REMARKS ON AN ALLEGED PROOF OF THE METHOD 

 OF LEAST SQUARES, CONTAINED IN A LATE 

 NUMBER OF THE EDINBURGH REVIEW. IN 

 A LETTER ADDRESSED TO PROFESSOR J. D. 

 FORBES* 



MY DEAR SIR, 



THE review of Quetelet's Letires h S. A, R. le Due regnant de 

 JSaxe Cobourg et Gotha, which appeared in the July Number 

 of the Edinburgh Review, contains a new demonstration of the 

 method of least squares which ought not, I think, to pass 

 unnoticed. If it is correct, it is so much simpler than those 

 which have hitherto been received, that it ought to supersede 

 them ; and if not, the sooner its incorrectness is pointed out the 

 better. 



Some years since, in a paper published in the Cambridge 

 Transactions for 1844, I made an analysis of all the demon- 

 strations, or professed demonstrations of the method of least 

 squares, with which I was then acquainted, and I therefore 

 read this new one with more attention than you perhaps have 

 given to it. 



The reviewer gives some account of the history of the sub- 

 ject, and remarks that the demonstration of the least squares 

 was first attempted by Gauss, but that his proof is no proof 

 at all, because it assumes that in the case of a single element 

 the arithmetical mean of the observed values is in all cases the 

 most probable value, " a thing to be demonstrated, not as- 

 sumed." Gauss afterwards gave another demonstration, which 

 is perfectly rigorous ; but of this the reviewer takes no notice, 

 though it i.3 mentioned in at least one of the works on the 

 theory of probabilities which he has recommended to the 

 attention of students. However, in the proof which the re- 

 viewer refers to, which is contained in the tract entitled Theoria 



* Philosophical Magazine, November, 1850. 



