Part IV

India, China, and Japan 500 BCE–1700 CE

In the six chapters that constitute this part, we shall survey a long period of development of mathematics in three cultures that grew up independent of the mathematics that flourished around the Mediterranean Sea. A different “flavor,” more numerically oriented and strongly algebraic, will be seen in all three places. This numerical orientation will be especially noted in geometry, where the approach is not the axiomatic, metric-free Euclidean system. Nonobvious relations among geometric figures are demonstrated using congruence, dissection, and the Pythagorean theorem.

The usual disclaimer applies in this part. It is a very small sample of what could be said; and for more details, the reader should consult the references cited in the corresponding chapters.

Contents of Part IV

1. Chapter 19 (Overview of Mathematics in India) contains a survey of some major achievements and outstanding mathematicians in India (including modern Pakistan) from the earliest times to the twentieth century. By looking at the prefaces to some of the great treatises, we gain some idea of the motivation for creating this knowledge. Again, the names mentioned are only a few of a large number that are worthy of mention.

2. Chapter 20 (From the Vedas to Aryabhata I) discusses the mathematics of the Hindu Vedas beginning around 500 BCE and the work of Aryabhata I (476–550).

3. Chapter 21 (Brahmagupta, the Kuttaka, and Bhaskara II) discusses the work of two more outstanding mathematicians from the seventh through twelfth centuries CE: Brahmagupta (598–670), and Bhaskara II (1114–1184). We end the story at that point, even though mathematics continued to flourish in India with no break at the end of the twelfth century, even anticipating some parts of the calculus.

4. Chapter 22 (Survey of Chinese Mathematics) is devoted to the Chinese development of arithmetic, algebra, and geometry to meet practical administrative needs. The treatises involved include the ancient Zhou Bi Suan Jing (Arithmetical Classic of the Zhou), which probably dates from a time earlier than 200 BCE, and the Han-Dynasty (200 BCE –200 CE) document Jiu Zhang Suanshu (Nine-Chapter Mathematical Treatise), which can be regarded as the fundamental text on classical Chinese mathematics.

5. Chapter 23 (Later Chinese Algebra and Geometry) discusses the study of higher-order equations by Chinese mathematicians and the advanced geometry of Liu Hui (220–280), Zu Chongzhi (420–501), and Zu Geng (ca. 450–ca. 520).

6. Chapter 24 (Traditional Japanese Mathematics) discusses the mathematics of Japan as it was developed from the Chinese classic works and elaborated during the Tokugawa Era from 1600 through 1867. This subject, called wasan (Japanese-style computation), is contemporaneous with a phenomenon that is apparently unique to Japan, namely, the hanging of votive plaques at Shinto and Buddhist shrines with worked-out mathematical problems on them. These plaques are called sangaku (computational framed pictures).