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"I 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 supplied 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 day's work."
"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 constructing 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 fact.
"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 inspiration of 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."