22 2. MATHEMATICAL CULTURES Iaccording to their awareness. Those having five senses are the highest, and those
having only one sense are the lowest.
Islam in India. The amazingly rapid Muslim expansion from the Arabian desert in
the seventh century brought Muslim invaders to India by the early eighth century.
The southern valley of the Indus River became a province of the huge Umayyad
Empire, but the rest of India preserved its independence, as it did 300 years later
when another Muslim people, the Turks and Afghans, invaded. Still, the contact
was enough to bring certain Hindu works, including the Hindu numerals, to the
great center of Muslim culture in Baghdad. The complete and destructive conquest
of India by the Muslims under Timur the Lame came at the end of the fourteenth
century. Timur did not remain in India but sought new conquests; eventually he was
defeated by the Ming dynasty in China. India was desolated by his attack and was
conquered a century later by Akbar the Lion, a descendant of both Genghis Khan
and Timur the Lame and the first of the Mogul emperors. The Mogul Empire lasted
nearly three centuries and was a time of prosperity and cultural resurgence. One
positive effect of this second Muslim expansion was a further exchange of knowledge
between the Hindu and Muslim worlds. Interestingly, the official adminstrative
language used for Muslim India was neither Arabic nor an Indian language; it was
Persian.
British rule. During the seventeenth and eighteenth centuries British and French
trading companies were in competition for the lucrative trade with the Mogul Em-
pire. British victories during the Seven Years War left Britain in complete control
of this trade. Coming at the time of Mogul decline due to internal strife among
the Muslims and continued resistance on the part of the Hindus, this trade opened
the door for the British to make India part of their empire. British colonial rule
lasted nearly 200 years, coming to an end only after World War II. British rule
made it possible for European scholars to become acquainted with Hindu classics
of literature and science. Many Sanskrit works were translated into English in the
early nineteenth century and became part of the world's science and literature.
We can distinguish three periods in the development of mathematics in the In-
dian subcontinent. The first period begins around 900 BCE with individual math-
ematical results forming part of the Vedas. The second begins with systematic
treatises concerned mostly with astronomy but containing explanations of math-
ematical results, which appear in the second century CE. These treatises led to
continuous progress for 1500 years, during which time much of algebra, trigonom-
etry, and certain infinite series that now form part of calculus were discovered, a
century or more before Europeans developed calculus. In the third stage, which
began during the two centuries of British rule, this Hindu mathematics came to
be known in the West, and Indian mathematicians began to work and write in the
modern style of mathematics that is now universal.
2.1. The Sulva Sutras. In the period from 800 to 500 BCE a set of verses of
geometric and arithmetic content were written and became part of the Vedas.^2
(^2) Most of this discussion of the Sulva Sutras is based on the work of Srinivasiengar (1967), which
gives a clear exposition but contains statements that are rather alarming for one who is forced to
rely on a secondary source. For example, on p. 6 we learn that the unit of length known as the
vyayam was "about 96 inches," and "possibly this represented the height of the average man in
those days." Indeed. Where, Mr. Srinivasiengar, have archaeologists discovered 8-foot-tall human
skeletons?