BEE-spoke-data/bees-v0 huggingface dataset:
But the space remaining between the two oblique furrows and the upper extremity of the cavity (bef), being limited on one side by the sides of the obtuse angle (bef) formed by these oblique furrows, and on the other by the sides of the obtuse angle (bef), formed by the upper edges of the cavity, and these sides and angles being equal, it produced a rhomboid (bef) similar to those of which the pyramidal bases are composed.
This piece, by its inclination, formed an angle with each of the trapezes, and consequently also, with the two trapezes united, a solid angle (fig. 19) the summit of which was placed at the junction of the three furrows, or similarly, behind the bifurcation of the opposite ridges (c, fig. 19 and 20%); but this solid angle was not a pyramidal base, it was a base composed of two trapezes and one rhomb.
This is the manner in which the bees work to form the base of the first anterior cell of the first row.
We have shown that they carved behind the projecting ridges, on the posterior face, two adjacent cavities separated only by a ridge (fig. 10); that they had determined the length and shape of these cavities by establishing two oblique ridges on their upper edge (fig. 12), and carved a furrow of the whole length of these cavities (fig. 18).
Thus they had divided them into two equal parts, and when the pieces to the right and left of the furrow were smoothed by their labor, these two pieces together made an angular plane (fig. 20).
They were of equal size, and as one of them was backed by one of the trapezes of the anterior cell, and as it was limited by the ridges, the shadow of which might have served as pattern for the bees working on the opposite side, it resulted that these two pieces, similar and equal to those of the anterior face, were themselves equal and similar to each other. The bases of the cells of the posterior face, in the first row, were then composed of two trapezes, such as we had already described, and this composition was a natural consequence of the method of the bees at the beginning of their work.
The three bases of cells which I have just described were the first built; but while they carved the furrows which separated them, others having lengthened the block in every direction, they were enabled to carve other cavities. They began their carving behind the vertical ridges of cells 2 and 3, and to the side of cavity Ndeg1, then on the posterior face, behind the opposite ridges: so that the trapezes were backed by other trapezes of similar shape and size (fig. 21 and 22). Usually, they dug out a face while others were making ridges on the opposite. So they formed cavities behind the lateral edges of the last built cells. Thus several bases, alternating on the two faces of the block, presented a first row of contiguous cells the tubes of which were not yet lengthened.
While these bees were polishing and perfecting these bases, other workers began the outlines of a second row of cells above the first, and partly behind the rhomb of the anterior cells; for their work follows a combined progress. We cannot say: _"When the bees had finished this cell they began new ones"_; but we can say: _"While some workers advance a certain piece, others outline adjacent cells."_ Moreover, the work made upon one face of the comb is a part of what is needed on the opposite face: it is a reciprocal relation, a mutual connection of the parts rendering them subservient to one another. It is therefore undoubted that a slight irregularity of their work on one side will affect the shape of the cells on the other side in a similar way.
Section III
#### Construction of the cells of the second row
The bottoms of the anterior cells of the first row, composed of two trapezes and one rhomb, were larger, as we noted, than those of the cells opposite or back of them, since the latter were composed of only two trapezes; so there was more space between the upper edge of the posterior cells and the edge of the block, than there was above the anterior cavities; this space was great enough for the base of an ordinary cell (fig. 20 and 22); but there was not room for a complete bottom above the anterior cells (fig. 19). The rough space between and above the posterior cavities in the opening of the angle of their oblique edges, extended much beyond their points (from be up to n, fig. 20: or be up to r, fig. 22). It was in that spot that several bees began the sketch of a new cell (fig. 22).
The first bee excavated a vertical fluting (fm, bp) in the space between the oblique edges (fc cb) of two neighboring cells and made edges for this new cavity, by accumulating the wax from this cavity to the right and to the left. In figures 23 and 27 the cell in question has been isolated so that we may better follow its development. The letters are the same as on figure 22. The contour (fmbp, fig. 23) shows the beginning of its outline. Its vertical ridges (fmbp) were exactly above the points of the two inferior cells Ndeg 2 and 3. These ridges started from these points and ascended vertically along the cavity up to a short distance from the edge of the block, which did not extend any farther than needed for the building of the entire base of the cell: the fluting was still terminated by a curved outline (prm, fig. 23). But other bees established two rectilinear margins on its curvature, which, united as two chords in the center of the arch, formed the obtuse angle (mrp, fig. 22). This cavity was then bordered by six ridges; the two lower ones (fc cb) belonging to the cells of the first row, Ndeg 2 and 3, between which the lower edge of this cell was placed; the lateral ridges (fm bp) parallel to each other ascending vertically, lastly the two upper ridges (rm pr) terminated the contour, joining one another and also joining the others at their lower extremity. These six ridges, equal in length, formed the hexagonal contour of the cell; but this contour was not uniform in its projection; it was higher at the angles (cpm) and depressed at the points (bfr) as may be seen more plainly in fig. 28, which displays the same cell at an angle.
