172 
Slew Variation, a Rejotrider 
(C) The hypotheses made to generalise the Gaussian law. 
(i) Poisson, Laplace and the early writers. 
(ii) Francis Galton and D. McAlister. 
(iii) Fechner. 
(iv) Edgeworth and Kapteyn. 
(v) General Results for Asymmetry*. 
(D) The criticisms of my theory by Dr Ranke and Dr Greiner and the reply 
to be made to them. 
A. The need for Generalised Frequency Curves, even in Anthropological 
Science. 
I have already pointed out that even Prof Ranke's measurements are not 
fully in accordance with the Gaussian theory — for the odds are great against a 
quantity exceeding its probable error more than five times. It is perfectly true 
that the English School have found that many characters, especially craniological 
characters, are for practical purposes sufficiently described by the Gaussian curve. 
But it is equally true that they have found other cases in which the deviation 
from the Gaussian curve is significant, and that they have only been able to 
measure this significance because they had a wider theory to base their researches 
upon. Dr Ranke entirely disregards the statements of Miss Fawcett and 
Dr Macdonell on this point. Both find a definite number of cases, the one 
in Egyptian skulls, the other in English skulls, in which the deviation from the 
Gaussian law is definitely significant -f*. Both conclude as I have done that in the 
case of many characters for a variety of practical purposes the Gaussian curve is 
sufficient; this is, however, not a ^/(eo?-eif2ca^ justification of the Gaussian curve, but 
an argument in favour of its empirical use in a certain definite number of cases. 
Dr Ranke may of course say that the exceptions that we have found are due to 
heterogeneity of our material. If so he must face the difficulty that the same set 
of crania can be homogeneous and give the Gaussian curve for their length and be 
heterogeneous for their breadth, deviating therein largely from the Gaussian curve. 
If he asserts that this is quite possible then he must meet the further difficulty 
that they can be homogeneous for their cephalic indices, which are based upon the 
ratio of the supposed heterogeneous to the homogeneous material ! The fact is 
that no unprejudiced observer can examine the constants by which we have defined 
the deviations from the Gaussian law without seeing that they present every 
variety of value, starting from the values to be expected on tlie Gaussian theory 
and rising to values which are absolutely incompatible with any Gaussian theory 
at all. In fact he must come to the conclusion that some theory is absolutely 
needful, which will provide a curve or series of curves capable of representing the 
* I have left out of consideration the general method of Thiele, followed in Germany by Lipps, 
because I have dealt with these authors in a recent memoir. 
t Biometrika, Vol. i. p. 443, and Biometrika, Vol. iii. p. 227. 
