386 THE PKINCIPLES OF SCIENCE. [CHAP. 



as if it were exactly the same thing as the average. 

 But the mean, for purposes of inductive inquiry, is not the 

 average, or arithmetical mean, though in a familiar illus- 

 tration of the theory the difference may be disregarded." 

 He goes on to say that, according to mathematical princi- 

 ples, the most probable result is that for which the sums 

 of the squares of the deviations is the least possible. It 

 seems probable that Mill and other writers were misled 

 by Whewell, who says 1 that " The method of least 

 squares is in fact a method of means, but with some 

 peculiar characters. . . . The method proceeds upon 

 this supposition : that all errors are not equally probable, 

 but that small errors are more probable than large ones." 

 He adds that this method " removes much that is arbitrary 

 in the method of means." It is strange to find a mathe- 

 matician like Whewell making such remarks, when there 

 is no doubt whatever that the Method of Means is only 

 an application of the Method of Least Squares. They are, 

 in fact, the same method, except that the latter method 

 may be applied to cases where two or more quantities have 

 to be determined at the same time. Lubbock and Drink- 

 water say, 2 " If only one quantity has to be determined, 

 this method evidently resolves itself into taking the mean 

 of all the values given by observation." Encke says, 3 that 

 the expression for the probability of an error " not only 

 contains in itself the principle of the arithmetical mean, 

 but depends so immediately upon it, that for all those 

 magnitudes for which the arithmetical mean holds good 

 in the simple cases in which it is principally applied, 

 no other law of probability can be assumed than that 

 which is expressed by this formula." 



Tlie Probable Error of Results. 



When we draw a conclusion from the numerical 

 results of observations we ought not to consider it suf- 

 ficient, in cases of importance, to content ourselves with 

 finding the simple mean and treating it as true. We 

 ought also to ascertain what is the degree of confidence 



1 Philosophy of the Inductive Sciences, 2nd ed. vol. ii. pp. 408, 409. 



2 Essay on Probability, Useful Knowledge Society, 1833, p. 41. 



3 Taylor's Scientific Memoirs, vol. ii. p. 333. 



