K. Pearson 
209 
Suppose an organ to require the conjunction of exactly n determinants of one 
kind to fix it, but that the size of the organ depends on how soon this conjunction 
takes place. Let D be the necessary kind of determinant and tt the chance that 
it is left in the right position after each operation, say a cell-division or cell-fusion. 
Let D' be any other kind of determinant and k its chance of appearing, so that 
/c-f- 7r = l. Then if D appears n times in the first n operations, we have a certain 
size for the organ, if in the first n + 1 operations another size, and if only in the 
first n + r operations a third size. But the chances of these respective aiDpearances 
are the terms of the series 
n (n + 1) , „ n(n + l)(n+'2) ...(n + r-\) 
2 \r 
= tt" (1 - a;)-" =(---] =(p- q)~'\ say, 
where p — q = l. Here p and q have lost the condition that they are both to be 
less than unity. I do not for a moment suggest that this is the real interpretation 
of a binomial with a negative q. I only assert that because we are as yet unable 
to certainly interpret such expressions, the absence of interpretation does not 
involve the absurdity which Ranke postulates. 
There are many other matters to which I might justifiably take exception in 
Ranke and Greiner criticism*, but I think I shall have said sufficient to convince 
the impartial reader of the following points : 
(i) The great bulk of modern statisticians are agreed that the Gaussian law 
is absolutely insufficient to describe observed facts. They may disagree as to the 
method of supplementing it. I do not think that the opinion of Ranke and 
Greiner can possibly weigh against those of Poisson, Quetelet, Galton, Edgeworth, 
Fechner and Kapteyn — all authorities who have had to deal for years with 
statistical data. 
(ii) The original use of the probability integral (the areas of the Laplace- 
Gaussian curve) as introduced by Laplace was to represent the sum of terms of 
a discontinuous series. To the mathematical mind there is no absurdity in this 
replacement of discontinuity by continuity ; it is the basis of the Eiiler-Maclaurin 
theorem. 
(iii) The dogmatic assertion of Ranke that variation in man is due to an 
infinite number of infinitely small fundamental cause-groups, simply neglects the 
* For example, all the discrete variates mentioned on p. 205 have been dealt with by biometric 
writers, and many others besides, yet Eanke speaks as if such writers had not dealt with discrete 
variation. He speaks of Ludwig's multimodal curves for flowers as if there had been no controversy 
as to the actuality of the " Fibonaccizahlen " modes, when due regard is paid to homogeneity of 
season and environment. He speaks as if Johannsen had demonstrated normal variation in his 
"Erbsenspopulation," when he has really applied no valid criterion whatever to test for asymmetry, etc. 
In short he seems to me to have neglected a great deal of the modern literature of the subject, and, 
if I may venture to say so, to write over-dogmatically ou what he has read. 
Biometrika iv 27 
