Leibniz, Darwin & the beehive

 1859

“Cell-making instinct of the Hive-Bee. I will not here enter on minute details on this subject, but will merely give an outline of the conclusions at which I have arrived. He must be a dull man who can examine the exquisite structure of a comb, so beautifully adapted to its end, without enthusiastic admiration. We hear from mathematicians that bees have practically solved a recondite problem, and have made their cells of the proper shape to hold the greatest possible amount of honey, with the least possible consumption of precious wax in their construction. It has been remarked that a skilful workman, with fitting tools and measures, would find it very difficult to make cells of wax of the true form, though this is perfectly effected by a crowd of bees working in a dark hive. Grant whatever instincts you please, and it seems at first quite inconceivable how they can make all the necessary angles and planes, or even perceive when they are correctly made. But the difficulty is not nearly so great as it at first appears: all this beautiful work can be shown, I think, to follow from a few very simple instincts.

I was led to investigate this subject by Mr. Waterhouse, who has shown that the form of the cell stands in close relation to the presence of adjoining cells; and the following view may, perhaps, be considered only as a modification of this theory. Let us look to the great principle of gradation, and see whether Nature does not reveal to us her method of work. At one end of a short series we have humble-bees, which use their old cocoons to hold honey, sometimes adding to them short tubes of wax, and likewise making separate and very irregular rounded cells of wax. At the other end of the series we have the cells of the hive-bee, placed in a double layer: each cell, as is well known, is an hexagonal prism, with the basal edges of its six sides bevelled so as to join on to a pyramid, formed of three rhombs. These rhombs have certain angles, and the three which form the pyramidal base of a single cell on one side of the comb, enter into the composition of the bases of three adjoining cells on the opposite side. In the series between the extreme perfection of the cells of the hive-bee and the simplicity of those of the humble-bee, we have the cells of the Mexican Melipona domestica, carefully described and figured by Pierre Huber. The Melipona itself is intermediate in structure between the hive and humble bee, but more nearly related to the latter: it forms a nearly regular waxen comb of cylindrical cells, in which the young are hatched, and, in addition, some large cells of wax for holding honey. These latter cells are nearly spherical and of nearly equal sizes, and are aggregated into an irregular mass. But the important point to notice, is that these cells are always made at that degree of nearness to each other, that they would have intersected or broken into each other, if the spheres had been completed; but this is never permitted, the bees building perfectly flat walls of wax between the spheres which thus tend to intersect. Hence each cell consists of an outer spherical portion and of two, three, or more perfectly flat surfaces, according as the cell adjoins two, three or more other cells. When one cell comes into contact with three other cells, which, from the spheres being nearly of the same size, is very frequently and necessarily the case, the three flat surfaces are united into a pyramid; and this pyramid, as Huber has remarked, is manifestly a gross imitation of the three-sided pyramidal basis of the cell of the hive-bee. As in the cells of the hive-bee, so here, the three plane surfaces in any one cell necessarily enter into the construction of three adjoining cells. It is obvious that the Melipona saves wax by this manner of building; for the flat walls between the adjoining cells are not double, but are of the same thickness as the outer spherical portions, and yet each flat portion forms a part of two cells.

Reflecting on this case, it occurred to me that if the Melipona had made its spheres at some given distance from each other, and had made them of equal sizes and had arranged them symmetrically in a double layer, the resulting structure would probably have been as perfect as the comb of the hive-bee. Accordingly I wrote to Professor Miller, of Cambridge, and this geometer has kindly read over the following statement, drawn up from his information, and tells me that it is strictly correct:-

If a number of equal spheres be described with their centres placed in two parallel layers; with the centre of each sphere at the distance of radius x √ 2 or radius x 1.41421 (or at some lesser distance), from the centres of the six surrounding spheres in the same layer; and at the same distance from the centres of the adjoining spheres in the other and parallel layer; then, if planes of intersection between the several spheres in both layers be formed, there will result a double layer of hexagonal prisms united together by pyramidal bases formed of three rhombs; and the rhombs and the sides of the hexagonal prisms will have every angle identically the same with the best measurements which have been made of the cells of the hive-bee.

