326 INDUCTION. 



see what degree of approximation can practically be made to the 

 necessary precision. 



The question falls within Laplace's sixth principle, of which a short 

 distance back, we gave the demonstration. The given fact, that is to 

 say, the series of coincidences, may have originated -either in a causal 

 conjunction of causes or in a law of nature. The probabilities, there- 

 fore, that the fact originated in these two modes, are as their ante- 

 cedent probabilities, multiplied by the probabilities that if they existed 

 they would produce the effect. But the particular combination of 

 chances if it occuiTed, or the law of nature if real, would certainly 

 produce the series of coincidences. The probabilities, therefore, 

 that the coincidences are produced by the two causes in question, 

 are as the antecedent probabilities of the causes. ■ One of these, the 

 antecedent probability of the combination of mere chances which 

 would produce the given result, is an appreciable quantity. The 

 antecedent probability of the other supposition may be susceptible 

 of a more or less exact estimation, according to the nature of the 

 case. 



;In some cases, the coincidence, supposing it to be the result of 

 causation at all, must be the result of a knowTi cause ; as the suc- 

 cession of aces, if not accidental, must arise from the loading of the 

 die. In such cases we may be able to form a conjecture as to the 

 antecedent probability of such a circumstance, from the characters 

 of the parties concerned, or other such evidence ; but it would clearly 

 be impossible to estimate that probability with anything like numerical 

 precision. The counter-probability, however, that of the accidental 

 ojigin of the coincidence, dwindling so rapidly as it does at each new 

 trial ; the stage is soon reached at which the chance of unfairness in 

 the die, however small in itself, must be greater than that of a causal 

 coincidence : and on this ground, a practical decision can generally be 

 come to without much hesitation, if there be the power of repeating the 

 experiment. ■ ' 



When, however, the coincidence is one which cannot be accounted 

 for by any knowTi cause, and the connexion between the two phenom- 

 ena, if produced by causation, must be the result of some law of nature 

 hitherto unknown ; which is the case we had in view in the last chap- 

 ter; then, although the probability of a casual coincidence may he 

 capable of appreciation, that of the counter-supposition, the existence 

 of an undiscovered law of nature, is clearly unsusceptible of even an 

 approximate evaluation. In order to have the data which such a case 

 would require, it would be necessary to know what proportion of aU 

 the individual sequences or coexistences occuiTing in nature are the 

 result of law, and what proportion are the result of chance. It being 

 evident that we cannot form any plausible conjecture as to this propor- 

 tion, much less appreciate it numerically, we cannot attempt any pre- 

 cise estimation of the comparative probabilities. But of this we are 

 sure, that the detection of an unknown law of nature — ^of some previ- 

 ously unrecognized constancy of conjunction among phenomena — is 

 no uncommon event. If, therefore, the number of instances in which 

 a coincidence is observed, over and above that which would arise on 

 the average from the mere concurrence of chances, be such that so 

 great an amount of coincidences from accident alone would be an 

 extremely uncommon event ; we have reason to concludo that the coiiv- 



