72: Elements of Translation: The Fusion of Eastern and Western Mathematics

Welcome back to the I Take History With My Coffee podcast where we explore history in the time it takes to drink a cup of coffee.


Xu Guangqi, Preface to the Chinese translation of Euclid’s Elements of Geometry, 1607
“The Jihe Yuanben (jee-her yahn-ben) is the Ancestor of Measure and Numbers it is that by which one exhausts all the aspects of the square, the round, the plane, and the straight, and by which one completely covers the use of compasses, carpenter’s square, water- level and measuring rope ... Starting from what is clearly perceptible, penetrates into what is most subtle; from what is doubtful certainly is obtained.” 

In the Forbidden City of Beijing, the Wanli Emperor had become obsessed with a foreign curiosity—a mechanical clock that chimed the hours without human intervention. Matteo Ricci had presented the device upon his arrival in Beijing. After Ricci introduced his famous mechanical clock to the emperor, it eventually broke down. When palace eunuchs brought it to the Jesuit residence for repairs, many of Ricci's high-ranking Chinese friends came to see the fascinating device.

Hearing about this attention, the emperor issued new orders: the clock must never leave the palace again. If repairs were needed, the Westerners should be summoned to fix it inside the palace walls. This imperial decree led to a surprising outcome. Rumors circulated throughout China that the emperor was granting special privileges to foreign missionaries, engaging in personal discussions with them, and regarding them as close advisors. The Chinese populace believed that the Jesuits had exceptional access to the withdrawn emperor.

The irony, as Ricci himself admitted, was that this was utterly false. While important mandarins couldn't even get audiences with the emperor, people assumed the foreign priests were his regular companions. The broken clock had inadvertently created a powerful myth of imperial favor that enhanced Jesuit prestige throughout the empire, even though Ricci and the emperor never actually met face-to-face.

This small mechanical marvel symbolized a much larger encounter that was unfolding in early seventeenth-century China. The decade from 1600 to 1610 marked a pivotal moment when European scientific knowledge first entered China through systematic collaborative translation, creating bridges between mathematical traditions that had developed independently for millennia. At the center of this historic exchange stood an unlikely partnership: Ricci, equipped with the latest European mathematical knowledge, and Xu Guangqi, a Chinese scholar-official seeking practical solutions to his empire's growing challenges.

Born in the vibrant city of Shanghai in 1562, Xu Guangqi experienced the rich cultural flourishing of the Ming Dynasty, a time when commercial vitality and urban elegance fostered an intellectual atmosphere that paralleled that of Renaissance Europe. Growing up in the Jiangnan region, he found himself in an academically competitive world, where prosperous merchant families nurtured many educated candidates, all vying for scarce positions in the imperial bureaucracy. This dynamic setting not only shaped Xu into an exceptionally talented scholar but also made him deeply attuned to the practical requirements of effective governance.

Xu's educational path reflected Ming literati excellence until a setback altered his course. In 1597, he achieved first place in Beijing's licentiate examination, marking him as one of China's promising young scholars. However, after completing the doctoral examination in 1601, he was disqualified due to a technical mishap during the lot-drawing ceremony for final rankings. This personal crisis—profoundly shameful in a culture that prioritized scholarly achievement—created a psychological opening crucial for China's intellectual future.

Xu failed his examination at a time when he was encountering Western learning through the efforts of Jesuit missionaries. His first contact was in 1596 with Lazaro Cattaneo in Shaozhou, but a more impactful encounter in 1600 with Matteo Ricci in Nanjing sparked his imagination. Ricci's maps, scientific instruments, and demonstrations of knowledge presented possibilities that traditional Chinese learning had not explored. For a scholar interested in practical mathematics—like surveying and agricultural improvement—Western science provided theoretical foundations lacking in Chinese tradition.

Xu's intellectual journey had always celebrated jingshi, the tradition of practical learning designed to "order the world" through the acquisition of valuable knowledge. Unlike some scholars who delve into abstract philosophical ideas, Xu felt that learning should truly serve the real-world needs of governance and society. His earlier research in Chinese mathematics convinced him that, while China's mathematical tradition was once quite sophisticated, it had unfortunately slipped into a pattern of rote computation, missing out on deeper theoretical understanding.