Hence we may safely conclude that if we could slightly modify the instincts already possessed by the Melipona, and in themselves not very wonderful, this bee would make a structure as wonderfully perfect as that of the hive-bee. We must suppose the Melipona to make her cells truly spherical, and of equal sizes; and this would not be very surprising, seeing that she already does so to a certain extent, and seeing what perfectly cylindrical burrows in wood many insects can make, apparently by turning round on a fixed point. We must suppose the Melipona to arrange her cells in level layers, as she already does her cylindrical cells; and we must further suppose, and this is the greatest difficulty, that she can somehow judge accurately at what distance to stand from her fellow-labourers when several are making their spheres; but she is already so far enabled to judge of distance, that she always describes her spheres so as to intersect largely; and then she unites the points of intersection by perfectly flat surfaces. We have further to suppose, but this is no difficulty, that after hexagonal prisms have been formed by the intersection of adjoining spheres in the same layer, she can prolong the hexagon to any length requisite to hold the stock of honey; in the same way as the rude humble-bee adds cylinders of wax to the circular mouths of her old cocoons. By such modifications of instincts in themselves not very wonderful,—hardly more wonderful than those which guide a bird to make its nest,—I believe that the hive-bee has acquired, through natural selection, her inimitable architectural powers." (Darwin)

1697

"Each thing that expresses essence or possible reality, strains towards existence; and these strainings are strong in proportion to the amount of essence or reality that the straining possibility contains. Or we could say: according to the amount of perfection it contains, for perfection is just the amount of essence. This makes it obvious that of the infinite combinations of possibilities and possible series, the one that exists is the one through which the most essence or possibility is brought into existence. A good rule to follow in practical affairs is:

always aim to get the most out of the least, that is, try for the maximum effect at the minimum cost, so to speak.

For example, in building on a particular plot of ground (the ‘cost’), construct the most pleasing building you can, with the rooms as numerous as the site can take and as elegant as possible. Applying this to our present context: given the temporal and spatial extent of the world—in short, its capacity or receptivity—fit into that as great a variety of kinds of thing as possible.

A different and perhaps better analogy is provided by certain games, in which all the places on the board are supposed to be filled in accordance with certain rules; towards the end of such a game a player may find that he has to use some trick if he is to fill certain places that he wants to fill. If he succeeds in filling them, but only by resorting special measures, he has achieved a maximal result but not with minimal means. In contrast with this, there is a certain procedure through which he can most easily fill the board, thus getting the same result but with minimal ‘cost’.

Other examples of the power of ‘minimal cost’: if we are told ‘Draw a triangle’, with no other directions, we will draw an equilateral triangle; if we are told ‘Go from the lecture hall to the library’, without being told what route to take, we will choose the easiest or the shortest route. Similarly, given that existence is to prevail over nonexistence, i.e. something is to exist rather than nothing, i.e. something is to pass from possibility to actuality, with nothing further than this being laid down, it follows that there would be as much as there possibly can be, given the capacity of time and space (that is, the capacity of the order of possible existence). In short, it is just like tiles arranged so as to get down as many as possible in a given area.

From this we can now understand in a wonderful way how the very origination of things involves a certain divine mathematics or metaphysical mechanism, and how the ‘maximum’ of which I have spoken is determined. The case is like that in geometry, where the right angle is distinguished from all other angles; or like the case of a liquid placed in something of a different kind—specifically, held by something solid but flexible, like a rubber balloon—which forms itself into a sphere, the most capacious shape; or—the best analogy—like the situation in common mechanics where the struggling of many heavy bodies with one another finally generates the motion yielding the greatest descent over-all. For just as all possibles strain pari jure for existence in proportion to how much reality they contain, so too all heavy things strain pari jure to descend in proportion to how heavy they are; and just as the latter case yields the motion that contains as much descent of heavy things as is possible, the former case gives rise to a world in which the greatest number of possibles is produced.

So now we have physical necessity derived from metaphysical necessity. For even if the world isn’t metaphysically necessary, in the sense that its contrary implies a contradiction or a logical absurdity, it is physically necessary or determined, in the sense that its contrary implies imperfection or moral absurdity. And just as the source of what essences there are is possibility, so the source of what exists is perfection or degree of essence (through which the greatest number of things are compossible). This also makes it obvious how God, the author of the world, can be free even though everything happens determinately. It’s because he acts from a principle of wisdom or perfection, which doesn’t make it necessary for him to act as he does but makes it certain that he will act in that way. It is only out of ignorance that one is in a state of indifference in which one might go this way and might go that; the wiser someone is, the more settled it is that he will do what is most perfect." (Leibniz)