The above are representations of the block and its excavations at this period. Supposing the parts at which the circles nearly come in contact with each other to be of the thickness proper for the partitions of the cells, the parts marked a in the front view and section (figs. 11 and 12) being more than the necessary thickness, the bees will (accor ding to the instinctive principle before mentioned) naturally remove what there is superfluous, thus forming an angle, determined by two intersecting vertical planes at the bottom of the cell, inasmuch as at the same time the parts marked b, in the back view and section (figs. 10 and 12), will also be removed. The partition between these two last-mentioned cells thus becomes perpendicular and of equal thickness, and is exactly opposed to the angle at the bottom of the first cell.
By this time the necessary secretion of wax has taken place in all the bees composing the festoons, and they are all anxious to dispose of their scales of wax. The sculpturer bees are also active, conse quently more wax is added to the margins of the original block, and more exca vations are formed. Supposing the block to have increased to double its original length and width, there would then be room for parts of four more excavations, on the side on which the first was made (fig. 13).
The same operation of reducing the wax in the thick parts marked c having taken place, the sides of the first cell also become straight and perpendicular, and by reducing the wax at the parts d to the -proper thickness in all the cells, the bottom of the first cell, and upper parts of the two cells beneath, in the diagram, become two sided. The work on the opposite side of the comb being in the same state of forwardness (for after the commencement it proceeds equally at all parts), will appear thus— In the above figure the angles at the bases of the cells are cut iuto the partitions of the opposing cells, and hence it is clearly seen that, from the position of those cells, perpendicular partitions of tho cells on this side must be longer than thoso of the other, and that the cells themselves must have three quadrilateral plates for their bases.
In carrying up the sides of the cell, the form is regulated by the intersection of the surrounding circles, as represented in fig. 15. But the circles described in fig. 15, parts of which are shown in most of the other figures, represent those which are inclosed by the hexagons ; whereas we believe the natural circumference of I each cell (supposing it to be cylindri cal) is that by which the hexagon is inclosed ; hence it will be necessary to imagine the circles partly intersecting each other.
It has now been demonstrated that the cells of the first tiers on each side are pentagonal ; that the bases of those on one side are each composed of two plates, while those of the other side are each composed of three plates ; and that, according to the laws laid down, they could not have been otherwise : now as this accords with all the accounts given of the proceedings in the construction of the comb, it seems to prove that the laws which we have laid down, as guiding their formation, are correct.
We have now followed the progress of the work until the com mencement of the second tiers of cells : it is unnecessary to describe the formation of these and the following tiers. It is shown that, according to certain laws, the first tiers of each side of the comb become pentagonal, and according to the same laws it is clear that the second and following tiers must become hexagonal; for the two sides forming the lower boundary of each cell of the first tier, also form the upper boundaries (or partitions) of two cells of the second tiers. As the upper part of the first tier is determined by the roof of the hive (represented by the horizontal line in diagram 13), so is the upper portion of the cells of the second tier determined by the lower portion of those of the first tier ; thus, the upper portion of each cell of the second tiers being composed of two planes meeting at an angle, and the work continuing, as in the progress of the first tier, four more planes will be constructed to form the lower portion, and complete the hexagon. It is thus that all the ordinary cells of a comb are hexagonal, and we believe it is clearly shown that they could not be otherwise, according to the mode of proceeding in their construction. Their form dependi entirely upon the commencement of the work, which necessarily throws the cells in such a position that each cell must be surrounded by six others, and consequently have six sides, each side being the common partition of two cells; and so long as are of equal diameter they must each be opposed to parts of three other cells on the opposite side of the comb, in such a way that supposing the external surface of the bottom of each cell were hemi spherical (which would be the ease were the wax not removed from the interstices), each hemisphere would touch three others ; but the wax being removed from the interstices and reduced to an equal thickness at all parts, and the bases of the sides of a cell not being all in the same plane, the bottom of each cell is thus formed into three equal rhomboidal pieces in three different planes, the three angles at their junction being respectively the lowest parts or the farthest removed from the mouth of the cell.