xvn.J THE LAW OF ERROR. 377 



tions is that which makes the sum of the squares of 

 the consequent errors the least possible. Let a, b, 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 - x) 2 + (b - x) z + (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 

 Mtthodes pour la Determination des Orbites des Cometes. It 

 is worthy of notice, however, that Eoger Cotes had long 

 previously recommended a method of equivalent nature in 

 his tract, " Estimatio Erroris in Mixta MathesL" l 



Herschel's Geometrical Proof. 



A second way of arriving at the Law of Error was 

 proposed by Herschel, and although only applicable to 

 geometrical cases, it is remarkable 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 

 principles of probability, 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 is infinitely more probable, diverge 

 from it by an amount which we must regard as error of 

 unknown origin. Now, to quote the words of Herschel, 2 

 " the probability of that error is the unknown function of 

 its square, i.e. of the sum of the squares of its deviations in 

 any two rectangular directions. Now, the probability of 

 any deviation depending solely on its magnitude, and not 

 on its direction, it follows that the probability of each of 

 these rectangular deviations must be the same function of 

 its square. And since the observed oblique deviation is 



1 De Morgan, Penny Cyclopedia, art. Least Squares. 



2 Edinburgh Review, July 1850, vol. xcii. p. 17. Reprinted Essays, 

 p. 399- This method of demonstration is discussed by Boole, Trans- 

 actions of Royal Society of Edinburgh, vol. rxi. pp. 627 630. 



