418 Prof. Forbes on the evidence for a Physical Connexion 



equal spacing, since unequal spacing corresponds to a dimi- 

 nished probability of occurrence. 



32. For further illustration of Mitchell's principle, let the 

 heavens be divided into p equal areas, and let the fall of a die 

 having p sides indicate the allotments into which a given star 

 is expected to fall " at random,'' there being as many dice 

 to throw as there are stars to distribute, then the question 

 resolves itself into this : — " Of all given possible distribu- 

 tions to assign that in favour of which the a priori proba- 

 bility is greatest." I answer, that a uniform (and therefore 

 generally symmetrical) distribution is more probable than 

 any other given (i. e. observed) distribution. For, to recur 

 to the case of the dice, let there be 100 dice, each having 

 100 faces inscribed with the uniform series 1 .... 100. It is 

 no doubt very improbable, a priori, that the 100 dice thrown 

 at random shall turn up exactly the arithmetical series 1 ... 100; 

 but it is at least twice as probable as any other definite sup- 

 position of distribution of numbers that can be made; — as for 

 instance that the number 100 should be wanting, and that 

 there should be two 99's. And the proof of this is sufficiently 

 simple ; the reason is the same as that there is a less chance of 

 throwing given doublets than a given pair of numbers not alike, 

 as A, B. For there are two combinations, A, B, and B, A, 

 in favour of the latter event, but only a simple combination, 

 as A A, in favour of the former*. Now the case of the 100 

 dice showing 100 consecutive numbers, represents the case of 

 100 stars, one of each of which falls into each of 100 different 

 allotments into which the sky may be divided. And this is 

 evidently a symmetrical distribution, or the nearest approach 

 to symmetry which is geometrically possible. Hence, if Mit- 

 chell's rules of reasoning be applicable in this case, the proba- 

 bility is greater for a distribution of the stars having occurred 

 " by mere chance, as it might happen," if such distribution 

 were (as an observed fact) mathematically symmetrical, than 

 if any given deviation from symmetry were observed in nature. 

 I think that it cannot be necessary to insist upon the error of 

 a conclusion so repugnant to common sensef . 



* See note B. 



f When 1 stated this result as that at which I had arrived in disproof of 

 Mitchell's theory, at the Meeting of the British Association in Edinburgh, 

 an able mathematician present entered into a calculation to show that the 

 uniformity of distribution of bodies falling at random would not be the most 

 probable result. Now the truth or error of his conclusion depends upon 

 the use of the words ** most probable." If by " most probable " we mean 

 that a uniform distribution, or (as in the case we have taken to refer to) the 

 turning up of all possible faces of the dice, is meant, the probability in favour 

 of it compared to all other possible cases taken together, which might occur, 



