BEE-spoke-data/bees-v0 huggingface dataset:
The memoir of Mr. Lhuilier contains not only the solution of the problem relative to the construction of the rhomboidal bases, so as to obtain, for a given cell, the least expense of wax, but also that of the problem relative to the "minimum minimorum," or to the shape of the cell of equal capacity that would cost the least, as well as other remarks regarding this subject. It ends with a memorandum from Mr. de Castillon, relative to the actual dimension of the cells of the bees.
This memoir and the latin work following that which I have mentioned, having been published for a long time and being consequently within reach of those who occupy themselves with these matters, it is sufficient to refer to them. But it may give them pleasure to find here an outline of the first elementary work which has been undertaken to solve the problem of the rhomboidal bases of the cells. A paper, written in Mr. Le Sage's own hand, and of very ancient date, presents this work in a very simple form. It was drawn from one of his portfolios in which he had gathered the material for a bulletin which is mentioned in the account of his life. We shall transcribe it here without change, up to the second note which we will replace by an explanation.
Notice of G. Le Sage on the base of the cells:
"Given the mutual inclination of the two planes, for instance 120 degrees; cut into them with a third plane, in such way that the three resulting angles be equal.
"This is a problem which even a very shallow artisan could solve with very simple instruments: for the only thing needed is to find the center of a proposed straight line; which even insects may do with their legs (a). This solves the famous problem of the _minimum,_ of which we are astonished to find the solution in the base of the bee's cell, and which consists in using the least possible quantity of wax, without diminishing the size of the cell;and for which they have needlessly used all the apparatus of the calculation of the infinite (b)."
(a) For the geometrical explanations.
"Problem. Given (Plate XI, fig. 2) the width AB of the face of a regular hexagonal prism; add to one of its edges or withdraw from it, the length AX, equal \(\sqrt{\mbox{AB}^{\mbox{ iny 2}}}\)
"Solution. Cut AB in two equal parts at C. Draw AD to equal AC. Draw CD; cut the line in equal parts at E. Draw AE, from A to X, upon AD or its prolongation.
Demonstration: (AE)'=1(AC)'=1(AB)'=1(AB)'=1i\(
rac{\mbox{(AB)}^{\mbox{ iny 2}}}{4}=
rac{\mbox{(AB)}^{\mbox{ iny 2}}}{8}\)
(b) "For the geometrical explanations."
This is the heading of the second note which we have suppressed. It was intended to demonstrate algebraically that the problem relating to the bases of cells amounts to the geometrical problem solved in the first note. Several reasons have induced us to substitute for this algebraical note a few detailed explanations. One may find, in the above mentioned works of Lhuillier (especially in the Berlin Memoirs for 1781, marginal note page 284), the method of Le Sage and his learned disciple, to determine, with the help of algebra, the minimum of expense in the construction of the bases.
Supposing the hexagonal cell to be a straight prism, the question is to cut the ridge conveniently at the desired spot. For this purpose, the rhomboidal base must be a minimum. A simple equation of the second degree, leads to the following formula: The distance from the intersecting point to the hexagonal base is equal to the half-side of this base, divided by the root of two, or, similarly, to the side divided by the root of eight.
## Chapter 5 Modifications in the Architecture of Bees.
The researches relative to the organization and development of animal productions, may not be, in spite of their importance, the most interesting in the eyes of the naturalist philosopher. Those which embrace the degrees, the resources and the limits of this faculty which serves as judgment for so numerous a class of beings, offer a still wider and more fertile field for his meditations.
The vulgar commonly believe that sensations and physical needs exercise an absolute empire over animals: unquestionably they have influence upon them in many circumstances; but it would be as difficult to explain the conduct of creatures subjected to instinct by the sole attraction of pleasure or fear of pain as it would be unfair to ascribe the virtues of feeling and reasoning beings to personal views alone, though it has often been asserted that self-interest was the only motive of our actions.
If there is a direct connection between the organization and the habits of living beings, this connection is so enigmatic in its characters that we cannot analyze it. We may notice, in physiology, a few salient points, such as the long beaks and webbed feet of some birds, by which we discern the locations they inhabit and the food which they secure; but there is a long reckoning from this to the cunning of animals, the recesses of their instinet; even if we should reason from their customary behavior, we might be drawn into error, for many of them have ingenious resources in difficult circumstances: they leave their usual routine and appear to act according to the situation in which they find themselves; that is indeed one of the most curious phenomenons of natural history.
