Abstract
This article reviews how historians of mathematics in China during the 1980s and 1990s overcame the preconceptions that ‘the history of mathematics in China has been fully studied’ and ‘it is an unproductive field’. It critically examines several major issues in the Nine Chapters on the Art of Mathematics (hereafter Nine Chapters) and Liu Hui's commentary that were either unsolved or incorrectly solved by previous scholars, thereby sparking a wave of research on the Nine Chapters and Liu Hui, both within China and internationally. The article argues that the Nine Chapters, compiled by Zhang Cang and Geng Shouchang in the Western Han Dynasty, established the foundational framework of classical mathematics in China but had limitations, such as the absence of mathematical definitions and derivations of mathematical formulas. In the 3rd century, under the influence of the vibrant academic debate during the Wei-Jin period, Liu Hui defined several mathematical concepts, comprehensively demonstrated the validity of the algorithms in the Nine Chapters, and further established the theoretical system of classical mathematics in China. Liu Hui's contributions still hold practical significance in the modern era.
Introduction
In November 2023, the 42nd session of the UNESCO General Conference approved an application submitted by the China Association for Science and Technology (CAST) to hold commemorative activities in honour of Liu Hui during 2024−2025. This marked the first time that China successfully hosted a scientist-themed commemorative event within UNESCO, and was seen as a great honour for both Liu Hui and China's scientific and technological community.
On 24 May 2021, the International Astronomical Union approved China's proposal to name eight lunar landscapes near the landing site of Chang’e-5, including Liu Hui (Baijiahao, 2021). On 2 September 2024, the International Committee on Small Planet Nomenclature officially named asteroid 361712, discovered by the Purple Mountain Observatory of the Chinese Academy of Sciences, as ‘Liu Hui Star’ (Xinhua News Agency, 2024). According to international practice, it is necessary to name the asteroid after a great mathematician. Many scholars advocated using ‘Zu Chongzhi’. However, the mathematical master Wu Wenjun argued that, ‘From a mathematical perspective, Zu Chongzhi cannot be regarded as the representative figure in the history of ancient mathematics in China. The true representative should be Liu Hui.’ He further added, ‘Measured by mathematical contributions, Liu Hui should be mentioned alongside Euclid and Archimedes’ (Wu, 1988).
In late September of 2024, an exhibition and academic seminar commemorating Liu Hui were held at the UNESCO headquarters in Paris. On 7 December of the same year, CAST, together with the Institute for the History of Natural Sciences of the Chinese Academy of Sciences, held a commemorative academic seminar on Liu Hui in Beijing.
The Nine Chapters on the Art of Mathematics established the basic framework of classical mathematics in China
Liu Hui is immortalized for his annotations on the Nine Chapters on the Art of Mathematics (Jiuzhang Suanshu九章算术, hereafter the Nine Chapters). The Nine Chapters is the most important mathematical classic in ancient China and has long been revered as the foremost work in Chinese mathematical literature. It consists of nine chapters: Fangtian, Sumi, Cuifen, Shaoguang, Shangong, Junshu, Yingbuzu, Fangcheng and Gougu, and is a product of the great social and economic transformations during the Spring and Autumn and Warring States periods. It is a compilation of mathematical knowledge accumulated from the Spring and Autumn, Warring States, Qin and Western Han dynasties, containing nearly a hundred highly abstract expressions of mathematical procedures (i.e., formulas, solutions and 246 example problems). It made significant contributions to the history of mathematics, pioneering in establishing rules for the four fundamental operations of arithmetic, proportion and proportional distribution algorithms, Yinbuzu algorithms, root extraction algorithms, solutions to linear equations, equation formulation techniques, addition and subtraction rules for positive and negative numbers, groups of Pythagorean triples, and partial methods for solving right-angled triangles, which were centuries or even millennia ahead of other cultural traditions. Thus, the Nine Chapters established the basic framework of classical mathematics in China.
However, the Nine Chapters also has obvious shortcomings. First, the classification standards are non-unified. Some chapter titles are based on applications, such as Fangtian, Sumi, Shangong and Junshu, while others are based on methods, such as Cuifen, Shaoguang, Yingbuzu, Fangcheng and Gougu. Second, the content is intermingled, and some texts do not correspond neatly to their titles. For example, the Cuifen chapter includes problems that can be solved using the ‘Jinyou’ method instead of the ‘Cuifen’ method. More importantly, it does not define mathematical concepts or provide derivations and proofs for mathematical formulas and solutions.
