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. [Illustration: FRONT AND REVERSE VIEW OF CELLS. ] [Illustration: METHOD OF JOINING CELLS. ]