Translating Euclid’s Elements into Chinese: Western Missionaries and the Enlightenment for Modern Chinese Mathematics During the Late Ming and Early Qing Dynasties

Abstract

During the late Ming and early Qing Dynasties, China underwent a period of broad-based economic and societal transformation. Among the cultural forces at play, the Christian culture has significantly impacted the trajectory of Chinese history. At the time, responding to a distinct socio-political environment, Western missionaries employed a variety of religious methodologies to pursue the goal of proselytizing. As part of missionary efforts, they introduced Western scientific and cultural knowledge into China alongside Christian doctrines, coinciding with a period of political and cultural transformation and development in China. Accordingly, this influx of new ideas from the West had a far-reaching impact on Chinese society. This paper focuses on the Chinese translation of Euclid’s Elements, examining the intercultural dissemination of Western mathematical knowledge through missionary activities. Furthermore, the study also elucidates the positive impact of Western mathematics carried with religious efforts on the Chinese traditional mathematical system via presenting a comparison of paradigms in mathematics. Finally, this study argues that the translation practice by Christian emissaries from the West in the natural sciences during the Ming and Qing Dynasties engendered novel intellectual currents, thereby facilitating the development of a contemporary Chinese knowledge framework and a shift in religious research toward comprehensive perspectives.

1. Introduction

The late Ming and early Qing period marked a significant historical juncture in the reintroduction of Western Christianity to China, following the earlier practice of Jingjiao 景教 (Nestorianism/the Church of the East) during the Tang dynasty and Yelikewenjiao 也里可溫教 during the Yuan dynasty. It was also a flourishing period of Sino-Western civilizational dialogue (T. Li 2022; Xiao 2015). In the sixteenth century, propelled by the Protestant Reformation and the expansion of Portuguese colonial power, Western missionaries, most notably the Society of Jesus, established their presence in China, with their primary objective being to propagate Christian beliefs through the peaceful conversion of the Chinese people (Gu 2013; Liu 2013). Furthermore, the advent of novel maritime pathways, coupled with the imperative to propagate Christian tenets overseas, accelerated this confluence, fostering the reciprocal dissemination of Sino-Western cultural paradigms, thereby introducing Western intellectual currents and cultural influences into Chinese socio-economic evolution.

The current scholarship on the history of Catholicism during the Ming and Qing dynasties has yielded substantial findings, primarily concentrating on the internal and external dimensions of the historical trajectory of religion. Those internal research efforts on Catholicism history has made notable contributions, including studies on the evolution of Catholicism (Bays 2011; Motte 2004; Tao and Wei 2024; Z. Xu 2015; Yao and Luo 2000), the role of church schools (He and Shi 1996; Lutz 1988), Christian doctrines (Berkhof 2000; Wang 2019), the interaction between Christian culture and traditional Chinese culture (Wickeri 2015; Xiao 2019), and Terms Questions (Huang 2019; Li et al. 2023). Such studies significantly contribute to the understanding of the acculturation and expansion of Christian culture, as well as the history of Christian values in China. Meanwhile, the external research on religious heritage has primarily focused on the interactions between Western missionaries and Confucian scholars (Ma 2021; Sun 2013), the dynamic between Christian traditions and Chinese social-cultural frameworks (X. Zhang 2005), the impact of Christianity on the Chinese sociopolitical landscape (Gernet and Demiéville 2011), the influence of Christianity on Chinese education (Li and Xiao 2018; Z. Wu 2003), the role of Christianity in the development of Chinese publishing industry (Zhao and Wu 2011), and so on. These comprehensive research outcomes provide a nuanced and in-depth understanding of the historical development of Christian culture in China, thereby establishing a solid foundation for reevaluating the cultural exchange between China and the West.

As we all know, the nexus between translation and society is indissoluble. Translation practices are circumscribed by societal elements, while simultaneously, translation exerts a formative influence on society (Fu and Li 2023; Fu and Zhang 2022; Tyulenev 2014; Venuti 2018). The translation endeavors of Western missionaries during the Ming and Qing dynasties were significantly constrained by the multifaceted social structures of China. To facilitate their proselytization objectives, diverse translation strategies were employed. Nevertheless, translation activities, as integral components of social dynamics and objective social realities, inevitably engender specific impacts on societal and cultural ideologies. In recent years, academia has reassessed the epistemic value of missionary translation, acknowledging its positive contributions to the modernization of Chinese knowledge. These research outcomes primarily concentrate on the following areas: missionary translation and the evolution of modern Chinese disciplines (Fu 2024); missionary translation and the advancement of Chinese mathematics (B. Zhang 2021, 2024); missionary translation and the dissemination of Chinese physics knowledge (Li et al. 2024; Li and Fu 2024); missionary translation and the expansion of Chinese geographical knowledge (He and Hou 2024); and so on. These investigations furnish a novel perspective on the course of Christian development throughout Chinese history, specifically, the role of missionary translation in the transformation and advancement of Chinese society and culture. If the history of religious development or the evolution and dissemination of doctrines constitutes a significant focus of religious studies, then the examination of religion’s role within the broader context of social and cultural development represents a novel approach. This perspective also reflects the current research findings, which situate the translation activities of missionaries within a historical framework, thereby expanding the scope of religious research.

