The honey-bee: its nature, homes and products

by William Hetherington Harris

Fig. 16.—Arrangement of Cells.

The next matter then to settle is, the magnitude of the angles at which the sides of the hexagon should slope towards each other, so as to be the most advantageous. Réaumur put the problem in mathematical language before M. König, a skilful geometrician, thus:—"To determine by calculations 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 matter possible might enter into its construction." The result of his investigations was that the angles of the rhombs must be 109° 26′ and 70° 34′. Cramer, professor of mathematics in the University of Geneva, also undertook the problem. His calculations, made on somewhat different principles from König's, gave for the angles 109° 28′ 16″, and 70° 31′ 44″. Maraldi, a third mathematician, assuming the equality of the angles of the trapezia forming the sides of the hexagon adjacent to the rhombs and those of the rhombs themselves, and that the solid angle at the apex of the pyramid, composed of equal obtuse angles, is precisely equal to each of the three angles at the base, also composed of three equal obtuse angles, came to the conclusion that the angles must be 109° 28′ and 70° 32′.

These three sets of results, so remarkably accordant, when we consider the minuteness of the differences between them, in figures so small as the actual honey-comb cells, show the closest correspondence to the actual measurements of the work of the bees. Maraldi found the angles of the latter to be 110° and 70°, as nearly as could be ascertained. We have dwelt at some length upon this point, because it illustrates, in a most marvellous manner, the power of that inborn faculty we call instinct, which arrives, without training, at results so precisely agreeing with those of the highest efforts of our intellectual reasonings. To the devout mind, the conclusion is inevitable that Divine Wisdom is the inspiring force which energizes the mental operations of the bees in their cell-building.

A further advantage of the actual shape of the honey-comb prisms is that, thereby, strength is combined with economy. No other form would so efficiently have carried the heavy weights constantly stored in the forms of honey, brood, and bee-bread.

The bottoms and sides of the cells are made of wax as thin as a sheet of writing-paper; but as walls of this thinness at the entrances would break down under the weight of the constantly passing insects, the margin at the opening of each cell is made four or five times thicker than the walls. Then, as the cells are lengthened, this thickness is reduced, always remaining the same, however, at the actual margins. Dr. Barclay also discovered that, though the tenuity of the divisions is so great, each, in point of fact, consists of two distinct layers agglutinated together. This gives, again, an increase of strength, as any practical builder would know who, in his "bressummers," adopts the same method of attaining lightness and power of sustaining great weights.

Fig. 17.—Diagram Showing Slope of Cells.

The actual size of the cells in a hive varies considerably, as we might expect. Without regarding those for queen-progeny, we should anticipate that those in which young drones are to be developed would be considerably larger than those prepared for workers. This is, indeed, the case. But as an abrupt change from the one kind to the other would be impossible without waste, the bees prudently graduate the difference by interposing a suitable series of intermediate sizes, whose bottoms, of course, have to depart from the normal conditions, and sometimes consist of two rhomboids and two hexagons, varying in size and form, and corresponding with four, instead of three, opposite cells. In these, stores are often found, instead of brood. If eggs are laid in them, they are generally those which will develop into males, and the space for development being smaller than usual, the drones occupying such cells are not so large as the average size.

As a rule, the hexagonal ends of twenty-seven worker cells, or nineteen drone cells, occupy a surface of one square inch. All the cells lie not quite horizontally, but sloping slightly downwards from the mouth towards the bases. This arrangement is designed to prevent the honey from easily flowing out. As the cells are filled with the liquid, the lower edge of each is first raised, and, in due time, the whole of the once open end is sealed over with a coating of wax mixed with a little propolis. This covering not only keeps the contents from running out, but prevents fermentation or candying, from contact with the air.