Part 8

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."

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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."

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"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.

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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.

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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."

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These reciprocal obstacles, it would seem, are capable of marvellous achievement; on the same principle, doubtless, that the vices of man produce a general virtue, whereby the human race, hateful often in its individuals, ceases to be so in the mass. We might reply, first of all, with Brougham, Kirby and Spence, and others, that experiments with peas and soap-bubbles prove nothing; for the reason that in both cases the pressure produces only irregular forms, and in no wise explains the existence of the prismatic base of the cells. But above all we might answer that there are more ways than one of dealing with rigid necessity; that the wasp, the humble-bee, the trigonae and meliponae of Mexico and Brazil achieve very different and manifestly inferior results, although the circumstances, and their own intentions, are absolutely identical with those of the bees. It might further be urged that if the bee's cell does indeed follow the law that governs crystals, snow, soap-bubbles, as well as Buffon's boiled peas, it also, through its general symmetry, disposition in opposite layers, and angle of inclination, obeys many other laws that are not to be found in matter. May we not say, too, of man that all his genius is comprised in his fashion of handling kindred necessities? And if it appear to us that his manner of treating these is the best there can possibly be, the reason only can lie in the absence of a judge superior to ourselves. But it is well that argument should make way for fact; and indeed, to the objection based on an experiment, the best reply of all must be a counter-experiment.

In order to satisfy myself that hexagonal architecture truly was written in the spirit of the bee, I cut off and removed one day a disc of the size of a five-franc piece from the centre of a comb, at a spot where there were both brood-cells and cells full of honey. I cut into the circumference of this disc, at the intersecting point of the pyramidal cells; inserted a piece of tin on the base of one of these sections, shaped exactly to its dimensions, and possessed of resistance sufficient to prevent the bees from bending or twisting it. Then I replaced the slice of comb, duly furnished with its slab of tin, on the spot whence I had removed it; so that, while one side of the comb presented no abnormal feature, the damage having been repaired, the other displayed a sort of deep cavity, covering the space of about thirty cells, with the piece of tin as its base. The bees were disconcerted at first; they flocked in numbers to inspect and examine this curious chasm; day after day they wandered agitatedly to and fro, apparently unable to form a decision. But, as I fed them copiously every evening, there came a moment when they had no more cells available for the storage of provisions. Thereupon they probably summoned their great engineers, distinguished sculptors, and wax-workers, and invited them to turn this useless cavity to profitable account.

The wax-makers having gathered around and formed themselves into a dense festoon, so that the necessary heat might be maintained, other bees descended into the hole and proceeded solidly to attach the metal, and connect it with the walls of adjacent cells, by means of little waxen hooks which they distributed regularly over its surface. In the upper semicircle of the disc they then began to construct three or four cells, uniting these to the hooks. Each of these transition, or accommodation, cells was more or less deformed at the top, to allow of its being soldered to the adjoining cell on the comb; but its lower portion already designed on the tin three very clear angles, whence there ran three little straight lines that correctly indicated the first half of the following cell.

After forty-eight hours, and notwithstanding the fact that only three bees at a time were able to work in the cavity, the entire surface of the tin was covered with outlined cells. These were less regular, certainly, than those of an ordinary comb; wherefore the queen, having inspected them, wisely declined to lay any eggs there, for the generation that would have arisen therefrom would necessarily have been deformed. Each cell, however, was a perfect hexagon; nor did it contain a single crooked line, a single curved figure or angle. And yet the ordinary conditions had all been changed; the cells had neither been scooped out of a block, according to Huber's description, nor had they been designed within a waxen hood, and, from being circular at first, been subsequently converted into hexagons by the pressure of adjoining cells, as explained by Darwin. Neither could there be question here of reciprocal obstacles, the cells having been formed one by one, and their first lines traced on what practically was a bare table. It would seem incontestable, therefore, that the hexagon is not merely the result of mechanical necessities, but that it has its true place in the plans, the experience, the intellect and will of the bee. I may relate here another curious instance of the workers' sagacity: the cells they built on the tin had no other base than the metal itself. The engineers of the corps had evidently decided that the tin could adequately retain the honey; and had considered that, the substance being impermeable, they need not waste the material they value so highly by covering the metal with a layer of wax. But, a short time after, some drops of honey having been placed in two of these cells, the bees discovered, in tasting it, that the contact of the metal had a deteriorating effect. Thereupon they reconsidered the matter, and covered over with wax the entire surface of the tin.

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Were it our desire to throw light upon all the secrets of this geometric architecture, we should have more than one curious question still to consider; as for instance the shape of the first cells, which, being attached to the roof, are modified in such a manner as to touch the roof at the greatest possible number of points.

The design of the principal thoroughfares is determined by the parallelism of the combs; but we must admire the ingenious construction of alleys and gangways through and around the comb, so skilfully contrived as to provide short cuts in every direction and prevent congestion of traffic, while ensuring free circulation of air. And finally we should have to study the construction of transition cells, wherein we see a unanimous instinct at work that impels the bees at a given moment to increase the size of their dwellings. Three reasons may dictate this step: an extraordinary harvest may call for larger receptacles, the workers may consider the population to be sufficiently numerous, or it may have become necessary that males should be born. Nor can we in such cases refrain from wondering at the ingenious economy, the unerring, harmonious conviction, with which the bees will pass from the small to the large, from the large to the small; from perfect symmetry to, where unavoidable, its very reverse, returning to ideal regularity so soon as the laws of a live geometry will allow; and all the time not losing a cell, not suffering a single one of their numerous structures to be sacrificed, to be ridiculous, uncertain, or barbarous, or any section thereof to become unfit for use. But I fear that I have already wandered into many details that will have but slender interest for the reader, whose eyes perhaps may never have followed a flight of bees; or who may have regarded them only with the passing interest with which we are all of us apt to regard the flower, the bird or the precious stone, asking of these no more than a slight superficial assurance, and forgetting that the most trivial secret of the non-human object we behold in nature connects more closely perhaps with the profound enigma of our origin and our end, than the secret of those of our passions that we study the most eagerly and the most passionately.

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And I will pass over too--in my desire that this essay shall not become too didactic--the remarkable instinct that induces the bees at times to thin and demolish the extremity of their combs, when these are to be enlarged or lengthened; though it must be admitted that in this case the "blind building instinct" fails signally to account for their demolishing in order that they may rebuild, or undoing what has been done that it may be done afresh, and with more regularity. I will content myself also with a mere reference to the remarkable experiment that enables us, with the aid of a piece of glass, to compel the bees to start their combs at a right angle; when they most ingeniously contrive that the enlarged cells on the convex side shall coincide with the reduced cells on the concave side of the comb.