*Nine Chapters on Mathematical Modernity* is an important work addressing the tensions that exist as a nation proud of its mathematical past interacts with foreign advances in mathematics. Cultural, linguistic, and political factors in China, present between the nineteenth and the middle of the twentieth century, provide the context for the developments in the Chinese mathematical culture which is the subject of this collection of essays by the prominent mathematician and sinologist, Andrea Bréard. Many passages that she cites are translations of primary sources, available here for the first time for non-Chinese readers. This includes, as an appendix, a translation of Li Shanlan’s 1872 work, *Methods for Testing Primality*.

The title and organization of the book into nine chapters pays homage to the traditional respect given by Chinese mathematicians to the classic *Jiu zhang suan shu*, the fundamental *Nine Chapters on Mathematical Procedures*. Bréard’s essays, while interrelated, discuss independent topics. The topics covered range from discursive modes to number theory to divination to statistics and data management. Chinese scholars of the period discussed in this book, while adapting to western advances in mathematics, were also eager to demonstrate the value of the centuries of native mathematical accomplishments by their Chinese predecessors.

Chapter 1 lays out the plan for the book: the author will use the case of China, primarily in the nineteenth and the first half of the twentieth century, to consider the global historical process of conceptual flows. One reaction to exposure to western mathematics was the insistence by Chinese scholars that Chinese mathematics and its history can provide a methodological alternative to foreign mathematics. As an example, Zu Chongzhi (429–500) became a national hero for, among other results, the calculation of \(\pi\) to an accuracy unsurpassed for centuries. (As a personal aside, when I participated in the 2006 MAA Study Tour of China, we enjoyed seeing a wax figure of Zu featured in the National Museum of China on Tiananmen Square in Beijing.) A well-known historian of mathematics, Quiao Baocong, wrote in a 1951 article in *Chinese Science Bulletin*, entitled “The Outstanding Achievements of Mathematics in Ancient China”: “When studying mathematics, we should venerate the outstanding achievements of our homeland!” Bréard also cites a fervent anti-imperialist newspaper article “Mathematics is the Discipline my Country’s People Excel in,” published in *People’s Daily* in 1951 and written by Hua Loo-keng, who had made an invited visit to Princeton University in 1946. This article mentions a number of results that were known in China prior to their introduction from the western tradition, including the Pythagorean Theorem, the value of \(\pi\), modular arithmetic, Pascal’s triangle, and systems of linear equations.

In Chapter 2, Bréard focuses on Xia Luanxiang’s work on the ellipse in the nineteenth century. Ellipses had not been considered in China before the arrival of the Jesuit Matteo Ricci in 1614. Technical material supporting the mathematical details of this chapter is offered in Appendix C. The essay includes an examination of the appearance of ellipses in the Chinese literature through 1900; most often the approach was through traditional Chinese mathematics.

Chapter 3 considers Li Shanlan’s attempt to complement Books VII–IX of Euclid’s *Elements* by offering tests for primality. Section 3.1 points out that there were no sources, either western or Chinese, available in Chinese until Li Shanlan and Alexander Wylie translated Books VII–XIII of *Elements* in the 1850s. (Again, a personal note: Bréard mentioned that it was Yu Xibao who established that the basis of this translation was Henry Billingsley’s *The Elements of Geometrie of the Most Ancient Philosopher Euclide of Megara* (1570); Yu was the leader of the MAA Study Tour of China mentioned previously.) Li and Wylie’s translation had to mediate between Greek and Chinese considerations of unity and number. Euclid’s proof of the infinitude of primes (Proposition IX.20 of the *Elements*) fits the Chinese algorithmic and calculational approach to mathematics: given a set of prime numbers, here is how to find another. Bréard sees Li’s interest in verifying that a newly constructed number is prime as the motivation for his work *Methods for Testing Primality*; her translation of this work appears in Appendix 2. Section 3.2 considers primality in Chinese sources, mining the ancients’ writings to develop procedures to test primality. Section 3.3 concludes this chapter by discussing the controversy regarding Li’s incorrect theorem to test primality, a converse to Fermat’s Little Theorem.

