350 



INDUCTION. 



of any one kind. If, therefore, after 

 such a number of trials that no fur- 

 ther increase of their number has 

 any material effect upon the average, 

 we find a preponderance in favour of 

 a particular throw, we may conclude 

 with assurance that there is some 

 constant cause acting in favour of 

 that throw, or, in other words, that 

 the dice are not fair ; and the exact 

 amount of the unfairness. In a 

 similar manner, what is called the 

 diurnal variation of the barometer, 

 which is very small compared with 

 the variations arising from the irre- 

 gular changes in the state of the 

 atmosphere, was discovered by com- 

 paring the average height of the 

 barometer at different hours of the 

 day. When this comparison was 

 made, it was found that there was 

 a small difference, which on the 

 average was constant, however the 

 absolute quantities might vary, and 

 which difference, therefore, must be 

 the effect of a constant cause. This 

 cause was afterwards 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 pre- 

 pared 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 causation, and to be received 

 therefore as one of the uniformities 

 of nature, though (until accounted 

 for d priori) only as an empirical 

 2aw. 



We will suppose the strongest case, 

 namely, that the phenomenon B has 

 never been observed except in con- 

 junction 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 co-exists with everything, 

 no number of instances of its co- 

 existence with B would prove a con- 

 nection ; 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 is to be reckoned as 

 evidence towards proving a connec- 

 tion between A and B, 



In addition to the question, What 

 is the number of coincidences which, 

 on an average of a great multitude 

 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 required for strik- 

 ing a fair average ? It is not only to 

 be considered what is the general 

 result of the chances in the long-run, 

 but also what are the extreme limits 

 of variation from the general result 

 which may occasionally be expected 

 as the result of some smaller number 

 of instances. 



The consideration of the latter 

 question, and any consideration of 

 the former beyond that already given 

 to it, belong to what mathematicians 

 term the Doctrine of Chances, or, in 

 a phrase of greater pretension, the 

 Theory of Probabilities. 



CHAPTER XVIII. 



OF THE CALCULATION OP CHANCES. 



§ I. "Probability," says Laplace,* 

 " has reference partly to our ignorance, 

 partly to our knowledge. We know 

 that among three or more events, one, 

 and only one, must happen ; but there 

 is nothing leading us to believe that 

 any one of them will happen rather 

 than the others. In this state of 



* Essai Philosophique sur let ProbabiliUs, 

 fifth Paris edition, p. 7. 



