378 INDUCTION. 



loaded, and the thi'ow were left to depend entirely on the changeable causes, 

 these in a sufficient number of instances would balance one another, and 

 there would be no preponderant number of throws of any one kind. If, 

 therefore, after such a number of trials that no further increase of their 

 number has any material effect upon the average, we find a preponderance 

 in favor of a particular throw ; we may conclude with assurance that there 

 is some constant cause acting in favor of that throw, oi*, 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 irregular changes in 

 the state of the atmosphere, was discovered by comparing the average 

 height of the barometer at different hours of the day. When this compari- 

 son 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 afterward 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 con- 

 junction between two phenomena, which has been observed a certain num- 

 ber 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 

 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 proba- 

 bility that they are connected is not measured by the total number of in- 

 stances in which they have been found together, but by the excess of that 

 number above the number due to the absolutely frequency of A. If, for 

 example, A exists always, and therefore co-exists with every thing, no num- 

 ber of instances of its co-existence with B would prove a connection ; 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 toward proving a con- 

 nection 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 de- 

 viation from that average is the occurrence credible, from chance alone, in 

 some number of instances smaller than that required for striking a fair av- 

 erage? 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 The- 

 ory of Probabilities. 