The lower part, still unfinished (fcbe, fig. 23) of the space comprised between the six ridges was backed by the rhomb of cell Ndeg 1, since cells 2 and 3, above which the hexagon was formed, were themselves backed partly by this cell. The rhomb, inclined from the horizontal, but whose main diagonal was horizontal shown on the side of cell Ndeg 1 (c, fig. 21), exhibited its lower side. After the bees had outlined and made edges on the bottom of the hexagonal cell, they busied themselves in smoothing down the other face of this rhomboidal piece, and lined it with the furrows (fe and eb) which corresponded to the ridges of the same name on the anterior face.
This piece was then a rhomb, and this rhomb (fcbe, fig. 22) seen from above, was the first and upper piece of a pyramidal base.
It occupied one-third of the surface of the cavity, for the obtuse angle (feb) being in the center, and its edges resting upon the two edges (fe and cb), which formed one-third of the circumference, it is easily conceived that the entire bottom of the cell was to be three times that of this rhomb. In fact there was above the rhomb, inside of the hexagon, enough space to admit of two rhombs similar to this one, but turned differently.
This part of the base of the cell remained in the same condition until the work upon the opposite face was far enough advanced to allow the bees to build a ridge on the reverse face of the cell in a vertical direction (er, fig. 21): (which could take place only after they had outlined the two cavities opposite the hexagonal cell). But after this ridge was established on the anterior face, behind the piece to be divided, one bee busied herself in making the base of the hexagonal cavity in that direction and made a furrow in the still unfinished space (er, fig. 22), from the upper edge of the rhomb, to the upper edge of the hexagon; when the two parts of this were finished, we saw that she had perfected two additional rhombs (ferm and erbb) equal to the rhomb (fcbe). Thus the six edges of the hexagonal contour enclosed three rhombs of equal size; or a pyramidal base complete: the first of them having been built on the lower edge of the block. It is easy to understand that, during this operation, other cells were outlined to the right and left, upon the adjacent cells of the first row, and there is no need of any further explanation in this regard since the work is similar to that which we have just described.
The block was still further enlarged by the wax producers and there was still more space above the cells of the first row for the construction of new cells (fig. 21). The space between the cells and the drawn line shows, on fig. 22, that this space was not yet filled during the building of the posterior cell of the second row, but it was filled when they built the anterior hexagonal cell.
A bee placed herself on the anterior face, so as to work in the unfinished space between the points of two cells of the first row, designated by numbers 1 and 4, between their oblique edges (fe fv). This bee immediately carved a furrow above the vertical edge that separated them, in a space equal to the diameter of an ordinary cell, but this space was already limited below by the cells of the first row. The bee gave to this cavity a fluting shape, its edges were made by two vertical ridges (er vn), and its upper edge, first made in a circle (fig. 25) was later changed by two other workers into straight ridges (on ov, fig. 21); making an obtuse angle; thus the cavity became hexagonal as with the opposite side, which was backed by it.
The dividing of this cell, (fig. 21 and 25) did not appear to cause the bees any trouble: the pieces which composed it were already partly shaped on the posterior face; on that side, two contiguous cells had between them a ridge (fm, fig. 22), which was to serve as guide: its shadow divided into two equal parts the inferior portion of the hexagon cavity. One could also
see the shadow of the oblique ridges of those two posterior cells, from the center m of the cell, one to the right and the other to the left, running towards vertical ridges at r and n (fig. 21).
This cavity appeared to be divided into three equal parts by the shadow of the posterior ridges. What we discerned in shadows was soon made real by the work of the bees; the shadows were converted into furrows opposite the ridges; the interval and the edge of the cell were smoothed until they showed distinct rhombs; but the bees, while shaping the vertical furrow divided the inferior part of the cavity, and the two first rhombs of the pyramidal base were formed to the right and left; then by following the oblique ridges of the posterior cells, they formed a third rhomb situated in the upper part of the cavity, and inclined similarly to the one of cavity N* 1.
This rhomb (onrm, fig. 21) was not opposite any outlined cell on the posterior face; it was backed by still crude material, comprised between the upper sides (rm mn) of two cells of the second row; so this space was to be later a part of the posterior face of a cell of the third row.
The result of the work of the bees, within the hexagonal cavity, was another pyramidal base; it differed in no way from the bases of the same row on the posterior face, which backed it, except in the position of its rhombs.
