A new perspective on Liu Hui's mathematical achievements

Abstract

Liu Hui is a seminal figure in the history of Chinese mathematics. In 263 CE, he commented on The Nine Chapters on Mathematical Procedures (Jiuzhang Suanshu 九章算术). Prior research has highlighted his contributions in formulating rigorous arguments for the correctness of mathematical procedures, employing logical reasoning through nearly infinite segmentation and systematizing Chinese mathematics. This article examines three other aspects of his work that have a profound impact on mathematics but have rarely been discussed before. First, it explores his discussion on the early development of mathematics, which profoundly influenced the cultural foundations of Chinese mathematics. Second, it analyzes his contributions to what is known as ‘geometrical algebra’, a precursor to the Confucian mathematical methods that emerged during the Southern and Northern dynasties (420–589 CE) and later evolved into the so-called Chinese algebra developed between the 11th and 13th centuries. Third, it examines his innovative concept and application of counting rods, which can be regarded as an early precursor to the textualization and symbolization of Chinese mathematics during the 13th century. Finally, the article argues that Liu Hui's achievements should also be understood in the context of his interactions with political power and his engagement with contemporary philosophical ideas.

Liu Hui is a central figure in the history of Chinese mathematics, and his mathematical achievements have been extensively studied by scholars (see e.g., Chemla and Guo, 2004; Guo, 1992; Li, 1998; Li and Du, 1987; Martzloff, 2006; Needham, 1959). In 2023, coinciding with the 1760th anniversary of Liu Hui's commentary (263 CE) on The Nine Chapters on Mathematical Procedures, UNESCO (the United Nations Educational, Scientific and Cultural Organization) initiated commemorative activities in his honour. I have also written an article to commemorate Liu Hui's life and mathematical accomplishments (Zhu, 2024). In this article, I further explore three of Liu Hui's achievements that have received relatively little attention in academic circles: his discourse on early mathematics, geometrical algebra and the use of counting rods, seeking guidance from experts in the field.

1. Liu Hui's discussion on the early development of mathematics

In the first paragraph of the preface to his commentary on The Nine Chapters on Mathematical Procedures (Jiuzhang Suanshu 九章算术), Liu Hui provides a brief overview of the early development of mathematics:

Long ago Pao Xi first devised the eight trigrams in order to ascertain the disposition of the supernatural spirits, to classify the natural properties of everything that exists, to create the nine–nine rule in accordance with the transformations of the six lines. By the time of [the] Yellow Emperor, their essence had changed, [the trigrams and nine–nine rule] were developed and extended, and thus calendars and records were established, musical scales were harmonized, and the origins of the dao were investigated, after which the subtle and profound essence [qi] of heaven and earth and everything in between could be understood and followed. It is recorded that ‘Li Shou created the numbers,’ but the details are not known. ¹ (Guo et al., 2013: 3–7)

In these sentences, Liu Hui traces the origins of mathematics back to Pao Xi, who devised the eight trigrams. He then mentions Li Shou—an official serving under the Yellow Emperor—crediting him with creating numbers. However, Liu Hui admits that he does not know the details of how Li Shou developed these numerals. The legend that Pao Xi (i.e., Fu Xi 伏羲) created the eight trigrams is recorded in the Book of Changes (Yi 易). Likewise, early Chinese writings—such as the bamboo slips held at Peking University and Sima Qian's (司马迁) Historical Records (Shiji 史记)—attribute the invention of numbers to the Yellow Emperor (黄帝) and his minister Li Shou (隶首) (see Han and Zou, 2015: 235; Sima, 1963: 1256). Therefore, it is clear that Liu Hui was simply reflecting the accepted lore of his time. Now, Liu Hui continues writing:

It is when the Duke of Zhou established rites that nine procedures existed, and it is the development of the nine procedures that generated The Nine Chapters. In the past the brutal Qin Dynasty burned the books, and the classic texts and procedures were destroyed or damaged. ² (Guo et al., 2013: 3–9)

The Rites of Zhou (周礼, Zhouli), originally titled Officers of Zhou (周官, Zhouguan), was rediscovered during the Western Han period and traditionally attributed to the Duke of Zhou. However, its content can be traced back to the Spring and Autumn period (see Shen and Li, 2004). According to the Rites of Zhou, the nine procedures (九数, Jiushu) formed part of the six arts (六艺, Liuyi), though the text itself never elaborates on the specifics of those nine procedures. ³ When the scholar Zheng Xuan (郑玄; 127–200) commented on the Rites of Zhou, he quoted a statement by the earlier scholar Zheng Zhong (郑众; ?–83):

