between Stars forming Binary or Multiple Groups. 421 



defined tendency to ultimate uniformity. The large dimen- 

 sions of the sieve, the symmetry of its pattern, and its position 

 vertically above the chess-board, are the conditions of this ap- 

 proximate uniformity of distribution, and they are evidently 

 far from casual circumstances. To generalize the experiment 

 with the sieve, we must imagine a great number of sieves 

 pierced with holes in every conceivable variety of pattern, one 

 of which has been selected at random for a given experi- 

 ment. By these considerations I believe that the ordinary 

 axioms of probability are not interfered with, but that the 

 problem of the distribution of stars " by chance as it may • 

 happen " receives as definite an illustration as so vague an 

 assumption admits of, and is shown to be incompatible with 

 the axiom that "each star is as likely [not hypothetically, but 

 actually] to be in one situation as another." This would only 

 be true on the almost infinitely improbable supposition of a set 

 of dice, not so formed expressly, being each and all perfectly 

 homogeneous and unbiassed. 



38. On the whole, the conclusions of this paper may be 

 thus summed up*. 



(1.) The fundamental principle of Mitchell is erroneous. 

 The probability expressed by it is an altogether different pro- 

 bability from what he asserts. His calculations are also ap- 

 parently inaccurate, in some instances at leastt. 



(2.) All the numerical deductions of his successors are 

 equally baseless. 



(3.) Were Mitchell's principle just, a perfectly uniform 

 and symmetrical disposition of the stars over the sky would 

 (if possible) be that which could alone afford no evidence of 

 causation, or any interference with the laws of " random ;" — 

 a result palpably absurd. 



(4?.) Special collocations, whether (a) distinguished by their 

 symmetry, or (J3) distinguished by an excessive crowding to- 

 gether of stars, or the reverse, inevitably force on the reason- 

 ing mind a more or less vague impression of causation; — an 

 impression necessarily vague, having nothing absolute, but 

 depending on the previous knowledge and habits of thought 

 of the individual, therefore incapable of being made the subject 

 of exact [i.e. mathematical] reasoning. 



(5.) The form of error into which those have been led who 

 had stated numerical probabilities against given arrangements 

 of stars being the result of accident, is two-fold. First, a coii- 



* These conclusions, excepting the last, are copied verbatim from the 

 paper read to the Physical Section of the British i\ssociation in August 1850. 



\ It would probably be more correct to say that all Mitchell's calculations 

 are wrongly deduced from his own premisses. 



