THE LAW Off ERROR. 437 



the necessary calculations would much reduce the utility 

 of the theory. 



By a process of reasoning, which it would be undesirable 

 to attempt to follow in detail in this place, it is shown 

 that, under these conditions, the most probable result of 

 any series of recorded observations is that which makes \ 

 the sum of the squares of the errors the least possible. * 

 Let a, 1), c, &c., be the results of observation, and x the 

 quantity selected as the most probable, that is the most 

 free from unknown errors : then we must determine x so 

 that (a - xf + (I- x) 2 4- (c - x) 2 + . . . . shall be the least 

 possible quantity. Thus we arrive at the celebrated 

 Method of Least Squares, as it is usually called, which 

 appears to have been first distinctly put in practice by 

 Gauss in 1795, while Legendre first published in 1806 an 

 account of the process in his work, entitled, ' Nouvelles 

 Methodes pour la determination des Orbites des Cometes.' 

 It is worthy of notice, however, that Roger Cotes had 

 long previously recommended a method of equivalent 

 nature in his tract, ' Estimatio Erroris in Mixta Mathesi d .' 



Herschel's Geometrical Proof. 



A second method of demonstrating the Principle of 

 Least Squares was proposed by Sir John Herschel, and 

 although only applicable to geometrical notions, it is re- 

 markable as showing that from whatever point of view 

 we regard the subject, the same principle will be detected. 

 After assuming that some general law must exist, and 

 that it is subject to the general principles of proba- 

 bility, he supposes that a ball is dropped from a high 

 point with the intention that it shall strike a given mark 

 on a horizontal plane. In the absence of any known 

 causes of deviation it will either strike that mark, or, as 



d De Morgan, 'Penny Cyclopaedia,' art. Least Squares. 



