Hundreds of years before a Greek scholar outlined his proof of the Pythagorean theorem（勾股定理）, ancient Chinese scholars proved it using just 17 characters, a renowned Chinese mathematician highlighted in a recent lecture.

Zhou Xiangyu（周向宇）, a member of the Chinese Academy of Sciences (CAS) and former director of itsInstitute of Mathematics, delivered the lecture at the National Science Communication Centre last Friday, where he spoke about China’s rich mathematical heritage.

“Ancient Chinese mathematics is a treasure trove, concise in language yet profound in meaning. Many more insights await discovery,” Zhou told Science and Technology Daily on Sunday.

He believes ancient Chinese books contain theories that significantly influenced modern mathematics and contributed to Chinese language and culture. Among these theories, the proof of the Pythagoreantheorem is particularly fascinating.

The theorem defines the relationship between the three sides of a right-angled triangle, stating that the square of the hypotenuse equals the sum of the squares of the other two sides.

In Western history, the earliest written proof appears in Elements（《幾何原本》）by Euclid, a Greek mathematician inthe 4th century BC.

But through his research, Zhou found that a complete definition and proof of the theorem existed as early as the Western Zhou dynasty in China (around 1000BC).

This proof appears in the opening of The Mathematical Classic of Zhou, known as the Zhoubi Suanjing（《周髀算經》）, in a dialogue between Zhou Gong（周公）, who was a foundational figure in Confucianism, and Shang Gao（商高）, a mathematician from the Western Zhou period.

In it, Zhou Gong（周公） asks Shang Gao（商高）: “The sky cannot be climbed step by step, nor can the Earth bemeasured by a ruler – so how can we measure the size of heaven and Earth?”（“夫天不可階而升，地不可得尺寸而度，請問數安從(cong) 出？”）

In response, Shang Gao（商高）suggests using mathematical methods to abstract and summarise things, taking the Pythagorean theorem as an example, offering both the definition and proof.

The original text contains just 17 characters: “Ji fang zhi, wai ban qi yi ju, huan er gong pan, de cheng san si wu,”（既方之，外半其一矩，環而共盤，得成三四五。） which roughly means: “Draw a square around them, halve one rectangle side, rotate it allaround, and you get three, four and five.”

An illustrated depiction of Shang Gao’s proof of Pythagorean theorem. This figure is the basis for the emblem of the Institute of Mathematics and Systems Science at CAS. Credit:Zhou Xiangyu

Zhou Xiangyu noted in the interview that he spent several years studying historical documents and reflecting on the meaning of each character. He eventually realised that “ji”（“既”） meant “all”（“全”） or “entire”（“都”）– shedding light on its use in this context.

In 2022, Zhou published a paper in Acta Mathematica Sinica outlining two interpretations of this phrase.（論文題目為(wei) 《中國古代數學的貢獻》，刊於(yu) 《數學學報》（中文版）2022年第4期）

The first interpretation uses three squares for each triangle side, dividing the second largest square inhalf, then rotating the resulting shapes to create a windmill-like pattern. Using a technique ofrearranging areas, the theorem can be proved visually.

The second approach follows the same steps initially, but changes the orientation of the triangleduring rotation, which makes the final calculation of area more intuitive.

Both interpretations demonstrate that ancient Chinese mathematical thought already integrated algebra with geometry. Remarkably concise, the classical Chinese proof uses just 17 characters, compared to about 10 lines of geometric formulas in Euclid’s explanation.

ToZhou, these characters provide a rigorous scientific explanation, challenging the misconception that Chinese mathematics lacked theoretical proofs.

“Shang Gao’s proof of the Pythagorean theorem is a scientific fact that deserves recognition,” Zhou told Science and Technology Daily. “Even when examined through the lens of modern mathematics, his proof holds up.”

An earlier article by Qu Anjing（曲安京）, history of science professor at Northwestern University in Xian, also argued that Shang Gao indeed proved the Pythagorean theorem.

He pointed out that in about the 3rd century, mathematician Zhao Shuang（趙爽）wrote an extensive commentary on The Mathematical Classic of Zhou and added an illustration confirming the theorem, which has since become widely known.

This illustration, widely admired for its elegance, inspired the emblem of the Institute of Mathematics and Systems Science at CAS and was used as the emblem for the 24th International Congress of Mathematicians held in Beijing in 2002.

Zhou was mentored by a student of the legendary mathematician Hua Luogeng（華羅庚）. Outside hisprofessional work, Zhou studies the history of mathematics and has given many public lectures.

Hua Luogeng（華羅庚）said, “Mathematics should be studied both broadly and deeply,” a view Zhou Xiangyu（周向宇）strongly agrees with.

“In studying mathematics, we should pay attention to the origins and development of mathematical concepts and methods, the roots and foundations, the source and flow.”

來源：數學史研究