Ricci, Xu Guangqi, and the Chinese vocabulary of Euclid

39.90° N, 116.37° E · Beijing · 1606

A Jesuit and a Hanlin Academy scholar sat down in Beijing to put Euclid into a language that had no word for "definition," "axiom," or "proof."

The Jesuit was Matteo Ricci, known in Chinese as Li Madou 利瑪竇. He had learned mathematics in Rome from Christopher Clavius, and he had carried Clavius's Latin edition, Euclidis Elementorum libri XV (Rome, 1574), to Macau in 1582 and north over the next two decades. The scholar was Xu Guangqi 徐光啟. He had passed the highest civil examination in 1604 and read no Latin at all.

Their method is printed at the head of the text: 利瑪竇口譯，徐光啟筆受. Ricci translated by mouth. Xu received it with the brush. Every afternoon from the autumn of 1606, Xu went to Ricci's residence near Xuanwumen. Ricci explained the Latin in spoken Chinese, and Xu turned what he understood into literary Chinese. Nothing was translated on paper; the Latin never met the page. Ricci's preface describes the loop: 求合本書之意，以中夏之文重複訂正，凡三易稿 (seeking to match the sense of the original, correcting it repeatedly in Chinese prose, three drafts in all). Euclid's text went in whole. Clavius's long commentary was sampled.

Ricci had tried once before. Around 1590, in Shaozhou, he worked through Book I with the scholar Qu Rukui. That manuscript is lost. His verdict on the difficulty survives: 言象之粗，而齟齬若是 (this is only the coarse layer of words and figures, and still it grinds like this).

The reason Chinese had no ready words for these things is that its mathematics was built on a different logic. The Nine Chapters on the Mathematical Art 九章算術, compiled by the first century CE, holds 246 problems. Each has three parts: 問, the problem (a field is fifteen paces by sixteen, how large is it); 術, the formula, stated as a procedure; and 答, the computed answer. The reader meets a rule by meeting its cases, and the generality is in the procedure, not in a statement about all fields. Liu Hui's commentary of 263 CE justifies the procedures, including the right-triangle rule, by cutting figures apart and reassembling them. So there was proof. What there was not was Euclid's direction of travel. Euclid works by deduction: he names each object first, states what he will assume about it, and only then proves anything, with every theorem resting on the ones before it. He never asks a question. Chinese mathematics was written from problem to rule. Euclid is written from rule to everything. Ricci and Xu had to build the form before they could fill it.

So they named the parts of a proof. A definition became 界說 jieshuo, "an account of boundaries." The book explains the coinage in its opening note: 凡造論先當分別解說論中所用名目，故曰界說 (whenever one builds an argument, one must first explain the names it uses; hence "boundary accounts"). The choice tracks the Latin: definitio comes from finis, a boundary, and Euclid's Greek horos means the same. Postulates became 求作 qiuzuo, "requested constructions." Common notions became 公論 gonglun, "public propositions."

Each proposition got a fixed layout: 題 (the statement), then 解 (the setup) for a theorem or 法 (the method) for a construction, then 論 (the demonstration). Margin references point back to numbered definitions and axioms. Book I, Proposition 1 ends 如所求, "as was required," a Chinese Q.E.F.

Diagram labels raised a separate problem. A, B, and C are sounds, and a Chinese reader had no use for them. Ricci and Xu used the ten Heavenly Stems, 甲乙丙丁戊己庚辛壬癸, a fixed ordinal series every literate reader could recite. Proposition 1 reads 甲乙直線上，求立平邊三角形: on the straight line jia-yi, construct an equilateral triangle. A note under Definition 12 sets a rule still in force: an angle is named by three characters, and the middle one is the vertex. Write ∠ABC in place of 甲乙丙 and you are following it.

Then the nouns. Point 點, line 線, straight line 直線, curve 曲線, surface 面, plane 平面, right angle 直角, acute angle 銳角, obtuse angle 鈍角, parallel lines 平行線, diagonal 對角線, and triangle 三角形 are all in the 1607 text. All of them are in a Chinese seventh-grade textbook today. The pattern is compression: a common character, one qualifier, and the compound behaves as a technical term.

The words for the architecture of proof did not last. 界說, 求作, 公論, 題, and 論 gave way to 定義, 公設, 公理, 命題, and 證明, which Masahiro Ogawa and An Tae-ok trace largely to Meiji-era Japanese usage. Ogawa argues that 論, an everyday word for "discussion," worked against the translators: it let later readers treat a Euclidean proof as one more argument among many. The names for things survived. The names for the moves did not.

Some survivors drifted. 平角 meant "plane angle" in 1607; today it means a straight angle of 180 degrees. 直線 covered what modern textbooks split into 直線, 射線, and 線段. 比例 was defined in Book V as the relation between two magnitudes, closer to "ratio"; it now means proportion.

And the title. 幾何 jihe is often explained as a sound-borrowing of "geo-." The evidence runs the other way. Neither Euclid's title nor Clavius's contains geometria. The book's opening note files its subject under the 幾何府, the "department of quantity," one of ten categories borrowed from Aristotle. And 幾何 was the ordinary word for "how much," the question that closes problem after problem in the Nine Chapters: 問…幾何. It was a translation of "magnitude." Because only Books I to VI, all plane geometry, circulated for 250 years, the word narrowed until it meant geometry.

The six books were printed in Beijing in 1607. Ricci died there in May 1610. In 1611 Xu revised the text with the Jesuits Sabatino de Ursis and Diego de Pantoja, working from Ricci's annotated copy. Books VII to XV waited until Li Shanlan and Alexander Wylie translated them from Henry Billingsley's 1570 English Euclid, published in 1857.

In this series the episode sits between two other rulebooks. A thousand years earlier, Xuanzang listed five cases in which a Buddhist term should be transliterated rather than translated, one of them being when China had no equivalent. Ricci and Xu met that case on nearly every page and chose to translate. Three centuries later, Yan Fu would reduce the job to 信、達、雅.

If you have worked with late-Qing or Republican textbooks: when did 甲乙丙 labels give way to A, B, C, and did 界說 disappear at the same moment, or separately?

31.19° N, 121.43° E · Xujiahui · Shanghai