Genesis of the haw of Error. 151 



of that law, and yet a member of that group may conceivably 

 deviate to infinity. Moreover (9) "infinite" in the definition 

 must be understood literally, not simply as an equivalent to 

 "immense."''' Thus, considering any frequency-function with 

 infinite mean powers of deviation, e.g. l/7r(lH-^' 2 ), if we have 

 to do only with values of x which, however large, are finite, we 

 can always take n (the number of the constituents) so large 

 that the aggregate, or average, will fulfil the law-of-error. 

 (10) As to the functions which extend to infinity without 

 violating the condition (8), it may be doubted whether outside 

 the law of error such frequency-groups have any concrete 

 existence. Even for the law-of-error considered as resulting 

 from a finite number of finite deviations the infinite deviation 

 is a limit never attained, probably not even by the velocities 

 in a molecular medley *. 



B. A right understanding of the theory is promoted, or, at 

 least, evidenced, by a correct appreciation of the writers to 

 whom it is due. The foundation of the theory is to be sought 

 in Laplace's classical treatise f. (11) The first section which 

 he devotes to the subject exhibits the essential features of 

 the law-of-error : namely, that, when the three conditions 

 above specified are present, " the peculiarities of the (con- 

 stituent) functions efface themselves in the final result"" }. 

 True, he does not advert to the fourth condition. But it was 

 not his wont to dilate upon conditions which might ordinarily 

 be taken for granted. Thus he repeatedly assumes that a 

 function may be expanded in powers of the variable, and 

 powers above the first neglected. In the next section, for 

 instance, he thus obtains a linear equation for the correction 

 of an observation. Of course he knew that there might 

 occur singular points. A caveat was the less necessary in 

 the present case, because in no application of the law which 

 he contemplated could an infinite deviation occur. He could 

 not suppose — norforsee that anyone would suppose — a really 

 infinite error-of-observation. 



It is, perhaps, remarkable that Laplace did not extend the 

 law which he demonstrated to account for the prevalence of 

 the law throughout Nature, as shown by Quetelet. (12) It 

 is still more remarkable that Laplace should not have applied 

 the law to demonstrate that the normal error-function would 

 " emerge from the mere superposition of the definite numbers 



* That the sum-total energy is finite is not conclusive proof.. 



f Theorie Analytique des Probabilites. Liv. II. cap. iv. 



t The words are those used by Professor Sampson in denying this 

 conclusion (Congress of Mathematicians 1912, Proceedings, p. L68, 

 par. 1). 



