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Funding for this project generously provided by Overdeck Family Foundation

attributed to 200–300 CE

Liu Hui's Exhaustion Method

Using recursively inscribed polygons to approximate π

A commentary prepared by Chinese mathematician Liu Hui around 300 CE used a clever method of exhaustion based on the Pythagorean theorem for determining the area of a disk and estimating the value of π.

Liu Hui's Exhaustion Method

This page from a sixteenth-century Ming dynasty edition of Jiuzhang suanshu (The Nine Chapters on the Mathematical Art) illustrates the method of exhaustion for determining the area of a disk. While no copies of the original work survive, it is preserved through extant copies of an edition and commentary prepared by Liu Hui around 300 CE. By inscribing regular polygons in a circle and computing their areas using a recursive procedure based on the areas of the "kites" made from pairs of triangles, Liu proved that 3.14103 < π < 3.14271 and gave an accurate estimate of the actual value as π = 3927/1250 = 3.1416.

Artifact dimensions

9.8 in. × 5.8 in.

Original artifact location

Cao Wei (historical name), China (current name)


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Additional Reading

  • Dauben, J. W. The Chinese "Euclid," Liu Hui. Ch.3, § V in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Ed. V. Katz.). Princeton, NJ: Princeton University Press, pp. 226–240, 2007.
  • Joseph, G. G. The Crest of the Peacock: Non-European Roots of Mathematics, 3rd ed. Princeton, NJ: Princeton University Press, pp. 193–198 and 266–270, 2011.
  • Liu, H. Jiu zhang suan shu. Various editions. 1896, 1899, 1975 photo-reprint, etc.
  • Mankiewicz, R. The Story of Mathematics. Princeton, NJ: Princeton University Press, p. 37, 2004.
  • Martzloff, J.-C. A History of Chinese Mathematics. Berlin: Springer-Verlag, pp. 278–282, 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.