Tsu Ch’ungChih
TSU CH’UNGCHIH
(b, Fanyang prefecture [modern Hopeh province], China, ca.a.d. 429; d, China, ca. a.d. 500)
mathematics.
Tsu Ch’ungchih was in the service of the emperor Hsiaowu (r. 454–464) of the Liu Sung dynasty, first as an officer subordinate to the prefect of Nanhsü (in modern Kiangsu province), then as an officer on the military staff in the capital city of Chienk’ang (modern Nanking). During this time he also carried out work in mathematics and astronomy; upon the death of the emperor in 464, he left the imperial service to devote himself entirely to science. His son, Tsu Keng, was also an accomplished mathematician.
Tsu Ch’ungchih would have known the standard works of Chinese mathematics, the Choupi suanching (“Mathematical Book on the Measurement With the Pole”), the Haitao suanching (“Seaisland Manual”),(“Mathematical Manual in Nine Chapters”), of which Liu Hui had published a new edition, with commentary, in 263. Like his predecessors, Tsu Ch’ungchih was particularly interested in determining the value of π. This value was given as 3 in the Choupi suanching; as 3.1547 by Liu Hsin (d.23); as or , by Chang Heng (78139); and as , that is 3.1547 by Wan Fan (219257).Since the original works of these mathematicians have been lost, it is impossible to determine how these values were obtained, and the earliest extant account of the process is that given by Liu Hui, who reached an approximate value of 3.14. Late in the fourth century, Ho Chēngtein arrived at an approximate value of , or 3. 1428.
Tsu Ch’ungchih’s work toward obtaining a more accurate value for π is chronicled in the calendrical chapters (Luli chih) of the Suishu, an official history of the Sui dynasty that was compiled in the seventh century by Wei Cheng and others. According to this work.
Tsu ch’ungchih further devised a precise method. Taking a circle of diameter 100,000,000, which he considered to be equal to one chang [ten ch’ih, or Chinese feet, usually slightly greater than English feet], he found the circumference of this circle to be less than 31,415,927 chang, but greater than 31,415,926 chang,[He deduced from these results] that the accurate value of the circumference must lie between these two values. Therefore the precise value of the ratio of the circumference must lie between theses two values. Therefore the precise value of the ratio of the circumference of a circle to its diameter is a 355 to 113, and the approximate value is as 22 to 7.
The Suishu historians then mention that Tsu Ch’ungchih’s work was lost, probably because his methods were so advanced as to be beyond the reach of other mathematicians, and for this reason were not studied or preserved. In his Chunsuan shih Lung’ung (“Collected Essays on the History of Chinese Mathematics” [1933]), Li Yen attempted to establish the method by which Tsu Ch’ungchih determined that the accurate value of π lay between 3.1415926 and 3.1415927, or .
It was his conjecture that
“As , Tsu Ch’ungchih must have set forth that, by the equality
one can deduce that
x=15.996y, that is that x=16y.
Therefore
For the derivation of
When a, b, c, and d are positive integers, it is easy to confirm that the inequalities
hold, If these inequalities are taken into consideration, the inequalities
may be derived.
Ch’ien Paotsung, in Chungkuo shuhsüehshih (“History of Chinese Mathematics“[1964]), assumed that Tsu Ch’ungchih used the inequality
S_{2n} < S < S_{2n} + (S_{2n} – S_{n}),
Where S_{2n} is the perimeter of a regular polygon of 2n sides inscribed within a circle of circumfernce S, while S_{n} is the perimeter of a regular polygon of n sides inscribed within the same circle. Ch’ien Paotsung thus found that
S_{12288} = 3.14159251
and
S_{24576} = 3.14159261
resulting in the inequality
3.10415926< π < 3.1415927.
Of Tsu Ch’ungchih’s astronomical work, the most important was his attempt to reform the calendar. The Chinese calendar had been based upon a cycle of 235 lunations in nineteen years, but in 462 Tsu Ch’ungchih suggested a new system, the Taming calendar, based upon a cycle of 4,836 lunations in 391 years. His new calendar also incorporated a value of fortyfive years and eleven months a tu (365/4 tu representing 360°) for the precession of the equinoxes. Although Tsu Ch’ungchih’s powerful opponent Tai Fahsing strongly denounced the new system, the emperor HsiaoWu intended to adopt it in the year 464, but he died before his order was put into effect. Since his successor was strongly influenced by Tai Fahsing, the Taming calendar was never put into official use.
BIBLIOGRAPHY
On Tsu Ch’ungchilh and his works see Li Yen, Chungsuanshih lunts’ung (“Collected Essays on the History of Chinese Mathematics”). I–III (Shanghai 1933–1934), IV (Shanghai, 1947), I–V (Peking, 1954–1955); Chungkuo shuhsüeh takang (“Outline of Chinese Mathematics” Shanghai 1931, repr. Peking 1958), 45–50; chunkuo suanhsüehshi (“History of Chinese Mathematics” Shanghai, 1937, repr. Peking, 1955); “Tsu Ch’ungchih, Great Mathematician of Ancient China,” in People’s China24 (1956), 24; and Chunkuo kutai shuhsüeh shihua (“Historical Description of the Ancient Mathematics of China” Peking, 1961), written with Tu Shihjan.
See also ch’ien Paotsung,Chungkuo shuhsüehshih(“History of Chinese Mathematics” Peking, 1964), 83–90;Chou Ch’ingshu, “Wokuo Kutai weita ti k’ohsüehchia; Tsu Ch’ungchih” (“A Great Scientist of Ancient China; Tsu Ch’ungchih”), in Li Kuangpi and Ch’ien Chünhua, Chungkuo K’ohs üeh chishu faming hok’ohsü chishu jēnwu lunchi (“Essays on Chinese Discoveries and Inventions in Science and Technology and the Men who Made Them” Peking, 1955), 270–282l Li Ti, Ta k’ohsüehchia Tsu Ch’ungchih (“Tsu Ch’ungchih the Great Scientist” Shanghai, 1959); Ulrich Libbrecht, Chinese Mathematics in the Thirteenth Century (Cambridge, Mass., 1973), 275–276; Mao I shēng, “Chungkuo Yüanchoulü lüehshih” (“Outline History of π in China”),in K’ohsüeh, 3 (1917), 411; Mikami Yashio, Development of Mathematics in China and Japan (Leipzig, 1912), 51; Joseph Needham Science and Civilization in China, III (Cambridge, 1959), 102; A.P. Youschkevitch, Geschichte der Mathematik im Mittelalter (Leipzig, 1964), 59; and Yen Tunchieh, “Tsu Keng Pieh chuan” (“Special Biography of Tsu Keng”) in K’ ohsüeh25 (1941), 460.
Akira Kobori
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