Chapter 4 discusses the challenges of translation of western mathematics, including the need to translate the symbolic notation. While Descartes’ notations for variables and constants were included in a 1712 Chinese translation of his work, their adoption faced the emperor’s opposition towards this “new” notation. In the nineteenth century, the notation was reintroduced but was transformed by cultural constraints, as of course, in the Chinese written language, characters were used for words rather than letters. Similar concerns arose with the translation of Greek letters. Also, texts were laid out vertically while computations were displayed horizontally in Chinese books. Algebraically, China had a well-established traditional celestial element algebraic method that had been rediscovered in the nineteenth century. Translations of some topics, including conics, probability, and calculus, proceeded without concern for tradition as there was nothing comparable to these topics in China’s mathematical past. Bréard points out Li Shanlan’s work in the 1880s as an example of a synthesis of the western and Chinese approaches to algebra. In Section 4.3.1, Bréard gives highlights of the varying sides of the question, “Should China adopt Arabic numerals and notations?” Finally, starting about 1905, official texts began systematically promoting the use of Arabic numerals.

The focus of Chapter 5 is a combinatorial identity discovered by Li Shanlan around 1850. Bréard argues that demonstration by analogy is a legitimate proof technique in the eyes of late Qing mathematicians. This inductive argument was both rhetorical and visual. In section 5.1, Bréard discusses the use of analogy more generally in Chinese mathematics. Section 5.2 provides a discussion of Chinese criticism of Li Shanlan’s claim that his work is a new pillar for Chinese mathematics. Section 5.3 compares the Chinese approach via analogies with the inductive arguments in Pascal’s 1665 *Traité du Triangle Arithmétique.*

Chapter 6 discusses the use of mathematics in divination, answering such questions as whether a sick person will recover or die and what will be the sex of an unborn child. Section 6.2 shows how hexagrams are related to Liu Hui and the lowest common denominator procedures in the *Nine Chapters*. Jiao Xun in 1797 pointed out the connection between the hexagrams and the sixth row of the arithmetic triangle, where the binomial coefficients sum to 64, which is the number of hexagrams. In section 6.3, Bréard notes that divination came under attack in the early Republican period. A defense strategy was to link divination techniques to mathematics and science, establishing that divination was “modern.” Another example of tying mathematics to divination is the use of the polar representation of complex numbers to relate the Five Elements (water, wood, fire, earth, and metal), the ten Heavenly Stems, the twelve Earthly Branches, and the sixty *Jiazi*. As late as 2003, Bréard reviewed a 1999 article for *Mathematical Reviews *that “shows the numerological and structural analogies between the Chinese classic [hexagrams] and biomechanics.” [MR1943948]

Chapter 7 addresses data management, in terms of making use of the statistical records that had been kept by Chinese governments throughout history. While the state had long collected data, it only began to use the data systematically after reforms in 1907. Bréard discusses the educational and political challenges to accomplish this. There were problems in collecting accurate and comparable data from the provinces. Missing data and the use of non-standardized monetary and metrical units provided some of the difficulties.

Chapter 8 discusses applied versus pure mathematics in China between 1850 and 1950, perhaps better cast as applications of mathematics versus fundamental principles. She offers statistics as a case study in the 1930s to the early 1950s, considering administrative statistics, social statistics, and mathematical statistics. Eventually Mao declared formalist approaches to statistical theory to be bourgeois.

In the brief chapter 9, Bréard considers the present. Section 9.1 includes a discussion of the revival of Nationalist Studies and its inclusion of problems from the *Nine Chapters* in education. Zhang Yitang and his work on the bound of gaps between primes is the subject of section 9.2. Finally, section 9.3 discusses the meaning of mathematical modernity in China.

This book is very useful. It is thoughtful and well researched. The inclusion of many, many translations of original source material, from Chinese into English, makes it a valuable reference in that sense as well.

Joel Haack is Professor of Mathematics at the University of Northern Iowa.