REVIEWS 



MATHEMATICS 



The Mathematical Theory of Probabilities. By Arne Fisher. Translated from 

 the Danish by C. Dickson and W. Bonynge. Vol. I. Second Edition. 

 [Pp. xxix + 289,] (New York : The Macmillan Co.) 



Mr. Fisher's book, the first volume, consists of three parts, the first dealing 

 with the mathematical theory of probability leading up to Baye's theorem, 

 the second dealing with the theory of dispersion and the application of various 

 criteria in considering whether a series of errors of observation are due to 

 chance or to certain definite laws, the third dealing with the analysis of 

 frequency distributions. He hopes in the second volume to deal with the 

 theory of correlation. 



The author has undertaken a great task to treat adequately the methods 

 of modern statistical analysis, and naturally, at a certain stage, he has to 

 abandon the general theory for the particular theory, and after the first 

 part he has followed the methods of the Scandinavian and Russian schools. 

 The application of the theory of probability to statistical problems has 

 developed rapidly in recent years on the Continent, in America, and in England, 

 and a great deal of work done in Scandinavia and in Russia has never become 

 well known to the English school owing to the language difiiculty. Mr. Fisher 

 is able to fill the role of liaison officer and present the work of Continental 

 writers in an understandable form for the benefit of English-speaking workers, 

 and on this account his book is of great service, and for this alone we are in 

 the author's debt. 



The first part of the book, being the development of the theory of 

 probability as the foundation of the theoretical basis of the modern statistical 

 calculus, does not suffer, as most works on probability suffer, from an over- 

 treatment of those interesting problems in probability which have academic 

 interest, but are not links in the chain of argument which leads up to Baye's 

 theorem, but it gives a careful and good treatment of the fundamental 

 theorems in probability. In these days, when the validity of Baye's theorem 

 is once again being assailed, it is good to be able to turn to Mr. Fisher's book, 

 where he gives a very full discussion of both sides of the argument concerning 

 this fundamental problem, and enables the practical statistician to feel sure 

 that there are good reasons why he should accept this theorem. It would have 

 been of interest to note his remarks on Prof. Pearson's paper in Biometrika 

 (1921) on "The Fundamental Problem of Practical Statistics." 



His connecting chapter on " The Law of Large Numbers " between the 

 theory of probability and the treatment of statistics serves a very useful 

 purpose, and enables the reader to see more clearly the connection between 

 classic probability and modern statistical analysis. In his discussion on the 

 " mean " we would have preferred more cogent reasons for his use of the 

 arithmetic mean as the best value to be obtained from a series of observations. 

 He frankly says : " Since we have no a priori reasons for choosing any one 

 particular value of the various a's . . . in preference to any other, we might 

 give equal weight to each set and take the arithmetic mean. . . ." 



In his treatment of frequency distributions he follows Gram, Thiele, 



478 " 



