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The Emperor's New Mathematics: Western Learning and Imperial Authority During the Kangxi Reign (1662-1722)

Catherine Jami
Oxford University Press
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Jiang-Ping Jeff Chen
, on

In the anecdotal account, King Ptolemy (323–283 BCE) would have liked to skip the hard work of learning mathematics and was told “there is no royal road to geometry.” By contrast, geometry and other mathematical subjects, including astronomy, were studied diligently by Emperor Kangxi (b. 1654, reigned 1662–1722) in China, who used the technical knowledge as part of statecraft in his long reign. The Emperor’s New Mathematics, Western Learning and Imperial Authority during the Kangxi Reign (1662–1722), which I will refer to hereafter as The Emperor’s New Mathematics, details the story of Kangxi, his life-long involvement with mathematics, the political implications of this endeavor at his court, the social and intellectual impacts in the empire, and the cultural influence on the rest of Qing dynasty (1644–1911).

Readers interested in the technical details of mathematics introduced and developed in 17th-century China might be disappointed in this otherwise excellent work because mathematical details are not among its goals. In the English literature, book-length monographs specifically on the history of Chinese mathematics are not numerous, especially considering the longevity of the country’s history and the richness of its culture. One survey, Benjamin Elman’s 2005 On Their Own Terms: Science in China, 1550–1900, treats the sciences (including mathematics), technology, and medicine in China. Specifically on mathematics, the table below summarizes the major works produced during the past 30 years. Interested readers might want to consult some of them for mathematical details.






Chinese Mathematics: A Concise History

Li Yan, Du Shiran

New York: Oxford University Press


The Sea Island Mathematical Manual: Surveying and Mathematics in Ancient China

Frank Swetz

University Park: Pennsylvania State University Press


Astronomy and Mathematics in Ancient China: Zhou Bi Suan Jing

Christopher Cullen

Cambridge: Cambridge University Press


A History of Chinese Mathematics

Jean-Claude Martzloff, (Stephen Wilson trans.)

Berlin: Springer-Verlag


Euclid in China: The Genesis of the First Translation of Euclid’s Elements in 1607 and Its Reception up to 1723

Peter Engelfriet

Leiden: Brill


The Nine Chapters on the Mathematical Art: Companion and Commentary

Shen Kangshen (ed.), John Crossley &  Anthony W-C Lun (trans.)

Oxford : Oxford University Press


The Chinese Roots of Linear Algebra

Roger Hart

Baltimore: Johns Hopkins University Press


The Emperor’s New Mathematics, Western Learning and Imperial Authority during the Kangxi Reign (1662–1722)

Catherine Jami

New York: Oxford University Press


Imagined Civilization, China, the West, and Their First Encounter

Roger Hart

Baltimore: Johns Hopkins University Press


Nine Chapters of the Art of Mathematics

Guo Shuchun (ed.) Joseph Dauben & Yibao Xu (trans.)

Shenyang: Liaoning Education Press



In many previous accounts of Emperor Kangxi’s involvement with mathematics, his study was often narrated without proper political or social context and separated from his study of other intellectual subjects, including classical studies. Correcting this deficiency by taking an inclusive approach, “calling upon the [diverse] sources and methods of ‘Chinese studies,’ venturing into the fields of cultural, political and social history,” Catherine Jami painstakingly weaves a rich and complex story of one of the most important periods in China and in the history of Chinese mathematics and science. The central figure is Kangxi Emperor of Qing China, a monarch from a minority ethnic group of the most populous empire, who ascended to the throne at the age of 7 and assumed personal rule at 15. He found himself confronting the challenging task of consolidating, building, and maintaining a vast empire, navigating and then manipulating troubled court politics with much shrewdness, reigning for 60 years. He had the second longest reign in Chinese history, only bested by his own grandson. Whether as a piece of statecraft, a political strategy, and/or a genuine personal interest, Kangxi Emperor’s studies of and engagement with mathematics and science certainly contributed to his success.

