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N order to store the supply of honey and lodge the
larvæ, the bees build with their wax little rooms
called cells, open at one end and closed at the other. They
are six-sided and arranged with perfect regularity. In
geometrical terms, each would be called a hexagonal prism,
or a prism with six facets.
"Do not be surprised at this introduction of terms belonging
to the beautiful and severe science of form—of geometry,
in short. Bees are geometricians of consummate skill. Their
constructions have required the exercise of the highest
intelligence. All the power of human reason was necessary to
follow, step by step, the insect's science. I will return
presently to this fine subject, a very difficult one, but I
will try to make it intelligible to you.
"The cells are placed horizontally, back to back and end to
end, in pairs, with the closed ends joining. Furthermore,
they are arranged side by side in greater or less number,
and they touch each other by their flat faces, each one of
which serves as partition wall for two contiguous cells. The
two layers of cells, back to back at their closed ends,
constitute what is called a comb or honey-comb. On one side
of this comb are found all the entrances to the cells
 of the
corresponding layer; on the other side the cells of the
second layer open. Finally, the honey-comb is suspended
vertically in the hive, with half its openings to the right
and half to the left. It adheres by its upper edge to the
roof of the hive, or to the bars that cross it inside.
"One comb is not enough when the population is numerous;
others are constructed like the first. The various combs,
ranged parallel to one another, leave free intervening
spaces. These are the streets, the public squares, the
thoroughfares, on which the openings of the two layers of
cells belonging to neighboring combs give, as the doors of
our houses open on the right and left of a street. There the
bees circulate, going from one door to another to deposit
their honey in the cells used as store-houses, or to
distribute nourishment to the young larvæ lodged one
by one in other cells. In these same public places they
assemble when necessary, hold consultations, and deliberate
on the affairs of the community. There, for example, among
the nurses going from door to door to see whether the
larvæ need feeding, and the wax-bees rubbing
themselves vehemently to extract the wax and begin to build,
is plotted the extermination of the drones; there, when the
birth of a new queen menaces the hive with civil war, the
project of emigration ripens. There—But let us not
anticipate. Let us return to the cells."
"I am longing to know the whole of the strange story of the
bees," Jules broke in.
"Patience! First of all let us see how the cells are
constructed. The bee that feels that it is
sup-  plied with the
materials for making wax rubs itself and extracts a sheet of
wax from the folds of its rings. With the little layer of
wax between its teeth, that is to say between its two
mandibles, it squeezes through the press of its comrades.
'Let me pass,' it seems to say; 'see, I have something to
work with.' The crowd makes way. The bee takes its place in
the middle of the workyard. The wax is kneaded between its
mandibles, pounded to pieces, then flattened out into a
ribbon, pounded again, and once more kneaded into a compact
mass. At the same time it is impregnated with a kind of
saliva that gives it flexibility. When the material is at
the proper stage, the bee applies it bit by bit. To cut off
the surplus, the mandibles serve as scissors; the
antennæ, in continual motion, serve it as probe and
measuring-compasses; they feel the wall of wax to judge of
its thickness; they plunge into the cavity to find out its
depth. What exquisite touch in this pair of living
compasses, to bring to successful completion a construction
so delicate and regular! Moreover, if the worker is a
novice, master-bees are there to watch it with an
experienced eye, to seize on the slightest fault at once and
hasten to remedy it. The maladroit worker modestly steps
aside and watches in order to learn. The trick learned, it
sets to work again. With thousands of wax-bees working
together, a comb two or three decimeters wide is often a
"You told us," said Claire, "that the cells are especially
remarkable for their geometrical arrangement."
 "I am just coming to that magnificent topic, but I shall
treat it briefly, I warn you. You are far from being able to
follow yet in its superior beauties the architecture of the
bees. Yes, my dear Jules, the wax house of a poor insect, to
be well understood, demands knowledge that very few persons
possess. Ah, you may study ever so long before you are able
fully to understand this marvel! For the present, here is
what I will tell you.
"The cells serve, some as store-rooms for the honey, others
as nests for the little ones. They are made of wax, a
material that the bees cannot procure in indefinite
quantities. They must wait until the stomach sweats a little
layer of it, and it forms very slowly, at the expense of the
insect's very substance. The bee builds with the materials
of its own body, it impoverishes itself in sweating the
wherewithal to construct the cells. You can judge from that
how precious a thing wax is to the bees, and with what
strict economy they must use it.
"And yet the innumerable family must be lodged, honey
store-rooms must multiply to supply the wants of the
community. Moreover, it is necessary that these store-rooms
and nurseries take up as little room as possible, so as not
to encumber the hive, and to permit free circulation to the
twenty or thirty thousand inhabitants of the city. In fine,
one of the hardest problems is presented to the bees: they
must make the greatest possible number of cells in the least
space and with the least wax possible. Well, friend Jules,
do you think you could solve the bees' problem?"
 "Alas! Uncle, I hardly understand the statement of it."
"To economize the wax, a very simple way suggests itself at
the outset: it is to make the partitions of the cells very
thin. You may be quite sure the bees are equal to this
elementary requirement. They make the wax walls scarcely as
thick as a sheet of paper. But that is not enough: it is
necessary above all to take the form into consideration and
to seek the most economical shape. Let us try. What shape
shall we give the cells to satisfy the conditions of economy
in space and wax?
"First of all let us suppose them to be round. Let us trace
on paper some circles of equal size and touching one
another. Between three of these contiguous circles there
will always be an unoccupied space. The round form will not
do, then, for the cells, since there will always be a waste
of space, or empty intervals.
"Let us make them square. We will trace equal squares on the
paper. In going about it properly we can arrange the squares
side by side without leaving any empty spaces between them.
Look at the inlaid floor of this room, composed of little
square red bricks. These bricks leave no intervening spaces;
they touch on every side. The square form, therefore, suits
the first condition, namely: to utilize all the space.
"But here is where another difficulty arises. Cells
fashioned on the square model would not hold enough honey
for the quantity of wax used in
con-  structing them. In order
to increase their capacity, you must increase as much as
possible the number of their facets. I will not try to
demonstrate to you this beautiful truth; it is beyond your
intelligence. Geometry affirms it; let us consider it a
"Starting from that, the choice is soon made. Among all the
regular figures that can be placed side by side without
leaving an unoccupied space, you must choose that which has
the greatest number of sides, for that is the one that will
hold the most honey for the same quantity of wax used.
"Geometry teaches that the only regular figures that can be
arranged without waste of space are: the three-sided figure,
or triangle; the four-sided, or square; and the six-sided,
or hexagon. That is all: no other regular figures touch all
around so as to leave no empty spaces between them.
"So it is, then, in the hexagonal form, or form with six
sides, that the cells can occupy, collectively, the least
space, use the least wax, and hold the most honey. Bees,
knowing these things better than any one else, make
hexagonal cells, never any other kind."
"Then bees have reason," remarked Claire, "like ours; even
superior, if they can solve such problems?"
"If bees constructed their cells after a premeditated,
considered, calculated plan, it would be something alarming,
my dear child: animals would rival man. Bees are profound
geometricians because they work, unconsciously, under the
 the sublime Geometrician. Let us stop this
talk, which I fear you have not wholly understood; but, at
any rate, I have opened your eyes to one of the greatest
wonders of this world."