between the spheres which thus tend to intersect. Hence each
cell consists of an outer spherical portion and of two, three, or more
perfectly flat surfaces, according as the cell adjoins two, three, or more
other cells. When one cell comes into contact with three other cells,
which, from the spheres being nearly of the same size, is very frequently
and necessarily the case, the three flat surfaces are united into a
pyramid; and this pyramid, as Huber has remarked, is manifestly a gross
imitation of the three-sided pyramidal bases of the cell of the hive-bee.
As in the cells of the hive-bee, so here, the three plane surfaces in any
one cell necessarily enter into the construction of three adjoining cells.
It is obvious that the Melipona saves wax by this manner of building; for
the flat walls between the adjoining cells are not double, but are of the
same thickness as the outer spherical portions, and yet each flat portion
forms a part of two cells.

Reflecting on this case, it occurred to me that if the Melipona had made
its spheres at some given distance from each other, and had made them of
equal sizes and had arranged them symmetrically in a double layer, the
resulting structure would probably have been as perfect as the comb of the
hive-bee. Accordingly I wrote to Professor Miller, of Cambridge, and this
geometer has kindly read over the following statement, drawn up from his
information, and tells me that it is strictly correct:--

If a number of equal spheres be described with their centres placed in two
parallel layers; with the centre of each sphere at the distance of radius x
[root]2, or radius x 1.41421 (or at some lesser distance), from the centres
of the six surrounding spheres in the same {227} layer; and at the same
distance from the centres of the adjoining spheres in the other and
parallel layer; then, if planes of intersection between the several spheres
in both layers be formed, there will result a double layer of hexagonal
prisms united together by pyramidal bases formed of three rhombs; and the
rhombs and the sides of the hexagonal prisms will have every angle
identically the same with the best measurements which have been made of the
cells of the hive-bee.

Hence we may safely conclude that if we could slightly modify the instincts
already possessed by the Melipona, and in themselves not very wonderful,
this bee would make a structure as wonderfully perfect as that of the
hive-bee. We must suppose