To understand his motivation to study technical fields such as mathematics and astronomy, we turn to Kangxi Emperor’s own writing. His later reflections indicate that the initial motivation for learning mathematics and astronomy was the ignorance of court officials. In the famous 1660s astronomy case, Kangxi defeated the regent Oboi 鰲拜 (1610–1669) and assumed personal rule. In his reflections, he downplayed the political implications of this power struggle, emphasizing instead his subjects’ incompetence in astronomy. He explained his personal endeavor and pursuit of studying mathematics and related learning. His knowledge of these technical disciplines allowed him to use the astronomy taught by the Jesuits to rebuke his Han Chinese official for “interpreting certain celestial phenomena as portents.” Several officials openly proclaimed that they were personally instructed by the emperor on the topics of cossic algebra (jiegen fang借根方, [methods] of borrowing root and powers). All these contribute to establish an image of a sage king of antiquity: a ruler that both reigns and teaches his subjects. At the same time, Kangxi Emperor’s enthusiasm for mathematics greatly elevated the status of mathematics within the body of knowledge as a whole in China. The mathematical compendium that resulted from one of his editorial projects records and makes available to Chinese scholars almost all the mathematics there was in China in the early 18th century. Much work has been done on various aspects of the compendium; but its far-reaching influence has been more assumed than properly evaluated and assessed, which The Emperor’s New Mathematics does a great job remedying.

The book is divided into five parts. It opens with a summary of the mathematics that the Jesuits brought into China between the 1580s and the aforementioned astronomy case. The Emperor’s New Mathematics strives to bring to our attention the actors’ perception of Jesuit mathematics: that it was part of Western Learning (xixue 西學), which includes all the knowledge that the Jesuits brought with them, including religious teaching. The book introduces the major actors along with the historical and cultural background, in order to give proper contexts and better understanding. It is worth noting that the division between Number (shu 數, traditional Chinese mathematical algorithms) and Magnitude (du 度 geometry, introduced by the Jesuits and related to measurement), and how scholars in certain regions perceived these concepts, inform us about the constructs scholars of the time built around mathematics of different natures and origins.

The second part demonstrates the complex multitude and diversity of the actors in this multifaceted story of an emperor and his life-long engagement with a difficult and technical subject. Beginning with the brief account of the astronomy case, in which the Jesuits lost their position in the astronomic bureau, Part Two is composed of four chapters, covering roughly the period from 1670 to early 1690s. Each of the four chapters focuses on one major player, a group of players, or a major event: Ferdinand Verbiest (1623–1688), Kangxi Emperor’s first Jesuit tutor; Mei Wending (1633–1722), a Chinese scholar well-versed in astronomy and mathematics whose specialty in these fields and disciples would be used later by Kangxi Emperor to replace the Jesuits at the court; the King’s mathematicians from France, who served as Kangxi Emperor’s tutors in early 1690s; and (last but not least) Kangxi’s Emperor’s visit to the Observatory in Nanjing during his second Southern Tour. Jami masterfully approaches the visit to Nanjing Observatory from multiple angles, citing various players’ accounts of the same event, each of which presents the visit in a way that suits his own purposes. Not only does the history come alive with these accounts, these descriptions supply excellent material for a stage play. No wonder Jami’s 7-year-old daughter thinks this chapter makes a great bedtime story.

Readers eager to know about the mathematics which Kangxi Emperor learned from the Jesuits will find answers in Part Three. After disseminating the important context and background of when, where, and how the lessons took place in Chapter 7, the three subsequent chapters discuss the contents of the lessons. The “new” Elements of Geometry (Jihe yuanben 幾何原本), vastly different in style and content from the treatises of the same title translated by Xu Guangqi (1562–1633) and Matteo Ricci (1552–1610) in 1607, was introduced to the emperor by the French Jesuits, who emphasized the topic’s “Frenchness.” Calculation by borrowed root and powers (jiegenfang 借根方) and the procedures of algebra for solving problems equivalent to modern-day polynomials of certain degrees were other topics in the Jesuit lessons based on Antoine Thomas’s (1644–1709) treatise, Synopsis Mathematica, the notes of which were compiled, revised, and expanded into the Outline of the Essentials of Calculation (Suanfa zongyao zuangang 算法纂要總綱), following the emperor’s annotations to the manuscript. The annotations in these manuscripts serve as evidence that the emperor indeed understood mathematics. These topics find their way into the mathematical compendium which was published in 1723, a year after Kangxi Emperor’s death.

