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 to stand from her fellow-labourers when several are making
their spheres; but she is already so far enabled to judge of distance,
that she always describes her spheres so as to intersect largely; and
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