162 Mr. Ivory on the Method of the Least Squares. 



Theory, we must choose in the general expressions those parti- 

 cular cases only that are capable of being applied practically, 

 and that are not too complicated for real use. The calcula- 

 tions would become impracticable if the equations to be solved 

 passed the first degree. If, therefore, we only retain the parts 

 of the foregoing equations which are of one dimension with 

 respect to the errors, we shall get 



rf.S.e- - d.'ii .e- ^ ^ 



and this proves that the value of S.e^ must be a minimum, 

 'he same simp! 

 -h^e^; whence 



The same simplification leads to the equation, log.-^^^-^^ — 



The foregoing investigations are at least clear and simple. 

 It follows as an unavoidable consequence, that if we adopt the 

 rule of the least squares as the most advantageous determina- 

 tion of a system of errors, the law of probability can be no- 

 thing else but the function kc~'^'''. On the other hand, when 

 we apply the doctrine of probabilities to find the most advan- 

 tageous method of combiniuii a set of errors, we shall fall 

 uj)on the method of the least squares, if the chance of an 



error be expressed by the function /jc"'^ ; but if the law of 

 the eiTors be different, the same rule will no longer be true. 

 The two things are necessarily connected, in so much that 

 the one leads exclusively to the other. The method of the 

 least squai'es cannot possibly be true in any other ]aw of pro- 

 bability than the one we have mentioned. 



Now the conclusion which has just been stated, is directly 

 at variance with what Laplace has determined in the Thcorie 

 'Analytique des Prohabilites, liv. 2. cap. 4. §.20. In a system 

 consisting of a great number of observations it is proved, in 

 the work we have cited, that the rule of the least squares 

 ought to be employed whatever be the law of the chance of 

 an error (p. 321, 3d edition). It is the generality of the con- 

 clusion that chiefly constitutes the merit of this demonstra- 

 tion. It must be added, that M. Poisson, who has lately 

 treated the same subject in the Connaissance des Terns 1827j 

 has arrived at the same conclusions with Laplace, which are 

 thus confirmed. All other authors likewise acquiesce in the 

 result of Laplace's investigation, and admit that the rule of 

 the least squares may subsist with different laws of probability. 

 Every authority is thus directly opposed to the opinion I have 

 ventured to express. The words of the poet, 



" Nullius addictus jiirare in verba niagistri," 



contain 