Invariable laws, concerning the actions of animals, appear to us a subject of admiration, for the mind becomes easily accustomed to ideas of order, and readily accepts a uniform plan; but in the designs of the author of nature, there is a flexibility, a freedom which bears the imprint of supreme power; the most opposite conditions are united without jar or confusion: can we conceive that beings subject to a common law and gifted with some intellect, should be able to swerve from the letter and act accordingly? That they should be able to change their processes and modify the prescribed rules when necessary? How can we imagine that there may be exceptions to the great laws of nature, and that animals be able to act, in some circumstances as if they understood the intentions of the law-maker? Those are phenomena which no theory can explain; but we must not get false views concerning the nature of animals; do not our prejudices blind us as to the distance from their faculties to ours? This would require profound investigations, and it is in that direction that the ultimate labors of the zoologists should tend. To acquit ourselves in part of our own debt in this regard, we will describe some anomalies which we have noticed in the behavior of the bees.
I will not yet develop the consequences which seem to result from these; it will be only after making known the bulk of their operations that I will take the liberty of making a few remarks on the true place which these insects occupy in the order of life.
Everything connected with the fabrication and use of combs has been skilfully combined; cells turned downwards like those of wasps would not have suited bees, since they have to store a fluid: each comb presents innumerable small honey-pots laid horizontally; they are provided on both sides; perhaps their shape, and the affinity between the wax and the honey, prevent the latter from escaping; the combs are on parallel planes and are separated from one another by lanes only a few lines in width. It was after measuring these fairly regular distances and the average thickness of the combs that I conceived the invention of book-hives, which I always used with success.
The parallelism of the combs is not one of the least difficult subjects of explanation, nor would it be explainable, did we conceive that their foundation was laid simultaneously by a number of workers. Experience teaches us, to the contrary, that one cannot see the bees begin different blocks of wax here and there at the same time. A single worker deposits materials in an apparently suitable direction; she departs; another replaces her; the block rises; the bees sculpture its opposite faces alternately; but scarcely are some rows of cells constructed when we perceive two other blocks similar to the first, established parallel to the first, which widen and lengthen on either side. These blocks soon become combs, for bees work with astonishing speed: shortly after, we notice two others still parallel and built in relative progression according to the date of their origin; the first one, being the oldest, exceeds those parallel to its two faces by some rows of cells, and the latter exceeds those following by a similar quantity; thus the two faces of a comb are in great part concealed by those next to it.
I shall not try to explain how bees take such accurate measurements and know the direction parallel to that of the first comb. But one perceives clearly that if they were permitted to place different blocks in the roof of their hive at the same time, these outlines would be neither properly spaced nor built parallel to one another.
We see an example of the same method in the manner of outlining the cells; it is invariably a single bee that selects and determines the place of the first cavity; this one, being established, serves to direct the ulterior labors. Did each of several workers sketch a cell at the same time, the symmetry of the cells resulting from this operation would be left to chance; for these insects have no discipline and know no subordination.
A great number of bees work upon the same comb, indeed, but they are not guided by a simultaneous impulse, as might be conceived if one did not observe their operations from the outset. This impulse is successive; a single bee begins each partial operation, and several others successively join their efforts with hers to the same end; each appears to act individually in a certain direction, directed by the bees which preceded her or by the state in which she finds the work which she is to continue; and the bee that commences a new operation is led to it by the effect of a harmony which rules the progression of the work. But if anything in the conduct of the bees
## 113 Modification in the architecture of bees.
could give the idea of almost unanimous consent (which we present as a doubtful thing) it is the inaction of the rest of the colony while a single bee determines the position of the comb. Immediately, others assist her and add to the height of the block; then they again cease acting and a single individual, of a different profession, if the word may be allowed in speaking of insects, traces the first outline of the base of a cell which, by its peculiar shape prepares a different kind of work; it is a base or a fundamental plan for establishing the entire edifice. A subtle sense of feeling shows to the bees, across the partition, the situation of the margins of the cavity, and it is from this that they direct their efforts for the division of the base of the new cells: but it is not only by the means of the ridges that they discover the direction to be followed; we have ascertained that they profit by divers circumstances to guide themselves in those excavations. The bee which forms the first cell is a remarkable exception; she works in a rough mass, and thus has nothing to point out the way, instinct being her sole conductor.
The workers outlining the cavities of the second row, on the contrary, can take advantage of the edges and angles previously made on the same face, and use them as basis or point of departure for subsequent operations. I shall soon give a singular example of the art with which they take advantage of these when they have no other resource, but I shall speak first of the ordinary labor of the workers: I had seen them sculpt only upwards; I explained all their maneuvers when they are working in that direction, but the explanation of their behavior and the results which they attain then might belong to a special case.
