“Besides the economy of wax,” says Reaumur, when considering this marvellous construction in its entirety,” besides the economy of wax that results from the disposition of the cells, and the fact that this arrangement allows the bees to fill the comb without leaving a single spot vacant, there are other advantages also with respect to the solidity of the work.  The angle at the base of each cell, the apex of the pyramidal cavity, is buttressed by the ridge formed by two faces of the hexagon of another cell.  The two triangles, or extensions of the hexagon faces which fill one of the convergent angles of the cavity enclosed by the three rhombs, form by their junction a plane angle on the side they touch; each of these angles, concave within the cell, supports, on its convex side, one of the sheets employed to form the hexagon of another cell; the sheet, pressing on this angle, resists the force which is tending to push it outwards; and in this fashion the angles are strengthened.  Every advantage that could be desired with regard to the solidity of each cell is procured by its own formation and its position with reference to the others.”

[55]

“There are only,” says Dr. Reid, “three possible figures of the cells which can make them all equal and similar, without any useless interstices.  These are the equilateral triangle, the square, and the regular hexagon.  Mathematicians know that there is not a fourth way possible in which a plane shall be cut into little spaces that shall be equal, similar, and regular, without useless spaces.  Of the three figures, the hexagon is the most proper for convenience and strength.  Bees, as if they knew this, make their cells regular hexagons.

“Again, it has been demonstrated that, by making the bottoms of the cells to consist of three planes meeting in a point, there is a saving of material and labour in no way inconsiderable.  The bees, as if acquainted with these principles of solid geometry, follow them most accurately.  It is a curious mathematical problem at what precise angle the three planes which compose the bottom of a cell ought to meet, in order to make the greatest possible saving, or the least expense of material and labour.* This is one of the problems which belong to the higher parts of mathematics.  It has accordingly been resolved by some mathematicians, particularly by the ingenious Maclaurin, by a fluctionary calculation which is to be found in the Transactions of the Royal Society of London.  He has determined precisely the angle required, and he found, by the most exact mensuration the subject would admit, that it is the very angle in which the three planes at the bottom of the cell of a honey comb do actually meet.”