Liu Hui's Annotations on the Nine Chapters on the Art of Mathematics was written in the fourth year of Jingyuan (263 CE) during the reign of Emperor Chenliu of the Wei Dynasty, and consists of ten volumes (Wei, 2010). The tenth volume, Chongcha was written and annotated by Liu Hui himself and later published separately as Haidao Suanjing. This work is considered alongside the Nine Chapters as one of the ten mathematical classics of China.
The upsurge of research on Liu Hui in the 1980s and 1990s
In the 1970s, the academic study of the history of mathematics in China faced two major issues. First, after the publication of the History of Mathematics in China edited by Qian Baocong (Qian, 1964; see also Li and Qian, 1998), one of the founders of the discipline, academics generally believed that ‘the history of mathematics in China had been thoroughly researched’ and regarded the field as a ‘depleted mine’. In reality, although Li Yan and Qian Baocong held Liu Hui's mathematical achievements in high regard, some important issues were not resolved, some were misunderstood, and others were not addressed. Overall, the evaluation of Liu Hui within China was inadequate. Second, the international academic community had little understanding of Liu Hui. For example, Joseph Needham's (1978) mathematical volume of the Science and Civilisation in China rarely mentions Liu Hui, and the evaluation of Liu Hui in the American Dictionary of Scientific Biography is also inadequate (Gillispie, 1973). When China entered the ‘Spring of Science’ in 1978, many key researchers in the history of mathematics in China gradually moved away from the field.
Inspired by the ongoing ‘Great Discussion on the Criterion of Truth’ during that time, Guo Shuchun believed that understanding Liu Hui should be based on the original text of his Annotations on the Nine Chapters on the Art of Mathematics. He therefore studied Liu Hui's annotations word by word and sentence by sentence. When he studied the ‘Yuantian’ method (圆田术, Method of Circular Field) section of the Fangtian chapter, he was surprised to find that Liu Hui's annotations stated: Cut the regular polygon inscribed in the circumference from each vertex to the centre [and divide it into an infinite number of small isosceles triangles]. Multiplying each side of a regular polygon by the radius of the circle, the product is always twice the area of each small isosceles triangle. So multiplying half of the circumference of a circle by its radius becomes the area of the circle. (Qian, 2023a; Li and Qian, 1998)
Liu Hui thereby proved the formula for the area of a circle in the Nine Chapters: ‘a half of the circumference of a circle being multiplied by its radius makes the area of the circle in bu of area’, which is
Subsequently, the Danish scholar Donald Wagner (1979) and Guo Shuchun (1984b, 2018) successively solved the longstanding proof of Liu Hui's principle, which had previously been unresolved and served as the foundation of Liu Hui's polyhedron volume theory.
In the autumn of 1980, several papers on Liu Hui presented at the Group for the History of Mathematics at the First National Congress of History of Science generated a strong response, and several researchers who had left the field of the history of mathematics in China gradually returned. In the 1980s, there was an upsurge in research on the Nine Chapters and Liu Hui's annotations both in China and internationally. This was marked by a surge in the number of research participants and published papers and books on the history of science in China.
Liu Hui and the atmosphere of academic debate in the Wei and Jin dynasties
Liu Hui's Annotations on the Nine Chapters on the Art of Mathematics is a product of the atmosphere of academic debate during the Wei and Jin dynasties. From the late Eastern Han Dynasty to the Wei and Jin dynasties, China underwent significant transformations in its economy, politics and cultural ethos. The manor economy became the dominant economic form, and an aristocratic system was established. Cao Cao advocated ‘appointing officials based on ability’, and Cao Pi implemented the Nine Grades Appraisal System, objectively allowing some scholars to focus on scientific and cultural creations. People sought ideological weapons from pre-Qin philosophers or heterodox thinkers of the Western and Eastern Han dynasties, leading to a great ideological emancipation. The most prominent feature was the rise of Xuanxue (玄学) and the atmosphere of academic debate, known as the social popular trend during the Zhengshi period. Xuanxue replaced Confucianism as the orthodox ideology and became a major social trend.
