xvii.j THE LAW OF EREOR. 377 



tions is that which makes the sum of the squares of 

 the consequent errors the least possible. Let a, h, 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 — ic)2 + (b — x)^ + (c — x)"^ + . . . . shall be the least 

 })Ossible 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 tirst 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 Eoger Cotes had long 

 jtreviously reconmiended a method of equivalent nature in 

 his tract, " Estimatio Erroris in Mixta Mathesi." ^ 



Hcrschel'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 

 jninciple 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 

 (hopped ii'om 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 bv an amount which we must regard as error of 

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

 " the probability of that error is the unknown function of 

 its square, i.e. of tlie 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 functi(m of 

 its sfjuare. And since the observed oblique deviation is 



' De M(n';.,';in, Penny Cyclopwdia, art. Lecust ISquares. 



2 JidiiiOiir>jk Jtcvicw, July 1850, vol. x(^ii. \). 17. Reprinted Esxays, 

 V- 399- f '''^ iiictliod of dcinoiistratioii is discussed hy Boole, TraiU' 

 actions of lioyul iSocidy of Jjdiuburyk, vol. xxi. pp. 627 — 6^0. 



