February 4, 19 15] 



NATURE 



613 



simultaneous equations; they are not baffled by 

 the occurrence of huge coefficients in the course 

 of elimination, and contrive to solve equations of 

 incredibly hig-h degrees. Finally, the elegant 

 designs of familiar things, such as toys, fans, etc., 

 suggested equally elegant problems of a quite dis- 

 tinctive kind, such as the Gion Temple problem, 

 an account of which is given, pp. 197—8. The 

 first solution involved an equation of degree 1024, 

 which was successively reduced (we are told) to 

 46 and 10 — the last by Ajima, who seems, so far 

 as we can judge, to have been the greatest m;ithe- 

 matician of his nation up to the present time. 



The fourteen chapters of this book are all inter- 

 esting, but we can only point out a few of the 

 topics. There is an excellent account of the way 

 of using sticks and abaci for calculation, and of 

 the early system of notation ; this is followed by 

 showing how the same implements were used to 

 solve equations, after the manner of Horner. 

 Coming now to the third period (1600-1675 or so) 

 we have circle-rectification of an Archimedean 

 type, magic squares and circles, and something 

 like a theory of quadrature. Seki Kowa is the 

 leading figure of the time, and chapter vi. is 

 entirely devoted to him, and gives abundant evi- 

 dence of his talent and ingenuity. The authors, 

 however, class him rather with C. Wolf or Barrow 

 than with Newton or Leibnitz. Chapter vii. deals 

 with Seki's contemporaries and the probable im- 

 portation of some western mathematics through 

 contact with the Dutch. 



Chapter viii. is on the yenri, or "circle prin- 

 ciple." Practically, this means the formulae con- 

 nected mainly with cyclometry, some of which are 

 essentially infinite series such as we have in 

 analytical trigonometry (c/. for instance, pp. 

 152—3). Matsunaga worked out ir to 50 figures 

 in the eighteenth century. 



Of the remaining chapters the most notable is 

 that on Ajima Chokuyen (1739-1798). Among 

 his achievements may be mentioned his (ana- 

 lytical) solution of Malfatti's problem, and his 

 anticipation of Steiner by discovering poristic 

 rings of circles touching each other and two given 

 fixed circles. He also dealt with repeating 

 decimals, diophantine problems, and quadrature ; 

 in the last he comes nearer than any of his pre- 

 decessors to the idea of a definite integral as the 

 limit of a sum. He may possibly have been 

 influenced by European work. 



A special feature of the book is the large 

 number of illustrations. Those which are dia- 

 grams for problems are always elegant and often 

 remarkably beautiful {e.g. pp. 96, 185, 186, 198, 

 240, 245) ; others are very instructive, like those 

 of the abaci, pp. 30-31. 



NO. 2.^62. VOL. Q4.1 



We cannot help asking ourselves : What is 

 likely to be the trend of Japanese mathematics for 

 the next generation or so? Japanese students are 

 now made acquainted with the vast structure of 

 European mathematics, and it is too massive and 

 too much based upon the fundamental nature of 

 things for them to ignore. If they are to add to 

 it, they must become acquainted w'ith it, unless 

 they go on making pretty little things that the 

 master-builders will put into their proper place. 

 But it would be a great pity if, in striving to con- 

 tribute to the substantial parts of the building, 

 the Japanese were to bury their special talents, 

 innate sense of asymmetrical beauty, and excep- 

 tional endurance and power of taking pains. Such 

 things as on one hand celestial mechanics, and 

 on the other diophantine analysis, seem admirably 

 suited to their genius ; if there be a planet of the 

 solar system beyond Neptune, or if there be a 

 proof of Fermat's last theorem, a Japanese is as 

 likely to discover it as anyone. G. B, M. 



PRACTICAL EDUCATION. 



(i) A Handbook of Vocational Education. By 

 Dr. J. S. Taylor. Pp. xvi + 225. (New York: 

 The Macmillan Co. ; London : Macmillan and 

 Co., Ltd., 1914.) Price 45. 6d. net. 

 (2) .4 Class-Book of Commercial Knowledge. By 

 E. J. Bailey. Pp. iii+125. (London: G. Bell 

 and Sons, Ltd., 1914.) Price 15. dd. 

 (i) " I ^HE aim of Dr. Taylor's book is to give 

 JL "a systematic survey of the general field 

 of vocational education, embodying both the his- 

 torical and logical aspects of the subject." But 

 the author has not done full justice either to the 

 subject or to himself by the short summary he has 

 produced. With the introduction, pleading for 

 equal opportunities for all, for education for 

 citizenship, for a due recognition of the bearing 

 of the industrial revolution on the teaching of 

 trades, English opinion will be in full sympathy. 

 So, too, with later chapters insisting on the need 

 for guidance in the choice of an occupation, and 

 on the part that trade schools should play in 

 making good the loss of the thorough, all-round 

 training afforded by the old apprenticeship system 

 at its best. But the historical account of indus- 

 trial education in Eurof)e, given in chapter ii., is 

 far too sketchy to be of real value. 



Germany, and in Germany the admirable con- 

 tinuation school system of Munich, designed and 

 built up by Dr. Kerschensteiner, holds pride of 

 place, and the constructive side of Dr. Taylor's 

 book may be said to be based on Dr. Kerschen- 

 steiner's principles. In his whole-hearted de- 

 votion to this master, who has exhibited in rare 



