Genesis of the Law of Error. 349 



penetrating logical instinct of Gauss or of Laplace. Though 

 Gauss wrote little to clear it up, it is quite evident through 

 his terse expressions that he saw it exactly in its right 

 position, and considered the postulated existence of a 

 frequency function not as an axiom but as an hypothesis, 

 while Laplace offered a proof that the known law of 

 frequency would emerge from the mere superposition of 

 indefinite numbers of small errors, which had arbitrary laws 

 of frequency of their own. Poisson gave the same proof in a 

 revised form. Such a theorem would relieve us of all difficulty, 

 for though we may seem to have got something out of 

 nothing, if the demonstration holds this paradox can only be 

 apparent. It is a theorem of convergence, and must be 

 judged so. It is either true or false. Such phrases as 

 " a tres-peu pres," u suivra sensiblement la loi de Gauss/' or the 

 charitable English equivalent " practically," with its power 

 to cover a multitude of logical sins, are not in the first place 

 admissible. It they are required to help the demonstration 

 out, that means, the theorem is false; for Poisson in particular 

 seems to have held that no conditions were necessary to 

 impose upon the frequencies of the elementary contri- 

 buting errors, — " la fonction fx aura telle forme que Von 

 voudraP 



I imagine that no one believes that the theorem is true in 

 the form that Poisson gave it. Certainly not Prof. Edgeworth, 

 who refers to instances given by himself in which it is falsified, 

 and states conditions under which it may be true. Such con- 

 ditions are an admission that the law does not exist unless the 

 errors possess a defined character. They constitute implicitly 

 a definition of errors as restricted to such a form as may be 

 necessary to produce the law. That is to say the law is a 

 consequence of limits tacitly imposed b}^ accepted notions as 

 to the nature of errors. 



Where then does the Law of Error come from, and why 

 does it apply, on the whole, so unerringly to the most diverse 

 and unselected material ? That it does not apply always and 

 of necessity, may be taken as admitted. That it does apply 

 very closely and very commonly is a matter of experience. 

 Without questioning that Laplace's theorem, subject to restric- 

 tions the precise character of which it is at the moment im- 

 material to specify, gives with great generality an account of 

 the origin of the law which is sufficient in the sense that on the 

 whole it is analytically convincing, can we add anything from 

 another point of view that will make its genesis and its pro- 