Examining the missionary endeavors of the late Ming and early Qing dynasties reveals that geometry, initially a Western text, was translated by Western missionaries with the collaboration of Chinese intellectuals. This played a crucial role in the construction of a Western-Confucian identity and the cultural missionary strategies of the West. Current scholarly research on the translation of geometric texts has yielded significant results, focusing on translator studies (B. Zhang 2024), translation studies (Lu 2013), and translation quality and linguistic quality assessment (Ji 2017), which aids in understanding the translation history of geometric texts. However, the original translation of geometry occurred during a period of societal transformation in China. The translation and dissemination of Western mathematical knowledge inevitably impacted the traditional Chinese mathematical knowledge system, fostering its evolution toward modern mathematical knowledge. This paper will, therefore, re-evaluate the original translation activities of geometry within the context of the transformation and development of Chinese social and cultural knowledge. It aims to elucidate the enlightening role of missionaries’ translation activities in the transformation and development of Chinese mathematical knowledge and to affirm the positive impact of missionaries’ translation activities as a social phenomenon on the development of Chinese social culture.

Therefore, based on the aforementioned research hypothesis, this paper examines the paradigm of the Chinese traditional mathematical knowledge system, building upon existing research. It analyzes the novel mathematical concepts introduced by Western mathematical knowledge, specifically focusing on geometry. Furthermore, it elucidates the evolving paradigm of the Chinese traditional mathematical knowledge system under the influence of integrated Western mathematical ideas. Finally, it underscores the impact of missionary mathematical translations on Chinese social culture, thereby demonstrating the correlation between Western missionaries’ translation activities and the advancement of the Chinese mathematical knowledge system. This analysis aims to validate the positive contribution of missionaries’ geometric translations to the development of the Chinese mathematical knowledge system.

2. The Pre-Euclidean Mathematical Paradigm in China: Practice-Oriented Arithmetical Traditions

Chinese traditional mathematics embraces a rich legacy, culminating in its zenith during the Song and Yuan dynasties (Jami 2012; Shi 2023). During this period, a multitude of notable accomplishments in both practical applications and theoretical frameworks were achieved. It even excelled in computational techniques, surpassing contemporary European studies by centuries (Guo and Kong 2007). Mathematics in that era prioritized addressing real-world issues, highlighting the strong integration of arithmetic theories with everyday life and production. Hence, a range of effective calculation techniques was generated at that time. The concept of paradigm was initially coined by Thomas Kuhn, a renowned philosopher of science. Paradigms are shared intuitions by a community of scholars, which are often tacitly communicated within the practice of science (Kuhn 1970). In essence, these constitute a practical–conceptual framework for cooperative research during periods of “normal science” (Rehg 2012). Subsequent scholars have expanded upon his work through deeper interpretations. For example, a paradigm is a set of conceptual systems and analytical approaches collectively accepted, used, and serving as common tools for intellectual exchange among theorists and practitioners (Fan 1995); a paradigm represents a shared constellation of group commitments (Anand et al. 2020). Considering this, alongside the continuity and closed nature of mathematics before the Ming and Qing dynasties contributed to a relatively solid mathematical structure, with minimal noticeable alterations in its overall framework over time (W. Wu 2020), this paper mainly examines ancient Chinese mathematical paradigms from three dimensions: practicality, theoretical features and form, and the computational method. The first dimension reflects a widely shared cognitive consensus, while the latter two represent the conceptual foundations and empirical techniques endorsed by both scholarly communities and the broader historical context. Additionally, these interconnected dimensions collectively reflect the integration of practical applications and the knowledge system of mathematics in society at the time.

2.1. Practicality

Chinese traditional mathematics naturally adhered to the ideology of “jingshizhiyong1 經世致用 (ordering the world and promoting utility),” acting as typical non-European mathematics (Joseph 2011). In the form of a practical technique or trick, mathematics during that period was closely driven by societal needs and continuously evolved through daily practice. Qin Jiushao 秦九韶 (1208–1261), a renowned mathematician of the Southern Song dynasty, depicted the power of mathematics in Jiuzhang suanshu2 九章算術 (The Nine Chapters), “da ze keyi tongshenming, shuntianming, xiao ze keyi jingshiwu, leiwanwu. 大則可以通神明, 順天命, 小則可以經世務, 類萬物 (If grand, it can communicate with divine wisdom and follow the mandate of heaven; if modest, it can govern worldly affairs and classify all things).” On the one hand, throughout China’s historical trajectory, the most significant aspect of mathematics lay primarily in its connection to the calendar system (Needham 1959). The calendar system, or a fortune-teller (Ding and Zhang 1989, p. 36), was consistently developed across successive dynasties in old Chinese civilization, as reflected in the saying “guancha tianxiang, jingshou minshi. 觀察天象, 敬授民時 (Observe celestial phenomena and reverently impart time to the people).” Even the decline in traditional mathematics in its later stages was closely tied to its practical utility. For example, when calendrical calculations began to produce errors and proved unreliable, traditional mathematics gradually lost its practical value and fell out of favor with the court. This fully underscores the fact that traditional mathematics was fundamentally driven by practical needs and closely linked to daily activities. On the other hand, an exhaustive review of Chinese classical mathematical volumes also reveals that they were intricately associated with the practical necessities of the era. It is evident not only in the fact that, since Jiuzhang suanshu 九章算術 (The Nine Chapters), most Chinese mathematical classics were compiled in the form of problems sets with solutions (Celia et al. 1999; Martzloff 2006), but also in the way these problems directly mirrored the practical needs of their respective eras, such as land surveying, granary capacity assessment, dike and embankment construction, taxation, and currency exchange (Liang et al. 1983).

