NA TORE 



603 



THURSDAY, AUGUST 14, 191; 



MATHEMATICS IN CHINA AND JAPAN. 

 The Development of Mathematics in China and 

 Japan. By Yoshio Mikami. Pp. x + 348. 

 Leipzig: B. G. Teubner; London: Williams 

 and Norgate, 1913.) Price 18 marks. 



THE time has not yet come for anything- like 

 a complete or final history of mathematics 

 in the Far East; meanwhile Mr. Mikami's work 

 will be welcomed as being- practically the first book 

 on the subject accessible to Europeans. Its con- 

 tents are unavoidably miscellaneous, but there are 

 two or three topics on which some remarks can 

 be made. 



First of all we find, as usual, that in the earliest 

 periods there is a special calculating apparatus, 

 which dominates not only methods of computation, 

 but forms of mathematical thought, for many 

 generations. In the case of China this consisted 

 of a board ruled in columns, and a set of calcula- 

 ting sticks or counters. So far we have the equi- 

 valent of the abacus ; but there is a very important 

 modification, apparently familiar at least as early 

 as 50 B.C. Red counters were used for additive, 

 and black for subtractive numbers, so there was 

 a visible distinction between +a and —a of a most 

 convenient kind. 



An abacus does not lend itself to calculation 

 with vulgar fractions, and strangely enough it 

 failed to suggest decimals. However, the Chinese 

 appear to have been in possession of all the rules 

 for working with vulgar fractions as early as the 

 middle or later half of the sixth century, although 

 it was confessed a difficult subject even by learned 

 men. In this respect they were 1000 years or so 

 ahead of Europe. Unfortunately Mr. Mikami does 

 not give any account of the notation (if any) used 

 for fractions at this date. 



Approximations to ir occupied the attention of 

 many mathematicians both in China and in Japan. 

 Perhaps the most remarkable fact in this con- 

 nection is that Tsu Ch'ung-chih (a.d. 430-501) 

 calculated w in an Archimedean mannpr, arriving 

 at upper and lower limits 2^4^5927 and 3 - i4i5g26. 

 In some unknown way he hit upon the values 22/7 

 and 355/113, which he called the inaccurate and 

 accurate values respectively. The appearance of 

 this celebrated value 355 '113 at so early a date 

 is very remarkable. It may be added that 7r= \/ 10 

 occurs before a.d. 139, and that many Chinese and 

 Japanese have calculated jt to a large number of 

 places of decimals. 



Another striking thing is that the Chinese seem 

 to have practised Horner's method of solving 

 NO. 228^, VOL. 91] 



numerical equations in the thirteenth century (see 

 pp. 74-8). In fact, both Chinese and Japanese 

 constantly use the principles of reversion of series 

 and successive approximation. We can venture 

 to smile at Mr. Mikami's hint that it is "not 

 impossible " that Europeans may have known of 

 the Chinese method ; but while doing so we must 

 be careful not to accuse our Eastern kinsmen of 

 borrowing without acknowledgment, unless due 

 evidence is at hand. 



There is, in fact, an extraordinary instance of 

 independent discovery, upon which Mr. Mikami 

 makes no remark, but which appears to be abso- 

 lutely certain, unless somebody has committed 

 an ingenious and elaborate fraud. Early in the 

 nineteenth century Steiner published some ex- 

 tremely elegant results about rings of touching 

 circles (or spheres) touching two given circles (or 

 spheres). His proofs depend partly upon using 

 the method of inversion, so as to change one of 

 the fixed circles (spheres) into a line (plane). 

 Under certain conditions we have a poristic ring 

 of variable touching circles. Now on pp. 238-46 

 of the present work Mr. Mikami gives a summary 

 of work by the Japanese Ajima Chokuyen, dated 

 1784, where Steiner's problem for circles is dis- 

 cussed without inversion, and the algebraic condi- 

 tions are given (in their simplest form) for poristic 

 rings of n circles when n = 3, 4, 5, ... 10 : and 

 the method is general enough for the condition to 

 be calculated in any case. 



Various problems are given from time to time. 

 Some of these are of a familiar type, and may be 

 of Indian or even Egyptian origin (e.g. we have a 

 variation of the sloping reed question). Others, 

 especially of the Japanese, are evidently of 

 native origin — suggested by toys, jugglers' tricks, 

 and so on. 



Matters of more general interest are a rule for 

 finding out whether an expected child is to be a 

 boy or a girl, the author's interview with one of 

 the last great Japanese mathematicians of the 

 older school, and lastly the title-page, which is a 

 very significant document. Written in English by 

 a Japanese, the book has been revised by an 

 American professor and published by a German 

 who has probably done more than any of his craft 

 for the spread of scientific literature. The lan- 

 guage is that of the lazy lion ; the rest belongs to 

 the lands of the two eagles and the rising sun. 

 Let the lion beware lest reflection show him that he 

 has an ass's head. 



The reviser. Prof. G B. Halsted, has shown 

 admirable taste in not converting Mr. Mikami's 

 idiom into standard English. In some cases it is 

 rather difficult to understand the author's explana- 



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