As foreigners in a vast self-sufficient empire, missionaries were considered untrustworthy to start with. Kangxi Emperor made abundantly clear that he valued the technical specialty offered by the Jesuits but not the rest of the Western Learnings, especially not their religion, though maintaining an attitude of tolerance. As more missionaries of various other orders came to China at the turn of the century, the Rites Controversy (about whether the worship of ancestors should be considered idolatry) was brewing in the early 1700s. It further deepened the Emperor’s distrust of the Jesuits. He then turned to Chinese scholars and Manchu/Mongol Bannermen to find talents in mathematics and astronomy.

An editorial project in the mathematical sciences, among several others, was underway, with the goal of compiling “all things mathematical” into a compendium. Part Four details the Office of Mathematics and its personnel, who carried out this enormous task. Also included in this part is the work of the French Jesuit Jean-François Fouquet (1665–1741), who wrote for the emperor a treatise on symbolic algebra, Aerrobala xinfa 阿爾熱巴拉新法 (the New method of Algebra). Kangxi Emperor’s initial enthusiasm quickly turned into disappointment, and he eventually dismissed it as laughable when another Jesuit either stopped explaining the ideas in the treatise to him, or perhaps failed to do so properly. As a result, this topic was not included in the editorial project.

The last part of this work is devoted to investigating in detail the mathematical compendium, Yüzhi shuli jingyun 御製數理精蘊 (Essence of Numbers and their Principles, Imperial Composed). On the surface, this compendium contains all the mathematics there was at the time, regardless of its origins. Such a construct in essence casts all mathematical knowledge, native or foreign, into the knowledge system familiar to the Chinese. No Jesuits were involved in the editorial project, as evidenced by the list of personnel involved, though the Chinese scholars in charge of the project had access to the material and lecture notes the emperor received from the Jesuits with the emperor’s annotations and commentary in red ink. Here we also learn that certain notes prepared for the emperor were first in Manchu and then translated into Chinese, which highlights the importance of Manchu sources in constructing the complete, fair and balanced story.

Although this superb work does not address the history of particular topics in mathematics, interested readers will benefit tremendously from the many pieces of evidence that are brought to the fore in Jami’s analyses. Take the topic of “symbolic algebra” as an example. Historians of Chinese mathematics generally consider that it did not appear in China until the 1850s, after the translation of Euler’s Elements of Algebra. The Emperor’s New Mathematics provides two important pieces of information that refute this widespread belief: the topic of Calculation by Borrowed Root and Powers introduced by the Jesuits in the 1690s and Jean-François Foucquet’s New Method of Algebra in 1711. In the former, the symbols for addition, subtraction, and equality are introduced, “polynomials” in one unknown are explained, and the four basic operations on polynomials discussed. In particular, the processes of computation are explicitly demonstrated and recorded, contrary to the general practice in the published treatises. It is worth noting that this topic finds its way into the Essence of Numbers and their Principles.

Foucquet’s manuscript is another interesting example. The “new” in the title is used to distinguish it from the Calculation by Borrowed Root and Powers introduced earlier. Foucquet did not even use the symbols introduced earlier and instead invented his own, for addition, subtraction, and equality. His claim that this new approach would be tremendously useful did not convince Kangxi Emperor in spite of his initial enthusiasm. As a result, the imperial printing office did not even produce a complete treatise from the manuscript. Consequently, no trace of this manuscript can be found in the Essence of Numbers and their Principles. Recently, the reviewer came upon the treatises composed and published in the early 19th century by Chinese scholars officials who were not directly linked to the court Jesuits. These treatises provide strong evidence that the practice of symbolic algebra by Chinese scholars might have been more common — and certainly earlier — than previously thought. These recently-discovered works demonstrate the richness of findings in The Emperor’s New Mathematics.

To describe The Emperor’s New Mathematics as a case study of knowledge transmission seems to make light of its contribution. Viewed from this perspective, however, The Emperor’s New Mathematics convinces us the importance of this particular period and invites participation in further investigation. The stories of the Jesuits in China are often simplified and romanticized as the first “East meets West.” The accounts in The Emperor’s New Mathematics amply demonstrate the intricate and complex nature of the transmission processes. The European actors in the book are mainly Jesuits. Not much is said about missionaries from other orders, as they did not interact much with the court in matters related to mathematics and the sciences. Among the Jesuits, different actors played different roles at different times in relation to the Chinese court and literati, not to mention the different factions within the Jesuit enclave in the late 17th century. For the knowledge being transmitted, it was not simply what the Jesuits had brought with them to China or what they had intended for the Chinese. The emperor and officials acted as a filter to determine what could be translated and discussed.