From this it is easy to conceive the manner of building of additional cells; they are always located between the upper oblique sides of the neighboring cells; above their points the bees will build vertical ridges which will be the right and left borders of the new cavity; then the contour will be made by erecting two other ridges horizontally, on the upper edge of the fluting, which will produce an hexagonal contour.
The inferior part of these cavities will always correspond with the walls of the opposite cells, that is why all the cells of that side will be divided by two rhombs in the lower part and one rhomb in the upper. (Of course it is the reverse in combs built downwards; so we must invert the figures in order to follow the natural order, as the work must be the same and the result similar: the building of pyramidal bases).
The posterior cells are all formed on the model of those which we have described; one rhomb in the lower part and two rhombs in the upper. The anterior hexagonal cells are all situated a little higher than the posterior ones, for their lower part corresponds with the upper rhombs of the opposite cells.
We still have to make a few remarks concerning the difference between the pyramidal bases and the bases of the first row: the latter are composed, as we have shown, of two trapezes and a rhomb, or two trapezes only, which ascend perpendicularly from the lat, so the position differs from those of the pieces of pyramidal bases; as the three pieces of a pyramidal base start from the top of the pyramid towards the edge which outlines the contour of its base, they are all three inclined forward in the same proportion. We have taken for granted, in our former explanations, that the furrow at the base of a cell was vertical. It was simply to show that it appeared to ascend vertically, when seen from its face. But when we cut the block vertically, we see that the furrow is oblique, since it starts from the base of thecavity to reach its edge; fig. 24, 28 represent a posterior and an anterior pyramidal base; none of their edges or furrows are vertical. It is not the same with the trapezes of the base of the first row, they are actually vertical no matter on which side we view them.
The result is that the union of the latter with the oblique rhomb of the cells of the anterior face must be in a slightly different angle from those of a pyramidal base.
Each of the six ridges which form the edges of a pyramidal base is to serve as basis for one of the six sides of a cell. The four sides of those of the first row (fig. 31) are also fixed in the same way around their base. The prisms which are a result of the union of the sides are fastened to the edges of the cavities made in the block.
At first sight, it would seem that there is nothing plainer than the adding of wax upon the edges of the contour of a cell base, but owing to the unevenness of this rim, which we have noticed, and which produces three projections and three recesses for the pyramidal bases, a projection for each of the bases of the anterior cells of the first row, and a recess for the posterior ones; on account of this unevenness the bees must begin by filling out what is lacking, by adding more wax on the shallower edges than they put on the projecting ones; in this way all the edges of the cells offer a uniform surface from the beginning and before the cells have acquired their proper dimensions. But the surface of a new comb is not entirely plane, for there is a progressive decrease in the work of the bees. The walls of the cells are prolonged in an order corresponding with the completion of the bases to which the cells belong (fig. 30), and the length of the tubes is so regulated that there is no gap or conspicuous irregularity among them. Thus the surface of a new comb is lenticular (fig. 29, 30, 31); the thickness of it decreasing to the edges, because the lately outlined cells are shorter than the older ones. This progression continues in the comb while it is increased in size; but as soon as the bees lack space for its prolongation, it loses this lenticular shape and becomes parallel. The bees form all the cells of the same depth by bringing the newest to the depth of the first built; the comb has then reached the shape that it is to preserve; but it is not yet fully completed; we will show later the termination of the work.
How can we explain the combination of their operations: why does instinct lead them to give a different form and different dimensions to the bottom of the first rows of either side, which have so great an influence on the remainder of the comb? How can the bees working on one face of a comb determine the space to be excavated for the invariable mutual relation of the bases? We should clear up this question, as the remainder of the problem depends upon it.
We do not see the bees visiting the two surfaces alternately, to compare the respective position of the cavities which they outline; nature did not instruct them to take measurements which would appear to us indispensable for the construction of symmetrical work: these insects content themselves with feeling the part to be hollowed out with their antennae, and appear by this means sufficiently apprised, to execute a very complicated work, in which all seems to be combined with great exactness.
Not a parcel of wax is removed before their antennae have felt (palped) the surface to be sculptured. Bees do not trust to their eyes for any of their operations; but with the help of their antennae, they can build, in darkness, those combs which are deservedly regarded as the most admirable production of insects. This organ is so flexible an instrument that it lends itself to the examination of the most delicate parts of the most distorted pieces; it may be used by them as compasses in the measurement of the smallest objects, such as the edge of a cell.
Bees, therefore, appear to be regulated in their work by some local circumstance; we noticed that, while sketching the bottoms of the first cells, before there was any ridge on the reverse, they sometimes caused such a ridge to appear on the opposite surface, by the simple pressure of their legs upon the wax still soft and flexible, or by the efforts made with their teeth to excavate the interior of the block. Sometimes it occasioned a breach of the partition, which was soon repaired, but a slight protuberance always remains on the opposite surface, which may serve as guide for the bees working upon that side. They place themselves to the right and left of this ridge to begin a new excavation, and heap up a part of the materials between the two flutings resulting from their work.