2.2. Theoretical Features and Form

In scholarly discourse, two contrasting perspectives persist regarding the theoretical framework of ancient Chinese mathematics. One school acknowledges its accomplishments but maintains that it was so predominantly empirical, lacking a systematic theoretical structure (Mikami 1913; Needham and Ronan 1978). The truth is, no complete and systematic algorithm can be founded solely on empirical experience (J. Li 2007, p. 7). Calculation-centric approaches and endorsing “Zhongfaqingli 重法輕理 (emphasizing methods over theory)” do not imply that mathematicians refrained from utilizing logical reasoning, nor does it imply ancient mathematics in China failed to include its theoretical basis (W. Wu 2020). Although traditional Chinese mathematics exhibited an uneven development between practice and theoretical constructions, its theoretical system nonetheless existed and should not be overlooked. Not only that, attributing the absence of theoretical foundations to Chinese mathematics solely based on its emphasis on practical applications is a one-sided perspective. Moreover, yuliyusuan 寓理於算 (integrating theory into calculation) and jinglian 精煉 (methodical refinement) are two distinctive attributes of the traditional mathematics of China (J. Li 2007; W. Wu 2020), which is exemplified by the fact that ancient Chinese mathematical classics, shibusuanjing3 十部算經 (The Ten Chinese Mathematical Canons), often emphasized the presentation of algorithms while providing minimal theoretical elaboration of the underlying mathematical principles. As a result, the traditional system of mathematical theory took on a distinctive form, namely represented by the exegesis of the canonical texts. Among them, zhoubi suanjin zhu4 周髀算經注 (The Commentary on the Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) is the most representative. In his commentary, Zhao Shuang 趙爽 (fl. 3rd century) gives details of the process of calculation leading to results stated in the main text without explanation (Cullen 1996). It is through this form of exegetical annotation that underlying mathematical principles are made explicit, reflecting not only the didactic function of commentary but also the theoretical foundation of ancient Chinese mathematics. Meanwhile, Zhao also proposed the earliest known visual proof of the Pythagorean theorem, using what he called the “gougu yuanfangtu 勾股圓方圖 (Hypotenuse diagram)” to demonstrate the relationship of a² + b² = c² through geometric dissection. His approach reflected an early understanding of geometric principles and spatial reasoning in classical Chinese mathematics. Liu Hui 劉徽 (ca. 225–295), on the other hand, was remarkable for his systematic and logical exposition of mathematical proof. The exegesis composed the first known mathematical proofs in ancient China (Chemla 2010). Liu’s use of “Geyuanshu 割圓術 (Polygonal approximation)” revealed an early grasp of continuity and geometric limit. By illustrating that an n-gon with an increasingly large number of sides could eventually coincide in shape and area with the circle, Liu provided a theoretical basis for the algorithm in Jiuzhangsuanshu 九章算術 (The Nine Chapters), transforming a computational rule into a conceptual proof.

2.3. Computational Method

Since ancient Chinese mathematics centered on “Suan 算 (Calculation)”, its development was inevitably accompanied by important computational tools. These tools not only acted as material carriers of mathematical operations but also reflected how the ancient understood numbers, space, and logic. A notable trait of the development of ancient Chinese mathematics was the use of bamboo rods to represent numbers as tangible counterparts, with calculation processes expressed through the arrangement and manipulation of these rods. This computational method, known as Suan 算 or Chousuan 籌算 (Counting-rods), occupied an important position in Chinese mathematics from the Han dynasty to the Yuan dynasty, as many techniques of numerical computation during this period relied on it (Martzloff 2006, p. 210). The origin of Chousuan 籌算 (Counting-rods) in ancient China can be traced to divinatory practices in the Yijing 易經 (Book of Changes), where stalks or rods were manipulated to interpret celestial phenomena. Over time, this mystical function gradually gave way to rational arithmetic, resulting in a practical system of numerical computation using counting rods. The counting method included manipulating the color, shape, and arrangement of rods through manual operations. It evolved from basic arithmetic operations such as addition, subtraction, multiplication, and division to more advanced techniques, including root extraction and exponentiation. By the Song and Yuan dynasties, this practical calculus had further developed into the “dayan qiuyishu 大衍求一術 (The Dayan Method for Finding Unity)” for indeterminate analysis and the “zhengfu kaifangshu 正負開方術 (the method of extracting roots with positive and negative numbers)” for solving higher-degree polynomial equations numerically. It can be observed that Suanchou 算籌 was a uniquely Chinese invention and the most effective computational tool before the advent of computers. The development of ancient Chinese computational techniques was largely influenced by bamboo-tokens calculus (W. Wu 2020). Moreover, the use of counting rods in ancient China was not merely a technical convenience but a fundamental part of a broader mathematical paradigm that prioritized algorithms, procedural reasoning, and the integration of practice with theoretical reflection.

