156 Prof. F. Y. Edgeworth on the 



difficulty of apprehending the relations of chance and law 

 (4) and (5) lightened by the introduction of the " holo- 

 morphic function" (Phil. Mag. p. 350), or other novel point ? 

 B. So far, perhaps, I may say with Professor Sampson, 

 ''the points of difference" between us appear "unsubstantial" 

 (Phil. Mag. p. 347). But with respect to his main criticism 

 of Laplace's proof, I cannot regard the difference as un- 

 substantial. I cannot retract, but would rather emphasize 

 what I have written on the point (Phil. Mag. vol. xxxv. 

 p. 425 (1918)). Professor Sampson argues: "The con- 

 clusion is, therefore, unwarranted, and there is no proof at 

 all that peculiarities of the functions efface themselves in the 

 final result. I do not. believe that the theorem is true " 

 (Congress, p. 167). I can only agree with these statements 

 in the case where the fourth condition is not complied with ; 

 that is, in a case which never occurs in fact. I cannot retract 

 the statement that the attack on the proof given by Poisson 

 and Laplace strikes at all the applications of the law (Phil. 

 Mag. May 1918, p. 423) ; for the objection to the reasoning 

 of Laplace and Poisson can only be admitted on a supposition 

 which would be fatal to every other proof of the law of 

 error *. The negative of that supposition, the axiomatic 

 fourth condition (8), forms the corner-stone of the edifice, 

 whether built according to the design of Laplace and Poisson 

 or some more modern plan. That fundamental support being 

 rejected, the whole edifice would collapse. All the applica- 

 tions of the theory — the Method of Least Squares, both as a 

 good method where the frequency-function for the errors of 

 observation is unknown, and as the best method f uhen the 

 frequency-function for the errors-of-observation is believed 

 by inference from the law-of-error to be normal J ; the 

 representation of statistics by a normal curve (or surface) in 

 the manner of Quetelet, or more exactly by the employment 

 of the second approximation given by Poisson §; the test of 

 Sampling as practised in social investigations by a Bowley 

 K'iar || : the splendid and useful results deduced from normal 



* The law being defined as above. Note % to p. 148. 



t See as to this distinction, Journal of the Statistical Society, 

 vol. 71, pp. 509,393 (1908), and references there given. Note too (loc. cit. 

 p. 499) that the law-of-error is required to determine the precision of a 

 result obtained by inverse probability from a set of observations for which 

 the error-function is known, but not normal. 



X Above, p. 149. > 



§ As employed with success by Bowlev, ' Elements of Statistics.' ed. 2 ; 

 and Journal of the Royal Statistical Societv passim. 



|l Referred to in the Journal of the Royal Statistical Society, 

 vol. 76, p. 192 (1912-13). 