The last part of the transmission, the reception of the knowledge by the educated population, is not among the focus points of this work and therefore is treated lightly. This shows that there are fruits ripe for picking in this rich and interesting field. Thanks to The Emperor’s New Mathematics, maybe more young scholars will enter the field or focus on this particular period for their own research work.

To find faults in this work is like “finding bones in an egg,” as the Chinese expression describes ill-meaning inspectors purposely picking on the smallest and most trivial mistakes in otherwise perfect work. And here are the little bones found in The Emperor’s New Mathematics. On the first page of the introduction, there is a typographic error: the third line from the bottom should read, “Kangxi’s contemporaries were…,” instead of “Kangxi’s contemporaries where….” Insignificant, but it does not make a good first impression.

The large margins are a peculiar feature. This reviewer ventures to guess that they were designed for readers to take notes. However considerate, this feature makes the pages look off-center and unbalanced. The citations are generally abundant and informative. At certain places, however, they are incomplete. Parts of Chapter 10, 10.3 and 10.4, can also be found in Han Qi’s article published in January, 2011. Considering its publication date in Great Britain, The Emperor’s New Mathematics might have already gone into production when Han’s article came out. Certain information in Sections 12.3 and 12.4 on the mathematical staff and the editorial project can also be found in Han Qi’s articles published in 1999 and 2003. (See the references below.)

The important story in The Emperor’s New Mathematics, expertly narrated by Catherine Jami, should be mandatory reading for college students interested in Chinese history or in the history of mathematics and sciences. The context in which mathematics and sciences were transmitted from Europe to China is intimately integrated with the overall politics and culture at the time. Jami’ incredible feat reminds us that even technical fields, traditionally marginalized in mainstream historical studies played, an integral and important role in shaping history as a whole. Although more books and scholarly works are needed, The Emperor’s New Mathematics provides an incredible contribution to the studies of Qing history and history of sciences in late imperial China.


Han Qi, “Kexue, zhishi yu quanli—riying guance yu Kangxi zai lifa gzhong de zuoyong 科学、知识与权力—日影观测与康熙在历法改革中的作用” (Science, Knowledge, and Power—The Roles of the Observation of the Sun and Kangxi in the Calendar Reform), in Ziran Kexueshi Yanjiu《自然科学史研究》, 2011, 30 (1), 1-18.

Han Qi, “Gewu qiongli yuan yu mengyangzhai—shiqi, shiba shiji zhi zhongfa kexuan jiaoliu 格物穷理院与蒙养斋——17、18世纪之中法科学交流” (The Institute of Studying Things and Pursuing Principles and The School for Tutoring the Young—the Sino-French Scientific Exchange in the 17th and 18th centuries), Sinology in France, 4, 1999, Beijing: Zhonghua shuju, 320-324. Also see Han Qi, “Kangxi shidai de shuxue jiaoyu ji qi shehui beijing 康熙时代的数学教育及其社会背景” (Teaching of Mathematical Sciences during the Kangxi Period and its Social Context), Sinology in France, 8, 2003, Beijing: Zhonghua shuju, 434-448.  

Jiang-Ping Jeff Chen is Professor of Mathematics at St. Cloud State University on St. Cloud, MN.

Part I Western learning and the Ming-Qing transition
1. The Jesuits and mathematics in China, 1582-1644
2. Western learning under the new dynasty
Part II The two first decades of Kangxi's rule
3. The emperor and his astronomer
4. A mathematical scholar in Jiangnan: the first half-life of Mei Wending
5. The "Kings' Mathematicians"
6. Inspecting the Southern sky: Kangxi at the Nanjing Observatory
Part III Mathematics for the emperor
7. Teaching "French science" at the court: Gerbillon and Bouvet's tutoring
8. The imperial road to geometry: new 'Elements of Geometry'
9. Calculation for the emperor: the writings of a discreet mathematician
10. Astronomy in the capital (1689-1693): scholars, officials and ruler
Part IV Turning to Chinese scholars and Bannermen
11. The 1700s: a reversal of alliance
12. The Office of Mathematics: foundation and staff
13. The Jesuits and innovation in imperial science: Jean-Francois Foucquet's treatises
Part V Mathematics and the empire
14. The construction of the 'Essence of numbers and their principles'
15. Methods and material culture in the 'Essence of numbers and their principles'
16. A new mathematical classic?