3. Mathematical Knowledge Transfer Across Cultures: Translating Euclid’s Elements into Chinese by Missionaries

The intercultural dissemination of Euclid’s Elements was a strategic choice by missionaries and a long-anticipated aspiration among Chinese intellectuals. On the one hand, missionaries sought to leverage the prestige of scholarly knowledge and the high regard for books in Chinese culture to facilitate their evangelical goals. On the other hand, some enlightened recognized not only the advancements in Western science and technology but also the limitations within China’s existing local mathematics, particularly its lack of systematic deductive reasoning in geometry (Martzloff 2006). They posited that the translation and dissemination of Western knowledge could contribute to the development of Chinese culture, so as to achieve the goal of maintaining the rule of the feudal dynasty.

3.1. Dual Dynamics in the Intercultural Transmission of Euclid’s Elements

The introduction of Euclid’s Elements into China was largely made possible through an unconventional approach to evangelization, which sought to promote religious ideas using scholarly engagement. During the Ming dynasty, China’s unique political structure and academic climate created formidable impediments to Western missionaries, restricting the pathways for their established preaching activities. For instance, the Ming government’s foreign policy of maritime prohibition and the prevailing Chinese perception of “foreigners” rendered modes of evangelization ineffective (Y. Wu 2017). Consequently, missionaries found it necessary to devise approaches distinct from those employed elsewhere (Henri 1964), which led them to gradually adopt more nuanced methodologies. Those Western religious scholars adopted flexible and adaptive missionary strategies, tailoring their methods to the specific cultural and social contexts of different ethnic groups to disseminate Christianity; among them, shujichuanjiao 書籍傳教 (Apostolate through books) was proved to be the most impactful (Xiao 2011). During the period, the strategy was evident in various ways: showcasing Western books to demonstrate the cultural origins of Christianity, producing Chinese writings and translating Western works to establish missionaries as knowledgeable role as Xiru 西儒 (Western Confucians), and directly spreading Christian doctrines through Chinese books (Li et al. 2024). Meanwhile, from the perspective of many Europeans at the time, Chinese mathematics was regarded as quite superficial. As reflected in A History of Chinese Mathematics, Martzloff (2006) stated that Chinese mathematics lacks systematic reasoning and is deemed of limited epistemic value. Precisely because of this perception, Matteo Ricci 利瑪竇 (1552–1610) sought to utilize the rigorous logical reasoning inherent in Western mathematics to impress Chinese scholars and officials, thereby enhancing the credibility of Western scholarship. This, in turn, enabled missionaries to gain access to elite social circles and indirectly advance their evangelical mission. As a canonical representative of European advanced mathematical thought, Euclid’s Elements was naturally included among the core texts utilized in evangelism.

Civilizations die by suicide, not by murder (Toynbee 1946). This reflection finds strong parallels in the intellectual and scientific decline experienced in late imperial China, where the Chinese educated patriots, upon recognizing the existential crisis confronting traditional natural sciences, began to express a strong desire for new intellectual vitality and scientific methodologies, hoping to revitalize academic life and halt the decline of their national civilization. According to the history of science, the Chinese mathematical traditions entered an unprecedented period of stagnation (Guo 2012; Hart 2013; Qian 2019; Shi 2023), to the point that its most significant achievements of traditional mathematics fell into oblivion during the mid-to-late Ming dynasty. One manifestation of this deterioration was that the highly sophisticated tianyuan (天元) place-value algebra, developed during the thirteenth century, had been forgotten, a fate shared by the calculating device—the counting rods (Jami 2012). Another notable expression of this deterioration was the inaccuracies of the Datongli 大統曆 (Datong calendar system), which stemmed from its reliance on outdated mathematical models. Even repeated attempts to reform the calendar proved ineffective (Song and Wang 2010). Accordingly, the aspiration for new epistemological resources became particularly acute at the time. For example, as Xu Guangqi 徐光啟 (1562–1633) famously states, “gujin zhongwai zhi xue, jie ke weiwo suoyong 古今中外之學, 皆可為我所用 (The knowledge of all times and all lands may serve my purpose).” (G. Xu 1984). These declarations underscored the conviction that integrating Western scientific knowledge was essential to rejuvenate Chinese intellectual traditions during their perceived decline. Furthermore, it is equally significant to consider the role of the religious conversion of Chinese intellectuals in shaping this epistemological shift. Before his conversion, although Xu valued shixue 實學 (concrete studies), his thinking was still driven by the Confucian principle of jingshi zhiyong 經世致用 (ordering the world and promoting utility), which encouraged him to instrumentalize knowledge, prioritizing its application in statecraft and the management of societal affairs. This orientation is evident in his pre-conversion writings. For instance, in the agricultural treatise Nongzheng quanshu 農政全書, G. Xu (1956) advocated “務實以利民 (practical efforts to benefit the people),” reflecting his pragmatic orientation toward agricultural concerns. However, following the conversion, the infusion of a theocentric worldview transformed Xu’s epistemological outlook, leading him to move beyond a purely utilitarian understanding of knowledge toward an integration of practicality with metaphysical inquiry and ethical cultivation. For example, Xu proposed the movement from mere comprehension of Western mathematics toward ‘surpassing’ the original, indicating an intellectual ambition beyond pragmatic application, one rooted in grasping a deeper rational order consistent with divine creation. Therefore, this transition from a focus on “圖存 saving the state” to the pursuit of “致知 attaining knowledge” embodies the cognitive shift and cultural repositioning of the literati during a period of multiple crises in late Ming China.

