108 THE PRINCIPLES OF SCIENCE. 



their periods of variation were the same even to infinitely 

 small quantities it would be certain, that is infinitely pro- 

 bable, that they were connected, however unlikely this 

 might be on other grounds. The general mode of esti- 

 mating such probabilities is identical with that applied 

 to other inductive problems. Thus, if the two periods of 

 variation were assigned by pure chance and entirely inde- 

 pendently of each other, the probability would be about 

 one in ten million that they would agree to the one ten- 

 millionth part ; but if the periods be observed to agree to 

 less than that part then there is a probability of at least 

 ten million to one in favour of the opposite hypothesis of 

 connexion. That any two periods of variation should by 

 chance become absolutely equal is infinitely improbable ; 

 hence if, in the case of the moon or any other change, we 

 could prove absolute coincidence, we should have certainty 

 of connexion b . With approximate measurements, which 

 alone are within our power, we must hope for approximate 

 certainty at the most. 



The general principles of inference and probability, ac- 

 cording to which we treat causes and effects varying in 

 amount, are exactly ~the same as those by which we 

 treated simple experiments. Continuous quantity, how- 

 ever, affords us an infinitely more extensive sphere of 

 observation, because every different amount of cause, 

 however little different, ought to be followed by a dif- 

 ferent amount of effect. If we can measure temperature 

 to the one hundredth part of a degree centigrade, then 

 even between o and 100 we have 10,000 possible dis- 

 tinct trials. If the precision of our measurements is 

 increased, so that the one thousandth part of a degree 

 can be appreciated, our trials may be increased tenfold. 

 The probability of connexion will be proportional to the 

 accuracy of our measurements. 



b Laplace, ' System of the World/ transl. by Harte, voL ii. p. 366. 



