CHANCE, AND ITS ELIMINATION. 



349 



aflPord an example of it when we re- 

 peat an experiment, in order, by tak- 

 ing the mean of different results, to 

 get rid of the effects of the unavoid- 

 able errors of each individual experi- 

 ment. Whtn there is no permanent 

 cause such a?, would produce a ten- 

 dency to error peculiarly in one direc- 

 tion, we are warranted by experience 

 in assuming that the errors on one 

 side will, in a certain number of ex- 

 periments, about balance the errors 

 on the contrary side. "We therefore 

 repeat the experiment, until any 

 change which is produced in the aver- 

 age of the whole by further repeti- 

 tion falls within limits of error con- 

 sistent 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 

 object of the eliminating process was 

 only to ascertain how much is attri- 

 butable 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 itself it escapes notice, and the 

 very existence of any effect arising 

 from a constant cause is first learnt 

 by the process which in general serves 

 only for ascertaining the quantity of 

 that effect. This case of Induction 



"* 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 arithmetical mean, though in a 

 familiar illustration of the theory the dif- 

 ference may be disregarded. If the devia- 

 tions 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 coin- 

 cide with the average, but will be either 

 below or above the average, the deviation 

 being toward* the side on which the 



may be characterised as follows. A 

 given effect is known to be chiefly, 

 and not known not to be wholly, deter- 

 mined by changeable causes. If it be 

 wholly so produced, then if the aggre- 

 gate be taken of a sufficient number 

 of instances, the effects of these dif- 

 ferent causes will cancel one another. 

 If, therefore, we do not find this tt» 

 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 aver- 

 age 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 oscil- 

 lation ; 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 dis- 

 covery 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 clumsilj 

 loaded that they must always throw 

 certain numbers ; otherwise the fraud 

 would be instantly detected. The 

 loading, a constant cause, mingles 

 with the changeable causes which 

 determine what cast will be thrown 

 in each individual instance. If the 

 dice were not loaded, and the throw 

 were left to depend entirely on the 

 changeable causes, these in a suffi- 

 cient number of instances would 

 balance one another, and there would 

 be no preponderant number of throws 



greatest number of 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 in- 

 variable elements from observation is that 

 in which the sum of the squares of the indi- 

 vidual aberrations," or deviations, "shall 

 be the least possible." See this principle 

 stated, and its grounds popularly explained, 

 by Sir John Herschel, in liis review of 

 Qu^telet on Probabilities, Essays, pp. 395 

 et seq. 



