208 
Skew Variation, a Rejoinder 
In this case the frequency of individuals with character x would be given by the 
curve 
y = y, (•'» - («2 - «)'■- 
This is a perfectly continuous curve, and one of my series of skew curves. 
Thus it is quite conceivable that a finite number of fundamental cause-groups 
should lead to an absolutely continuous distribution*. 
Now how does Ranke treat this illustration ? He first states that all continuity 
must involve an infinite number of cause-groups, or variation in man being 
continuous, must be associated with an infinite number of cause-groups. He had 
this very case before him, and yet he writes : 
Die Analyse dor Elomontarursachen ergibt uns also Tinweigerlich die bishej- immer angenom- 
mene imendliche Anzahl derselben, die unendliche Kleinhoit dor Wirkung jeder einzelnen Ursache 
und die Kontimiitiit der moglichen Wirkungsgrade. 
Sie ergibt also wiiklich die Vcrhaltnis.se, die wir zum Verstiindniss der kontinuerlichen 
Variationskurve ganz uimniganglicli notig haben. Denn wio soil eine kontiimerliche Kiu've sich 
aus der Kombination endliclier Baiisteine ergeben (S. 321) ? 
Ranke only gets out of the difficulty by asserting that since the number of 
causes is finite, but must be infinite for variation, my continuous curve based on 
a finite number of cause-groups cannot represent variation ! A more remarkable 
specimen of circular reasoning can hardly be conceived. The fact is that Ranke 
suffers from the old third Gaussian axiom, i.e. the supposition that the increments 
that go to build up the variate are independent of each other. The fundamental 
cause-groups are by no means Bausteine in the sense that the total variate is the 
sum of these Bausteine placed on top of each other ! The causes determine the 
magnitude of the variate, but not at all necessarily by their sum. 
(c) Ranke asserts that some of my curves have constants which if we 
endeavour to interpret them from the standpoint of the binomial give impossible 
or improbable values for the constants. 
The answer to this is that the series were only the scaffolding to deduce the 
curves. The differential equation to the curves contains the limit to a good many 
other frequency systems which directly diverge from the fundamental axioms of 
Gauss. I used the original series as a means of dispensing with the Gaussian 
axioms in familiar cases, but the result reached involves a good deal more than can 
be interpreted by the original series. Ranke can only see absurdity in a binomial 
with a negative p or q. But the nature of the sources of variation is so little 
known to us that we cannot possibly assert the absurdity of such values. We may 
not indeed be able to directly interpret them in the case of man, say, but they 
occur and recur in chance investigations. I will illustrate this in one case only, 
but such will demonstrate the required possibility and dispose at once of Ranke's 
argument as to absurdity. 
* Making )• and a.j infinite, but s-r finite, we get the curve I have deduced as the Hmit to a 
binomial of finite power. In other words, that curve is also shown to correspond to a possible 
continuity. 
