78 SCIENCE AND METHOD. 



phenomena obey the laws of chance when small 

 differences in the causes are sufficient to produce 

 great differences in the effects. The probabilities of 

 these small differences can then be regarded as 

 proportional to the differences themselves, just be- 

 cause these differences are small, and small increases 

 of a continuous function are proportional to those 

 of the variable. 



Let us pass to a totally different example, in which 

 the complexity of the causes is the principal factor. 

 I imagine a card-player shuffling a pack of cards. 

 At each shuffle he changes the order of the cards, 

 and he may change it in various ways. Let us take 

 three cards only in order to simplify the explanation. 

 The cards which, before the shuffle, occupied the 

 positions i 2 3 respectively may, after the shuffle, 

 occupy the positions 



123, 231, 312, 321, 132, 213. 



Each of these six hypotheses is possible, and their 

 probabilities are respectively 



/i. A. A. A. A> A- 

 The sum of these six numbers is equal to i, but that 

 is all we know about them. The six probabilities 

 naturally depend upon the player's habits, which we 

 do not know. 



At the second shuffle the process is repeated, and 

 under the same conditions. I mean, for instance, 

 that /4 always represents the probability that the 

 three cards which occupied the positions i 2 3 after 

 the n'" shuffle and before the n^\'\ will occupy the 

 positions 321 after the «+i'" shuffle. And this re- 

 mains true, whatever the number n may be, since the 



