68 INDUCTION. 



wards ascertained, deductively, to be the rarefaction of 

 the air, occasioned by the increase of temperature as 

 the day advances. 



$ 5. After these general remarks on the nature of 

 chance, we are prepared to consider in what manner 

 assurance may be obtained that a conjunction between 

 two phenomena, which has been observed a certain 

 number of times, is not casual, but a result of causa- 

 tion, and to be received therefore as one of the uni- 

 formities in nature, although (until accounted for a 

 priori) only as an empirical law. 



We will suppose the strongest case, namely, that 

 the phenomenon B has never been observed except in 

 conjunction with A. Even then, the probability that 

 they are connected is not measured by the total 

 number of instances in which they have been found 

 together, but by the excess of that number above the 

 number due to the absolute frequency of A. If, for 

 example, A exists always, and therefore coexists with 

 everything, no number of instances of its coexistence 

 with B would prove a connexion ; as in our example 

 of the fixed stars. If A be a fact of such common 

 occurrence that it may be presumed to be present in 

 half of all the cases that occur, and therefore in half the 

 cases in which B occurs, it is only the proportional 

 excess above half, that are to be reckoned as evidence 

 towards proving a connexion between A and B. 



In addition to the question, What is the number 

 of coincidences which, on an average of a great mul- 

 titude of trials, may be expected to arise from chance 

 alone ? there is also another question, namely, Of 

 what extent of deviation from that average is the 

 occurrence credible, from chance alone, in some 

 number of instances smaller than that which consti- 



