7 i2 THE POPULAR SCIENCE MONTHLY. 



ises. They are different facts, as when one sees that the tide rises 

 m times and concludes that it will rise the next time. These are 

 the only inferences which increase our real knowledge, however use- 

 ful the others may be. 



In any pi-oblem in probabilities, we have given the relative fre- 

 quency of certain events, and we perceive that in these facts the rela- 

 tive frequency of another event is given in a hidden way. This being 

 stated makes the solution. This is therefore mere explicative reason- 

 ing, and is evidently entirely inadequate to the representation of syn- 

 thetic reasoning, which goes out beyond the facts given in the prem- 

 ises. There is, therefore, a manifest impossibility in so tracing out 

 any probability for a synthetic conclusion. 



Most treatises on probability contain a very different doctrine. 

 They state, for example, that if one of the ancient denizens of the 

 shores of the Mediterranean, who had never heard of tides, had gone 

 to the bay of Biscay, and had there seen the tide rise, say m times, 

 he could know that there was a probability equal to 



m + l 



m + 2 



that it would rise the next time. In a well-known work by Quetelet, 

 much stress is laid on this, and it is made the foundation of a theory 

 of inductive reasoning. 



But this solution betrays its origin if we apply it to the case in 

 which the man has never seen the tide rise at all ; that is, if we put 

 m = 0. In this case, the probability that it will rise the next time 

 comes out , or, in other words, the solution involves the concep- 

 tualistic principle that there is an even chance of a totally unknown 

 event. The manner in which it has been reached has been by con- 

 sidering a number of urns all containing the same number of balls, 

 part white and part black. One urn contains all white balls, another 

 one black and the rest white, a third two black and the rest white, 

 and so on, one urn for each proportion, until an urn is reached con- 

 taining only black balls. But the only possible reason for drawing 

 any analogy between such an arrangement and that of Nature is the 

 principle that alternatives of which we know nothing must be con- 

 sidered as equally probable. But this principle, is absurd. There is 

 an indefinite variety of ways of enumerating the different possibilities, 

 which, on the application of this principle, would give different results. 

 If there be any way of enumerating the possibilities so as to make them 

 all equal, it is not that from which this solution is derived, but is the 

 following : Suppose we had an immense granary filled with black and 

 white balls well mixed up ; and suppose each urn were filled by taking 

 a fixed number of balls from this granary quite at random. The rela- 

 tive number of white balls in the granary might be anything, say 

 one in three. Then in one-third of the urns the first ball would 