In 249 CE, Sima Yi launched a coup that led to the elimination of many leading figures of the Cao Wei period, including legitimate scholars such as He Yan. Some scholars were forced to further pursue the path of mysticism and detachment. Ji Kang (223−262) and the other Seven Worthies of Bamboo Grove broke through the Discussion of the Zhengshi period and attempted to reconcile the views of Confucianism and Taoism, further liberating their thought. The Xuanxue advocates emphasized conformity to the essence of nature, which is certainly a favourable factor for the development of mathematics and science and technology. They attached great importance to ‘victory of reasoning’ and explored the laws of thinking through a process known as ‘analysing and identifying the reason’. People's abstract thinking ability during this period was significantly higher than in the Han Dynasty, and even surpassed that of the Hundred Schools of Thought Controversy during the Spring and Autumn and Warring States periods (Hou et al., 1957; Li, 1985).
Liu Hui's purpose in annotating the Nine Chapters was to ‘use pictures for disassembly and words for analysing reason’ (Liu, 2014). Mathematics is often used by scholars of Xuanxue as a tool for rigorous analysis. The development of mathematics was deeply influenced by the Xuanxue of the Wei and Jin dynasties. Liu Hui defined mathematical concepts, proved or refuted propositions in the Nine Chapters, and pursued correctness in reasoning and rigor in proof. His commitment to ‘The reason and principle should make sense’ in mathematics was consistent with the principle of analysing reason in the intellectual world that was practiced at that time.
On the principle of analysis, Liu Hui, along with Ji Kang, Wang Bi, He Yan and others, upheld the principle that one should ‘be succinct’, should be good at ‘drawing inferences about other cases from one instance’, and be opposed to ‘using excessive language’ (Ji, 1956; Wang, 1986). Liu Hui's analysis of mathematical principles was deeply influenced by the ‘analysing and identifying the reason’ and the atmosphere of academic debate. Liu Hui not only shared similarities in thought with the thinkers such as Ji Kang, Wang Bi and He Yan, but many of his linguistic and syntactical choices are also similar to those of these thinkers (Guo, 1984c, 2018). Therefore, Liu Hui was likely born in the late 1920s or later (Guo, 1992a), and the online assertion that he was born in 225 CE is inaccurate.
The pre-Qin philosophers who became active in these debates also provided important ideological material for Liu Hui's mathematical work. Although Confucianism weakened during the Wei and Jin dynasties, it remained an important school of thought. Liu Hui directly quoted Confucius and Confucian doctrines in his work. Liu Hui was also influenced by Taoism. Among the academic debate, two Taoist works stood out, namely Laozi and Zhuangzi, and the Book of Changes is a classic revered by all families. Liu Hui believed that mathematicians should understand mathematical principles, like Pao Ding understands the body structure of cows, and apply mathematical methods as flexibly as Paoding uses a knife. The phrase ‘Paoding jieniu’ is taken from Zhuang Zi. When proving what is now called Liu Hui's principle, he proposed that ‘the finest is the smallest, and the smallest is the intangible’, which originated from the idea that ‘the finest is intangible’ and ‘the intangible cannot be separated by numbers’ in Zhuang Zi (Guo, 1961).
However, among the pre-Qin philosophers, Liu Hui revered the Mohist school most highly. Liu Hui's preface and his annotations on the Nine Chapters cited several original texts from pre-Qin classics, but only Zhou Li, Zuo Shi Zhuan and Mo Zi had titles. In fact, the idea of ‘cutting again and again until it is impossible to cut’ in Liu Hui's method of circle division was derived from the same origin of the concept that ‘something is impossible to be further divided’ in the Mo Jing (see Sun, 1954).
The upsurge in research on the Nine Chapters and Liu Hui has restored the true features of history
The upsurge in research on the Nine Chapters and Liu Hui has resolved many longstanding issues, correcting previous misunderstandings and restoring the true historical picture.
