CH. XVII.] THE LAW OF ERROR. 375 



tions should follow the same law," ^ and the special Laws 

 of Error which will apply to certain iustruineiits, as for in- 

 stance the repeating circle, have been investigated by 

 Bravais.'^ He concludes that every distinct cause of error 

 gives rise to a curve of possibility of errors, which may 

 have any form, — a curve which we may either be able or 

 unable to discover, and which in the first case may be 

 determined by a 'priori considerations on the peculiar 

 nature of this cause, or wliich may be determined d 

 iwsteriori hy observation. Whenever it is practicable and 

 worth the labour, we ought to investigate these special 

 conditions of error ; nevertheless, when there are a great 

 number of different sources of minute error, the general 

 resultant will always tend to obey that general law which 

 we are about to consider. 



Establishment of the Law of Error. 



Mathematicians agree far better as to the form of the 

 Law of Error than they do as to the manner in which it 

 can be deduced and proved. They agree that among a 

 number of discrepant results of observation, that mean 

 quantity is probably the best approximation to the truth 

 which makes the sum of the squares of the errors as small 

 as possible. But there are three principal ways in which 

 this law has been arrived at respectively by Gauss, by 

 Laplace and Quetelet, and by Sir John Herschel. Gauss 

 proceeds nmch upon assumption ; Herschel rests upon 

 geometrical considerations ; while Laplace and Quetelet 

 regard the Law of Error as a development of the doctrine 

 of combinations. A number of other mathematicians, such 

 as Adrain of New Brunswick, Bessel, Ivory, Donkin, Leslie 

 Ellis, Tait, and Crofron have either attempted indejjendent 

 jn'oofs or have modified or commented on those here to be 

 described. For full accounts of the literature of the 

 subject the reader should refer either to Mr. Todhunter's 

 Hidory of the Theory of Probability or to the able memoir 

 of Mr. J. W. L. Glaisher.^ 



' Philosophical Magazine, 3rd Series, vol. xxxvii. p. 324. 



2 Letters on the. Theory of I'rohahilities, by Qiietek't, translated by 

 0. G. DowucK, NoU'.s to Letter XXVL jip. 286—295. 



3 O71 tlie Law of Facility of Krrors of Observations, and on the 

 Method of Least Squares, Meiuoirs of the Royal Astronomical Society, 

 vol. xxxix. p. 75. 



