The first translation of Euclid’s Elements into Chinese

東西文理，又自絕殊；字義相求，仍多闕略。

The patterns of writing in East and West are utterly different, and when we seek words to match meanings, much is still missing.

— Matteo Ricci, preface to Jihe yuanben, 幾何原本 (1607)

Nearly 2,000 years after Euclid wrote his mathematical treatise Elements, a Jesuit and a Ming Dynasty scholar finally began to translate the work into Chinese.

The former was a missionary from the Papal States, now the Marche region of modern-day Italy. Matteo Ricci carried a Latin edition of Euclid’s Elements with him for nineteen years as he worked his way inland from Macau in 1582, eventually becoming the first European allowed to enter the Forbidden City. His copy was edited by his former teacher in Rome, Christopher Clavius, and had itself been translated by way of Arabic.

The latter was a Chinese scholar-official, Xu Guangqi (徐光啟), who had recently become a member of the Hanlin Academy (翰林院). He was also newly baptized as a Roman Catholic on account of Ricci’s influence. Xu would later lead the Chongzhen calendar reform, systematically introducing European mathematics and astronomy to China.

The pair began working together on a translation of Elements in the autumn of 1606. They worked by discussion. Ricci’s grasp of literary Chinese was not equal to Elements, and Xu could not read Latin. So Ricci explained each Latin passage in spoken Chinese, and Xu then wrote down the translations in classical Chinese.

They finished the first six books on plane geometry and published Jǐhé Yuánběn, 幾何原本, in 1607. It was the first Chinese translation of Euclid. Albeit an incomplete one. The two had planned to complete the remaining nine books, but circumstances intervened; Ricci died.

The complete translation of Euclid’s Elements took another 250 years to finish. It was again translated by discussion, this time by a British missionary and Chinese mathematician in Shanghai, using an English translation brought over from London.

We think of math as universal, a study of immutable truths. History makes clear, however, that methods vary. At the first translation of Euclid’s Elements, the Chinese language did not have terms for axiom, proof, definition, or theorem as mathematical concepts. The foundational text of Chinese mathematics was written in problems and procedures, rather than theorems; it was concerned with methods and numerical applications rather than theory and geometric figures.

Very few texts have been passed to the modern world through a more serpentine path than Euclid’s Elements. We know little about the author, who hand-wrote on papyrus scrolls in Alexandria. The Greek was first translated into Arabic in Baghdad, under the patronage of the caliph, then Latin by way of an Arabic edition likely obtained by a traveler in Antioch, then carried by sea to Macau and overland to Peking over nineteen years, where it required two people from the Western and Eastern traditions to translate the nuanced mathematical concepts into Chinese.

This is the story of this enchanting text and its many translations.

First, explore the first eight definitions from Book I, in all the languages this piece will cover.

Στοιχεία

Euclid wrote Στοιχεία, known as Elements, circa 300 BCE in Ancient Greece. It was a hand-copied scroll, only fragments of which survive. We know very little about the author. Elements is undoubtedly one of the most important works of Western history. It established a system of definitions, axioms, and propositions, “which has served as a model of rigorous mathematical reasoning for more than two thousand years, and perhaps will last forever,” as mathematician Chuanming Zong writes.

Elements is a 13-book treatise on plane and solid geometry and number theory which includes the Pythagorean theorem, Thales's theorem, the Euclidean algorithm, the proof that primes are infinite, and the five regular solids.

Very few texts have been passed to the modern world through such a complex path of editions, translations, revisions, and abridgements.

The original text is particularly hard to recover, even more so than other ancient Greek works of literature, philosophy, and history. Because Euclid's Elements was used as a textbook (at that, the world’s oldest continuously-used math textbook), teachers felt free to make liberal additions, corrections, and omissions.

The first translation of Elements was made in Arabic by the mathematician al-Ḥajjāj (الحجّاج بن يوسف بن مطر) in 9th-century Baghdad, around 800 CE.

