ON THE METHOD OF LEAST SQUARES*. 



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. 



Something, 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 Theoria Motus, 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 investi- 

 gation gave his second demonstration in the Theoria Combina- 

 tionis 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 pro- 

 duce 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 



* Transactions of the Camlridge Philosophical Society Vol. vra. [Read 

 March 4, 1844] 



