154 



NATURE 



[December 13, 1900 



the Board of Education, South Kensington. The two 

 little books under notice have been written in the same 

 spirit, and contain some sections from the previous 

 volume, but the treatment is more elementary and many 

 new exercises are given. 



In Part i. the subjects dealt with belong to arithmetic, 

 algebra and the mensuration of parallelograms, triangles 

 and polygons. Prominence is given to contracted 

 methods, use of decimals, and explanations of algebraical 

 expressions. Scales, calipers, and other simple measuring 

 instruments are described in the chapters dealing with 

 mensuration, and their use is well exemplified. Part ii. is 

 devoted to logarithms, the slide rule, mensuration of 

 circles, ellipses and irregular plane figures, volumes and 

 surfaces of solids, more difficult algebraic expressions 

 than are given in Part i., and the graphic representation 

 of varying quantities. Among noteworthy points in this 

 part may be mentioned the clear account of uses to 

 which a slide rule may be put, the descriptions of plani- 

 meters, and the ingenious uses made of squared paper 

 in the section on graphic representation. 



The books are full of exercises illustrating the appli- 

 cations to every-day problems of the principles de- 

 scribed, and at the end of Part ii. a set of tables of 

 logarithms and anti-logarithms is given, to enable the 

 student to work out problems by logarithms when con- 

 venient. It would be too much to say that the books 

 contain an ideal course of mathematics for technical 

 students, but they may fairly claim to provide far more 

 inspiring information and serviceable exercises than 

 can usually be found in text-books designed for use 

 in schools. 



Exercises in Natural Philosophy, with Indications how 

 to Answer them. By Prof. Magnus Maclean, D.Sc, 

 F.R.S.E. Pp. x -H 266. ■ (London: Longmans, Green 

 and Co., 1900.) 



The ability to deal with quantitative results is an 

 essential qualification of a student of physical science. 

 Laboratory work provides some material for the exercise 

 of this faculty, but it is often necessary to use data 

 obtained by others, and to work out problems other than 

 those which are afforded by the student's own practical 

 work. Dr. Maclean's book contains numerous exercises 

 of this character, covering most of the subjects studied 

 in courses of physical science, and many worked-out 

 examples of typical cases suggesting methods of solution 

 for those which follow. Wisely used, the book will 

 provide teachers with useful exercises in mathematics 

 applied to physics, and will make a convenient supple- 

 ment to text-books in which such exercises are not 

 given. Many text-books do contain questions upon the 

 subjects dealt with, but even in these cases some good 

 additional problems for solution could be selected from 

 the book under notice. 



Tables of useful data and physical constants are 

 printed at the end of the volume. 



Memoirs of the Countess Potocka. Edited by Casimir 

 Stryienski. Authorised translation by Lionel Strachey. 

 Pp. xxiv + 253. (New York : Doubleday and McClure 

 Company, 1900.) 



These memoirs cover the period from the third parti- 

 tion of Poland to the incorporation of what was left of 

 that country with the Russian Empire. They deal with 

 episodes — more or less romantic and interesting — in 

 Countess Potocka's career, referring to journeys, Court 

 balls, and Napoleon I., between 1812 and 1820. The 

 authoress died, at the age of ninety-one, in Paris, where 

 her brilliant salon held no insignificant place in the 

 gilded pleasures of the Second Empire. There is little 

 of interest to scientific readers in the memoirs ; but one 

 or two incidents referring to astrologers are amusing. 



NO. 1624, VOL. 63] 



LETTERS TO THE EDITOR. 



\The Editor does not hold himself responsible for opinions ex- 

 pressed by his correspondents. Neither can he undertake 

 to return, or to correspond with the writers of, rejected 

 manuscripts intended for this or any other part of Nature. 

 No notice is taken of anonymous communications. \ 



Inverse or "a posteriori" Probability 



The familiar formula of Inverse Probability may be stated as 

 follows : — 



Let the probabilities of a number of mutually exclusive causes 

 or conditions Cj C2 . . . C^ be Pj Pg . . . P, respectively, and 

 the probabilities that if Ci C2, &:c., are realised, an effect or 

 result E will happen be /j ^2 • • • Pr respectively ; then if E 

 happens, the probability that it happened as a result of C,. is 



Vrpr 



2P/' 



The current proofs of this are unsatisfactory, more especially 

 one based on a theorem of James Bernoulli ; for even if the 

 ordinary statements of the principle of this theorem were cor- 

 rect, which must be disputed, the argument by which it is applied 

 to Inverse Probability is demonstrably erroneous. 



In consequence of the difficulty felt about the usual proofs, 

 there seems to be a tendency to drop the subject, as unsound, 

 out of mathematical theory. 



Now it would not be hard to show that there is no essential 

 difference of principle between problems of Inverse Probability 

 and those of ordinary Probability, and therefore it can hardly 

 be doubted that the former should admit of as accurate mathe- 

 matical treatment as the latter. 



The following is offered as a proof which can claim the same 

 rigour as the theorems of ordinary Probability, and illustrates 

 the identity of principle in both kinds of Probability : — 



Let A and B be contingencies which are not independent, 

 then'.by a known theorem 



Prob. concurrence of A and B = Prob. A x Prob. of B 

 if A happens. 



Or, as it may be shortly expressed, 



Prob. A with B = Prob. A x Prob. B if A. 



Similarly 



Prob. A with B = Prob. B x Prob. A if B. 

 . • . Prob. A X Prob. B if A = Prob. B x Prob. A if B. 



Prob. Aif B = 



Prob. A X Prob. B if A 

 Prob. B 



and this is really our theorem. For put A = Cr and B = E. 



. •. Prob. C. if E = Pro b.C.x Prob. EifC. 

 Prob. E 



But Prob. C^=P^, Prob. E if Cr=/r, and obviously Prob. E 

 = 2P/ by a known theorem. 



.-. Prob. C,.ifE = ?^. 

 2P/ 



Another demonstration may be given which, though a little 

 longer, is quite simple. 



If the whole number of " equally likely " cases with reference 

 to a given contingency E is b, and the number of these in favour 

 of E is a, then the mathematical probability of E is, of course, 



-=/, suppose. 

 b 



Considered as a fraction, /= — , where n is any quantity 

 nb 

 whatever. 



Suppose n an integer, as a fractional value does not here con- 

 cern us. We may consider each of the original " equally likely " 

 cases as including n " equally likely " sub-cases; and then we 



can interpret the fraction — f as we interpreted -, and say that 



7ib 



there are nb new cases equally likely, and of these na are in 

 favour of E. 



Obviously, if x is the total number of equally likely cases, the 

 number in favour of the event or contingency is/;r. Again, if q 

 is the probability that E happens if C happens, this means that 

 q of the equally likely cases of C's happening are in favour of 



