CHANCE, AND ITS ELIMINATION. 377 



This kind of elimination, in which we do not eliminate any one assignable 

 cause, but the multitude of floating unassignable ones, may be termed the 

 Elimination of Chance. We afford an example of it when we repeat an ex- 

 periment, in order, by taking the mean of different results, to get rid of the 

 effects of the unavoidable errors of each individual experiment. When there 

 is no permanent cause, such as would produce a tendency to error peculiar- 

 ly in one direction, we are warranted by experience in assuming that the 

 errors on one side will, in a certain number of experiments, about balance 

 the errors on the contrary side. We therefore repeat the experiment, un- 

 til any change which is produced in the average of the whole by further 

 repetition, falls within limits of error consistent with the degree of accuracy 

 required by the purpose we have in view.* 



§ 4. In the supposition hitherto made, the effect of the constant cause A 

 has been assumed to form so great and conspicuous a part of the general 

 result, that its existence never could be a matter of uncertainty, and the ob- 

 ject of the eliminating process was only to ascertain how much is attribu- 

 table to that cause ; what is its exact law. Cases, however, occur in which 

 the effect of a constant cause is so small, compared with that of some of the 

 changeable causes with which it is liable to be casually conjoined, that of it- 

 self it escapes notice, and the very existence of any effect arising from a 

 constant cause is first learned by the process which in general serves only for 

 ascertaining the quantity of that effect. This case of induction may be char- 

 acterized as follows : A given effect is known to be chiefly, and not known 

 not to be wholly, determined by changeable causes. If it be wholly so pro- 

 duced, then if the aggregate be taken of a sufiicient number of instances, the 

 effects of these different causes will cancel one another. If, therefore, we 

 do not find this to be the case, but, on the contrary, after such a number of 

 trials has been made that no further increase alters the average result, 

 we find that average to be, not zero, but some other quantity, about which, 

 though small in comparison with the total effect, the effect nevertheless 

 oscillates, and which is the middle point in its oscillation ; we may conclude 

 this to be the effect of some constant cause ; which cause, by some of the 

 methods already treated of, we may hope to detect. This may be called 

 the discovery of a residual phenomenon by eliminating the effects of 

 chance. 



It is in this manner, for example, that loaded dice may be discovered. 

 Of course no dice are so clumsily loaded that they must always throw cer- 

 tain numbers ; otherwise the fraud would be instantly detected. The load- 

 ing, a constant cause, mingles with the changeable causes which determine 

 what cast will be thrown in each individual instance. If the dice were not 



* In the preceding discussion, the mean is spoken of as if it were exactly the same thing 

 with the average. But the mean, for purposes of inductive inquiry, is not the average, or ar- 

 ithmetical mean, though in a familiar illustration of the theory the difference may be disre- 

 garded. If the deviations on one side of the average are much more numerous than those on 

 the other (these last being fewer but greater), the effect due to the invariable cause, as dis- 

 tinct from the variable ones, will not coincide with the average, but will be either below or 

 above the average, the deviation being toward the side on which the greatest number of the 

 instances are found. This follows from a truth, ascertained both inductively and deductively, 

 that small deviations from the true central point are greatly more frequent than large ones. 

 The mathematical law is, "that the most probable determination of one or more invariable 

 elements from observation is that in which the sum of the squares of the individual aberra- 

 tions," or deviations, '■^ shall he the least possible." See this principle stated, and its grounds 

 popularly explained, by Sir John Herschel, in his review of Quetelet on Probabilities, Essays, 

 p. 395 et seq. 