The compilation of the Nine Chapters has been a subject of intense academic debate within the academic community for over 200 years. Dai Zhen, relying on limited historical sources, concluded that Zhang Cang (?‒152) did not participate in the compilation of the Nine Chapters (Dai, 2014). Qian Baocong dated the completion of the Nine Chapters to the early Eastern Han Dynasty (Qian, 2023b; see also Li and Qian, 1998). However, in the 1980s, through the careful examination of various historical materials and an analysis of prices reflected in the Nine Chapters (Hori, 1988), the erroneous views of Dai, Qian and others were overturned, confirming Liu Hui's argument that: If the nine types of mathematical knowledge are like this, then the Nine Chapters on the Art of Mathematics are correct. In the past, the Qin Dynasty burned books, and the classics were destroyed. Since then, Zhang Cang, the Marquis of Beiping in the Han Dynasty, and Geng Shouchang, a chief finance official, have both been good at calculating. Zhang Cang and others, due to the remnants of old texts, each referred to deletion and supplementation.
This restored Zhang Cang and Geng Shouchang's historical status as famous mathematicians. Furthermore, the academic misunderstanding that the Nine Chapters is merely a set of applied problems with one question, one answer and one method was corrected, and it is recognized that the main part of the Nine Chapters adopts standardized examples that form part of mathematical theory.
According to Liu Hui's own account, his annotations on the Nine Chapters include two types of content: his own mathematical innovations (i.e., those that ‘comprehended the meaning’) and previous mathematical research achievements (i.e., those that ‘adopted what was seen’). The latter is of great significance for filling in the gaps in the historical development of mathematics in China since the completion of the Nine Chapters, and for accurate textual collation.
Liu Hui extended the concept of ‘lü’ from a limited set of algorithms and problems in the Nine Chapters to most of the algorithms and more than 200 problems. He believed that with the help of Qitong Theory, which is a method for handling ratio issues, ‘lü’ would become the ‘guiding principle of arithmetic’ (Guo, 1984d, 2018).
After examining Liu Hui's reasoning and proofs, it can be concluded that Liu Hui mainly used deductive reasoning, refuting the misconception in local and international academic circles that ancient mathematics in China relied solely on intuition and lacked formal logic. Consequently, Liu Hui is regarded as the founder of the classical theoretical system of mathematics in China (Guo, 1983b, 2018). Wu Shoukang (巫寿康), a graduate student of the master of logic Shen Youding (沈有鼎), reached a similar conclusion. This achievement was reported by the Xinhua News Agency, which interviewed the Institute for the History of Natural Sciences of the Chinese Academy of Sciences on 22 April 1982. The following day, it was broadcast on the ‘Morning News and Summary of Capital Newspapers’ of China National Radio and reprinted by the People's Daily (Zhang, 1982).
Liu Hui's mathematical theoretical system was not only an inheritance of the mathematical framework of the Nine Chapters but also a fundamental transformation of it. Thus, it laid the theoretical foundation of classical mathematics in China, forming a ‘mathematical tree’ that originated from the ‘guiju’ (规矩, compass and carpenter's square) and ‘duliang’ (度量, measurements), with ‘branches that, although divided, share the same trunk’ (Guo, 1987, 2018).
Based on the record in the History of the Song Dynasty discovered by Yan Dunjie that Liu Hui was conferred the title of Baron of Zixiang, it is believed that Liu Hui was from Zixiang, which is located in the present-day Zouping, Binzhou City, Shandong Province. The Qilu region has always been regarded as one of the centres of Chinese civilization. This discovery is of great significance for further understanding Liu Hui (Guo, 1992b, 2018).
Regarding the formulation of the chongcha method, Qian Baocong deduced it based on relationships involving lü (Qian, 1964; see also Li and Qian, 1998), while Wu Wenjun believed it was derived using the Out-in Complementary Principle (Guo, 1992a; Wu, 1982, 1988). Given that Liu Hui used these two methods to deduce the complex solutions in the Gougu chapter of the Nine Chapters, it is reasonable to assume that he would have adopted these two methods for the even more complex method of double differences (Guo, 1992a).
