- CHINA^27
Srinivasa Ramanujan. The topic of power series is one in which Indian mathemati-
cians had anticipated some of the discoveries in seventeenth- and eighteenth-century
Europe. It was a facility with this technique that distinguished Ramanujan, who
taught himself mathematics after having been refused admission to universities in
India. After publishing a few papers, starting in 1911, he was able to obtain a
stipend to study at the University of Madras. In 1913 he took the bold step of
communicating some of his results to G.H. Hardy. Hardy was so impressed by
Ramanujan's ability that he arranged for Ramanujan to come to England. Thus
began a collaboration that resulted in seven joint papers with Hardy, while Ra-
manujan alone was the author of some 30 others. He rediscovered many important
formulas and made many conjectures about functions such as the hypergeometric
function that are represented by power series.
Unfortunately, Ramanujan was in frail health, and the English climate did not
agree with him. Nor was it easy for him to maintain his devout Hindu practices so
far from his normal Indian diet. He returned to India in 1919, but succumbed to
illness the following year. Ramanujan's notebooks have been a subject of continuing
interest to mathematicians. Hardy passed them on to G.N. Watson (1886-1965),
who published a number of "theorems stated by Ramanujan." The full set of
notebooks was published in the mid-1980s (see Berndt, 1985).
3. China
The name China refers to a region unified under a central government but whose
exact geographic extent has varied considerably over the 4000 years of its history.
To frame our discussion we shall sometimes refer to the following dynasties:^6
The Shang Dynasty (sixteenth to eleventh centuries BCE). The Shang rulers con-
trolled the northern part of what is now China and had an extensive commercial
empire.
The Zhou Dynasty (eleventh to eighth centuries BCE). The Shang Dynasty was
conquered by people from the northwest known as the Zhou. The great Chinese
philosophers known in the West as Confucius, Mencius, and Lao-Tzu lived and
taught during the period of disorder that came after the decay of this dynasty.
The Period of Warring States (403-221 BCE) and the Qin Dynasty (221-206 BCE).
Warfare was nearly continuous in the fourth and third centuries BCE, but in the
second half of the third century the northwestern border state of Qin gradually
defeated all of its rivals and became the supreme power under the first Qin emperor.
The name China is derived from the Qin.
The Han Dynasty (206 BCE-220 CE). The empire was conquered shortly after the
death of the great emperor by people known as the Han, who expanded their control
far to the south, into present-day Viet Nam, and established a colonial rule in the
Korean peninsula. Contact with India during this dynasty brought Buddhism to
China for the first time. According to Mikami (1913, pp. 57-58), mathematical and
astronomical works from India were brought to China and studied. Certain topics,
such as combinatorics, are common to both Indian and Chinese treatises, but "there
(^6) Because of total ignorance of the Chinese language, the author is forced to rely on translations of
all documents. We shall write Chinese words in the Latin alphabet but not strive for consistency
among the various sources that use different systems. We shall also omit the accent marks used to
indicate the pitch of the vowels, since these cannot be pronounced by foreigners without special
training.