MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 359 
ef, the axis of symmetry parallel to the axis of the cylinder. Then, if the cylinder 
be rotated, we shall have a series of grey tints from a darkish e to a lighter f. 
Now, if we ask several hundred persons to select a tint which would result from 
mixing the tints at e and f, we shall obtain a continuous frequency curve, falling, 
however, entirely within the range e to fi Or, again suppose a frequency curve 
obtained by plotting up the frequency of a given ratio of leg-length to total body- 
length, or of carapace to body-length. Here the range must lie between 0 and 1. 
It is not that other values are excessively improbable, they are by the conditions 
of the problem absolutely impossible. Hence, it is clear that the curves obtained 
by Professor WreLpon and Mr. H. Tompson in the case of shrimps, crabs, and 
prawns, can only be approximately normal curves, even if it were possible for the 
ratios to run from 0 to 1. But as a matter of fact, the possible range is very 
much smaller. We may not be able to assert, @ priori, what it is, but for an 
adult prawn to have a carapace 3 or yolgo Of its body-length, or a man a leg 
3 or zo of his body-leneth, may be regarded as impossibilities ; they are abnor- 
malities, which could hardly survive to the adult condition. Precisely the same 
remarks apply to skull indices, and probably to the relative size of all sorts of 
organs in the adult condition. We may not know the range, @ priori, but we are 
quite certain that one exists, and it is a quantity to be determined—just as the mean or 
the standard deviation—from our measurements themselves. We may take it that in 
most biological measurements of adults there is a range of stability, so to speak, 
organs not falling within this range are inconsistent with the continued existence of 
the individual, with the assumption that he has lived to be an adult.* Nor is this 
question of range confined to biological statistics. A barometric frequency curve 
must show the same peculiarity ; there are excessively low and excessively high 
barometric heights which would be not only inconsistent with the survival of any 
meteorological observer, but also with the existing features of physical nature on 
this earth. In vital statistics we find precisely the same thing, a curve of percent- 
ages of mothers of different ages for the children born during any year in a country 
would be definitely limited by the ages of puberty and the climacteric, which cannot 
be pushed indefinitely towards childhood and senility respectively. Again in disease 
and mortality curves, while the lower limit of life is clear, it is highly probable that 
an upper limit exists, if we can only fix it by investigation of our statistics them- 
selves. A man of the present day, as now organised, may be able to live 120 years, 
perhaps, but we have exceeded his vital possibilities if we take, say, 200 years. 
Thus the problem of range seems a very important one, it theoretically excludes the 
use of the normal curve in many classes of statistics; it is quite true that, for 
many practical purposes, frequency curves of limited range may be sensibly identical 
either with unlimited curves, or even with normal curves, but, in other cases, this 
* Absolute malformations, congenital, or due to post-natai accident are excluded. Abortions or 
amputations would be naturally excluded from our measurements, 
