338 11. POST-EUCLIDEAN GEOMETRYof the Mongol expansion, which brought the conquest of China in the early part of
the century, then the conquest of Kievan Rus in 1243, and finally, the sack of Bagh-
dad in 1254. Despite the horrendous times, the astronomer-mathematician Nasir
al-Din al-Tusi (1201-1274) managed to produce some of the best mathematics of
the era. Al-Tusi was treated with respect by the Mongol conqueror of Baghdad,
who even built for him an astronomical observatory, at which he made years of
accurate observations and improved the models in Ptolemy's Almagest.^17 Al-Tusi
continued the Muslim work on the problem of the parallel postulate. According
to Gray (1989, pp. 50-51), al-Tusi's proof followed the route of proving that the
summit angles of a Thabit quadrilateral are right angles. He showed by arguments
that Euclid would have accepted that they cannot be obtuse angles, since if they
were, the summit would diverge from the base as a point moves from either summit
vertex toward the other. Similarly, he claimed, they could not be acute, since in
that case the summit would converge toward the base as a point moves from either
summit vertex toward the other. Having thus argued that a Thabit quadrilateral
must be a rectangle, he could give a proof similar to that of Thabit ibn-Qurra to
establish the parallel postulate.
In a treatise on quadrilaterals written in 1260, al-Tusi also reworked the trigo-
nometry inherited from the Greeks and Hindus and developed by his predecessors
in the Muslim world, including all six triangle ratios that we know today as the
trigonometric functions. In particular, he gave the law of sines for spherical trian-
gles, which states that the sines of great-circle arcs forming a spherical triangle are
proportional to the sines of their opposite angles. According to Hairetdinova (1986)
trigonometry had been developing in the Muslim world for some centuries before
this time, and in fact the mathematician Abu Abdullah al-Jayyani (989-1079), who
lived in the Caliphate of Cordoba, wrote The Book on Unknovm Arcs of a Sphere,
a treatise on plane and spherical trigonometry. Significantly, he treated ratios of
lines as numbers, in accordance with the evolution of thought on this subject in
the Muslim world. Like other Muslim mathematicians, though, he does not use
negative numbers. As Hairetdinova mentions, there is clear evidence of the Mus-
lim influence in the first trigonometry treatise written by Europeans, the book De
triangulis omnimodis by Regiomontanus, whose exposition of plane trigonometry
closely follows that of al-Jayyani.
Among these and many other discoveries, al-Tusi discovered the interesting
theorem that if a circle rolls without slipping inside a circle twice as large, each
point on the smaller circle moves back and forth along a diameter of the larger
circle. This fact is easy to prove and an interesting exercise in geometry. It has
obvious applications in geometric astronomy, and was rediscovered three centuries
later by Copernicus and used in Book 3, Chapter 4 of his De revolutionibus.
5. Non-Euclidean geometry
The centuries of effort by Hellenistic and Islamic mathematicians to establish the
parallel postulate as a fact of nature began to be repeated in early modern Europe
with the efforts of a number of mathematicians to replace the postulate with some
other assumption that seemed indubitable. Then, around the year 1800, a change
(^17) The world's debt to Muslim astronomers is shown in the large number of stars bearing Arabic
names, such as Aldebaran (the Follower), Altair (the Flyer), Algol (the Ghoul), Betelgeuse (either
the Giant's Hand or the Giant's Armpit), and Deneb (the Tail).