January 8, 1920] 



NATURE 



465 



orders in response to attractive offers of hundred- 

 pound prizes the expectation value of which is not 

 one-tenth of the price paid. Yet it was only re- 

 cently that the Central Welsh Board excluded 

 probabilities from an examination syllabus in 

 algebra which was simply loaded up with ques- 

 tions on collections of letters and symbols that 

 could convey no meaning- to the victims of the 

 examination system. 



Other applications are to such problems as 

 life assurance and statistics. In the former 

 the calculations must, of course, be largely 

 left to experts, but the public ought to 

 acquire an intimate familiarity with the nature 

 and meaning of probability and expectation and 

 their numerical representation in order properly to 

 appreciate the transactions. This elementary 

 knowledge should come under arithmetic, not 

 algebra. As for statistical applications, the 

 systematic way in which parliamentary electors 

 are misled for lack of understanding these things 

 is evident. They have not realised that when 

 wages go up prices also go up. 



Now Prof. Castelnuovo's treatise strikes the re- 

 viewer as just the kind of book of which it would 

 be worth while to publish an English translation. 

 It is a long time since we had a standard work 

 on the subject on similar lines, and in the 

 interval our notions on teaching mathematics 

 have certainly moved in a practical direction. 

 Prof. Castelnuovo's book well meets the situation. 

 Of course, the treatment is mathematical and the 

 calculus is freely used, but the formula are intro- 

 duced as statements of principles rather than as 

 purely algebraic relations, and the whole treat- 

 ment centres largely round practical applications. 

 Any B.Sc. candidate would find the book quite 

 easy reading, and the subject very useful in con- 

 nection with physics, biology, philosophy, or, 

 indeed, any branch of science, even including that 

 all-embracing subject, aeronautics. A special 

 feature on the more advanced side is the account 

 given at the end of the discoveries of Tchebychef, 

 who is quoted as having made the greatest con- 

 tributions to the subject after Laplace. 



Of course, the initial difficulty lies in the defini- 

 tion of probability, regarding which Prof. Castel- 

 nuovo's treatment both in the preface and at the 

 beginning of the text is probably as good as the 

 circumstances permit. There is no unique defini- 

 tion of probability, and in most cases no unique 

 measure of its value. The old definition by an 

 event which may happen in m ways and fail in 

 n ways postulates a preconceived condition of 

 "equal probability " for the m and n ways. There 

 are, as Prof. Castelnuovo points out, many cases, 

 NO. 2619, VOL. 104] 



in particular in games of chance, in which this 

 postulate is admissible and the measure of proba- 

 bility has something like a unique value. But in 

 most cases the estimated probability of an event 

 depends on the extent of knowledge possessed by 

 the person making the estimate, and, indeed, is a 

 continually varying quantity depending on the pro- 

 gress of previous events. No two people would 

 assign the same measure to the probability of a 

 certain candidate passing an examination, and, 

 indeed, the estimated chance varies continuously 

 until the appearance of the list (sometimes even 

 afterwards !). All that we can do in place of a 

 (definition is to substitute numerous examples in 

 which the measure of probability is free from 

 ambiguity. The nearest approach to a definition 

 is given by the rules for compounding probabili- 

 ties, of which the above-mentioned old definition 

 is a particular case, with the additional convention 

 that the probability always lies between o and i, 

 and that after an event has happened we must 

 substitute i for the probability of its happening 

 and o for the probability of its failing in our 

 future estimates of the probabilities of dependent 

 events. In fact, the theory of probability owes its 

 existence to ignorance of future, and partial igno- 

 rance of past, events. 



Attention has been frequently directed by the 

 reviewer to energy running to waste among our 

 mathematicians which could be utilised in con- 

 nection with aeroplanes. In our universities a 

 great deal of waste energy in the departments of 

 pure mathematics could also be utilised by turning 

 out graduates with a knowledge of probabilities 

 and statistics which would filter down through the 

 teachers to the elementary schools and thus to 

 the citizens of the future. And for a start at the 

 top of the ladder, Prof. Castelnuovo's book seems 

 excellent. G. H. Bryan. 



THE STUDY OF THE FAMILIAR. 



A Source Book of Biological Nature-Study. By 

 Elliot Rowland Downing. (The University 

 of Chicago Nature-Study Series.) Pp. xxi-t-503. 

 (Chicago, Illinois : The University of Chicago 

 Press ; London : The Cambridge University 

 Press, 1919.) Price 3 dollars net. 



IT is encouraging to read that never before has 

 there been in America " so insistent a demand 

 for a more thorough and more comprehensive 

 system of instruction in practical science." To 

 direct this demand, Mr. Downing is editing a 

 Nature-study series, and has written a source 

 book for the biological side. It aims at showing 

 students in schools of education and teachers at 

 work what materials are readily available and 



