is not a merely pedantic point; it would theoretically still be
possible to prove non-Euclidean geometry inconsistent, and so deduce postulate 5 after all. A wider
variety of geometries (more or fewer dimensions, varying rules of measurement) were outlined
by Riemann in his groundbreaking paper of 1854, and publicized in the years which followed by
Helmholtz; and in particular, the meaning of ‘non-Euclidean’ was clarified. Proof of consistency - Who should have much more than a passing reference; he was the dominant mathematician in almost all fields in the years
from 1800 to 1840. - To be precise: Lobachevsky’s first memoir, ‘Theory of Parallels’ in Russian, was in theKazanMessengerin 1829; Bolyai’s ‘Science
Absolute of Space’, in Latin, was published in 1831 as an appendix to his father’sTentamen.