The convergence of personal crisis and intellectual opportunity led to Xu's religious conversion in 1603. After intensive study with missionary João da Rocha in Nanjing, he was baptized as "Paul," becoming what historians call the prototype "Confucian Christian"—a scholar who embraced Catholic faith while maintaining his cultural identity as a Chinese literati. This dual identity would prove essential for his later role as a cultural bridge-builder.

Xu's vision transcended personal advancement. He crafted a strategy for knowledge integration called "supplement Confucianism and supplant Buddhism" (buru yifo)(boo roo yee for). Instead of seeing Western learning as a replacement for Chinese tradition, he viewed it as a way to enhance Confucian scholarship with theoretical rigor and mathematical foundations. His concept of Dushu zhixue (doe shoe she sway)(Studies of measures and numbers) provided an intellectual framework for merging foreign mathematical knowledge with traditional Chinese academic categories.

This preparation would prove crucial when the calendar crisis struck the empire. Xu's collaboration with Ricci had given him unmatched expertise in Western mathematical and astronomical methods, while his 1604 doctoral success and position in the Hanlin Academy provided institutional credibility. The personal crisis that had seemed devastating in 1601 ultimately positioned him to lead China's most crucial scientific reform.

Matteo Ricci arrived in China in 1583 with advantages that previous missionaries lacked. His education under Christopher Clavius at the Collegio Romano provided him with a mastery of advanced European mathematics and astronomy. At the same time, his humanistic training offered the cultural flexibility essential for engaging with Chinese intellectual circles. Unlike missionaries in other regions dependent on European military or economic power, Ricci understood that success in China required earning respect through demonstrated intellectual excellence.

Ricci understood Chinese psychology precisely. He noted that Chinese scholars were "arrogantly self-important, believing that China was the world, but the more self-important they were, the more inferior they felt when the truth was revealed." This insight shaped his missionary strategy: instead of attacking Chinese learning, he demonstrated the superiority of Western knowledge through practical applications, enabling scholars to conclude the limitations of traditional methods themselves.

Ricci's "upper-class route" marked a strategic shift from conventional missionary methods. Rather than aiming for mass conversions among the general populace, he focused on the educated elite, whose impact could foster broader cultural transformation. This approach necessitated that he act not only as a religious instructor but also as a learned scholar, engaging with China's intellectual leaders on their own grounds. His residences featured Western books, scientific instruments, and maps as conversation starters, showcasing Europe's dedication to knowledge and offering tangible proof of scientific progress.

Ricci's methodology was highly effective. He considered mathematics a "door-knocking stone" and a "key to the gate" that could eventually enlighten Chinese minds regarding religious truths. His logic progressed as follows: if Western scholars showcased their superior understanding of the visible world through mathematical accuracy, then Chinese thinkers would likely be more receptive to Western teachings about the unseen spiritual domain. This approach entailed framing Christian ideas "in Confucian colors," leveraging local conceptual frameworks to render foreign ideas more relatable and palatable.

Instead of literal translations that confused Chinese readers, he partnered with Chinese scholars to create culturally adapted versions of European texts. This involved Ricci explaining mathematical concepts orally while his collaborator transcribed and adapted the content into suitable Chinese literary forms.

This collaborative approach required creating new Chinese vocabulary for European mathematical concepts without indigenous equivalents. The joint terminology development was not only linguistic but also conceptual, necessitating both partners to understand how different intellectual traditions organized knowledge. Ricci's willingness to embed Western learning within traditional Chinese categories, such as gewu qiongli (ge woo she ong lee)(investigation of things, exhaustively mastering principles), demonstrated his recognition that successful cultural transmission required working within existing frameworks.


The translation of Euclid's Elements of Geometry into Chinese stands as one of history's key milestones in cross-cultural intellectual exchange. In 1605, Ricci and Xu Guangqi began a year-long collaboration that brought systematic deductive geometric reasoning to Chinese mathematics for the first time. Their combined efforts culminated in Jihe Yuanben (jee-her yahn-ben), released in 1607, which significantly altered Chinese mathematical discourse and laid the groundwork for centuries of scientific advancement.

The source material originated from Christopher Clavius's 1574 annotated Latin edition, which represented the most advanced European mathematical knowledge available at the time. This choice was not accidental—Clavius had been Ricci's teacher and was widely regarded as "Euclid of the sixteenth century" for his contributions to mathematical education. The direct intellectual lineage from Clavius to Ricci to Chinese mathematics ensured that China received not only ancient geometric knowledge but also contemporary European mathematical thought at its highest level.

