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around 560 CE

Zhoubi Suanjing Pythagorean Theorem Proof

One of the first recorded proofs of the Pythagorean theorem

The Chinese text Zhoubi Suanjing gives reasoning for proving the Pythagorean theorem for a (3, 4, 5)-triangle. The original work is attributed to the period of the Zhou Dynasty (1046–256 BCE). Compilation and addition of materials continued into the Han Dynasty (202 BCE–220 CE), and the oldest surviving copy dates from roughly 100 BCE. The pages highlighted here originate from a Ming-Dynasty copy of the work printed in 1603.

Zhoubi Suanjing Pythagorean Theorem Proof

The figure used to prove the Pythagorean theorem in Zhoubi Suanjing is usually referred to as the "hypotenuse diagram," or xian tu. It is ascribed to Zhen Luan (535–566), a Chinese mathematician and astronomer who mentions the diagram while discussing Zhao Shuang's third-century commentary of Zhoubi Suanjing. The xian tu provides an elegant dissection proof of the Pythagorean theorem, which is known in Chinese as the gou-gu theorem.

Artifact dimensions

5.5 in. × 9.9 in.

Artifact origin


Current artifact location

Rare Book & Manuscript Library, Butler Library, Columbia University in the City of New York


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Computational Explanation

Other Resources

Additional Reading

  • Beckmann, P. A History of Pi. New York: Barnes & Noble, pp. 174–177, 1993.
  • Cullen, C. "Zhao Shuang and Pythagoras's Theorem." Appendix One in Astronomy and Mathematics in Ancient China: The Zhou bi suan jing. Cambridge University Press, pp. 206–217, 1995.
  • Dauben, J. W. "Chinese Mathematics." Ch. 3 in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Ed. V. J. Katz). Princeton, NJ: Princeton University Press, pp. 193–194, 214, 216–226 and 283, 2007.
  • Gillon, B. S. "Introduction, Translation and Discussion of Chao Chun-ch'ing's [=Zhao Junqing = Zhao Shuang] 'Notes to the Diagrams of Short Legs and Long Legs and of Squares and Circles." Historia Mathematica, Vol. 4, pp. 253–293, 1977.
  • Joseph, G. G. The Crest of the Peacock: Non-European Roots of Mathematics, 3rd ed. Princeton, NJ: Princeton University Press, pp. 248–261, 2011.
  • Liu, H. Jiu zhang suan shu. 1896. [Photoreprint ed. of the Wenyuan Ge copy. Taipei: Taiwan shang wu yin shu guan, 1975.]
  • Maor, E. The Pythagorean Theorem: A 4000-Year History. Princeton, NJ: Princeton University Press, pp. 62–66, 2007.
  • Martzloff, J.-C. A History of Chinese Mathematics. Berlin: Springer-Verlag, pp. 296–299, 1987.
  • Shen, K.; Crossley, J. N.; and Lun, A. W.-C. The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford and Beijing: Oxford University Press, 1999.
  • Swetz, F. and Kao, T. I. Was Pythagoras Chinese? An Examination of Right Triangle Theory in Ancient China. University Park, PA: Pennsylvania University Press, 1977.