The Arabic translation was then used for the Latin translation, made by Adelard of Bath in 1130 (a copy he likely obtained while travelling in Antioch, in modern-day Türkiye). Medieval Europe thus became acquainted with Euclid through Latin translations of Arabic translations of the original ancient Greek. The Latin from Arabic translation was also used for the first printed edition of Elements, which included a note that the geometrical diagrams were particularly difficult to print.

Conveying the mathematical depth of Euclid’s work, with its koan-like sentences and dense logical intricacy, is a challenge in itself. The earliest English translation in 1570 by Henry Billingsley (a character who will appear later) included hand-glued, three-dimensional paper pop-up models to help readers visualize geometric solids in Book XI.

And Oliver Byrne’s 1847 edition became a landmark of Victorian book design, using primary colors to visually demonstrate geometric proofs. The elegant designs look incredibly modern and even seem to prefigure the work of decades-later Bauhaus and De Stijl artists, Piet Mondrian and László Moholy-Nagy among them.

Journey to Peking

While Marco Polo is more famous for his travels in China, Matteo Ricci was the most important person to bring Western culture, science, and religion to China. He was the first Westerner invited into the Forbidden City. He drew the first world map in China, brought mechanical clocks to the court, and taught astronomy and science. After his death, he was the first foreigner to be granted a burial site in Peking by imperial decree.

Yet his entry to the cloistered country was not easy. It took him nearly twenty years to be allowed to settle in the capital. When he landed at Macau, a Portuguese trading post at the mouth of the Pearl River, in 1582, foreigners were not permitted to live in the country’s interior. For every step inland, he needed an official’s patronage, which lasted only as long as that official’s posting. He moved slowly towards Peking, adopting the Chinese language, dress, and cultural customs along the way, as well as the Chinese name 利瑪竇 (Li Madou). He moved from Macau (澳門) to Zhaoqing (肇慶), Shaozhou (韶州), Nanchang (南昌), Nanjing (南京), and finally Peking (北京) in 1601.

Over these twenty years, Ricci traveled with his precious books — among them his copy of the 1574 Latin translation of Euclid’s Elements, edited by his former mathematics and astronomy teacher Christopher Clavius.

He first met Xu Guangqi (徐光啟) in Nanjing. In 1604, Xu succeeded in the Imperial Examination (科舉) and became a member of the Hanlin Academy (翰林院), after which the two became close friends.

In the autumn of 1606, they began to translate Ricci’s Latin translation of Euclid’s Elements.

Even as a text alone (to say nothing of the mathematical concepts), translating Latin to classical Chinese posed a huge challenge. The two languages exist on far branches of the language tree with divergences in grammar, morphology, style, and structure. The patterns of writing in East and West are utterly different from each other (東西文理又自絕殊), wrote Ricci.

The pair's method is printed under the title of the text: 泰西利瑪竇口譯，吳淞徐光啟筆受 (orally translated by Li Madou of the Far West, received with the brush by Xu Guangqi of Wusong.) Ricci explained the Latin in spoken Chinese, which Xu then formulated into classical Chinese.

They finished the first six books of Euclid’s Elements and published them under the Chinese name Jihe Yuanben (幾何原本) a year later. They decided to study the Chinese public's reception before translating the rest of the text. But Ricci died in 1610 before they could complete their translation.

It took another 250 years to translate the remaining nine books into Chinese. The British missionary Alexander Wylie and the Chinese mathematician Li Shanlan finally completed the translation in 1857, in Shanghai. They used the same method of oral transmission and discussion to translate the text. This time, they used a copy from London printed in English, which was translated by Henry Billingsley from the original Greek.

(Billingsley himself is a fascinating case. He was a wealthy merchant, chief Customs Officer, and later Lord Mayor of London. Scholars originally doubted he made the landmark translation himself. But his original study with his own handwriting, an expertly-annotated Greek edition of Euclid’s Elements, was identified in 1879 in the Princeton University Library.)

The translation of Euclid's Elements was thus finally completed in Chinese more than two thousand years after it was originally written.

幾何原本

Of course, the Chinese had their own mathematics system. They developed arithmetic and elementary geometry more or less at the same time as Euclid. In fact, even as early as the fourth century BCE, Chinese reckoners were laying out counting rods in columns, where a rod’s position fixed its value and an empty column designated zero, effectively implementing a decimal place-value system. No other ancient civilization is known to have employed such a method. The ancient Chinese were also the first to use negative numbers, a practice not adopted in Europe until the 1400’s.