The nine procedures refer to ‘Rectangular fields’ (Fangtian 方田), ‘Unhusked and husked grain’ (Sumi 粟米), ‘Sharing according to the degree’ (Cifen 差分), ‘Reducing the length and adding the width’ (Shaoguang 少广), ‘Discussion works’ (Shanggong 商功), ‘Collecting tax in a fair way’ (Junshu 均输), ‘Measures in square’ (Fangcheng 方程), ‘Excess and deficit’ (Ying buzu 赢不足), and Pangyao (旁要 meaning not certain). Today, there are ‘Repeated difference’ (Chongcha 重差), Xijie (夕桀 meaning not certain), and ‘Base and height of a right triangle’ (Gougu 句股). ⁴ (Zheng and Jia, 1980: 731)

The versions of the nine procedures recorded by Zheng Zhong and Zheng Xuan closely mirror those in The Nine Chapters, with the sole exception that Pangyao differs from the ninth chapter's Gougu section. Zheng Xuan also noted that today we possess the Gougu text. This observation led Qian Baocong (钱宝琮; 1893–1974) to conclude that The Nine Chapters must have been finalized after Zheng Zhong (Qian, 1963: 84). However, this also represents the earliest extant account of The Nine Chapters being derived from the nine procedures. When Liu Hui composed his commentary, he adhered precisely to the records of Zheng Zhong and Zheng Xuan. Now, Liu Hui continues writing:

From that time on, the Marquis of Beiping of the Han dynasty, Zhang Cang, and Deputy Grand Minister of Agriculture [of the Han Dynasty] Geng Shouchang both are renowned for [being] good at mathematics. Cang and others, based upon what survived of the old damaged works, evaluated each [of the texts], deleting [parts] and repairing [others]. Therefore, upon comparing their contents these [reconstructed works] and the old [texts] may differ. Furthermore, most of the words and terms belong to times closer to ours. ⁵ (Guo et al., 2013: 9–10)

In the Book of Han (汉书), Ban Gu (班固; 32–92 CE) records the story ‘Zhang Cang determined the rules and norms’. Liu Hui (刘徽; 225–295 CE), in his commentary on The Nine Chapters, expanded this narrative by crediting Zhang Cang and Geng Shouchang with reconstructing surviving and damaged mathematical works. By doing so, Liu Hui implied that The Nine Chapters reached their completed form as a result of Zhang Cang's and Geng Shouchang's editorial and preservational efforts.

In these passages, Liu Hui effectively traces an early mathematical tradition through three successive narratives: (1) ‘Li Shou created numbers’, describing the very origin of numeration; (2) ‘The Duke of Zhou established ritual systems’, which gave rise to the Nine Procedures; and (3) ‘Zhang Cang determined the rules and norms’, by which The Nine Chapters were compiled. During the early Tang dynasty (618–907), when Li Chunfeng (李淳风; 602–670) selected Liu Hui's commentary as the official exegesis of The Nine Chapters, these three origin stories were endorsed by the government. A 12th-century mathematical compendium titled The Source and Development of Mathematics (Suanxue Yuanliu, 算学源流)—appended to the Records on Forgotten Mathematical Procedures (Shushu Jiyi, 数术记遗), a textbook of the Imperial University's School of Mathematics—preserves precisely the same three narratives recorded by Liu Hui (Bao, 1981: 1a–2b). In short, Liu Hui's three narratives formed the classical account of early mathematical history, until a fourth tale—about numbers embodied in the He Diagram (河图, Hetu) and the Luo Shu Square (洛书, Luoshu)—was added in the 13th century by two southern scholars, Yang Hui (杨辉; active 1261–1276) and Qin Jiushao (秦九韶; 1208–1268). ⁶

2. Liu Hui's contribution to ‘geometrical algebra’

Extracting square roots was a standard mathematical procedure in ancient China. In the fourth chapter of The Nine Chapters, a method called Kaifang (开方, ‘opening’ or ‘setting up a square’)—implemented with counting rods—is recorded. ⁷ Previous studies—including my own—have consistently read Liu Hui's commentary on ‘Kaifang Shu’ as both a technique for finding the side length of a square whose area equals a given product and as a geometric process of successively subdividing that square to determine those side lengths (see e.g., Lam, 1970; Wang and Needham, 1955). In other words, the geometric basis of extracting a square root is determining the length of the sides of a given square. In this section, I offer an alternative reading of Liu Hui's commentary—one that both captures the original intent of the procedure in The Nine Chapters and harmonizes with later Confucian methods.

