pyramidal basis 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 the square root of 2 or radius x 1.41421 (or at some lesser
distance), from the centres of the six surrounding spheres in the
same 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 the Melipona to make her cells
truly spherical, and of equal sizes; and this would not be very
surprising, seeing that she already does so to a certain extent, and
seeing what perfectly cylindrical burrows in wood many insects can
make, apparently by turning round on a fixed point. We must suppose the
Melipona to arrange her cells in level layers, as she already does her
cylindrical cells; and we must further suppose, and this is the greatest
difficulty, that she can somehow judge accurately at what distance