Historians in general agree that the decimal place-value system represents an important chapter in the early history of mathematics. Some even compare its importance in mathematics to the Copernican heliocentric theory in astronomy [Löffler 1912, p.79]. Because of its significance in mathematics, scholars and historians alike have investigated the origins of this numeration system. There is a consensus that the system was transmitted to Europe from Islamic countries where in turn it was borrowed from India. That is why the system is often referred to as the Hindu-Arabic numeration system [Smith and Karpinski 1911; Hill 1915]. Whether the system was invented independently in India, or was influenced by Babylonian, Egyptian, Greek, or even Chinese numeration systems, is still an unsettled historical question.

At one time Europeans thought that the Chinese numeration system had no positional values, and that the decimal place-value system was introduced to China from Europe by Jesuit missionaries. Such incorrect views were the result of ignorance of the rod numeration system used for arithmetic calculations in ancient Chinese mathematics. As early as 1839 the French sinologist Édouard C. Biot pointed out that the decimal place-value system can indeed be found in Chinese texts compiled as early as the Mongolian Empire (Yuan Dynasty, 1279-1368) [Biot 1939]. Biot's position was reinforced thirteen years later by the British missionary, Alexander Wylie, who indicated that the *Shushu Jiuzhang* (*Mathematical Treatise in Nine Sections*), written in 1247, not only used a decimal place-value system, but also included a symbol for zero [Wylie 1966, 169]. Since the 1850s, scholars have been pondering the relation between the Indian and Chinese numeration systems [Smith and Karpinski 1911, 32]. Thanks to such pioneering works on ancient Chinese mathematics as those by Mikami Yoshio, Li Yan, Qian Baocong and others, the priority of the Chinese numeration system is now acknowledged among historians of mathematics.

Considering the fact that a rod numeral system was used in China at least by the Warring States Period (475-221 BCE), and the uncertain and relatively late dating of ancient Indian mathematical texts, the similarity of problems and methods for solving them found in both Chinese and Indian texts, some scholars, for instance Wang Ling [Wang 1958] and Joseph Needham [Needham and Wang 1959, 5-17], have argued that the Hindu numeration system was influenced by the Chinese decimal place-value system. Lam Lay Yong and Ang Tian Se add considerably more evidence to this position. Lam Lay Yong, in fact, was encouraged to study the history of Chinese mathematics by Needham when she was a graduate student at Cambridge University.

The book under review is an enlarged edition of the same title published in 1992. The revised version appeared two years after the primary author, Lam Lay Yong, was presented the Kenneth O. May Medal, the highest honor to be conferred by The International Commission on the History of Mathematics in 2002. In addition to a forward by Joseph W. Dauben, the new edition includes the text of an invited lecture, "Ancient Chinese Mathematics and Its Influence on World Mathematics," which Lam delivered at a special International Symposium on History of Chinese Mathematics during the International Congress of Mathematics in Beijing on August 27, 2002. Otherwise, the contents of this new edition remain the same as that of the first edition.

The authors of the book set out to achieve three objectives: "The first is to show the development of arithmetic and the initial stages of algebra in China. The second is to conduct a detailed study of *Sun Zi suanjing*" (p. xxi), and the third is to establish that "the Hindu-Arabic numeral system has its origins in the [Chinese] rod numeral system" (p. xxii).

The first two goals, especially the second one, seem to this reviewer to have been achieved. Besides providing the first full English translation of the *Sun Zi suanjing* 孫子算經 (*The Mathematical Classic of Sun Zi*), one of the ten classics of ancient Chinese mathematical texts, the book discusses the major extant versions of the *Sun Zi suanjing* and its possible dating, estimated to around 400 CE. Various problems in the text are discussed, including the famous Chinese remainder theorem, as well as the socioeconomic aspects of the problems. Substantial parts of the book are devoted to explaining Chinese rod numerals and how they were used to perform the four basic arithmetic operations, to handle fractions, and to extract square roots.