The collaborative process itself was revolutionary. Working together daily, Ricci explained geometrical concepts orally while Xu transcribed and adapted the material into classical Chinese. This was not mere translation but a cultural transformation, requiring the creation of entirely new Chinese mathematical vocabulary. Xu invented permanent terms for point (dian)(dee ann), line (xian)(see ann), surface (mian) (me ann), and angle (jiao) (jee ow), as well as concepts like parallel lines, acute angles, and obtuse angles that had no equivalents in traditional Chinese mathematics.

The linguistic innovation encompassed notational systems and geometric styles standard in Chinese mathematics. More fundamentally, the translation introduced Aristotelian organization of mathematical knowledge through definitions, postulates, axioms, theorems, and proofs. This marked a departure from traditional Chinese texts, which presented problems and algorithmic solutions without systematic foundations.

The six translated books encompassed essential concepts of plane geometry, geometric algebra, circles, regular polygons, and the arithmetic theory of proportion. Each theorem featured thorough logical proofs, highlighting not only mathematical methods but also the reasoning that supported them. For Chinese readers familiar with computational techniques that focused on "how" to solve problems, this focus on logically demonstrating "why" mathematical relationships were valid marked a significant conceptual transformation.

Xu Guangqi's evaluation of the work underscores his recognition of its revolutionary significance. In his foreword, he praised the logical completeness of Euclidean geometry, noting that "four things in this book are not necessary: it is not necessary to doubt, to assume new conjectures, to put to the test, to modify. In addition, four things in this book are impossible: it is impossible to remove any particular passage, to refute it, to shorten it, or to place it before that which precedes it." This represented a completely foreign approach to knowledge for most Chinese readers, who were accustomed to more flexible and practical mathematical traditions.

The encounter between Chinese and European mathematical traditions revealed fundamental organizational differences. Chinese mathematics developed as a practical discipline focused on solving real-world problems through algorithmic computation, whereas European mathematics emphasized deductive reasoning from abstract principles through systematic geometric proof.

By the late Ming period, Chinese mathematics had significantly declined. Advanced Song-Yuan methods had become incomprehensible to contemporary scholars; many classical texts were either incomplete or lost, and the examination system's focus on literary composition had caused educated elites to overlook mathematical learning. Practical computation was relegated to merchants using abacus calculations.

European mathematics, in contrast, developed from Greek geometric foundations that prioritized logically demonstrating "why" mathematical relationships held true. Ricci's Euclidean tradition organized knowledge through systematic progression from definitions and postulates to complex theorems, treating mathematics as a theoretical science that reveals universal truths through rigorous proof.

Chinese responses to Euclidean geometry reflected a mix of appreciation and skepticism. While scholars acknowledged the logical precision of Western techniques, many considered the lengthy geometric proofs overly complex compared to the more efficient Chinese algorithms. Despite these differences, the translation efforts led to significant integrations, incorporating explanatory reasoning into Chinese literature and uncovering practical applications for European ideas in fields such as surveying, engineering, and astronomy—really important areas, considering the threats from Japanese pirates and the need for better defenses.

The selective reception reflected deeper cultural preferences that persisted throughout the early modern period. Chinese scholars adopted the operational aspects of European mathematics while overlooking the philosophical implications of systematic deductive reasoning, thereby enabling practical advancements without fundamental changes to Chinese intellectual culture.

Ricci's decision to halt the translation at six books reflects his evangelical motivation. When Xu expressed eagerness to continue, Ricci responded, "First disseminate this, let like-minded people study it, and if it proves useful, then we'll consider the rest." However, Ricci's reasoning was strategic: the six books already demonstrated Christianity as a "rational religion" rooted in logic. Using mathematical truth for religious credibility, he found no need to complete the translation.

Supporting translation works complemented Elements by demonstrating practical applications. Celiang Fayi (sir lee ong figh yee)(Explanations on Measurement, 1608) presented fifteen surveying problems with geometric methods, while Celiang Yitong (sir lee ong ee tong)(Similarities and Differences in Measurement, 1608) provided a comparative analysis of Western methods' advantages and Chinese traditions. These works aimed to "create a more solid basis for Chinese mathematics" by providing theoretical foundations for practical applications.