In The Nine Chapters, the problem of extracting a square root is presented as follows:

Suppose we have a product of 55,225 bu.

One asks: Into what square can this be transformed?

The answer is: 235 bu.

Kaifang procedure reads: … ⁸

According to Li Jimin's research (李继闵; 1938–1993), the bu (步) is normally a unit of length; here, however, it denotes an area—specifically, a very slender rectangle measuring 55,225 bu in length and 1 bu in width (see the right part of Figure 1), and in other cases, the length unit can be also used to denote a volume, that is, a cuboid with a square cross-section (Li, 1998: 768–778). Therefore, the technical term Ji 积 is better to be translated into ‘product’ instead of ‘area’. When The Nine Chapters asks ‘Into what square can this be transformed?’, it is asking for a square whose area equals the rectangular product—namely, 55,225 bu. Hence, the original meaning of extracting the square root in The Nine Chapters is simply to find a square whose area equals the given product.

Figure 1.

Liu Hui's interpretation for square-root extraction.

When Liu Hui comments on the Kaifang procedure, he employs specific technical terms and figures—Yellow jia (黄甲), Yellow yi (黄乙), Vermillion region (朱幂) and Blue-green region (青幂)—to illustrate how the square is built up. The mathematics is straightforward. First, he constructs a square of side length a, so a² corresponds to the Yellow jia (the large square). Next, he considers a square of side length a + b. Since (a + b)² = a² + 2ab + b², the small square b² corresponds to Yellow yi, and each rectangular part ab corresponds to one Vermillion region. In the third step, he extends the construction to a square of side length a + b + c, where the Blue-green region represents the additional area c (a + b). See the left part of Figure 1. Karine Chemla (2010) has argued that the diagrams (Tu 图) in Liu Hui's commentaries were originally material objects rather than ideal figures, and that by the 13th century they had been converted into paper diagrams.

The key point is that the procedure of extracting the square root is always interpreted as a process of disaggregating a square; conversely, the procedure may also be understood as a method for constructing one. These two seemingly contradictory explanations can both be valid. For example, when Liu Hui writes that one ‘first obtains the sides of the yellow jia’ (see Guo, 2004: 134), ⁹ previous studies have understood this operation as cutting the yellow jia square from the whole square (i.e., the radicand), continuously removing pieces from the whole square (see the left part of Figure 1). However, it can also be interpreted as cutting the yellow jia from the product and then transforming it into a square, in which case the process involves continuously cutting pieces from the product to form the square (i.e., transform the right part into the left part of Figure 1). I believe the earlier misunderstanding stems largely from the fact that previous scholars—such as Yang Hui and Dai Zhen (戴震; 1724–1777)—only presented the completed square composed of various pieces, without indicating how the square was obtained (see Figure 2). In fact, Liu Hui writes: ‘For any procedure in which one disaggregates the product and makes a square of it, multiplying the square's side by itself returns the original product in fen’ (see Guo, 2004: 136). ¹⁰ This sentence clearly indicates that Liu Hui preserved the algorithm as a method for constructing a square rather than deconstructing one. Therefore, Liu Hui provides a dynamic geometric basis for the procedure of extracting square roots carried out with counting rods. ¹¹

Figure 2.

Geometric basis for square-root extraction by Yang Hui and Dai Zhen. Note. ^aSee Yongle Dadian 永乐大典 [Great Compendium of the Yongle Era], volume 16344, 8a, preserved in the Library of the Cambridge University. ^bSee Dai (1992: 668).

During the Southern and Northern dynasties (420–589 CE), Huang Kan (皇侃; 488–545 CE) devised a geometric procedure for extracting square roots in his commentary on the Analects (论语). His approach was later refined by Jia Gongyan (贾公彦; active 650–655 CE) and other Confucian scholars, with Jia explicitly employing the term ‘make a square of it’ (方之, fangzhi). ¹² In essence, this Confucian technique for square-root extraction represents a further development of Liu Hui's methods in The Nine Chapters, although without the use of counting rods. ¹³

Between the 11th and 13th centuries, scholars devised the ‘section of pieces’ method (tiaoduan fa条段法). This approach uses geometric constructions to formulate equations and later served as the geometric foundation for the celestial source method (tianyuan shu 天元术), which uses counting rods to set up equations, and is known as Chinese algebra. ¹⁴ In this sense, Liu Hui's understanding can be regarded as a forerunner to the ‘section of pieces’ method.