3.2. The Translation of Euclid’s Elements

Books I–VI of Euclid’s Elements were translated during the Ming dynasty by Matteo Ricci 利瑪竇 and Xu Guangqi 徐光啟, marking a significant milestone in introducing systematic Western geometry to China. More than two centuries later, the remaining books were translated through the collaboration between the Qing mathematician Li Shanlan 李善蘭 (1811–1882) and the British missionary Alexander Wylie 偉烈亞力 (1815–1887). Their efforts completed the remaining books (Books VII–XV), thereby offering Chinese readers a more comprehensive understanding of the Euclidean system. In addition, prior to the Xu-Ricci translation, Chinese intellectuals such as Qu Taisu 瞿太素 (fl. 16th century) and Zhang Yangmo 張養默 (fl. 16th century) had undertaken exploratory attempts to translate the first book of the Elements (Hsia 2010). Because these were motivated by personal interest and practical knowledge concerns, their early efforts were fragmented and did not produce systematic results. Similarly, the translation of Euclid’s Elements was first assigned to Diego Pantoja 龐迪我 (1571–1618) and an unidentified friend initially commissioned by Xu Guangqi 徐光啟, yet the collaboration proved to be less effective, because Xu’s friend was not up to the task (Jami et al. 2001, p. 33). The collaborative translation of Xu and Ricci was based on Christoph Clavius’s Euclidis elementorum libri XV (Ogawa 2011). Embodying the fusion of Chinese and Western wisdom, this pioneering mathematical translation was achieved by the oral accounts of Matteo Ricci 利瑪竇 and written down by Xu Guangqi 徐光啟, exemplifying their remarkable collaboration and intercultural exchange. Xu recorded the operation in the prelude of Jihe Yuanben 幾何原本, the Chinese name of Euclid’s Elements: “Xiansheng jiugong, ming yu kouchuan, zi yibi shouyan. 先生就功, 命餘口傳, 自以筆受焉.” It is the same method that was used from the fifth to the eighth century to translate Buddhist texts (Zürcher 1972).

Nevertheless, despite their joint involvement in the translation project, their respective objectives exhibited subtle differences—religion outreach on one side and national salvation on the other. Ricci, recognized as the father of the Jesuit China mission (Jami et al. 2001, p. 21), fully appreciated the resistance to direct evangelization in China due to its distinctive institutional and intellectual context; therefore, he positioned Western-origin science as an intermediary for evangelisation, due to the fact that Jesuits’ science at the time had a much more pervasive influence on China than their religion (Jami 2012, p. 13). Furthermore, Ricci was more interested in utilizing Euclidean mathematics for the propagation of Christianity than in diffusing the scientific work itself (Ogawa 2011). Ricci’s translation of Western mathematics was thus only regarded as a useful tool of his indirect evangelism. Conversely, Xu Guangqi 徐光啟, as an imperial officer and minister (Ogawa 2011), perceived his engagement in the translation of Western knowledge as a patriotic duty-one of the numerous efforts he devoted to saving and reinforcing the Ming dynasty in the face of national crisis. This motivation was much more evident in his repeated emphasis on strengthening military defense and agricultural productivity through Western technology (Jami et al. 2001). For example, Xu argued for the adoption of Western calendrical and mathematical systems to improve governance and national security (Elman 2005). In addition, this initiative also reflected Xu Guangqi’s commitment to actualizing the Confucian ideal of jingshizhiyong 經世致用, which advocates applying knowledge to address real-world challenges. Meanwhile, Xu also regarded Euclidean geometry as a foundational tool for accessing the broader Western scientific system, in that the knowledge structure and theoretical system established in Western science were based on the Elements (Shi 2023). Hence, his primary objective was to introduce more systematic and practical knowledge through it, with the aim of saving and strengthening the Ming dynasty in a time of crisis. In addition, even though Xu’s status as a convert inevitably incorporated some missionary intent in his collaborative translation efforts, he always manifested much more interest in the social problems of the present world (Ogawa 2011). In light of this, these divergent goals gave rise to subtle differences in their translation philosophies. Matteo Ricci 利瑪竇 emphasized adaptability and cultural accommodation, ensuring the content aligned with local rhetorical and cognitive expectations (Gianni 2003), so as to facilitate the audience’s understanding and acceptance. Xu Guangqi 徐光啟, on the other hand, prioritized Huitong 會通 (Comprehension and integration) and Chaosheng 超勝 (Surpassing), reflecting a strong sense of translational subjectivity. Xu also emphasizes the integrated understanding of Chinese and Western scientific cultures. It involves introducing Western learning on the foundation of traditional Chinese science and technology and absorbing the strengths of heterogeneous cultures.

