598 THE POPULAR SCIENCE MONTHLY 



■wrote in Ujjain in the seventh century, was indebted for at least one of 

 his problems to the great Chinese classic, the Chiu-chang Suan-shu. 



Of the scholars whose contributions to mathematics were note- 

 worthy, it will suffice to mention only a few, although it should be 

 stated that these are the greatest of their respective periods so far as we 

 now know. 



No one knows how far back the Chow-pi goes. It purports to be a 

 dialogue between Chow Kung and Kao (or Shang Kao) and to have 

 been written c. 1100 B.C. Wylie* translates part of it, and shows that 

 it contains a reference to the Pythagorean proposition and to a primi- 

 tive trigonometry. It has long been known that the discovery of Pytha- 

 goras was the proof and not merely the fact, for the latter is mentioned 

 in Egyptian writings before his time, and in Hindu works that probably 

 antedated him, so that it is not surprising to find it in China. 



Chang T'sang, who died in 153 B.C., restored the Chiu-chang Suan- 

 shu, or Arithmetical Eules in Nine Sections, for the antiquity of which, 

 in its original form, great claims are made.^ 



From the standpoint of mathematics the most interesting features 

 of this work are the use of negative numbers, the trigonometry of the 

 right triangle, and the fang-ch'eng process. The last named constituted 

 one of the nine sections and concerned the solution of simultaneous 

 linear equations. This would hardly be worth mentioning except for 

 three reasons: (1) "We have nothing of this kind in the algebra of 

 Europe as early as this; (2) from this method of the Chinese came, by 

 direct descent, the early Japanese method that led by obvious steps to 

 the invention of determinants by Seki before the idea occurred to Leib- 

 nitz;® (3) the method for the extraction of roots led Ch'in Chiu-shao, 

 in 1247, to anticipate Horner's method, as will presently be shown. 



Sun-tsu, whose date is unknown, but who probably lived in the 

 third century of the Christian era, wrote a work on arithmetic in which 

 he set forth the process known as t'ai-yen ch'iu-yi-shu, a form of inde- 

 terminate analysis that was afterwards employed with much success. It 



*" Chinese Eesearches," Shanghai, 1897, Part III., p. 163. 



^From the preface of the edition by Liu Hui (third century a.d.) we have 

 this statement : ' ' After the terrible Ching had burned the books, the classics and 

 (works on) arts were dispersed and lost. Later, in the Han Dynasty, the Duke 

 of Pei-ping, Chang T'sang, and also Ken Shou-Ch'ang (Ching Ch 'ou-eh 'ang) 

 a Commissary agent, were well known because of their talents in (the domain 

 of) number. T 'sang and this other, because of the incompleteness of the ancient 

 writings, revised (by adding to and taking from) the work. So did each claim. 

 Upon comparing the contents their works were seen to differ from the original, 

 but the subject matter is much alike." 



The origin of the work is often asserted to go back to c. 2650 B.C. 



* See T. Hayashi, ' ' The Fukudai and Determinants in Japanese Mathe- 

 matics," in the Tolcyo SugaTcu-BulurigakJcwai Kisi, Vol. V. (2), p. 254. 