This ridge, converted into a real rectilinear rib, becomes a guide for the bees to recognize the direction to be followed in making the vertical furrow of the anterior cell.
We have often conceived, when seeing these insects follow so accurately the reverse of the ridges to excavate the corresponding furrows, that they perceive the greater or less thickness of the partition, from its flexibility, elasticity, or from some other physical property of beeswax: whatever may be the case, it is certain that they give the bases of the cells a uniform thickness, without having any mechanical means of measuring them; for the same reason, they may perceive very distinctly whether there is a ridge behind the partition, and excavate into it until they have reached the point which must not be exceeded.
I do not wish to present these explanations with more positiveness than as simple hypotheses. I desired to show the linking of the operations of the bees, but I have not undertaken to discover the secret causes of their actions.
I believe however that they may be explained without resorting to extraordinary causes. The length of the cavitles, their respective position and the thickness of the block once determined, the inclination of the oblique sides of the trapezes of the first row, to which that of the lozenges of the second is subordinate, is established of itself, without any need of instruments for measuring angles or without calculation on the part of the bees.
What we need to understand is the manner in which they establish the relation between the unequal cells of the first row. One of the contributing causes for securing these dimensions, upon which so many important conditions depend, is the manner in which the block is enlarged.
Its original height determines about the vertical diameter of the posterior cavities, which is equal to two-thirds that of a common cell. But theycannot complete the bottom of the anterior cell until the block is enlarged, which on that account is extended more than necessary for finishing the cell, but just enough to give sufficient space for the entire base of a posterior cell of the second row; for the rhomb making part of it is already comprised in the interval of the cells formed of trapezes. The bees, adding to the block two-thirds of the diameter of a cell, are enabled to construct, on the anterior face, the bottoms of cells of the second row, a part of which is already intercepted between the upper edges of the first cells; but there will not be room yet for the construction of a third row, until the block shall be enlarged anew.
The bees cannot deviate from the prescribed rule, unless particular circumstances alter the bases of their work, for the block is always lengthened in uniform quantity; and what is admirable, this is done by the wax producers, the depositories of its first elements, who do not have the faculty of sculpturing the cells.
Thus in dividing the functions between the wax producers and the small bees, the author of nature appears to have distrusted the exclusive lights of instinet.
What simplicity and depth in the means, what linking of causes and effects! It is a diminutive image of this harmony which strikes us in the great works of creation.
Such processes could not be foreseen. One does not divine the ways of nature, it lays out methods that confound our science, and it is only by studying it carefully that we may succeed in unveiling some of its mysteries.
From the facts just described, should we not draw the conclusion that the geometry which shines in the work of the bees is rather the result of their operations than the principle that causes them?
Our readers will doubtless share the pleasure which we experienced when we received the following communication, in which one perceives a singular connection between the geometrical solution supplied by a skilful mathematician, and the work of the bees, such as we have exhibited from our observations.
The bases of the cells of the first row, which determine the inclination of the rhombs of the entire comb, represent two sides of a prism cut so as to form three angles equal to the rhomboidal plane which they intercept. One might believe that the bees succeed in constructing their cells by the sole knowledge that they have of the required section of the prism, and the solution supplied by Mr. Le Sage shows it to be more simple than one had thought.
We find great satisfaction in recalling here the labors which are too little known, of a savant who is beloved of his countrymen, and we are authorized to state that the project of publishing his principal works has not been overlooked by Professor Prevost, of Geneva, who mentioned it in the preface of his "Notice upon the life and writings of Le Sage, 1 vol. in-18 at J. J. Paschoud, Paris and Geneva.
## Part 1 Introduction
In 1781, Mr. Lhuilier sent to Mr. DeCastillon a memoir upon the "minimum of wax of the bees", which was read at the Academy of Berlin and inserted in his Memoirs for the same year. This learned mathematician gives in it, in a few words, the history of the researches made upon this subject, by Maraldi, Reaumur, Koenig, &c., and treats the subject by a more simple method than had yet been done in previous works, since he reduces the problem to a few elementary propositions.
In this memoir he gives honorable mention to Mr. Le Sage. He mentions him also in a later work and gives more details on the mathematical process through which this philosopher solved the problem relating to the shape of the base of the cells constructed by the bees. He informs us, that Le Sage was, to his knowledge, the first to treat this matter in an elementary way; that he had treated it algebraically; that he had employed, for that purpose, a method easily applicable to all the problems which do not exceed the second degree; a method which had been kindly communicated to Mr. Lhuilier ten years previous to the time when he published his own.