July 19, 1888] 



NATURE 



273 



theory, and accepts Micl.e'.l's views : " If Michell be in 

 error, it is in the methods of calculation, not in the general 

 validity of his reasoning and conclusions." 



On the other hand, Leibnitz, Kant, Forbes, Boole, and 

 Mill (" Logic," xvii., xviii., xxv.), while allowing some 

 value to the theory, doubt if it can be rigorously applied 

 to obtain definite numerical results. 



The interest and importance of the subject, and the 

 length of time which has elapsed since any detailed dis- 

 cussion of it has been undertaken, furnish an excuse for 

 the following suggestions, which are made in the hope 

 that they may elicit more valuable arguments and 

 opinions. 



More than a century ago, Michell (Phil. Trans., 1767, 

 p. 243) attempted to find the probability that there is some 

 cause for the fact that the stars are not uniformly distributed 

 over the heavens, but frequently form binary combina- 

 tions or larger groups. Michell's results are quoted with 

 approval by Laplace ("Theorie des Prob," p. 63), and by 

 Herschel ("Astronomy," p. 607), though the latter men- 

 tions that Michell's data are too small, and immediately 

 afterwards quotes Struve's solution of the same problem, 

 which seems to be inconsistent with Michell's. I select 

 Michell's problem for discussion, since it has been 

 accepted by high authority and vigorously attacked, and 

 for the sake of simplicity in the calculations shall confine 

 my remarks to binary combinations. 



Michell's statements are not very clear, and his arith- 

 metical methods are cumbrous, but his argument may 

 be condensed as follows : " What, it is probable, would 

 have been the least apparent distance of any two or more 

 stars anywhere in the whole heavens, upon the supposition 

 that they had been scattered by mere chance ?" Imagine 

 any star situated on the surface of a sphere (S = \t:r % ) of 

 radius r, and surrounded by a circle of radius a (= r sin 0, 

 •where 6 is che angle subtended by a at the centre of the 

 sphere), the area of this small circle is s — no? = rrr 2 sin 2 #. 

 The probability that another star, " scattered by mere 



s 

 chance," should fall within this small circle is -~, and that 



it should not fall within it 



S 



But there is the same 



chance for any one star as for any other to fall within the 

 circle, hence we must multiply this fraction into itself as 

 many times as the whole number of stars («) of equal 

 brightness to those in question. " And farther, because 

 the same event is equally likely to happen to any one 

 star as to any other, and therefore any one of the whole 

 number of stars (n) might as well have been taken for the 

 given star as any other, we must repeat the last found 



chance n times, and consequently ( r — - J will repre- 

 sent the probability that nowhere in the whole heavens 

 any two stars among those in question would be within 

 the given distance (a) from one another, and the com- 

 plement of this quantity to unity will represent the 

 probability of the contrary." 



In the case of the two stars, /3 Capricorni, Michell takes 

 n = 230, 6 = 3' 20". Hence 



A B S ( sm 3' 2 °") 2 = 



1/42545 19, 



1/804 ; 



1 that no 

 fall so near 



•which Michell takes as 1/4254603 ; and 



Q=(i - 1/4254603 Y' = 1 - 5 2 9Q° = 

 V >+ ^ oj 4254603 



or, according to Michell, the probability is 



two stars equal in size to (3 Capricorni shall 



to one another as they do. 



Prof. J. D. Forbes {Phil. Mag., December 1850) 



objects to the entire principle upon which Michell's work 



is based, and has pointed out some errors in detail. 



Todhunter (" Theory of Prob.," p. 334) and Boole (" Laws 



of Thought," p, 365) countenance these objections ; but 



before discussing them it will be well to mention other 

 attempts to solve the same problem. 



Struve ("Cat. Nov.," p. 37) has used an entirely 

 different method. The possible number of binary com- 

 binations of n stars is n ^ n 1 V ; and the chance that 



1 . 2 

 such a pair should fall on a small circle of area s is j/S, 

 where S is the surface of the portion of the sphere in 

 which n has been counted. Hence the chance that any 

 pair of stars should fall within the circle is n(n - i)s/2S. 



Taking S as the surface from — 1 5 of declination to 

 the North Pole, n = 10229, an d & — 4*, Struve finds 

 p — o , oo78i4. 



Herschel (" Ast.," p. 607), either in error or by a re- 

 calculation from different data, quotes Struve as finding 

 that the probability is 1/9570 against two stars of the 

 7th magnitude coming within 4" by accident. 



Applying Struve's formula to Michell's data for £ 

 Capricorni, we have 



230 X 229 



2 



4254603 



1/161-5, 



or 161/162, as the probability that no two such stars fall 

 within the given area. 



Forbes, with the aid of a mathematical friend, offers 

 the following solution : — Suppose the n stars are repre- 

 sented by dice, each with v(>ri) sides, where v repre- 

 sents the number of small circles in the spherical surface, 

 or S/s. The chance of two stars falling into one circle is 

 the same as that two dice show the same face. 



The total number of arrangements without duplication 

 is — 



v . v — 1 .v — 2....V — n-\-i, 



and the total number of falls is v* ; hence the probability 

 of a fall without duplication is — 



v — 1 . v — 2 



v — n -+- l/v" ; 



and the chance that two or more dice show the same 

 face is — 



1 — [ v I \ v -n . v". 



In the case of /3 Capricorni v = 4254603, and n = 230. 

 Evaluating by Stirling's theorem, Forbes gives p = o 00617 

 = 1/160 nearly, which does not differ much from ri z /2v. 



A recalculation has given me p = 1/162. The result 

 then agrees with that of Struve and differs from that of 

 Michell. 



The following suggestions are due in substance chiefly 

 to Boo'e and Forbes, but their language has been freely 

 altered, and misapprehension of their meaning may 

 therefore be feared. 



In all such cases an hypothesis (" the random distribu- 

 tion of stars ") is assumed, and the probability of an 

 observed consequence (" the appearance of a double 

 star ") calculated. The small probability of this result of 

 the assumed hypothesis is held to imply that the prob- 

 ability of the hypothesis is equally small, and therefore 

 the probability of the contrary hypothesis is very large. 



According to Boole, " the general problem, in whatever 

 form it may be presented, admits only of an indefinite 

 solution," since in every solution it is tacitly assumed 

 that the a priori probability of the hypothesis has a 

 definite value, generally o or 1, and also a definite prob- 

 ability is assigned to the occurrence of the event observed 

 if the assumed hypothesis were false. 



In Michell's problem it is assumed that the stars are 

 either scattered at random or obey a general law j no 

 notice is taken of the possible case that a general law 

 holds for stars within a certain distance from our system, 

 beyond which an entirely different law may obtain. 

 Again, the subjection of each system to a separate 

 intelligence is tacitly ignored. 



