CHAPTER 2

THE GROWTH OF MATHEMATICS TO 1600

[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.

GALILEO GALILEI (1564–1642)

So far, we have looked at the ancient roots of mathematics, primarily around the Mediterranean Sea. In this chapter we will look further east, to the two great Asian cultures of China and India, before returning to Western Asia and Europe. The mathematics developed in these cultures was in each case influenced by others, although less so in the case of China. We will trace some of the connections among the different traditions, but must acknowledge up front that much remains unknown about this fascinating topic.

2.1 China

The Master said: Learning without thinking is useless. Thinking without learning is dangerous.

CONFUCIUS (551–479 BCE)

China.1

Major Chinese dynasties, from the Qin to the Ming.

History to 1600: An Overview

China is dominated by two great river systems. The Yellow River and the Yangtze River both originate in the Tibetan Plateau, and flow eastward to the Pacific Ocean. The Yellow River is in northern China, the Yangtze further south.

The earliest known Chinese civilizations arose in the Yellow River valley. By 4500 BCE villagers there were growing millet and raising pigs. The earliest known writing dates from the Shang dynasty (c. 1550–1046 BCE), on “oracle bones”—animal bones and turtle shells sometimes used for divination—and on bronze. The scribes also wrote on bamboo strips, but these didn’t last very long. The Shang dynasty only encompassed a small area in northern China.

The successor to the Shang was the Zhou dynasty (c. 1046–256 BCE), a semifeudal society with a somewhat larger geographical extent. Work on an early version of the Great Wall of China, intended to protect the northern border against invasion by nomads, commenced in the 8 th century BCE. Culture flourished in the 6 th century BCE, with the founding of several academies of scholars. The most famous philosopher of this time was Confucius (Kung Fu-zi, “Master Kung”), who valued education, loyalty, moderation, and his version of the Golden Rule: “What you do not wish for yourself, do not do to others.” Also dating from this period is the Taoism of Lao-Tzu, a more mystical philosophy and an important influence on early Chinese science.

After the Zhou dynasty was the short-lived but important Qin dynasty (221–206 BCE). This dynasty was initiated by one of the most remarkable rulers of any time and place, Qin Shi Huangdi (“First August and Divine Emperor of Qin”). Qin united all of China for the first time. He reorganized the government, destroying the power of the feudal lords, and he established a central bureaucracy. He engaged in massive infrastructure projects, building roads and irrigation systems, constructing a canal linking the Yangtze and Pearl River systems in the south, and completing the Great Wall in Inner Mongolia. The current Great Wall was built later on the foundations of this one. The construction of the wall cost the lives of thousands of workers and helped lead to the end of the Qin dynasty a few years after its founder’s death.

Qin Shi Huangdi standardized the Chinese system of weights and measures, and the Chinese written language too. After this time, regardless of the local spoken language, educated Chinese used the same written characters. The emperor distrusted scholars and banned private ownership of books. For his tomb, he employed about 700,000 workers over a period of more than 30 years to build a massive underground palace in wood, clay, and bronze. In 1974 local farmers made a remarkable discovery at his tomb. He had “protected” his underground palace with an army of 8000 life-size soldiers and horses modeled in clay. The army was complete with generals and chariots, and each soldier was unique. Qin Shi Huangdi died at age 49 in 210 BCE. His dynasty was overthrown four years later.

The Han dynasty (206 BCE–220 CE) succeeded the Qin. The most notable period in this dynasty was the rule of Emperor Wu, which lasted over 50 years, from 140-87 BCE. During this time, Wu extended his empire from Vietnam in the south to northern Korea, and westward into central Asia. The Silk Road became a major trading route, reaching all the way to Rome. Chinese culture flourished. In the year 2 CE, a census put the population of China at almost 60 million people. Sometime in the first century paper was invented. Education, based on Confucianism, became the route to advancement in the bureaucracy, via a new civil service examination system. This system was maintained, with brief disruptions, into the 20th century.

The period after the Han dynasty (220–598 CE) was a time of political disunity, characterized by a succession of short-lived dynasties and invasions from the north. In fact, much of Chinese history was characterized by invasions from the north; the Great Wall was built and rebuilt in an effort to protect against northern invaders. Although the wall was not always successful, the invaders did not replace the native Chinese culture, instead typically adopting it themselves.

There were several notable developments in this period, which comprised the Three Kingdoms Period (220–265) and the Northern and Southern Dynasties (265–589). One was the invention of paired stirrups in western China, which made the cavalry a more effective fighting force. This was also the time that the first porcelains were made. Although Buddhism was first imported from India in the Han dynasty, it became more important in these centuries. Also in this time, there was a great migration of people from northern China to the south, many as refugees. Thus the southern, rice-growing region of China became more important.

The empire was restored in the short-lived Sui dynasty (589–618). Like the Qin dynasty, the Sui is noted for its huge projects and its unpopularity. In addition to work on the Great Wall, the Sui reportedly conscripted five million peasants to work on the Grand Canal linking the Yangtze valley to northern China. This canal, over 1 0 0 0 miles long and still the longest in the world, was used mainly to transport grain from the south to the north.

Block printing was invented around 600. In this method, a wooden block was carved in such a way that the desired characters stood out. The block was then coated with ink and pressed upon paper or other substance. With this method the same page could be printed repeatedly.

Next up was the T’ang dynasty (618–907), one of the high points in Chinese civilization. Poetry and the fine arts flourished. The capital at Chang’an was the largest city in the world, with almost two million people. The Imperial Academy was founded in 754 to prepare scholars for public service. The curriculum was based on classical Confucian literature. Gunpowder was invented in the 8 th century.

After the T’ang ensued a half-century of disunity, ending with the establishment of the Song dynasty (960–1279), another brilliant period. The Song rule of northern China was somewhat tenuous; a northern invasion forced them to move their capital south to Hangzhou. Nonetheless, the arts, trade, and urban culture flourished. Agricultural production doubled. Tea and cotton cultivation expanded. Neo-Confucianism expanded its influence at the expense of Buddhism.

A couple of important advances in shipping date from the Song dynasty. One is the stempost rudder, which provided increased maneuverability for sailing ships. Another is the magnetic compass, which may actually have been invented earlier but was first widely used in shipping in this time.

Movable type printing also dates from the Song dynasty. In block printing an entire page is carved at one time. The new method involved carving individual characters, which then could be assembled into a page for printing. The characters could be reused.

Starting around 1200 the Mongols under Genghis Khan built one of the greatest empires ever, covering most of northern and western Asia, stretching into Eastern Europe and the Middle East. His empire was larger in area and population than the Roman empire. Kublai Khan, Genghis’s grandson, moved the capital of his empire to what is now Beijing, in 1264. From there, he completed the conquest of southern China, and formed the Yuan dynasty (1279–1368).

Chinese contacts with the West increased during Mongol rule. Moslems, Tibetan Buddhists, Nestorian Christians, and Roman Catholics were all invited to China. The most famous European visitor was the Venetian Marco Polo, who visited Kublai Khan in Beijing 1275–1292 and wrote about this amazing place upon his return to Europe, which was at the time backward compared to China. Chinese mathematics attained its greatest achievements at this time.

The foreign rule of the Yuan was never popular, and it was overthrown by an ex-Buddhist monk named Zhu Yuanzhang, who established the Ming dynasty (1368–1644). The Ming rulers rebuilt the Great Wall and built a new southern capital at Nanjing.

In the early 14th century, there was a remarkable series of naval expeditions launched from China, seven in all, led by the eunuch Zheng He (c. 1371–1433). These preceded by several decades the more famous European voyages of exploration. Zheng He traveled throughout the...