Right Triangle Concepts in Ancient China: From Application to Theory

Bronowski J. , The ascent of man (Boston, 1973), 160–1. Bronowski notes, “To this day, the theorem of Pythagoras remains the most important single theorem in the whole of mathematics. That seems a bold and extraordinary thing to say, yet it is not extravagant; because what Pythagoras established is a fundamental characterization of the space in which we move, and it is the first time that is translated into numbers”.

Plimpton 322 contained in the G. A. Plimpton Collection, Columbia University Library. The tablet is written in Old Babylonian script and dates from the period 1900–1600 B.C. It was first described by Neugebauer O. and Sachs A. , Mathematical cuneiform texts (New Haven, Conn., 1945) and has been the subject of much interpretation and speculation, for example see: Buck R.C. , “Sherlock Holmes in Babylonia”, American mathematical monthly, lxxxvii (1980), 335–45; Friberg J. , “Methods and tradition of Babylonian mathematics”, Historia mathematica, viii (1981), 277–318.

Many traditional people still reckon their agricultural seasons or migrations by solstice observations, for example, the tribes people of Northern Borneo. See Hose C. and McDougall W. , Pagan tribes of Borneo (London, 1912).

See the discussion given in Seidenberg A. , “The ritual origin of geometry”, Archive for history of exact sciences, i (1962), 488–527.

Sima Qian (145–86 B.C.), in his discussion on the history of the Xia dynasty (c. 21st–16th century B.C.) notes in his Chronicles, Book 2: “Through the four seasons [Emperor Yu] rode chariots on land, rode in boats on water, skied on mud and in hilly places, travelled in a sedan chair, in his left hand he held a plumb-line, in his right [hand] a gnomon and compass, in order to rid the land of floods and open up the nine districts and the nine roads [i.e. China]”.

The Chinese believed in a dualistic system of natural harmony. This system depended on the interaction and balancing of two primal forces, the Yin and the Yang. Yin and Yang attributes could be assigned to any object. Yin was passive, dark, female, etc.; Yang was active, light, male, etc. In the case of numbers, odd numbers were Yang and even numbers, Yin. In their mystical number configurations, if harmony was to be preserved, an even number had to be paired with an odd number. For example, in the Chinese Luoshu diagram (the first known magic square of order 3), 1 and 6 occupy the north position, 3 and 8 the east position, 2 and 7 the south and 4 and 9 the west. For more detailed discussions of Yin-Yang numerology see Ho P. Y. , Li, Qi and Shu: An introduction to science and civilization in China (Hong Kong, 1985).

Loomis E. S. , The Pythagorean proposition (Washington, D.C., 1968), 3–4.

Needham J. , Science and civilisation in China, iii (Cambridge, 1954), 284.

The Zhoubi presents the Gai Tian (covering heaven) cosmology which held that the heavens formed a “conical hat” covering over the earth. For a detailed discussion on the origins of the Zhoubi see Li Y. and Du S. , Chinese mathematics: A concise history, translated by Crossley J. and Lun A. (Oxford, 1987), 25–28.

Needham , op. cit. (ref. 8), 22–23.

ibid., 23.

Gillon B. S. , “Introduction, translation, and discussion of Chao Chun-Ch'ing's ‘Notes to the diagrams of short legs and long legs and of squares and circles’”, Historia mathematica, iv (1977), 253–93.

This “hypotenuse diagram” partitions the square on the hypotenuse of a given right triangle into four images of the triangle and a square the length of whose sides is the difference between the legs of the triangle. A contemporary mathematician, Liu Hui, described this figure as “the diagram giving the relation between the hypotenuse and the sum and difference of the other two sides whereby one can find the unknown from the known”. Bronowski attributed this diagram to Greek origins (op. cit. (ref. 1), 158–60) and it is frequently associated with the Hindu mathematician Bhaskara (1114–c. 1185) (e.g. Eves H. , An introduction to the history of mathematics (Philadelphia, 1990), 228).

Lam L. Y. and Shen K. , “Right-angled triangles in Ancient China”, Archive for history of exact sciences, xxx (1984), 87–112.

In Babylonian mathematics, algebraic relationships were conceived geometrically. This tradition of geometric algebra passed to the Greeks and is illustrated by the propositions given in Euclid's Elements, Book II. For a fuller discussion of Babylonian and Greek geometric algebra see van der Waerden B. L. , Geometry and algebra in ancient civilizations (New York, 1983). For a further discussion of Zhao's geometrically based algebraic demonstrations see Lam and Sheng, op. cit. (ref. 14).

Ho P. Y. , “Liu Hui”, Dictionary of scientific biography , viii (New York, 1973), 418–25. Liu Hui and his work have been the focus of recent research activities in China; see, for example, Wu W. , Jiuzhang suanshu yu Liu Hui (Arithmetic in nine sections and its commentator Liu Hui) (Beijing, 1982).

Swetz F. J. and Kao T. I. , Was Pythagoras Chinese? An examination of right triangle theory in Ancient China (University Park, Penn., 1977).

ibid., 29–30, 36.

ibid., 48–51.

Wu W. , “The out-in complementary principle”, Ancient China's technology and science (Beijing, 1983), 66–69.

The solution prescription given differs from other solutions in the problem set in that it does not detail extensive solution procedures, rather it refers to a particular procedure, dingfa, a computing technique for extracting square roots. This procedure is carried out on a counting board employing a set of computing rods. Thus one is instructed to extract the square root of an expression which in contemporary algebraic terms would be x² + 34x − 71000 = 0. For further discussion on the Chinese solution of quadratic equations see: Swetz F. , “The evolution of mathematics in Ancient China”, Mathematics magazine, lii (1979), 10–19; Lam L. Y. , “The geometric basis of the Ancient Chinese square-root method”, Isis, lxi (1969), 96–102. The techniques of the Chinese counting rod mathematics is discussed in Ang T. S. , “Chinese computation with the counting-rods”, Kertas-Kertas Pengajian Tionghoa, i (1977), 97–109.

Swetz and Kao , op. cit. (ref. 17), 55–56.

Translated and discussed in Ang T. S. and Swetz F. , “A Chinese mathematical classic of the third century: The Sea Island mathematical manual of Liu Hui”, Historia mathematica, xiii (1986), 99–117.

ibid., 105.

See Wu W. , “Investigations into the original proofs of the Sea-Island mathematical manual”, in Wu, op. cit. (ref. 16), 162–81.

Ricci Matteo (1552–1610), the first Jesuit missionary to enter China (1582), collaborated with the Chinese scholar Xu Guangqi (1562–1633) to translate the first six books of Euclid's Elements of geometry into Chinese. They published their work in 1607. The translation was based on the 1574 Euclidis elementorum libri xv of Christopher Clavius (1537–1612). It was not until 1857 that the complete translation of Euclid was accomplished in China by the British missionary Alexander Wylie and his Chinese colleague, the scholar Li Shanlan. For a discussion of Wylie and Li's collaboration, see Martzloff J.C. , “Li Shanlan (1811–1882) and Chinese traditional mathematics”, The mathematical intelligencer, iv (1992), 32–37.

A complete analysis of the problems is provided in Swetz F. , The Sea Island mathematical manual: Surveying and mathematics in Ancient China (University Park, Penn., 1992).

Ang and Swetz , op. cit. (ref. 23), 114.

This work has been studied by Chemla K. , “Reflects des mesures du cercle sur la mer”, unpublished doctoral dissertation Paris, 1982.

See discussion in Li and Du , op. cit. (ref. 9), 242–4.