- NON-EUCLIDEAN GEOMETRY 339
in attitude took place, as a few mathematicians began to explore non-Euclidean
geometries as if they might have some meaning after all. Within a few decades the
full light of day dawned on this topic, and by the late nineteenth century, models
of the non-Euclidean geometries inside Euclidean and projective geometry removed
all doubt as to their consistency. This history exhibits a sort of parallelism with
the history of the classical construction problems and with the problem of solving
higher-degree equations in radicals, all of which were shown in the early nineteenth
century to be impossible tasks. In all cases, the result was a deeper insight into the
original questions. In all three cases, group theory came to play a role, although a
much smaller one in the case of non-Euclidean geometry than in the other two.
5.1. Girolamo Saccheri. The Jesuit priest Girolamo Saccheri (1667-1733), a
professor of mathematics at the University of Pavia, published in the last year of
his life the treatise Euclides ab omni ncevo vindicatus (Euclid Acquitted of Every
Blemish), a good example of the creativity a very intelligent person will exhibit
when trying to retain a strongly held belief. Some of his treatise duplicates what
had already been done by the Islamic mathematicians, including the study of Thabit
quadrilaterals, that is, quadrilaterals having a pair of equal opposite sides and equal
base angles and also quadrilaterals having three right angles. Saccheri deduced with
strict rigor all the basic properties of Thabit quadrilaterals with right angles at the
base.^18 He realized that the fundamental question involved the summit angles of
these quadrilaterals—Saccheri quadrilaterals, as they are now called. Since these
angles were equal, the only question was whether they were obtuse, right, or acute
angles. He showed in Propositions 5 and 6 that if one such quadrilateral had obtuse
summit angles, then all of them did likewise, and that if one had right angles,
then all of them did likewise. It followed by elimination and without further proof
(Proposition 7, which Saccheri proved anyway) that if one of them had acute angles,
then all of them did likewise. Not being concerned to eliminate the possibility of
the right angle, which he believed was the true one, he worked to eliminate the
other two hypotheses.
He showed that the postulate as Euclid stated it is true under the hypothesis
of the obtuse angle. That is, two lines cut by a transversal in such a way that the
interior angles on one side are less than two right angles will meet on that side of
the transversal. As we now know, that is because they will meet on both sides of the
transversal, assuming it makes sense to talk of opposite sides. Saccheri remarked
that the intersection must occur at a finite distance. This remark seems redundant,
since all distances in geometry were finite until projective geometers introduced
points at infinity. But Saccheri, in the end, would be reasoning about points at
infinity as if something were known about them, even though he had no careful
definition of them.
It is true, as many have pointed out, that his proof of this fact uses the exterior
angle theorem (Proposition 16 of Book 1 of Euclid) and hence assumes that lines
are infinite.^19 But Euclid himself, at least as later edited, states explicitly that
(^18) It is unlikely that Saccheri knew of the earlier work by Thabit ibn-Qurra and others. Although
Arabic manuscripts stimulated a revival of mathematics in Europe, they were apparently soon
forgotten as Europeans began writing their own treatises. Coolidge (1940) gives the history of
the parallel postulate jumping directly from Proclus and Ptolemy to Saccheri, never mentioning
any of the Muslim mathematicians.
(^19) Actually, the use of that proposition is confined to elaborations by the modern reader. The
proof stated by Saccheri uses only the fact that lines arc unbounded, that is, can be extended to