and kindred sciences. Mathematics must be the handmaid. Unfortunately in some cases
it appears to have become the tyrant.
Fallacious Lastly, one caution must be given as to the employment of the Probability Curve as
Of purity S a test From what has been already said it will be understood that when the Probability
“ T Curve and the Frequency Curve differ by a large amount there is every reason to suppose
really that this difference expresses heterogeneity in the series. The converse, however, is not
composite. universai|y true. This is an extremely important point, for it is possible to obtain what
appears to be a pure curve by the blending of two or more curves. Sometimes the Resultant
Curve will present few points of distinction from a pure curve, but as a general rule this
deception is indicated by a skewness in the Composite Curve. The possibility of this arises
from the fact that two pure curves may be blended so as to produce varying results
according to the relative positions of their mean lines. When these are some distance
apart the Resultant Curve will have two apices. As the mean lines approach one another
the two apices will merge into one, and a broad-topped curve be the result When the
mean lines approach nearer still this distinctive mark vanishes and, if the Component
Curves are of about the same size, the Resultant Curve will appear very like a pure curve.
If, however, as generally happens, the two Component Curves differ in size the Resultant
Curve will present a skewness towards the larger Component Curve.
CH A P T E R V II I
A LIMITED APPLICATION OF TH E MATHEMATICAL THEO RY TO
OUR SERIES.
I n the preceding chapter an account has been given of the general theory of the is it certain
mathematical treatment of series. It has now to be considered how far this theory can be S^rrorefeW
of assistance for the purposes of the present inquiry. In the first place it must be amhropo- l°
remarked that Quetelet devoted himself rather to illustrating the employment of the Law metry ?
of Errors than to actually demonstrating its universal validity. He proceeded upon the
assumption that the Law of Errors if applicable to one branch of science would be
applicable to all, and the assumption has been acted upon by all those who have since
advocated the utilization of such a method in anthropological researches. But although
acted upon, the assumption still remains unproved, and even Professor Karl Pearson in
his Huxley Lecture1 admits ‘ that it is certainly not true for organs and characters in all
types of life,’ adding further that he would ‘ be the last to assert that no human characters
can be found which do not diverge sensibly from the Gaussian distribution/ ‘ But/ he
says, ‘ it really does describe in a most remarkable manner the distribution of most
characters in mankind/ In evidence of this statement he advances three series of
observations, viz. upon the cephalic index of 1982 boys, upon the stature of 1000 women,
and upon the head breadth of 3000 criminals. But to our mind these examples are not
convincing. There is a possible explanation of their conformity to the Gauss-Laplacian
law of errors, which would by no means entail the inference that all other series of
observations would be equally amenable. That explanation is in brief that the illustrations
selected are taken from series in which the indices or dimensions are of medial character.
Thus the cephalic index of the boys ranges from 67 to 92, and the arithmetical
mean falls at 78-79, which, as the curve is quite symmetrical, shows that the subjects
belong to a mesatz-cephalic type of mankind. And for this very reason, viz. that the
cephalic indices are just those which stand midway in the whole possible range of variation
in mankind, there is nothing to prevent the deviations from being distributed equally on
either side of the mean. But let it be supposed that not English schoolboys but negroes
of a pronounced dolichocephalic type or Mongols of extreme brachycephalic head form
had been the subjects of examination. In that case would it have been possible for the
swing of the variations to be equal on either side of the mean? The occurrence of
extreme forms is necessarily much rarer than the occurrence of those which tend to fall
within the medial (i.e. the ‘ meso ’) group. Or, to put it in another way, the dolichocephalic
stock would tend rather to become less than more dolichocephalic, while on the other hand
the brachycephalic stock would vary more in the direction of mesaticephaly than of
sphericity. There is a limit at each extremity of the scale of possible variation, and this
limit cannot be passed. And yet there are races which stand almost at the respective
extremities of the scale, the very dolichocephalic and the very brachycephalic, which
1 Jo u rn . A nthrop. In s t., vol. xxxiii, p. 189.