206 
Skew Variation, a Rejoinder 
and the further statement that variation in man is essentially continuous, is at all 
valid. Our units of grouping for the numbers available are not very fine, we can 
hardly successfully classify a few hundred observations into more than 20 groups, 
and with this unit of grouping it would bo practically impossible to distinguish 
between the apparently continuous distribution and a discrete distribution of a 
similar number of classes with the variates modified by growth, nurture or any 
other scattering tendencies. Ranke appears entirely to have overlooked the current 
biological theory of inheritance summed up in the words inheritance by deter- 
minants. Such theories, whether they be those of Weismann or Mendel, lead us 
directly to discrete variation*. The discreteness of the variation will be more or 
less, in many cases probably entirely, obscured by the environmental influence. In 
such cases the number of fundamental cause-groups is not infinitely great, and 
Ranke is overlooking current biological views when he asserts that we must take 
" die Anzahl der Elementarursachen selbst als imendlich gross und die Grosse der 
Wirkung der einzelnen Ursache als unendlich klein." If the number of determinants 
which fix a character is finite, that character would correspond to a discrete 
variation of limited range. If the number of determinants be very large, the 
distribution would by Laplace's theorem be represented more and more closely by 
the normal curve. 
I toss ten coins into the air and for every head in the result I pay a gramme of 
gold-dust, the frequency distribution of gold-dust would closely be given by the 
terms of the binomial (| -1- A)^" as in the points of our Fig. 1. But suppose instead 
of weighing niy gramme of gold-dust accurately, I give a " handful " of sugar. If 
6 heads turn up I give six handfuls of sugar, but each of these will not be exactly 
my standard mean handful. I am unlikely to give either five or seven standard 
handfuls as my six approximate handfuls, but in some cases even these might be 
possible ; we pass in fact from discrete to continuous variation, and the nmltimodal 
character of the discrete variates will disappear with the roughness of the handfuls, 
or have the pet'ky appearance of random sampling. The total area up to any 
midpoint between two discrete groups s and s + 1 will be given by the continuous 
integral which represents the first s terms of the binomial. If we have two such 
total ai-eas, one up to the midpoint between groups s and s + 1 and the other up 
to the midpoint between groups s + 1 and s + 2, then an interpolated area between 
these values as given by the continuous integral will be sensibly the same as 
if, c being the vmit of discrete difference, we determined a curve corresponding 
to the mean binomial frequency in the Spielrawm c, i.e. 
c 
by simply fractionising r, i.e. we replaced the factorials by Stirling's theorem or 
used r functions, and supposed r to change continuously from s to s + 1. This is 
* Thus I have shown that a generalised Mendehan theory leads directly to skew binomial 
distributions of characters in the general population. Phil. Trans. Vol. 203 A, pp. 63 — 86. 