So it was necessary to learn whether they always act in the same manner and pass through the same proceedings: they sculpt upwards with much less speed than downwards; but this circumstance was favorable to the observing of the diverse labors which the formation of the cells requires; otherwise it would have been impossible to follow the detail of their proceedings; yet the tardiness of the bees in this occurrence had its inconveniences; their work was sometimes interrupted for several hours; the wax was wanting when needed or it was not sculpted soon as deposited, or several blocks were deposited on the same lath. Their work was evidently relaxed or impeded, and it was only through the number of small combs which they built that we were enabled to overlook the irregularities of their operations and form an accurate opinion of their architecture; so it was important to learn whether the process which we had seen them follow was the same in ordinary circumstances; it was to clear this doubt that I had a hive constructed of a new shape (Plate I, fig. 6).
In order to fill my requirements, the top of this hive must be composed of several pieces, removable without disturbing the bees; it was also necessary that each part be separable and removable each time that we desired to ascertain the progress of their work. A ceiling, composed of glass and wooden slips placed alternately in a horizontal plane, was used for this purpose. A screw at each end of the slips permitted us to raise the combs above the top of the hive, so that we might examine them and return them conveniently without deranging the bees; in this way we could remove such of them as we wished to preserve and compel the bees to build others.
When they were established in their new home, they built combs alongthe wooden laths following the direction of the intersecting line between the slips and the laths.
The first block established by them gave us nothing new; we removed it and the bees built another at once; they again established it on the edge of a lath; we gave them time to carve their first cells: then by turning the screws upon which the work rested, the latter was brought up and allowed us to observe the formation of the new outlines; they showed flutings similar to those seen in the combs built upwards; we again lowered the comb and the bees continued their work. A few minutes later, the comb was again observed, the outlines had been advanced, the faces of the cells on both sides were unlike each other; they showed vertical trapezes; but only the anterior cells had a rhomb at their lower extremity; we then saw the bees proceed with the cells of the second row and became convinced that the march of their operations was in every point similar to what we had witnessed in the upward building.
We compelled the bees to begin a great number of small combs, the outlines of which, more or less advanced, showed us that they were built upon the same principles and with the same gradations as those that had been built upwards.
It is clear, in my opinion, therefore, that the particular configuration of the first cells, on both sides, determines invariably the shape of the pyramidal bases of all the subsequent cells.
One could not have foreseen that the bees, in beginning their masonry, could have taken other measurements and a different method from that which they follow for the rest of the comb. This already proves that these insects do not act altogether mechanically; however, as it might be taken for granted that there is a necessity in this order of work, I will cite a frequent example of an altogether different action.
When I compelled the bees to work upwards, they laid blocks and built combs, upon the horizontal part of the laths; but they were not always so docile. I have often seen them use wax from under their rings to stretch and extend some old combs into the space in which I desired them to build new ones (Plate IX, fig. 2 shows the lath with comb above and below).
The manner in which they do this deserves attention. To extend a comb placed under a lath and bring it up into the space above it, they begin by lengthening forward the upper edges of the cells of the first row, perpendicularly to the comb, and so that their extremities will project a little beyond the lath; after they have thus staked the ground, and fixed the points to work from, they pile wax on the vertical edge of the lath and with this wax they shape curves from the two parallel ridges of a cell and transform them into the walls of a cell. Thus they succeed in outlining regular hexagons on the vertical side of the lath, each with six ridges which will serve as bases to the cells which they will erect at that spot; these cells are flat bottomed, since they are built on the edge of the lath (Plate IX, fig. 2), but their diameter is equal to that of cells which are built upon a block of wax; if the lath is thicker than required for a single row of cells, the bees continue with additional ridges above those previously traced, until they reach the upper edge of the lath; when they build upon its surface a block which
## Modification in the architecture of bees.
they sculpt in accord with the hexagons outlined on the wood; they will then give the first row of cells the ordinary shape of cells of the first row, and three rhombs to all subsequently built cells.
We thus perceive that bees may shape cells upon wood and give them hexagon contours without pyramidal bases; thus they deviate from their usual routine, but not in the measurement of the cells and the shape of their walls, which they outline upon the wood with a symmetry that guides them in their interior work. But they do this by taking advantage of the previously built cells to form their ridges and give the curves a proper base. These flat bottom cells are less regular than the ordinary cells; we see in them some cells the contours of which are not angular, or the dimensions of which are not exact; but in all of them we notice the hexagons more or less marked.