Unlike the pioneering Ricci-Xu collaboration that operated without a preexisting technical lexicon, necessitating extensive terminological innovation, Li-Wylie’s collaborative translation act followed the previously established translation conventions (Han 1998). Similarly, the translation of the latter nine volumes (Books VII–XV) of Euclid’s Elements was also carried out against the backdrop of a national crisis. In the face of the Western powers’ invasion during the Opium War, Li Shanlan 李善蘭 keenly recognized the importance of introducing advanced Western science and sought to revitalize the nation through the translation of Western texts. His translation philosophy was marked by a strong sense of national consciousness, and his translation practices embodied the characteristics of state-driven translation efforts.

4. The Advent of the New Paradigm for Chinese Mathematics: Axiomatisation and Logical Deductive Systems

The logical deductive systems imported by Jihe Yuanben 幾何原本 constituted a fundamental epistemological shift in the landscape of traditional Chinese mathematical thought. Previously centered on practical calculation and rule-based problem solving, Chinese mathematics began to incorporate the axiomatic method and logical deduction, leading to a breakthrough. This missionary-translated book functioned as a significant turning point in the later development of Chinese mathematics, marking a milestone. This new paradigm introduced Chinese scholars to a more abstract, theoretical framework for mathematics and laid important groundwork for the modernization and formalization of mathematical thought in China.

4.1. Axiomatization

One of the most distinctive features of Jihe Yuanben 幾何原本 is its foundation upon a clearly defined set of definitions, axioms, and postulates, elements largely absent from traditional Chinese mathematics, which had long prioritized procedural “Shu 術 (techniques)” over theoretical structures. According to Euclidean theory, each theorem or conclusion in geometry requires one or more foundational principles as its logical basis throughout the deductive process. These fundamental principles are self-evident and do not require proof. Euclid classified them into two categories: axioms, which apply to mathematics in general, and postulates, which are specific to geometry. Hence, Jihe Yuanben 幾何原本 brought about the establishment of a basic axiomatization system in traditional Chinese mathematics, transforming it from an example-based approach to problem-solving into a more coherent and logically derived system of propositions. At the heart of this axiomatic system lies the construction of mathematical theory through precise definitions. Following the translation of Jihe Yuanben 幾何原本, most of these new definitions were expressed through imported terminologies, which helped to initiate a more structured and formalized method of mathematical reasoning within the Chinese intellectual tradition. Through the translations of geometric works during the Ming and Qing dynasties, a systematic terminology for plane geometry, solid geometry, and analytic geometry gradually took shape (B. Zhang 2024). For example, the terms “dian 點 (point)”, “xian 線 (line)”, “zhixian 直線 (straight line)”, “quxian 曲線 (curve line)”, “jiao 角 (angle)”, “ruijiao 銳角 (acute angle)”, “zhijiao 直角 (right angle)”, “dunjiao 鈍角 (obtuse angle)”, “zhou 軸 (axis)”, “hengzhou 橫軸 (axis of abscissas)”, “heng zuobiao 橫坐標 (abscissas)”, “zong zuobiao 縱坐標 (ordinate)”, “tuoyuan 橢圓 (ellipse)”, “sanjiaoxing 三角形 (triangle)”, “dengbian sanjiaoxing 等邊三角形 (equilateral triangle)”, and “sibianxing 四邊形 (quadrilateral)” were first determined by the translation. Moreover, the introduction of geometric terminology not only had a profound impact on the development of geometry itself but also acted as a model for later translations in other branches of mathematics. In subsequent works on algebra and analysis, for instance, one can also observe the emergence and systematic definition of new mathematical terms. This axiomatic approach, characterized by clear definitions, logical reasoning, and deductive structure, originated with Jihe Yuanben 幾何原本 and gradually permeated various aspects of traditional Chinese mathematics. Over time, it contributed to the initial formation of a more structured, theoretical, and standardized system of mathematical knowledge in China.

