almost
constantly baffled. In the end he is obliged to exercise his
judgment as to the limit at which he should cease to subdivide. If
he subdivides very frequently, the groups become too small to have
statistical value; if less frequently, the groups will be less truly
specific.

A species may be defined as a group of objects whose individual
differences are wholly due to different combinations of the same set
of minute causes, no one of which is so powerful as to be able by
itself to make any sensible difference in the result. A well-known
mathematical consequence flows from this, which is also universally
observed as a fact, namely, that in all species the number of
individuals who differ from the average value, up to any given amount,
is much greater than the number who differ more than that amount,
and up to the double of it. In short, if an assorted series be
represented by upright lines arranged side by side along a
horizontal base at equal distances apart, and of lengths
proportionate to the magnitude of the quality in the corresponding
objects, then their shape will always resemble that shown in Fig. 1.

The form of the bounding curve resembles that which is called in
architectural language an ogive, from "augive," an old French word
for a cup, the figure being not unlike the upper half of a cup lying
sideways with its axis horizontal. In consequence of the multitude
of mediocre values, we always find that on either side of the
middlemost ordinate _Cc_, which is the median value and may be
accepted as the average, there is a much less rapid change of height
than elsewhere. If the figure were pulled out sideways to make it
accord with such physical conceptions as that of a row of men
standing side by side, the middle part of the curve would be
apparently horizontal.

[Illustration: Fig. 1.]

[Illustration: Fig. 2.]

The mathematical conception of the curve is best expressed in Fig. 2,
where PQ represents any given deviation from the average value, and
the ratio of PO to AB represents the relative probability of its
occurrence. The equation to the curve and a discussion of its
properties will be found in the _Proceedings of the Royal Society_,
No. 198, 1879, by Dr. M'Alister. The title of the paper is the
"Law of the Geometric Mean," and it follows one by myself on
"The Geometric Mean in Vital and Social Statistics."

We can lay down the ogive of any quality, physical or mental,
whenever we are capable