K. Pearson 
189 
All the leading statisticians, from Poisson to Qiietelet, Galton, Edgeworth, and 
Fechner, with botanists like de Vries, zoologists like Weldon have realised that 
asymmetry must be in some way described before we can advance in our theory of 
variation. In innumerable cases the important quantities measured by rj, % and d 
actually exist ; these have each their physical significance and they must be 
found. It is perfectly open to Ranke and Greiner to criticise my method of 
determining these quantities, but that they should shut their eyes to their existence 
appears to me only compatible with a very small acquaintance with the data of 
variation. 
Let us see now how various authorities have met this difficulty of skewness. 
B. The Gaussian Curve*. 
Gauss proceeds from the axiom that : The arithmetical mean of a series of 
observations gives their most probable value, i.e. the mean is the value of maximum 
frequency. This result is not axiomatic. It can only be a result of experience, 
and if it were true it would make the normal curve as much a result of experiment, 
i.e. an empirical result, as any other proposed curve of frequency. Gauss's proof 
demands, however, something more than this first statement. It involves (i) the 
equal probability of errors in excess of the mean and of errors in defect, (ii) the 
continuity of magnitude in the errors, and (iii) the independence of all the small 
contributions to the total error. 
Experience shows that Gauss's fundamental axiom as to the mode and mean 
coinciding is not universally true. It is not true of eri-ors of observations, it is not 
true of variations in living forms. Gauss reaches a differential equation which 
leads to the normal curve. His proof seems to me, as it has done to many others, 
quite invalid, because the equal probability of errors in defect and excess of the 
mean is not demonstrated, the possible dependence of contributory elements is not 
discussed, and the question of continuity of errors is not considered. 
C. (i) Laplace and Poisson. 
Laplace and after him Poisson took, I venture to think, much firmer ground. 
They did not assume (i) and (ii), but they did not realise the importance of (iii). 
They proceeded by evaluating the terms of the binomial : 
* In writing for Germans I naturally spoke of the Gaussian curve. But I am not clear that 
precedence is to be given to Gauss. Gauss first gave a proof of the well-known equation y = y„e~^^'/''' 
in his Theoria Motvs; Corporum Coelestinin of 1809. This was three years before the publication of 
Laplace's Thtorie Analytiqne des Prohahilite^ of 1812. But to give absolute priority to Gauss is to 
disregard Laplace's earlier memoirs, particularly those of 1782, " Sur les approximations des Formules 
qui sent fonctions des trfes-grands nombres," and its Sxtite du Memoire of 1783. On p. 433 of the latter 
memoir Laplace actually suggests the importance of forming a table of the probability integral \e ' '^dt. 
The Theorie des Prohahilites reproduces the substance of this memoir, and on this account some writers 
have post-dated Laplace's work. Gauss stated that he had used the method of least squares in 1795, 
but this does not necessarily involve a knowledge of the probability integral, and if it did, it is ten 
years after Laplace. On the whole my custom of terming the curve the Gauss-Laplacian or noniud 
curve saves us from proportioning the merit of discovery between the two great astronomer 
mathematicians. 