4.2. Logical Deductive Systems

When translating Western mathematical texts, Xu Guangqi 徐光啟 placed great emphasis on theoretical transmission. He not only brought the axiomatic method into China but also systematically introduced the Western system of deductive logic (B. Zhang 2021, p. 74). According to the preface to the translation of Euclid’s Elements, the whole book’s ingenuity can be described in one word, 明 Ming (clarity): this is to point out the special characteristic of logical deduction (Li and Du 1987). The so-called deductive system, or deductive reasoning, refers to a mode of thinking in which one infers unknown aspects of a phenomenon based on established theoretical knowledge that reflects the objective laws governing that phenomenon. It is a reasoning process that moves from the general to the particular and from the universal to the specific. The use of reductio ad absurdum, analysis, and synthesis in its reasoning process undoubtedly introduced a new mode of thinking to the Chinese audience, who were generally less accustomed to logical reasoning and abstract thought. Rather than remaining at the level of proposing propositions and performing empirical calculations, this approach pursued the underlying rationale of propositions through rigorous logical deduction, thereby enhancing cognitive flexibility. The logical deductive framework (definition—axiom—theorem—proof) introduced through Euclid’s Elements catalyzed paradigmatic reforms among Chinese mathematicians. Unlike the traditional approach in Chinese mathematics, which directly presented conclusions and simplified the process, the new system emphasized deriving theorems through rigorous logical reasoning starting from basic axioms and definitions. This method focuses on clear organization and the rigor of reasoning, encouraging Chinese mathematicians to move beyond relying solely on empirical calculations and intuitive conclusions and instead to place greater importance on the structure of reasoning and the process of proving. Hence, with Euclidean geometry as a model, Chinese scholars began to value the role of logic and proof in mathematical thought and actively adopted and promoted this logical deductive mode of thinking, writing numerous works on geometry. For example, in his “Tongwen suanzhi 同文算指 (A Guide to Mathematical Calculation in Common Terms)”, collaborated by Matteo Ricci 利瑪竇, Li Zhizao 李之藻 applied Western arithmetic as a tool to rework the solutions to various types of general quadratic equations in traditional Chinese mathematics with a more systematic and symbolic way of thinking instead of empiricism (Pan 2006). This methodology exhibited structural isomorphism with the deductive logic and axiomatic architecture characteristic of Western mathematical epistemology. Other examples include “Ceyuanhaijing 測圓海鏡 (Sea Mirror of Circle Measurement)” by Li Rui 李銳 (1768–1817), which revisited and extended traditional circle-squaring methods under the influence of geometric rigor, and “Shuli jingyun 數理精蘊 (Essence of Mathematical Principles)” by Mei Wending 梅文鼎 (1633–1721), which emphasized logical structure and proof in the formulation of mathematical arguments. These works signaled a methodological rupture from intuitive or empirical calculations to a reasoning-based system that stressed the importance of definitions, axioms, and deductive sequences in mathematical exposition.

5. The Enlightenment of Modern Chinese Mathematics Knowledge in China

Prompted by the pragmatic needs to reform the traditional Chinese calendar and the academic evangelization by Western missionaries, the translation of Euclid’s Elements was carried out through collaboration between missionaries and enlightened Chinese scholars. This endeavor not only introduced the axiomatic and deductive structure of Western mathematics but also provided empirical validation of its theoretical rigor, thereby reshaping Chinese perceptions of mathematical knowledge. The dissemination of Elements marked a pivotal moment in the transformation of China’s traditional mathematical paradigm, facilitating a shift from algorithmic and utilitarian approaches to more abstract, logical reasoning. This section thus explores how the introduction of novel mathematical ideas contributed to the enlightenment of mathematical knowledge in China, and recognizes the role of missionary translators in catalyzing the social progress and cultural development of China.

5.1. The Dissemination of Western Mathematical Ideas

Initially, Euclid’s Elements gained widespread circulation following its translation, serving as a conduit for the dissemination of novel Western mathematical concepts. Subsequent to its initial publication, Euclid’s Elements underwent multiple revisions and reprints in China, driven by the significance attributed to Western mathematical knowledge. This process significantly propagated contemporary Western mathematical ideas while also intentionally supplementing the existing framework of ancient Chinese mathematical thought and algorithmic systems. Furthermore, the publication and distribution of Euclid’s Elements spurred the application of Western mathematical concepts in the composition of subsequent scholarly works, including Suanfa Quanshu 算法全書 (Algorithms, edited by Edward T. R. Moncrieff, 1852), Baxianbiao Yijuan 八線表一卷 (Trigonometric Function, edited by Giacomo Rho, period of the second Qing emperor (1644–1662)), Cesuan 策算 (Tables de logarithms et les Usage, translated by Ignatius Kgler, 1722), and so on. Prior to the translation and publication of the Elements, the Chinese mathematical knowledge predominantly emphasized practical experience. The introduction of Euclid’s Elements marked a shift, integrating logical reasoning with empirical observation, thereby fostering the scientific establishment of new concepts and a greater emphasis on logical rigor. Ultimately, Euclid’s Elements extended its influence to Japan, impacting the Japanese mathematical knowledge system (Sa 2017). Given the historical reliance on traditional Chinese cultural knowledge in Japan, its mathematical understanding exhibited empiricism and a relative lack of scientific rationality. Under the influence of Euclid’s Elements, the Japanese mathematical knowledge system also initiated a transition toward a modern framework.