After seeing the bees build upwards and downwards, it was natural to investigate whether we could compel them to build their combs in some other direction. We tried to confuse them by placing them in a hive with the upper and lower sides entirely glazed, so that they had no place of support except on the side walls of their dwelling.
They clustered in one of the angles of the hive, in an impenetrable mass; we were compelled to disturb them in order to see their work and found that they had built combs perpendicular to one of these walls: they were just as regular as those which they usually build under a horizontal ceiling. This was a remarkable result, since the bees, habitually sculpting downwards, were compelled to place their foundations on a plane which is unusual to them. But the cells of the first row were similar to those which they construct ordinarily, with the only difference that the ridges were built in a different direction: the other cells were none the less fit for the common uses and distributed on both faces, with the bottoms alternately corresponding with the same symmetry.
I put these bees under still greater trial: having observed that they try to build their combs, in the shortest way, towards the opposite side of the hive, I covered the latter with a pane of glass, in order to find whether they would be content with a surface which they do not usually trust, unless their cluster can hang in the close vicinity to a substance less slippery than glass. I knew that they prefer fastening their combs to wood, and that they accept glass only when they have been deprived of any other substance to strengthen their constructions. I had no doubt however that they would fasten the comb to this pane, taking chances to later strengthen it by securing more stable attachments, but I was far from suspecting what they would do.
As soon as the board was covered with this smooth and slippery surface, they deviated from the straight line which they had hitherto followed, and continued their work by bending their comb at right angle and so that the forward edge would reach one of the walls left uncovered.
Varying this experiment in several ways, I saw the bees constantly change the direction of their combs whenever I approximated a surface too smooth to admit of their clustering at the ceiling or on the sides of the hive; they always selected the direction which would bring them to thewoodens sides; I thus compelled them to curve their combs in the strangest shapes, by placing a pane at a certain distance in front of their edges.
These results indicate an admirable instinct; they denote even more than instinct; for glass is not a substance against which bees may be warned by nature; there is nothing as polished as glass or resembling glass in their natural abodes, the interior of trees. The most singular part of their work was that they did not wait till they arrived at the surface of the glass to change the direction of the combs, they selected the suitable spot beforehand; did they anticipate the inconvenience that might result from any other mode of construction? The manner in which they made an angle in the combs was no less interesting; they necessarily had to alter the ordinary fashion of their work and the dimensions of the cells; therefore those on the convex side were enlarged to two or three times the diameter of the others on the opposite face. Can we understand how so many insects occupied at once on both sides would concur in giving them the same curvature, from one end to the other; how they could decide to build small cells on one face, while upon the other face they built cells of so exaggerated dimensions; and is it not still more wonderful that they should have the art of making cells of such great discrepancy correspond between them? The bottom of the cells being common to both sides, the tubes alone assumed a taper form. Perhaps no other insect has ever supplied a more decisive proof of the resources of instinct, when compelled to deviate from the ordinary courses.
But let us observe these bees in natural circumstances, for it is not even necessary to test their instinct in order to see them modify the order of their architecture: by comparing what nature has required of them with the means employed by them in unexpected cases, we will better judge of the extent of their faculties.
Since the cells of bees are to serve as cradles to individuals of different sizes, the caliber of those tubes must be proportional to the intention of their use. Thus the workers which were to build drone cells, were to follow a larger pattern than when they build ordinary worker cells; but they gave them the same shape, their bases are also composed of three rhombs, their prisms of six walls, and their angles are equal to those of the small cells. The diameter of worker cells is 2 2/5 lines, that of drone cells is 3 1/3 lines; those dimensions are so fairly constant that some authors believed that they might be used as invariable patterns of measurement.
The drone cells rarely occupy the higher part of the combs; it is usually in the middle or in the lateral parts that they are found, and they are not isolated; they are built together and correspond with one another on the two faces of the comb.
No observations have been made of the art with which they manage to construct alternately large and small cells without too flagrant inequalities in their work. The manner in which the drone cells are surrounded can alone explain how the transition is effected: when they are to sculpt drone cells below worker cells, they build several rows of intermediate cells, whose diameter augments progressively until gaining that necessary for drone cells; for the same reason when the bees wish to return to worker size they change by progressive gradation down to the size of cells of that class. We usually see three or four rows of intermediate cells; the first drone cells still participate in the irregularity of the ridges which adjoin them, some bases corresponding to four cells instead of three. Their furrows are always in the line of the ridges, but one of the sides of the base, instead of being immediately opposite to the center of the cell on the reverse, divides it unequally, which alters the shape of the bottom, so that it no longer presents three uniform rhombs, but consists of pieces more or less irregular (See the appendix which follows).