58 3. MATHEMATICAL CULTURES II
Al-Tusi. Nasir al-Din al-Tusi (1201-1274) had the misfortune to live during the time
of the westward expansion of the Mongols, who subdued Russia during the 1240s,
then went on to conquer Baghdad in 1258. Al-Tusi himself joined the Mongols and
was able to continue his scholarly work under the new ruler Hulegu, grandson of
Genghis Khan. Hulegu, who died in 1265, conquered and ruled Iraq and Persia
over the last decade of his life, taking the title Ilkhan when he declared himself
ruler of Persia. A generation later the Ilkhan rulers converted from Buddhism to
Islam. Hulegu built al-Tusi an observatory at Maragheh, a city in the Azerbaijan
region of Persia that Hulegu had made his seat of goverment. Here al-Tusi was
able to improve on the earlier astronomical theory of Ptolemy, in connection with
which he developed both plane and spherical trigonometry into much more sophis-
ticated subjects than they had been previously. Because of his influence, the loss of
Baghdad was less of a blow to Islamic science than it would otherwise have been.
Nevertheless, the constant invasions had the effect of greatly reducing the vitality
and the quantity of research. Al-Tusi played an important role in the flow of math-
ematical ideas back into India after the Muslim invasion of that country; it was his
revised and commented edition of Euclid's Elements that was mainly studied (de
Young, 1995, p. 144).
4. Europe
As the western part of the world of Islam was growing politically and militar-
ily weaker because of invasion and conquest, Europe was entering on a period of
increasing power and vigor. One expression of that new vigor, the stream of Eu-
ropean mathematical creativity that began as a small rivulet 1000 years ago, has
been steadily increasing until now it is an enormous river and shows no sign of
subsiding.
4.1. Monasteries, schools, and universities. From the sixth to the ninth cen-
turies a considerable amount of classical learning was preserved in the monasteries
in Ireland, which had been spared some of the tumult that accompanied the decline
of Roman power in the rest of Europe. From this source came a few scholars to the
court of Charlemagne to teach Greek and the quadrivium (arithmetic, geometry,
music, and astronomy) during the early ninth century. Charlemagne's attempt to
promote the liberal arts, however, encountered great obstacles, as his empire was
divided among his three sons after his death. In addition, the ninth and tenth cen-
turies saw the last waves of invaders from the north—the Vikings, who disrupted
commerce and civilization both on the continent and in Britain and Ireland until
they became Christians and adopted a settled way of life. Despite these obstacles,
Charlemagne's directive to create cathedral and monastery schools had a perma-
nent effect, contributing to the synthesis of observation and logic known as modern
science.
Gerbert. In the chaos that accompanied the breakup of the Carolingian Empire
and the Viking invasions the main source of stability was the Church. A career
in public life for one not of noble birth was necessarily an ecclesiastical career,
and church officials had to play both pastoral and diplomatic roles. That some of
them also found time for scholarly activity is evidence of remarkable talent. Such
a talent was Gerbert of Aurillac (ca. 940 1002). He was born to lower-class but
free parents in south-central France. He benefited from Charlemagne's decree that
monasteries and cathedrals must have schools and was educated in Latin grammar