Guo Shuchun reflected on Liu Hui's statement in the preface, ‘both heaven and earth can be measured, not to mention the height of Mount Taishan and the area of rivers and seas’, and believed that Liu Hui actually observed and measured Mount Taishan. The prototype of the ‘Observing the Sea Island’ problem in Haidao Suanjing refers to Mount Taishan rather than an island along the coast. The Jade Emperor Peak of Taishan is 1524 metres high, and the area to the southwest of it is extremely steep. The altitude of the two banks of the Dawen River, which is more than 40 kilometres from the Jade Emperor Peak, is between 70 and 90 metres, with no obstacles in between. In October 1990, with the support of the Taishan Management Committee, Guo Shuchun observed and measured Mount Taishan from the north bank of the Dawen River. Using the chongbiao (重表, double poles) method within the chongcha method, he calculated the height of Mount Taishan to be 1976 metres. The margin of error was small, and the result was much more accurate than the measurement made by Ruan Yuan in the Qing Dynasty (Guo, 1992c, 2018).
Prompted by Yan Dunjie, Guo Shuchun collated the Yuzantang and Weiboxie editions of the Nine Chapters, and found Qian Baocong's assessments of these two editions to be inappropriate, thus dispelling the blind faith in Qian's research on these editions. Subsequently, he discovered the Juzhen version, which had been viewed by the emperor of the Nine Chapters in the Nanjing Museum. Then, he collated almost all the available editions of the Nine Chapters that he could find. He found that there were several copies with basically the same content but subtle differences in the mid-Tang Dynasty. The copy that became the base text for the Southern Song edition and Yang Hui edition was the closest or might even be the same. The copy that later became the base text for the Yongle Canon edition was different, and it corrected the academic assumption that the Yongle Canon edition copied the Southern Song edition. Dai Zhen's work of compiling the Nine Chapters from the Yongle Canon edition was of great significance, but was rather rough, containing many errors and misattributions. He even passed off his incorrect collations as the original text of the Southern Song edition. The version collected by Qian Baocong corrected a large number of the incorrect collations made by Dai Zhen and Li Huang. Qian made important contributions by continuing to collate the Nine Chapters. However, it was found that the base text of Qian's collated version was a re-engraved edition of the Weiboxie edition in 1890, which he undervalued, and there were also many incorrect collations. Guo Shuchun concluded that the Nine Chapters warranted a full re-collation (Guo, 1990, 2018, 1991, 2014: Appendix3). This was endorsed by Wu Wenjun. Subsequently, with the support of Wu Wenjun, Li Xueqin and Yan Dunjie, Guo Shuchun began to collate the Nine Chapters.
In the 1980s and 1990s, a large number of papers on the Nine Chapters and Liu Hui's annotations were published by Chinese and international scholars. Since 1990, about 30 books on the Nine Chapters and Liu Hui's annotations have been published. The following are the main works highly regarded by scholars.
Huijiao Jiuzhang Suanshu (汇校九章算术, Collective Collations of the Nine Chapters on the Art of Mathematics), written by Guo Shuchun, was published by Liaoning Education Press in 1990. It corrected a number of incorrect collations by Dai Zhen, Li Huang and Qian Baocong, and restored a large amount of original text that was correct in the Southern Song edition and the Yongle Canon edition. It also re-collated some words that were inappropriately collated or missed by in previous publications. In 2004, the supplementary edition of the Huijiao Jiuzhang Suanshu (增补版汇校九章算术, Supplementary edition of Collective Collations of the Nine Chapters on the Art of Mathematics was published (Guo, 2004), and was selected as one of the 91 excellent ancient book collations in 2013 by the National Press and Publication Administration and the National Leading Group for the Collation and Publication of Ancient Books. In 2014, Jiuzhang Suanshu Xinjiao (九章算术新校, New Collation of the Nine Chapters on the Art of Mathematics) was published, which was actually the third edition of the Huijiao Jiuzhang Suanshu. The third edition of the Encyclopedia of China: History of Science and Technology Volume included an entry ‘Huijiao Jiuzhang Suanshu’.
Liu Hui, a Preeminent Mathematician in the Ancient World, written by Guo Shuchun, was published by Shandong Science and Technology Press in 1992, and was later revised and published several times by publishers of Taiwan Province.
A Chinese‒French bilingual edition of the Nine Chapters on the Art of Mathematics, co-written by Guo Shuchun and Karine Chemla, was published by Dunod in Paris in 2004 and 2005. In 2006, it was awarded the Hirayama Ikuo Prize by the Academy of Inscriptions and Belles-Lettres of the Institute of French. In 2018, it was selected for display in the major exhibition to commemorate the 40th anniversary of China's reform and opening up. It was also included as an entry in the Volume of History of Science and Technology in the third edition of the Encyclopedia of China.