3. Liu Hui's use of counting rods

All calculations in The Nine Chapters are performed with counting rods, using two alternating notation systems: the vertical form for units, hundreds and ten-thousands, and the horizontal form for tens, thousands and hundred-thousands (see Table 1). In many instances, the written procedures closely reflect the physical manipulations with the rods—textual descriptions were directly shaped by the material operations. A prime example is the eighth chapter, Fangcheng (方程), which in modern terms deals with systems of linear equations.

Table 1.

Numerals by counting rods.

Source: Li and Du (1987: 8).

The core technique in the Fangcheng procedure for eliminating unknowns is called zhichu (直除), or ‘direct removal’. However, Liu Hui later refined this approach by introducing hucheng xiangxiao (互乘相消), which is often translated as ‘mutual multiplication and elimination’. In my 2010 analysis, I examined the pivotal role of counting rods in this transformation (Zhu, 2010). Below, I present the key points, illustrated with the example of Problem 7 in the eighth chapter. The problem reads as follows:

Suppose there are 5 oxen and 2 sheep, and they are worth 10 liang in gold; [if there are] 2 oxen and 5 sheep, they are worth 8 liang in gold.

One asks: How much is each ox and sheep worth in gold?

The answer is: 1 ox is worth 13/21 liang in gold; 1 sheep is worth 20/21 liang in gold.

The procedure reads: use Fangcheng … ¹⁵

In modern notation, the problem becomes solving the system of equations $\{\begin{array}{c}5x+2y=10\\ 2x+5y=8\end{array}$ —where x denotes the price of an ox and y the price of a sheep—and the solution is: $\{\begin{array}{c}x=1+13/21\\ y=20/21\end{array}$ . Using counting rods, the procedure can be represented as follows:

$$\begin{array}{cc}2& 5\\ 5& 2\\ 8& 10\end{array}\to \begin{array}{cc}10& 5\\ 25& 2\\ 40& 10\end{array}\to \begin{array}{cc}5& 5\\ 23& 2\\ 30& 10\end{array}\to \begin{array}{cc}& 5\\ 21& 2\\ 20& 10\end{array}$$

Here, we apply the direct removal method: we continuously subtract the right column from the left column until the top entries in the left column are eliminated. Hence one obtains y = 20/21. However, Liu Hui offers the following commentary:

In order to equalize and homogenize [the oxen and sheep], [put the number of] oxen in the top position [of each column], which should be mutually multiplied one by the other. Keep the right column unchanged, additionally putting down 10 for the oxen, 4 for the sheep, and 20 for the value of the gold in liang. In the left column, [put down] 10 for the oxen, 25 for [the] sheep, and 40 for the value of the gold in liang. The number for oxen is exactly the same [in both columns], but the gold is greater by 20 liang because there is a difference of 21 sheep. Subtract the column with the smaller [numbers] from the column with the larger [numbers]. The number for oxen is eliminated, and only the number of sheep and the value of the gold appears, from which the results can be known. Subtracting the smaller from the larger, although there may be 4 or 5 columns, there is no difference. (Guo et al., 2013: 952–953) ¹⁶

In this commentary, Liu Hui employs the mutual multiplication and elimination method. The key here is to additionally put down (10, 4, 20), which keep the right column unchanged. Liu Hui does this because he intends to reuse the original right-hand values in subsequent operations, rather than replacing them with their doubles (10, 4, 20). Therefore, Liu Hui's computations can be represented as follows:

$$\begin{array}{cc}2& 5\\ 5& 2\\ 8& 10\end{array}\to \begin{array}{ccc}10& 2& 5\\ 4& 5& 2\\ 20& 8& 10\end{array}\to \begin{array}{ccc}10& 10& 5\\ 4& 25& 2\\ 20& 40& 10\end{array}\to \begin{array}{cc}& 5\\ 21& 2\\ 20& 10\end{array}$$

When we compare Liu Hui's procedure to that of The Nine Chapters, two main differences emerge. First, Liu Hui's method requires fewer steps—especially as the number of columns increases. Second, it requires more counting rods: to keep the right-hand column intact, Liu Hui must place extra rods, whereas The Nine Chapters avoids this by using direct removal. Clearly, Liu Hui sought a balance between the simplicity of a procedure and the number of counting rods required. This explains why he employed the mutual multiplication and elimination method only in Problem 7 of the eighth chapter—a case involving just two unknowns and two columns.

