the " Method of Least Squares" 323 



have taken place if the wafer really occupied this position 

 is more probable than that of those which are similarly implied 

 in any other hypothesis as to its place. If you find anybody 

 sceptical as to this, pray ask them to point out the passage, 

 either in the introductory essay, or in the work itself, or in the 

 supplements. 



What, then, did Laplace demonstrate? something so un- 

 like this, that one is disposed to wonder how he can have been 

 thus misunderstood. The method of least squares is simply a 

 method for the combination of linear equations, of which the 

 unknown quantities are the elements to be determined ; the 

 constant term of each being a direct result of observation, and 

 therefore affected by an unknown error, while the coefficients 

 are supposed absolutely known. 



If there are more equations than requisite, that is, more than 

 elements to be determined, what is the best way of combining 

 them? In the first place, they must clearly be combined by 

 some system of constant multipliers, else the resulting equa- 

 tions, not being linear, would generally be insoluble. This 

 condition, however, though absolutely necessary in practice, 

 is in no way derived from the theory of probabilities. It is a 

 merely practical limitation. The question thus narrowed is 

 simply to determine the system of factors to be employed for 

 obtaining the value of any particular element. The factors 

 must of course be such that, in the final equation, the coeffi- 

 cient of this element may be unity, and those of the others 

 severally equal to zero. 



These conditions being fulfilled, we get a value for the ele- 

 ment in question which is affected by an unknown error,namely 

 the sum of the errors of observation multiplied respectively 

 by the corresponding factors. The mean arithmetical value 

 of this sum may in theory at least be determined, if we know 

 the law of probability of error for each observation ; and 

 Laplace calls that system of factors the most advantageous 

 which makes this mean value a minimum. If, however, the 

 law of probability of error is unknown, the mean value of the 

 error cannot be determined. Nevertheless, if the number of 

 observations is very large, this mean value approximates to a 

 certain limit, the form of which is independent of the law of 

 probability. The essence of Laplace's demonstration consists 

 in its enabling us to determine this limit. When this is done, 

 it may easily be shown that the most advantageous system of 

 factors, those, namely, which make this limiting mean value 

 of the error a minimum, will give the same value to the ele- 

 ment to be determined as the system of final equations ob- 

 tained by employing the method of least squares, provided 



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