The cells being formed of wax, a substance secreted by the bees in no great abundance, it is important that as little as possible should be consumed. Bees, therefore, in the formation of their cells have to solve a problem in geometry, namely, “a quantity of wax being given, to form of it similar and equal cells of a determinate capacity, but of the largest size in proportion to the quantity of matter employed, and disposed in such a manner as to occupy in the hive the least possible space.” Every part of this problem is practically solved by bees. If their cells had been cylindrical, which form seems best adapted to the shape of a bee, they could not have been applied to each other without leaving a number of useless vacant spaces. If the cells had been square or triangular, this last objection would be removed; but a greater quantity of wax would have been required, and the shape would have been inconvenient to a round-bodied animal. Hexagonal cells are admirably fitted to the form of the insect, at the same time that their sides apply to each other without the smallest vacant intervals. Another important saving in materials is gained by making a common base serve for two layers of cells. Much more wax as well as room would have been required, had the combs consisted of one layer only. But this is not all. The base of each cell is not an exact plane, but is usually composed of three lozenge-shaped pieces, placed so as to form a pyramidal concavity. From this form it follows that the base of a cell on one side of the comb is composed of portions of the bases of three cells on the other. By this arrangement a greater degree of strength is obtained, and also a more roomy cell, with less expenditure of wax. This has been clearly proved, as also that the angles of the base of the cell are exactly those which require the smallest quantity of wax. It is obvious that these angles might vary infinitely; but by a very accurate measurement Maraldi found that the great angles were in general 109° 28′, the smaller ones 70° 32′. Réaumur, suspecting that the object of choosing these angles was to spare wax, proposed to M. König, a skilful geometrician, to determine by calculation what ought to be the angle of a hexagonal cell with a pyramidal bottom, formed of three similar and equal rhomboid plates, so that the least possible matter might enter into its construction. After an elaborate calculation, the geometrician found that the great angles of the rhombs should be 109° 26′, and of the small angles, 70° 34′, a surprising agreement between the solution of the problem, and the actual measurement.
FRONT AND REVERSE VIEW OF CELLS.
METHOD OF JOINING CELLS.
The bees have also another contrivance for saving wax. They form the bottoms and sides of the cells of wax not thicker than writing paper; but as walls of this thinness at the entrance would be perpetually injured by the going in and out of the workers, they make the margin at the opening of each cell three or four times thicker than the walls.
FESTOON OF WAX-MAKERS.
It has already been said that wax is a secretion naturally formed in certain membranous bags in the body of the bee. As the secretion goes on, the wax oozes through the membrane, and forms in thin plates on the outside. The position adopted by the insects during this process is strange and almost ridiculous. Their proceedings are as follows:—The wax-makers, having taken a quantity of honey or sugar into the stomach, suspend themselves to each other, the claws of the fore legs of one being attached to those of the hind pair of another, until they form themselves into a cluster, consisting of a series of festoons or garlands, which cross each other in all directions, until they form a dense curtain, and in which most of the bees turn their back upon the observer. In this position the wax-makers remain immovable for about twenty-four hours, during which period the secretion of wax takes place. At last one of them is seen to detach itself from the rest, and to make its way to the top of the hive, where it turns itself round, and clears a space of about an inch in diameter. It then seizes one of the plates of wax with a pincer, formed at the joints of the leg, and drawing it forwards, one of the fore legs takes it with its claw and carries it to the mouth. The insect then proceeds by means of its mandibles and its proboscis to reduce the plate to a riband of wax, which it softens with a frothy liquor. During this operation the proboscis is sometimes flattened like a spatula, then like a trowel, at other times it is like a pencil, terminating in a point. The liquor mixed with the wax gives it a whiteness and ductility which it had not before, the object being to make it fit for working into any shape.
