[50]
Let us go on, then, with the story of our hive; let us take it up where we left it; and raise, as high as we may, a fold of the festooned curtain in whose midst a strange sweat, white as snow and airier than the down of a wing, is beginning to break over the swarm. For the wax that is now being born is not like the wax that we know; it is immaculate, it has no weight; seeming truly to be the soul of the honey, that itself is the spirit of flowers. And this motionless incantation has called it forth that it may serve us, later--in memory of its origin, doubtless, wherein it is one with the azure sky, and heavy with perfumes of magnificence and purity--as the fragrant light of the last of our altars.
[51]
To follow the various phases of the secretion and employment of wax by a swarm that is beginning to build, is a matter of very great difficulty. All comes to pass in the blackest depths of the crowd, whose agglomeration, growing denser and denser, produces the temperature needful for this exudation, which is the privilege of the youngest bees. Huber, who was the first to study these phenomena, bringing incredible patience to bear and exposing himself at times to very serious danger, devotes to them more than two hundred and fifty pages; which, though of considerable interest, are necessarily somewhat confused. But I am not treating this subject technically; and while referring when necessary to Huber's admirable studies, I shall confine myself generally to relating what is patent to any one who may gather a swarm into a glass hive.
We have to admit, first of all, that we know not yet by what process of alchemy the honey transforms itself into wax in the enigmatic bodies of our suspended bees. We can only say that they will remain thus suspended for a period extending from eighteen to twentyfour hours, in a temperature so high that one might almost believe that a fire was burning in the hollow of the hive; and then white and transparent scales will appear at the opening of four little pockets that every bee has underneath its abdomen.
When the bodies of most of those who form the inverted cone have thus been adorned with ivory tablets, we shall see one of the bees, as though suddenly inspired, abruptly detach herself from the mass, and climb over the backs of the passive crowd till she reach the inner pinnacle of the cupola. To this she will fix herself solidly, dislodging, with repeated blows of her head, such of her neighbours as may seem to hamper her movements. Then, with her mouth and claws, she will seize one of the eight scales that hang from her abdomen, and at once proceed to clip it and plane it, extend it, knead it with her saliva, bend it and flatten it, roll it and straighten it, with the skill of a carpenter handling a pliable panel. When at last the substance, thus treated, appears to her to possess the required dimensions and consistency, she will attach it to the highest point of the dome, thus laying the first, or rather the keystone of the new town; for we have here an inverted city, hanging down from the sky, and not rising from the bosom of earth like a city of men.
To this keystone, depending in the void, she will add other fragments of wax that she takes in succession from beneath her rings of horn; and finally, with one last lick of the tongue, one last wave of antennae, she will go as suddenly as she came, and disappear in the crowd. Another will at once take her place, continue the work at the point where the first one has left it, add on her own, change and adjust whatever may seem to offend the ideal plan of the tribe, then vanish in her turn, to be succeeded by a third, a fourth, and a fifth, all appearing unexpectedly, suddenly, one after the other, none completing the work, but each bringing her share to the task in which all combine.
[52]
A small block of wax, formless as yet, hangs down from the top of the vault. So soon as its thickness may be deemed sufficient, we shall see another bee emerge from the mass, her physical appearance differing appreciably from that of the foundresses who preceded her. And her manner displays such settled conviction, her movements are followed so eagerly by all the crowd, that we almost might fancy that some illustrious engineer had been summoned to trace in the void the site of the first cell of all, from which every other must mathematically depend. This bee belongs to the sculptor or carver class of workers; she produces no wax herself and is content to deal with the materials others provide. She locates the first cell, scoops into the block for an instant, lays the wax she has removed from the cavity on the borders around it; and then, like the foundresses, abruptly departs and abandons her model. Her place is taken at once by an impatient worker, who continues the task that a third will finish, while others close by are attacking the rest of the surface and the opposite side of the wall; each one obeying the general law of interrupted and successive labour, as though it were an inherent principle of the hive that the pride of toil should be distributed, and every achievement be anonymous and common to all, that it might thereby become more fraternal.
[53]
The outline of the nascent comb may soon be divined. In form it will still be lenticular, for the little prismatic tubes that compose it are unequal in length, and diminish in proportion as they recede from the centre to the extremities. In thickness and appearance at present it more or less resembles a human tongue whose sides might be formed of hexagonal cells, contiguous, and placed back to back.
The first cells having been built, the foundresses proceed to add a second block of wax to the roof; and so in gradation a third and a fourth. These blocks follow each other at regular intervals so nicely calculated that when, at a much later period, the comb shall be fully developed, there will be ample space for the bees to move between its parallel walls.
Their plan must therefore embrace the final thickness of every comb, which will be from eighty-eight to ninety-two hundredths of an inch, and at the same time the width of the avenues between, which must be about half an inch, or in other words twice the height of a bee, since there must be room to pass back to back between the combs.
The bees, however, are not infallible, nor does their certainty appear mechanical. They will commit grave errors at times, when circumstances present unusual difficulty. They will often leave too much space, or too little, between the combs. This they will remedy as best they can, either by giving an oblique twist to the comb that too nearly approaches the other, or by introducing an irregular comb into the gap. "The bees sometimes make mistakes," Reaumur remarks on this subject," and herein we may find yet another fact which appears to prove that they reason."
