ON THE METHOD OF LEAST SQUARES. 13 



essentially the same. Thus Mr Ivory conceived that if Laplace 

 arrived at the same result as Gauss, it was because in the process 

 of approximation he had introduced an assumption which re- 

 duced his hypothesis to that on which Gauss proceeded. In this 

 I think Mr Ivory was certainly mistaken ; it is at any rate not 

 difficult to show that he had misunderstood some part at least of 

 Laplace's reasoning: but that so good a mathematician could 

 have come to the conclusion to which he was led, shows at once 

 both the difficulty of the analytical part of the inquiry, and also 

 the obscurity of the principles on which it rests. Again, a recent 

 writer on the Theory of Probabilities has adopted Poisson's 

 investigation, which, as I have said, is the development of La- 

 place's, and which proves in the most general manner the supe- 

 riority of the rule of least squares, whatever be the law of pro- 

 bability of error, provided equal positive and negative errors are 

 equally probable. But in a subsequent chapter we find that he 

 coincides in Mr Ivory's conclusion, that the method of least 

 squares is not established by the theory of probabilities, unless 

 we assume one particular law of probability of error. 



These two results are irreconcilable ; either Poisson or 

 Mr Ivory must be wrong. The latter indeed expressed his 

 dissent from all that had been done by the French mathematicians 

 on the subject, and in a series of papers in the Philosophical 

 Magazine gave several demonstrations of the method of least 

 squares, which he conceived ought not to be derived from the 

 theory of probabilities. In this conclusion I cannot coincide ; 

 nor do I think Mr Ivory's reasoning at all satisfactory. 



From this imperfect sketch of the history of the subject, we 

 perceive that the methods which have been pursued may be 

 thus classified. 



1. Gauss's method in the Theoria Motfis, which is followed 

 and developed by Encke and other German writers. 



2. That of Laplace and Poisson. 



3. Gauss's second method. 



4. Those of Mr Ivory. 



I proceed to consider these separately, and in detail. 

 For the analysis of Laplace and Poisson, I have substituted 

 another, founded on what is generally known as Fourier's theo- 



