XVI. On the Method of Least Squares. By R. L. Ellis, Esq., M.A., Felloiv of 



the Cambridge Philosophical Society. 



[Read March 4, 1844.] 



The importance attached to the method of least squares is evident from the attention it has 

 received from some of the most distinguished mathematicians of the present century, and from the 

 variety of ways in which it has been discussed. 



Sometliing, however, remains to be done — namely, to bring the different modes in which the 

 subject has been presented into juxta-position, so that the relations which they bear to one another 

 may be clearly apprehended. For there is an essential difference between the way in which the 

 rule of least squares has been demonstrated by Gauss, and that which was pursued by Laplace. 

 The former of these mathematicians has in fact given two different demonstrations of the method, 

 founded on quite distinct principles. The first of these demonstrations is contained in the Tlieoria 

 Motfis, and is that which is followed by Encke in a paper of which a translation appeared in the 

 Scientific Memoirs. At a later period Gauss returned to the subject, and subsequently to the 

 publication of Laplace's investigation gave his second demonstration in the Thenria Comhinationis 

 Observationum. 



The subject has been also discussed by Poisson in the Connaissance des Terns for 1827, and by 

 several other French writers. Poisson's analysis is founded on the same principle as Laplace's : it is 

 more general, and perhaps simpler. It is not, however, my intention to dwell upon mere differences 

 in the mathematical part of the enquiry. 



The consequence of the variety of principles which have been made use of by different writers 

 has naturally been to produce some perplexity as to the true foundation of the method. As the 

 results of all the investigations coincided, it was natural to suppose that the principles on which 

 they were founded were 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 reduced his hypothesis to that on whicli 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 Laplace's, and which proves in the most general manner the superiority of the rule of least 

 squares, whatever be the law of probability 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 tiiethod 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 pro- 

 babilities. In this conclusion I cannot coincide; nor do I think Mr. Ivory's reasoning at all 

 satisfactory. 