[54]
We know that the bees construct four kinds of cells. First of all, the royal cells, which are exceptional, and contrived somewhat in the shape of an acorn; then the large cells destined for the rearing of males and storing of provisions when flowers super-abound; and the small cells, serving as workers' cradles and ordinary store-rooms, which occupy normally about four-fifths of the built-over surface of the hive. And lastly, so as to connect in orderly fashion the larger cells with the small, the bees will erect a certain number of what are known as transition cells. These must of necessity be irregular in form; but so unerringly accurate are the dimensions of the second and third types that, at the time when the decimal system was established, and a fixed measure sought in nature to serve as a starting-point and an incontestable standard, it was proposed by Reaumur to select for this purpose the cell of the bee.*
*It was as well, perhaps, that this standard was not adopted. For although the diameter of the cells is admirably regular, it is, like all things produced by a living organism, not _mathematically_ invariable in the same hive. Further, as M. Maurice Girard has pointed out, the apothem of the cell varies among different races of bees, so that the standard would alter from hive to hive, according to the species of bee that inhabited it.
Each of the cells is an hexagonal tube placed on a pyramidal base; and two layers of these tubes form the comb, their bases being opposed to each other in such fashion that each of the three rhombs or lozenges which on one side constitute the pyramidal base of one cell, composes at the same time the pyramidal base of three cells on the other. It is in these prismatic tubes that the honey is stored; and to prevent its escaping during the period of maturation,--which would infallibly happen if the tubes were as strictly horizontal as they appear to be,--the bees incline them slightly, to an angle of 4 deg or 5 deg.
"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."
*Reaumur suggested the following problem to the celebrated mathematician Koenig: "Of all possible hexagonal cells with pyramidal base composed of three equal and similar rhombs, to find the one whose construction would need the least material." Koenig's answer was, the cell that had for its base three rhombs whose large angle was 109 deg 26', and the small 70 deg 34'. Another savant, Maraldi, had measured as exactly as possible the angles of the rhombs constructed by the bees, and discovered the larger to be 109 deg 28', and the other 70 deg 32'. Between the two solutions there was a difference, therefore, of only 2'. It is probable that the error, if error there be, should be attributed to Maraldi rather than to the bees; for it is impossible for any instrument to measure the angles of the cells, which are not very clearly defined, with infallible precision. The problem suggested to Koenig was put to another mathematician, Cramer, whose solution came even closer to that of the bees, viz., 109 deg 28 1/2' for the large angle, and 70 deg 31 1/2' for the small.
[56]
I myself do not believe that the bees indulge in these abstruse calculations; but, on the other hand, it seems equally impossible to me that such astounding results can be due to chance alone, or to the mere force of circumstance. The wasps, for instance, also build combs with hexagonal cells, so that for them the problem was identical, and they have solved it in a far less ingenious fashion. Their combs have only one layer of cells, thus lacking the common base that serves the bees for their two opposite layers. The wasps' comb, therefore, is not only less regular, but also less substantial; and so wastefully constructed that, besides loss of material, they must sacrifice about a third of the available space and a quarter of the energy they put forth. Again, we find that the trigonae and meliponae, which are veritable and domesticated bees, though of less advanced civilisation, erect only one row of rearing-cells, and support their horizontal, superposed combs on shapeless and costly columns of wax. Their provision-cells are merely great pots, gathered together without any order; and, at the point between the spheres where these might have intersected and induced a profitable economy of space and material, the meliponae clumsily insert a section of cells with flat walls. Indeed, to compare one of their nests with the mathematical cities of our own honey-flies, is like imagining a hamlet composed of primitive huts side by side with a modern town; whose ruthless regularity is the logical, though perhaps somewhat charmless, result of the genius of man, that to-day, more fiercely than ever before, seeks to conquer space, matter, and time.
[57]
There is a theory, originally propounded by Buffon and now revived, which assumes that the bees have not the least intention of constructing hexagons with a pyramidal base, but that their desire is merely to contrive round cells in the wax; only, that as their neighbours, and those at work on the opposite side of the comb, are digging at the same moment and with the same intentions, the points where the cells meet must of necessity become hexagonal. Besides, it is said, this is precisely what happens to crystals, the scales of certain kinds of fish, soap-bubbles, etc., as it happens in the following experiment that Buffon suggested. "If," he said, "you fill a dish with peas or any other cylindrical bean, pour as much water into it as the space between the beans will allow, close it carefully and then boil the water, you will find that all these cylinders have become six-sided columns. And the reason is evident, being indeed purely mechanical; each of the cylindrical beans tends, as it swells, to occupy the utmost possible space within a given space; wherefore it follows that the reciprocal compression compels them all to become hexagonal. Similarly each bee seeks to occupy the utmost possible space within a given space, with the necessary result that, its body being cylindrical, the cells become hexagonal for the same reason as before, viz., the working of reciprocal obstacles." [58]
