Chinese Extraction of Cube Roots 1 Paul Yiu December, 1996 1. As I have been enjoying for the most part of the year many of the enlighten- ing discussions in this list on various topics in the history of mathematics, I would like to respond to several recent postings on ancient Chinese methods of extraction of square and cube roots. I am not a historian of mathematics, nor a Chinese classicist, but a student of mathematics keenly interested in the pedagogical insights in the Chinese classics Jiuzhang Suanshu, especially its commentary by LIU Hui in the third century. So, please take this series of postings as an amateur’s finding prompted by interesting questions posed recently in this list. I am sure most of these are well known among historians of mathematics. In his posting on December 6, Professor J. Gonzalez Cabillon quoted two passages in the works of Professor LAM Lay Yong of Singapore: The Chinese also developed a similar method to find the cube root of a number and in so doing were able to solve a cubic equation of the form x3 + ax2 + bx = c,wherea, b and c are pos- itive. Through the method of finding a side of a square or cube with the rod numeral system, the Chinese invented a notation to represent quadratic and cubic equations and were able to solve numerically aparticular2 type of such equations. One could say that at this stage there were indications that arithmetic was forging into the yet to be known field of algebra. Arch. Hist. Exact Sci., 47 (1) 1994, 1-51. The ancient Chinese performed the four fundamental operations on fractions and the extractions of square and cube roots as 1Minor revision of 5 postings to the Math-History-List, Mathematical Association of America, December 14, 15, 1996. 2Underscore mine. 1 YIU: Chinese extraction of cube roots 2 easily as we perform these operations using our numeral system. More important, in spite of the two thousand years’ gap and the difference in media, the procedures that they used bear parallel resemblance to the ones we use. [...] Arch. Hist. Exact Sci., 37 (4) 1987, 365 - 392. My library here does not have Arch. Hist. Exact Sci., nor have I seen Professor LAM’s book (with ANG Tian Se) Fleeting Footsteps, Tracing the Conception of Arithmetic and Algebra in Ancient China. 3 But I happened to have a copy of her 1994 paper on an overview of Jiuzhang Suanshu (Nine Chapters of the Mathematical Art). To me, these paragraphs are terse summaries of a long tradition of Chinese mathematics. In the first passage I underline the words numerically and particular. It should not be taken to mean that ancient Chinese knew how to solve cubic equations in radicals. It means that the principles and methods of extraction of square roots and cube roots as explained in Jiuzhang Suanshu have gradually evolved into a method of solving polynomial equations, a method known as zeng cheng kai fang (extraction of roots by addition and multiplication) in the Song period (9th – 11th century), finally capable of dealing with equations of arbitrarily high degrees and coefficients positive, negative, and zero. Such numerical methods are indeed spigot algorithms which give a positive root of a polynomial equation digit by digit. Here, I am borrowing the term ‘spigot algorithm’ from an article of Stan Rabinowitz and Stan Wagon in the American Mathematical Monthly. 4 A spigot algorithm “pumps out digits one at a time and does not use the digits after they are computed”. Professor SIU Man Keung has explained the extraction of square root in an early posting 5 In practice, each of the digits is determined by deliberation. In Jiuzhang Suanshu and other Chinese texts, the digits obtained in this way are called ‘shang’, the same character for the word ‘quotient’. This character shang (deliberation) captures the essence of the division process, (unless one insists on performing division by repeated subtraction). In a posting on November 28, SIU wrote If by ”trial and error” one means to first guess an answer and see how good or bad it is and then proceed to the next step, then I agree that the ancient Chinese method for calculating square roots, cube roots, etc. is of that category. But if by ”trial 3World Scientific, Singapore, 1992. 4vol. 102 (1995) 195 – 203. 