But classical Chinese mathematics developed along quite a different trajectory compared to its Western counterparts. Chinese mathematics was largely motivated by problems of calendar, trade, land, agriculture, government, and architecture. It was more focused on algebra and practical applications as opposed to geometry and theoretical reasoning. And like the language, it was very concise, even spare.

The foundational text, occupying a position equivalent to Euclid’s Elements, was Jiǔzhāng suànshù (九章算術, Nine Chapters on the Mathematical Art). It was compiled over the Han dynasty and in the first century CE, with no recorded author. The text contains 246 problems structured as problem (問), answer (答) and general procedure (術).

For example, the right-triangle relation (the theorem we now know as the Pythagorean theorem) is there as the gōugǔ (勾股) rule:

問 (problem): 今有句三尺，股四尺，問為弦幾何？
A right triangle has a short leg of 3 feet and a long leg of 4 feet. How long is the hypotenuse?

答 (answer): 答曰：五尺。
Answer: 5 feet.

術 (procedure): 句股術曰：句股各自乘，并，而開方除之，即弦。
The leg-and-leg procedure: square each leg, add the squares, take the square root. That is the hypotenuse.

The procedure works for any right triangle, but is given to the reader as an instruction — not as a statement or claim about all right triangles. A later commentary by Liu Hui (劉徽) supplied the reasoning by slicing the two smaller squares and reassembling them into a larger square.

Euclid’s version, Proposition I.47, runs very differently.

A general statement comes first (in every right triangle the square on the hypotenuse equals the two squares on the legs), then a proof built on the propositions before it. No side is given a specific length or computed value; every magnitude is only a line labeled by a letter. While the Nine Chapters move from a question to a procedure, Euclid presents a definition and then theorem of universality.

It’s mind-bending to consider that the deductive method is not self-evident, as we tend to treat it centuries later, but a system of conventions. It starts by asking the reader to agree to a set of terms and definitions which are essentially unproven assumptions, and then uses these pieces to build towards mathematical certainty. Before a single proof appears in Elements, the reader is first required to accept 23 definitions, 5 postulates, and 5 common notions.

Ricci and Xu translated Euclid thus not only as a text but as a genre of radically new mathematical reasoning. The pair coined a few dozen new technical terms, including jièshuō (界說) for definition, qiúzuò (求作) for postulate, and gōnglùn (公論) for axiom (although these were later replaced by terms developed in Meiji Japan and then carried back into Chinese). Their names for figures, like 點 for point, 直角 for right angle, 平行線 for parallel lines, and others, survived. These terms are still found in primary-school textbooks today.

The word chosen to translate definition as a mathematical term, 界說, illustrates how meaning was carried in surprising ways. Euclid used ὅρος, horos, meaning a boundary or limit. The Latin definitio carries the same image, from finis for border. Ricci and Xu coined an assemblage of jie 界, meaning boundary, and shuo 說, meaning explanation. Their first note explains that when constructing an argument, one must first mark the limits of the names used in it, hence “boundary-explanations.” The original Greek etymology was thus preserved in this Chinese translation.

For the title, on the other hand, they repurposed native Chinese terms. Jihe 幾何, was actually the Chinese term for “how much.” It was used because geometry and arithmetic were both studies of quantity. In fact, 幾何 is a formulation used in every problem in the Nine Chapters. The second half of the title, yuanben 原本, is yuan 原, source, and ben 本, root, their rendering of Elements. So 幾何原本 meant something like “the origins of quantity.”

Only Ricci and Xu’s translation of the first six books, all plane geometry, circulated in China for the next 250 years. So the word jihe 幾何 (simplified Chinese: 几何) has come to refer to geometry alone.

Translation is sometimes seen as simply moving meaning between one language and another. Yet language is often a record of what people have thought thus far, not merely a code for universal understanding. The Chinese translation of Euclid’s Elements, a singular text, reveals how translation is often a much older and stranger discipline.

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