K. Pearson 
199 
no less than eight of Duncker's scries in the case of Gelasimus purjilator and in 
various other distributions. In all these cases the platykurtosis is significant, and 
the double Gaussian curve fails us hopelessly. 
C. (iv) Tlie Edgewurtli-Kapteyn Curves. 
Kapteyn*, without recognizing Edgeworth's priority, has proceeded in the 
manner indicated on p. 178 above. He assumes that some quantity x obeys the 
normal distribution 
N 
He then takes x = F{X)- M and reaches the frequency distribution : 
F = F' (Z) e-i^-^/r/- (xxxi). 
Thus far (as we have already shown) nothing has been achieved, because this 
equation may by a proper choice of F {X) represent any curve whatever. As 
Kapteyn himself says, following Edgeworth, "as F{x) may represent any function, 
we see that the equation may be made to represent any curve whatever. Therefore 
it must be the most general form of frequency curve possible " (p. 17). There is, 
however, one point to be raised here. What is x of which the observed character 
is a function ? Is it, as in the explanatory illustrations cited by Kapteyn, 
another characteristic of the organism ? If so we ought in some cases to be able 
to determine it. What is the character which obeys the normal law ? For 
example, sagittal arc in English women is almost exactly normal in its distribution, 
and nasal breadth is very asymmetrical. Shall we take a; = sagittal arc and 
X = nasal breadth and make 
x = F{X)-M'i 
Now every biologist knows that such a relation is not in the least true. No 
two characters in an organism are in any way connected by a mathematical 
function, such that when one is given the other is determined. The relation is 
always of the loose kind that we term association or correlation. A'^ does not 
fix X, and a multitude of xa with varying degree of probability are associated with 
a given x. This correlation is often of a very low order. Between any two 
characters of a given organism, no such relation of perfect correlation as that 
involved in Edgeworth or Kapteyn's relation has ever been discovered. Very 
imperfect correlation or at any rate all degrees of correlation have been invariably 
demonstrated to exist. The function x has no real existence as a biological entity. 
It is only a mechanism for introducing the normal curve, and is not a true character 
of the organism at all. Supposing, as in English female crania, nasal breadth is 
asymmetrical, what is the quantity which is symmetrically distributed of which 
nasal breadth is a function ? It has no reality in the organism at all, and Kapteyn 
proceeds to make it still more impossible in the following manner. If x has 
* Skew Frequency Carves in Biolo<jij and Statistics, Groningen, 1903. 