On May 3, 1610, Matteo Ricci suddenly fell seriously ill in Beijing. Despite treatment from the city's best doctors, his condition worsened rapidly. His room was filled with visiting Christians and Chinese friends, which prompted him to move to larger quarters to accommodate everyone. After receiving last rites and conversing with fellow Jesuits and Chinese associates on May 10-11, he peacefully died at 7 PM on May 11, blessing his companions as he passed.

For two days after his death, a steady stream of visitors, mainly Beijing's officials, came to honor this foreign priest who seemed fully Chinese to them. Many of his friends, backed by the Board of Rites, petitioned the emperor for a burial site. The emperor granted this unprecedented request, providing imperial land near Beijing's western wall—the first time a foreign missionary received such honors.

Ricci was entombed on November 1, 1610. His premonition that his death might serve Christianity's interests proved correct, as the imperial burial grant provided official recognition of both his status and protection for the religion he taught. Years later, his friend Wang Yingming erected an official stone proclamation at his grave, listing the extraordinary array of high-ranking scholar-officials who had been his close friends, demonstrating his profound impact across Chinese society.

The network of relationships and institutional foundations that survived Ricci's death demonstrated the sustainability of his collaborative approach. Chinese converts like Xu Guangqi possessed both the technical expertise and cultural authority necessary to continue the transfer of knowledge. The translation methodologies and terminology developed through the mathematical projects provided established frameworks for future collaboration, while the imperial recognition of Ricci's contributions legitimized the continued Jesuit presence in China.

Just seven months after Ricci's death, a dramatic astronomical event validated everything that the Italian missionary and his Chinese collaborators had worked to achieve. On December 15, 1610, a solar eclipse exposed the fundamental inadequacy of traditional Chinese astronomical methods, creating a crisis that would elevate Xu Guangqi to the leadership of the empire's most significant scientific reform.

The Astronomy Bureau's prediction for the eclipse failed spectacularly, missing the actual timing by 2 to 3 marks (15 to 43 minutes)—an error so significant that it threatened the very foundations of imperial authority. Eclipse prediction was one of the most politically sensitive astronomical tasks, directly linked to the emperor's mandate of heaven and the legitimacy of governmental rule. Accurate calendrical calculation was not just technical but essential for maintaining cosmic harmony.

In stark contrast to the Chinese failure, Diego de Pantoja and Sabbathin de Ursis, Ricci's Jesuit successors, accurately predicted the timing of the eclipse with remarkable precision. When their precise prediction became known to their Chinese associates, it provided undeniable evidence of Western astronomical superiority that could not be dismissed as cultural prejudice or religious propaganda.

The failure of the eclipse highlighted ongoing issues that had affected Chinese astronomy during the Wanli period. The Ministry of Rites recorded persistent errors in eclipse forecasts, leading to a loss of faith in conventional techniques. Instead of dismissing foreign insights, the Ministry recognized the advantages of Western astronomy and advised prompt measures to obtain European knowledge.

One of the most important steps taken by the Ministry was the decision to appoint Xu Guangqi to oversee the translation of foreign astronomical works held by the Jesuits. This choice demonstrated a remarkable acknowledgment that the Chinese converts possessed a unique blend of Western technical knowledge and Chinese cultural authority, both essential for effectively implementing the necessary reforms. Thanks to a decade of mathematical translation, they had developed the expertise and credibility that this challenging situation truly required.

All elements were now in place for comprehensive calendar reform: the crisis had been recognized, Western solutions had been identified, qualified leadership was available, and institutional support had been secured. When Xu Guangqi eventually led the calendar reform project from 1629 to 1633, he drew directly on the foundation of knowledge, relationships, and methodologies established through his collaboration with Matteo Ricci.

Part of Ricci’s success was due to his character, temperament, and personality. Following his death, the question arose of whether his successors could continue building upon Ricci’s foundation. In the next episode, we will discuss the period from Ricci’s death up to the next major astronomical crisis, the solar eclipse of 1629.


As always, maps and other supporting resources for all episodes are listed in the episode description. In the meantime, for more historical content, please visit the “I Take History With My Coffee” blog at itakehistory.com.  Become a free or paid subscriber on Substack and get related historical content delivered to your inbox.  We’re on Facebook, Instagram, and Threads.  Feedback and comments are welcome at itakehistory@gmail.com.  Or you can leave a review on Apple Podcasts or on Spotify.  You can also help support this podcast by buying me a coffee at buymeacoffee.com/itakehistory.  If you know anyone else who would enjoy this podcast, please let them know.  And thanks for listening.