Comparing the Chinese methods with those using Hindu-Arabic numerals as described in the English translations of three Arabic texts on arithmetic by al-Khwārizmī, al-Uqlīdisī, and Kūshyār ibn Labbān, the authors contend that the methods are identical. They offer this as the most convincing evidence to argue that "If the Hindu system was not an original invention, it follows that it must be transmitted from the [Chinese] rod system" (p. 185). Additional supporting arguments are: 1) The Chinese counting rods display numbers in descending order from left to right, the same as the Indian way of writing numbers; 2) In rod numerals, zero is represented by a vacant place. A Chinese character, *kong* 空, is used to denote this value in written form. The Indian word, *sunya*, and the Arabic word, *sifr*, reflect this same idea; and 3) Fractions are represented in the three Arabic texts in exactly the same way as they appear in the Chinese system using counting rods.

Some might say that these arguments, nevertheless, are insufficient to warrant the third and most important objective of the book. But the evidence presented by Lam and Ang makes clear that it is a distinct possibility that the Chinese rod numerals were in fact the origin of the decimal place-value system. The book not only stimulates the reader's interest in this centuries-old issue concerning the origins of the decimal place-value system, but also provides an excellent introduction to one of the important ancient Chinese mathematical texts for English readers, along with a full translation of *The Mathematical Classic of Sun Zi*. Moreover, the book has been carefully proofread; this reviewer found only one printing mistake — the Chinese character for "*hao*," 豪, on page 190, should be 毫 .

Many textbooks aimed at general education students include a section on numeration systems. I have used Mathematical Ideas, by Charles Miller, Vern Heeren and John Hornsby as the textbook for students who take Math 100 at BMCC for two years. The description of the Chinese numeration system in this book (and in many similar books) is really not satisfactory. Those who teach from this book and similar ones would benefit by having *Fleeting Footsteps* on their desk as a corrective.

In short: *Fleeting Footsteps* should be read by anyone interested in the history of Chinese mathematics or in the origins of number systems.

**References:**

Biot, Édouard C. "Note sur la connaissance que les Chinois ont eue de la valeur de position des chiffres," *Journal Asiatique* **8** (1839): 497-502.

Hill, G. F. *The Development of Arabic Numerals in Europe*. Oxford: Oxford University Press, 1915.

Löffler, Eugen. *Ziffern und Ziffernsysteme der Kulturvölker in alter und neuer Zeit*. Leipzig: B. G. Teubner, 1912.

Miller, Charles, Vern Heeren, and John Hornsby. *Mathematical Ideas*. Harlow, Essex, UK: Pearson Education, Expanded 12th Edition, 2004.

Needham, Joseph. *Science and Civilisation in China* vol. 3. Cambridge: Cambridge University Press, 1959 (with the assistance of Wang Ling).

Smith, David E., and Louis C. Karpinski. *The Hindu-Arabic Numerals*. Boston: Ginn, 1911.

Wang Ling. "The Chinese Origin of the Decimal Place-Value System in the Notation of Numbers and the Possibility of Its Transmission to India." *Proceedings of the International Congress of the History of Science, Adelaide, Australia, 20-27 August, 1958*, p. 5.

Wylie, Alexander. "Jottings on the Science of the Chinese — Arithmetic." *North China Herald,* nos. 108-113, 116, 117, 119-121 (1852). Reprinted in *Shanghai Almanac for 1853 and Miscellany*, Shanghai, 1853; in *Chinese and Japanese Repository* **1** (1864): 411-17, 448-57, 494-500; **2** (1864): 69-73; and in *Chinese Researches* (Shanghai, 1897; London: K. Paul, Trench, Trubner & Co., 1937; Taipei: Ch'eng-wen, 1966, pp. 159-194).

**Yibao Xu** ([email protected]) teaches in the Department of Mathematics at Borough of Manhattan Community College of the City University of New York. His primary research interests are history of Chinese mathematics and transmission of mathematical knowledge between cultures.