In his commentary on the last problem of the eighth chapter, Liu Hui raises a new Fangcheng method (fangcheng xinshu 方程新术). After saying ‘Record one example demonstrating each procedure's execution, including the number of counting rods used; a single instance for each case will suffice’ (Guo, 2004: 367), ¹⁷ Liu Hui directly compared the number of counting rods required by the original procedure (77 counting rods) with those used in his revised method (124 counting rods) (Wei and Guo, 2016). Clearly, Liu Hui continues to strike a balance between procedural simplicity and the number of counting rods required. Liu Hui remarks: ‘In all major problems addressed in The Nine Chapters, none require computations exceeding one hundred counting rods. Though the counting rods are few, they suffice to compute much’ (Guo, 2004: 367). ¹⁸ Here, he emphasizes the epistemological value of minimality in procedural methods, asserting that problems should be solved by striking a balance between using the fewest counting rods and the simplest mathematical operations.

Liu Hui's ideas and practices have had a profound and enduring impact on the development of mathematics. During the Southern and Northern dynasties, mathematical works produced in northern China—such as the Mathematical Canon by Master Sun (Sunzi Suanjing 孙子算经), the Mathematical Canon by Zhang Qiujian (Zhangqiujian Suanjing 张丘建算经) and the Mathematical Canon by Xiahou Yang (Xiahou Yang Suanjing 夏侯阳算经)—introduced fundamental concepts not found in The Nine Chapters, including the use of counting rods to represent numbers and methods for multiplication and division. By the Tang Dynasty in the 8th century, rod calculations had been refined so that calculations once requiring three parallel rows of rods could be carried out with a single row, greatly reducing the number of rods needed. On one hand, this trend transformed counting rods into a purely computational tool, as seen in Yang Hui's mathematical works, eventually leading to their replacement by the abacus, which became widely popular during the Ming Dynasty (1368–1644). On the other hand, in the 13th century, scholars across both northern and southern China developed a numeral system based on counting rods and made extensive use of rod numeral diagrams to illustrate procedures. This culminated in an intermediate phase of the textualization and symbolization of Chinese mathematics, most notably reflected in Qin Jiushao's Mathematical Book in Nine Chapters (Shushu Jiuzhang 数书九章, 1247). ¹⁹

4. Conclusion

Previously, Liu Hui's mathematical achievements were primarily recognized for providing a rigorous foundation for The Nine Chapters, introducing early concepts of limits and systematizing Chinese mathematics. The Nine Chapters offers only procedural texts without any formal proofs. In his commentary, Liu Hui systematically establishes the correctness of each procedure—what Karine Chemla (2012) interprets as proofs. In his commentary to the circular area formula, rectangular-pyramid (called yangma阳马) and right triangular pyramid (called bienao鳖臑) area formula and square and cubic root extraction, Liu Hui employs logical reasoning through nearly infinite segmentation. ²⁰ In Liu Hui's commentary, every procedure is articulated through the concept of lü 率, which Guo Shuchun (1992) regards as embodying the systematic structure of Chinese mathematics. ²¹ This article, however, explores Liu Hui's contributions in three additional areas: early mathematical history, geometric algebra and the use of counting rods. In summary, Liu Hui made a comprehensive contribution to ancient Chinese mathematics.

Liu Hui's contributions are closely tied to his personal background. First, he was deeply influenced by the Wei–Jin intellectual tradition of dialectical debate and was himself a renowned scholar of that era (Guo, 1992: 321–345). Second, evidence from his works—such as his method of measuring the height of Mount Taishan (referred to as the ‘sea island’) using the principle of repeated differences (chongcha 重差), ²² his verification of the volume of Wang Mang's bronze measures using π, ²³ his own preface, ²⁴ and the fact that there is no official bibliography for him ²⁵ —suggests that he was not accepted by the Jin dynasty (Zhu, 2024). Overall, Liu Hui was a scholar who distanced himself from worldly affairs to focus on mathematics, leading to his extraordinary achievements in the field.