an inch in length, about two-thirds of a cell, or two lines, high, and declining towards the extremities. We have seen other foundation walls from an inch to an inch and a half long, the form being always the same; but none ever of greater height.

66

The vacuity in the centre of the cluster had permitted us to discover the first manoeuvres of the bees, and the art with which they laid the foundations of their edifices. However, it was filled up too soon for our satisfaction; for workers collecting on both faces of the wall obstructed our view of their further operations *."

Curtain of Wax-workers.-(See p. 114.)

p. 358.

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CHAPTER VI.

ARCHITECTURE OF THE HIVE-BEE CONTINUED.-FORM OF THE CELLS.

THE obstruction of which M. Huber complains only operated as a stimulus to his ingenuity in contriving how he might continue his interesting observations. From the time of Pappus to the present day, mathematicians have applied the principles of geometry to explain the construction of the cells of a bee-hive; but though their extraordinary regularity, and wonderfully selected form, had so often been investigated by men of the greatest talent, and skilled in all the refinements of science, the process by which they are constructed, involving also the causes of their regularity of form, had not been traced till M. Huber devoted himself to the inquiry.

As the wax-workers secrete only a limited quantity of wax, it is indispensably requisite that as little as possible of it should be consumed, and that none of it should be wasted. Bees, therefore, as M. Réaumur well remarks, have to solve this difficult geometrical problem:-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 manuer as to occupy the least possible space in the hive. This problem is solved by bees in all its conditions. The cylindrical form would seem to be best adapted to the shape of the insect; but had the cells been cylindrical, they could not have been applied to

* Réaumur, vol. v., p. 380.

each other without leaving a vacant and superfluous space between every three contiguous cells. Had the cells, on the other hand, been square or triangular, they might have been constructed without unnecessary vacancies; but these forms would have both required more material and been very unsuitable to the shape of a bee's body. The six-sided form of the cells obviates every objection; and while it fulfils the conditions of the problem, it is equally adapted with a cylinder to the shape of the bee.

M. Réaumur further remarks, that the base of each cell, instead of forming a plane, is usually composed of three pieces in the shape of the diamonds on playing cards, and placed in such a manner as to form a hollow pyramid. This structure, it may be observed, imparts a greater degree of strength, and, still keeping the solution of the problem in view, gives a great capacity with the smallest expenditure of material. This has actuaily, indeed, been ascertained by mathematical measurement and calculation. Maraldi, the inventor of glass hives, determined, by minutely measuring these angles, that the greater were 109° 28', and the smaller, 70° 32'; and M. Réaumur, being desirous to know why these particular angles are selected, requested M. Konig, a skilful mathematician, (without informing him of his design, or telling him of Maraldi's researches,) to determine, by calculation, what ought to be the angle of a six-sided cell, with a concave pyramidal base, formed of three similar and equal rhomboid plates, so that the least possible matter should enter into its construction. By employing what geometricians denominate the infinitesimal calculus, M. Koenig found that the angles should be 109° 26' for the greater, and 70° 34' for the smaller, or about two-sixtieths of a degree, more or less, than the actual angles made choice of by bees. The equality of inclination in the

angles has also been said to facilitate the construction of the cells.

M. Huber adds to these remarks, that the cells of the first row, by which the whole comb is attached to the roof of a hive, are not like the rest; for instead of six sides they have only five, of which the roof forms one. The base, also, is in these different, consisting of three pieces on the face of the comb, and on the other side of two: one of these only is diamond-shaped, while the other two are of an irregular four-sided figure. This arrangement, by bringing the greatest number of points in contact with the interior surface, ensures the stability of the comb.

It may, however, be said not to be quite certain, that Réaumur and others have not ascribed to bees the merit of ingenious mathematical contrivance and selection, when the construction of the cells may more probably originate in the form of their mandibles and other instruments employed in their operations. In the case of other insects, we have, both in the preceding and subsequent pages of this volume, repeatedly noticed, that they use their bodies, or parts thereof, as the standards of measurement and modelling; and it is not impossible that bees may proceed on a similar principle, M Huber replies to this objection,

that bees are not provided with instruments corresponding to the angles of their cells; for there is no more resemblance between these and the form of their mandibles, than between the chisel of the sculptor and the work which he produces. The head, he thinks, does not furnish any better explanation. He admits that the antennæ are very flexible, so as to enable the insects to follow the outline of every object; but concludes that neither their structure, nor that of the limbs and mandibles, are adequate to explain the form of the cells, though all these are employed in the operations of building,— the effect, according to him, depending entirely on the object which the insect proposes.

We shall now follow M. Huber in the experiments which he contrived, in order to observe the operations of the bees subsequent to their laying a foundation for the first cell; and we shall again quote from his own narrative: :

"It appeared to me," he says, "that the only method of isolating the architects, and bringing them individually into view, would be to induce them to change the direction of their operations, and work upwards.

"I had a box made twelve inches square and nine deep, with a moveable glass lid. Combs full of brood, honey, and pollen, were next selected from one of my leaf hives, as containing what might interest the bees, and being cut into pieces a foot long, and four inches deep, they were arranged vertically at the bottom of the box, at the same intervals as the insects themselves usually leave between them. A small slip of wooden lath covered the upper edge of each. It was not probable that the bees would attempt to found new combs on the glass roof of the box, because its smoothness precluded the swarm from adhering to it; therefore, if disposed to build,