5November . YIU: Chinese extraction of cube roots 3 and error” one means that it is just empirical guesswork without theory behind it, then I cannot agree that the ancient Chinese method is of that category. Then he went on to explain the geometry given in LIU Hui’s commentary of the method. In a paper (written in Chinese) on the constructive and mechanistic aspects of traditional Chinese mathematics, 6 Professor WU Wentsun has incorporated the method of zeng cheng kai fang into a computer programme for the determination of the real roots of a cubic equation of the form 3 2 x + a1x + a2x = a3 with positive a1, a2,anda3. WU remarks that “the principle of such a method [like the extraction of square root] is self evident from the proce- dures, and that it is precise to any desired degree of accuracy”. Perhaps it is appropriate to read a passage by Joseph Needham on this subject. In his Science and Civilisation in China, 7 Needham wrote: 8 The solution of numerical higher equations for approximate val- ues of roots begins, as far as we know, in China. It has been called the most characteristic Chinese mathematical contribu- tion. That it was well developed in the work of the Sung al- gebraists has long been known, but it is possible to show that if the text of the Han Jiuzhang [Suanshu], (Nine Chapter of the Mathematical Art) is very carefully followed, the essentials of the method are already there at a time which may be dated as of the −1 century. ... Among the problems in the Jiuzhang, one, the finding of a cube root, may be written x3 =1, 860, 867, and another, involving thesquareandthesimpleterm,x2 +34x +71, 000 = 0. Natu- rally, these equations were not written as such, but set up with counting rods on a board. The uppermost line in the table is to contain the radical solution sought, the second line (shi)is 6QIN Jiushao and Shushu Jiuzhang, pp.73 – 87, Beijing, 1987. 7volume III (1959), Chapter 19 on Mathematics, pp. 126 ff. 8I have replaced Needham’s transliterations of Chinese characters by Pinyin. Again, emphasis mine. YIU: Chinese extraction of cube roots 4 the constant term before the operation, the third (fa,orlaterin the operation, ting fa) is for numbers obtained in its successive stages, the fourth (zhong) receives another number found at an intermediate state, and the fifth (jie suan) is the coefficient of x3 before the operation. unnamed line for the root 1860867 shi (dividend) fa (divisor) zhong (middle) jie suan (borrowed rod) The process starts by raising the 1 in the jie suan line to 106 so that 3 1, 000, 000x1 − 1, 860, 867 = 0. After this the text, not at all explicit, uses the word yi, ‘discus- sion’, which must mean the choosing of a first root figure a (e.g. 1). This is then inserted in its proper place on the top line. Then follows a set of operations by which the numbers on the counting - board are converted into a pattern corresponding to Horner’s transformed equation: 3 2 1000x2 +30, 000x2 + 300, 000x2 − 860, 867 = 0, Here x2 = 10(x1 − a). With the transformed equation, ‘discus- sion’ (yi) again takes place, and the next digit of the solution is inserted. The same process is repeated (with modification) to find the third digit, at which point the Jiuzhang procedure ends. In this particular case, the answer was 123. LIU Hui in the +3rd century, however, pointed out that the process could be continued past the decimal point (as we should say) as long as required. The Jiuzhang author himself seems to have carried square-root extraction to one decimal place. The system can be found in all the later mathematical books with terminological variations, and minor changes in the steps, gen- erally not improvements. Numerical equations of degrees higher than the third occur first in the work of QIN Jiushao around +1245. He deals very clearly with equations such as YIU: Chinese extraction of cube roots 5 −x4 + 763, 200x2 − 40, 642, 560, 000 = 0. ... In considering the long lead of Chinese mathematics in this field, one may suggest that the counting - board, with its lines repre- senting increasing powers, was particularly convenient for the purpose, though of course no attempt was made, even in the Song, to produce a generalized theory of these equations. 2. This last paragraph in Needham’s naturally brings me back to the second passage of LAM’s cited by Professor Gonzalez Cabillon. 9 But I would like to first address Professor Domingo Gomez Morin’s question on December 12: [A]part from trivial examples on perfect cubes, I think it would be equally easy to show here many numerical examples on ancient Chinese approximations to the solution of cube roots. In this way, could anyone bring me any ancient Chinese approximation to the solution of cube roots, as for example, the cube root of 2?” My resources in Chinese literature is extremely limited. I am sure other scholars more knowledgeable and resourceful in Chinese mathematics can help. But here I shall report on my findings. Ididnot expect to be able to find informations on the cube root of 2.