Annotations and Translations of the Nine Chapters on the Art of Mathematics, written by Guo Shuchun, was published or revised and published nine times in 11 years after 2009 by Shanghai Classics Publishing House, and is known as a ‘constant-selling book’.
Interpretation of the Nine Chapters on the Art of Mathematics, written by Guo Shuchun, was published by Science Press in 2019 and 2020.
The Collation and Annotation of the Nine Chapters on the Art of Mathematics completed by Guo Shuchun will be published by Zhonghua Book Company.
The major project of the National Social Science Foundation of China, ‘The bilingual version of the Nine Chapters on Art of Mathematics annotated by Liu Hui, Li Chunfeng, Jia Xian, and Yang Hui in Chinese and English’ (2017−2023), was led by Guo Shuchun as the chief expert. The final report written by Guo Shuchun, Joseph W Dauben, Chen Jianping, Zou Dahai, and Hong Wansheng is soon to be published by Science Press of China and EDP Sciences of France. The Chinese edition written by Guo Shuchun, Chen Jianping, Joseph W Dauben, Duan Yaoyong and Zou Dahai will be published by Science Press.
The modern value of Liu Hui's theoretical system of mathematics
The integration of classical mathematics in China into unified global mathematics in the early 20th century marked a significant historical advancement. However, completely discarding classical mathematics in China is not advisable. Many ideas and methods presented in the Nine Chapters and Liu Hui's annotations surpass those used in current primary and secondary school textbooks. For example, mastering key features such as the place-value system, the mechanical thought of mathematics and the algebraization of geometric problems, which are well established in the Nine Chapters and Liu Hui's annotations, would enable students to more easily grasp mathematical concepts and methods. The concept of ‘lü’ still holds practical significance for the reform of primary and secondary school mathematics textbooks. Although the main works of Guo Shuchun on the Nine Chapters and Liu Hui's annotations have been widely reprinted and republished, the most frequently reprinted copies are the modern Chinese translation and annotation versions for general readers, as well as the explanatory versions or coloured-picture versions written for children.
The Nine Chapters and Liu Hui's annotations can inspire modern mathematical research. Most of the achievements in the Nine Chapters and Liu Hui's annotations are characterized by constructiveness, algorithmization and mechanization, and can be easily transformed into programs and implemented by computers. Based on this understanding, Wu Wenjun pioneered the theory of mathematical mechanization.
The achievements of the Nine Chapters and Liu Hui's annotations refute the widespread misconception that there was no science in ancient China. Liu Hui's proficient use of deductive logic and his superb mathematical proofs strongly challenge the prejudice in the academic community that ancient mathematics in China only had applications without theory. Classical mathematics in China is also part of the mainstream development of world mathematics. The Nine Chapters and Liu Hui's annotations provide excellent reading materials for patriotic education, highlighting China's rich mathematical heritage.
The Nine Chapters and Elements are two great mathematical works produced in the ancient world, which have profoundly influenced the mathematics of the East and the West for 2000 years. Among Western academic circles, the prejudices against ancient mathematics in China, except for a few Eurocentric scholars, are due to their lack of understanding of the Nine Chapters and Liu Hui's annotations. The source of these prejudices within the Chinese academic community originates from Europe and America. Therefore, it is an important task for Chinese scholars to introduce the Nine Chapters and Liu Hui's annotations to foreign academic communities, especially those in Europe and America, in their original form. This is also an important part of cultural exchange between China and foreign countries, which would ultimately enable foreigners to understand ancient Chinese civilization.
Footnotes
Acknowledgements
I would like to express my gratitude to Dr Zhang Wen at the Institute for the History of Natural Sciences, Chinese Academy of Sciences, who provided great help in English polishing in the preparation of this paper.
Funding
The author received no financial support for the research, authorship, and/or publication of this article.
Declaration of conflicting interests
The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Author biography
Shuchun Guo is a research professor at the Institute for the History of Natural Sciences, Chinese Academy of Sciences. His research interest is in the history of Chinese mathematics.