676 



SCIENCE 



[N. S. Vol. XL. No. 1036 



ods of the calculus is found in tte yenri, or 

 circle principle, whicli tradition states was 

 devised by Seki Kowa. 



This work should appeal to a wide circle 

 of readers, to the students of the history of 

 science, to all interested in Japanese civiliza- 

 tion and even to the general reader, for much 

 of the work is non-technical. Certainly this 

 book will contribute to a juster and broader 

 appreciation of the Japanese genius. 



Louis C. Karpinski 



Uniyeksity or Michigan 



The Development of Mathematics in China and 



Japan; Abhandlungen zur Geschichte der 



mathematischen Wissensehaften, Vol. XXX. 



By YosHio MiKAMi. Teubner, Leipzig, 1913. 



G. E. Stechert and Co., New York. Pp. x + 



347. 



The activity of Mr. Mikami in making the 

 mathematics of China and Japan known to the 

 western world is highly to be commended. Be- 

 sides many articles dealing with particular 

 problems of the history of mathematics, Mr. 

 Mikami has an earlier work, " Mathematical 

 Papers from the Far East," in the same series 

 as this volume under discussion, and also 

 another book jointly with Professor David 

 Eugene Smith, " A History of Japanese Mathe- 

 matics," published by The Open Court Pub- 

 lishing Company. The more active coopera- 

 tion of some English-speaking historian of 

 mathematics would have been desirable in the 

 two volumes which were published in Ger- 

 many. Professor G. B. Halsted has, indeed, 

 prefatory notes in the volumes which imply 

 that the task of correcting the English was 

 entrusted to him, but the literary charm of 

 Professor Halsted's own works is lacking here, 

 and even unintelligible as well as non-idiomatic 

 English mars the excellence of these works. 

 Errors are too numerous to be listed. 



The book is divided into two parts : the first 

 21 chapters discuss the Chinese mathematics, 

 and the following 26 chapters the Japanese. 

 Three chapters which are of great value to 

 the student of the history of science are en- 

 titled, A General View of the Japanese Mathe- 

 matics, A Chronology of the Japanese Mathe- 



matics, and A Short Notice of the Historical 

 Studies of the Japanese Mathematics. Some- 

 what similar treatment of the Chinese portion 

 would have added much to the value of the 

 work. An omission in the bibliography of the 

 historical works is Souciet (Pere), Observa- 

 tions mathematiques, astronomiques, etc., 

 tirees des anciens livres Chinois, ou faites 

 nouveilement aux Indes et a la Chine par les 

 peres de la Camp, de Jesus (Paris, 1Y29), to 

 which my attention has been called by Pro- 

 fessor W. W. Beman. 



Considerable uncertainty attaches to the 

 dating, and even the content, of the ancient 

 Chinese and Japanese mathematical treatises, 

 but this, we may say, seems somewhat charac- 

 teristic of our knowledge of the early Orient, 

 particularly India. An evidence of this uncer- 

 tainty is the fact that Mikami's description of 

 the early " Arithmetic in Nine Sections " is 

 quite different (footnote, p. 10) from that 

 given by T. Hayashi in his " Brief History of 

 Japanese Mathematics " which appeared in 

 the Nieuw Archief, Tweede Reeks, Deel VI. 

 (not accessible to me). 



To the student of mathematics the most 

 striking feature of this history will doubtless 

 be the processes of solution of equations of 

 higher degree than the second, by means of the 

 sangis or calculating pieces. These solutions 

 require a great amount of detail and approach 

 closely the methods of Horner and Newton. 

 The attention paid to the " squaring of the 

 circle " is of interest, and the approach to a 

 determinant notation is truly striking. The 

 student of the history of mathematics wiU 

 doubtless be most impressed by the description 

 of the early Chinese process of multiplication 

 of an integer of several places by an integer of 

 the same kind, for the process corresponds in 

 many details to the methods taught in the 

 early works on the Hindu art of reckoning. 



Some allowance for the enthusiasm of a 

 Japanese writer must be made by the reader. 

 However, to compare the Japanese Seki with 

 Newton, " If Seki did not surpass Newton in 

 his achievements, yet he was no inferior of 

 the two," is quite beyond the bounds of allow- 

 able enthusiasm, for no evidence is presented 