5.2. The Generation of a Modern Mathematical Knowledge System in China

Prior to the translation and publication of Euclid’s Elements by Matteo Ricci 利瑪竇 and Xu Guangqi 徐光啟, Western mathematical ideas had already entered China in scattered forms, primarily through astronomical texts and calendar reforms. However, these ideas and concepts were neither framed as a coherent body of knowledge nor distinguished as part of a separate mathematical system. The translation of Euclid’s Elements in 1607 marked the first systematic and explicit introduction of modern Western mathematical thought into China (Engelfriet 1998). In contrast to traditional Chinese mathematics, which emphasized practical algorithms and empirical solutions, Euclid’s Elements was fully grounded in axiomatic logic and deductive reasoning. This distinction highlighted a fundamentally different epistemological approach—one that aligned with the scientific rationality emerging in Europe. Accordingly, Elements initiated a paradigm shift in the Chinese mathematical knowledge system, laying the groundwork for its gradual Westernization. The theoretical structure and logical rigor embedded in Euclidean geometry not only challenged existing mathematical practices but also influenced the evolution of disciplinary mathematics in China. Moreover, the translation introduced a set of mathematical terminologies—many of which remain in use today—that helped establish a foundation for modern mathematical education in China. Thus, the dissemination of Euclid’s Elements catalyzed both the modernization of Chinese mathematical thought and the institutionalization of a modern mathematical knowledge system.

5.3. The Promoting Modern Mathematics Education in China

The primary objective of Western missionaries in translating and disseminating Western scientific knowledge mirrored that of translating other scientific theories: to exert a proactive influence on the intellectual and ideological landscape of the Chinese people, thereby establishing a conducive environment for their missionary practice. Regardless of their approach, whether active or passive, Western missionaries prioritized the introduction of science. The propagation of religious doctrine constituted the paramount goal of Western missionaries in the East, with the translation and dissemination of Western science serving as a strategic means. Despite the instrumental nature of mathematical translation practices in advancing religious objectives, the scientific knowledge introduced by Western missionaries garnered attention and acceptance among Chinese people. The translation and publication of Euclid’s Elements, for instance, exerted a profound impact on the Chinese mathematical knowledge system, particularly in the realms of science education and knowledge dissemination. The logical rigor and scientific principles inherent in Western mathematical knowledge prompted enlightened Chinese intellectuals to recognize the significance of innovation in mathematical learning, leading to the deliberate integration of mathematics into school curricula for the study of Western mathematical knowledge. The establishment of Jingshi tongwenguan 京師同文館 (Imperial Tung Wen College) in 1862 exemplified a pivotal moment in modern educational reform in China. The inclusion of mathematics in the curriculum of Jingshi tongwenguan 京師同文館 signified the formal integration of mathematics into the modern education system, laying the groundwork for its evolution as an independent subject in China. Following the establishment of the Republic of China, the continuous influx of updated Western mathematical knowledge and theories, coupled with ongoing innovations, continued to shape the trajectory of China’s mathematical education system.

6. Conclusions

The dissemination of Christian culture in China during the late Ming and early Qing dynasties, a significant episode in the cultural exchange between East and West, has furnished a wealth of historical materials for comprehending the trajectory of Christian development, thereby stimulating considerable scholarly findings. Nevertheless, contemporary research predominantly emphasizes the ontological dimensions of religious study. As an objective phenomenon or fact in societal evolution, the propagation of religion inherently intersects with other elements of the prevailing social structure, consequently influencing economic and social development, which constitutes a crucial area of focus for religious studies.

The initial introduction of Western mathematical knowledge into China was facilitated by the collaborative translation of Euclid’s Elements by Western missionaries and Chinese intellectuals. This study examines the traditional Chinese mathematical knowledge system, highlighting its pragmatic aims, which contrast with the logical reasoning of Western mathematics. The decline of Chinese mathematical knowledge, evident in astronomical calendar projection errors during the late Ming Dynasty, underscores this divergence. The translation and dissemination of Western mathematical knowledge, driven by the academic missionary approach of Western missionaries, aimed to validate Western scientific advancements and legitimize Christian cultural influence. Despite the missionaries’ objectives, the translation of Western mathematical concepts significantly impacted Chinese mathematical knowledge, contributing to social development and intellectual enlightenment (Li and Gao 2025).

Consequently, this paper discusses how the translation and dissemination of Euclid’s Elements not only introduced Western mathematical ideas but also spurred the development of modern Chinese mathematical knowledge. The widespread acceptance of Western mathematical concepts coincided with the gradual formation of Chinese modern mathematical education, largely influenced by the intellectual impact of missionary translation efforts. Therefore, from a socio-historical perspective, the translation activities of Western missionaries are intrinsically linked to the transformation and development of Chinese society, offering a comprehensive understanding of the historical contributions of Western missionaries and the evolution of religious history. However, this study acknowledges limitations, such as the publications and dissemination of Euclid’s Elements and the translation and variation in mathematical terms, which warrant further studies. Nevertheless, the analysis of the relationship between the translation of Euclid’s Elements and the formation of Chinese modern mathematical knowledge provides new insights into religious propagation and the transformation of modern disciplinary knowledge in other areas of Asia, thereby expanding the scope and enriching the contents of religious studies.