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quired being such as theory and observation “ nearly agree” in giving to the cells, Lord Brougham replies: “The nearly' is quite incorrect : there is an absolute and perfect agreement between theory and observation.” *

Mathematicians appear to be of one accord in this ; viz. if economy of space and wax is sought, that the form of the cell should be the one alleged to have been ascertained by Maraldi, and which was really calculated by Koenig, and by hundreds of others since his time. Careful observations, however, tend to prove that such a cell is rarely, perhaps never, realized. For, while the deviations from the true form do not exceed a certain limit, a piece of comb, ten cells square, can hardly be found in which one or more irregularities do not occur, of such magnitude, that, however they may look to the bee's eye, can be readily detected by man's. The best observers, such as Reaumer, Hunter, the Hubers, and others, have noticed some of these, but as their investigations had for their chief object the clearing up of other points relating to the habits of the bee, the irregularities of the cells were passed by, for the most part, with merely a mention.

Worker Cells. — These will be treated of first, because they are the most numerous. The drones of a hive only amount at the most to a few hundreds, while the workers are estimated at many thousands, and the number of cells is proportional to the number of young reared. All the varieties found in the worker are repeated in the drone and honey cells, though in the last-mentioned kind the variations are the most marked, and some are introduced which are not found in either of the others.

The average diameter of a worker cell, measured on a line perpendicular to its sides, as deduced from the following table, is 0.201, or one fifth of an inch, but it may be increased or diminished in different parts of the same comb.f Reaumer expresses his belief that this was the case, but he gives no measurements. The table given below is the result of the examination of four pieces of comb, which were in all re

* Op. cit., p. 350.

| Reaumer found that twenty worker cells measured four inches less half of a line ; “neglecting the half of a line, the diameter of a single cell would be 2.4 lines" (French); and Huber gives the same dimensions, as also Kirby and Spence, who quote their description of everything relating to the bee from Reaumer and Huber. Latreille found that 76 millimeters comprised 14 cells, when measured in one direction, and 14.5 in another.

spects good average specimens. First, a line of ten cells,* arranged in the direction of the diameter, perpendicula to one of the sides, and then two other sets of the same number, similarly arranged in the direction of the other two diameters, and crossing the first, were carefully measured. Three series of such measurements were made from different parts of each comb. The columns marked I., II., III. give the measurements in the direction of the three diameters.

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The greatest aggregate diameter of any one series of ten cells was 2.10 inches, and the least 1.85 inches, making a difference of 0.25 inch, or the diameter of a cell and a quarter. The average difference is, however, a little less than 0.10 inch. These irregularities do not accumulate beyond a certain amount, and those of one portion are often counteracted in another portion of the same row. In a large piece of comb, sixty cells occupied the space of one foot, which would make the diameter of a cell equal to 0.20 inch ; nevertheless, ten cells taken from either end, and ten taken from the middle of this same comb, when compared, gave marked differences. This correction is not, however, a constant condition, for we have, perhaps in most instances, found Hunter's statement correct, viz. that the cells gradually increase in size, the last formed being the largest.f

* Ten cells were measured, in order to avoid the accumulating error resulting from the measurement of a series of single cells. The error in the measurement of ten cells is no greater than that of measuring one, and divided among the ten becomes inappreciable.

| Works of John Hunter, Palmer's edition, Vol. IV. p. 436.

It may be asked, if the comb was not built with all its diameters equal, but afterwards accidentally disturbed. The comb is suspended mostly from the uppermost portion, the lowermost hanging free until considerable progress is made, when it is more or less attached by the sides; taking into consideration the material of it, and the weight, when filled with honey, or covered with crowds of bees, it seems quite probable that in a hot day the softened wax would be stretched by its own weight, thus making the transverse diameter of the cells shorter, and the others proportionally longer. To test this, cells from six different pieces of comb were measured in the direction of their three diameters; the result was, that the aggregate transverse diameters of 570 cells was 38.94 inches, and that of the other two was 38.84 and 38.90 inches respectively. The transverse diameter, the one liable to be shortened, was absolutely a little the longest.

A variation in the diameters does not necessarily bring with it an inequality in the breadth of the sides, or a difference in the angles. If, however, one of the sides is wider or narrower than the others, which it often is, the angle which it makes with the adjoining ones must be greater or less than 120°, the normal angle. In order to be able to measure the sides as accurately as possible, cross sections were made midway between the mouth and base of the cell, where they are thinnest and the angles sharpest. These sections were obtained by filling the cells with plaster of Paris, and after this had hardened, cutting them down to the required point. In this way, all distortion was prevented. The following table gives the result of the measurement of the different sides of a series of twelve cells.

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Of all the parts of the cell there is none where the variation is more striking than in the rhombic faces of the base. This fact is the more noteworthy, since it is upon these, and the angles they make with each other and the sides, that rests the nicest part of the problem relating to the adaptation of the cell to the contained bee. The relative size of the faces may be so changed that two of them make nearly the whole of the base, while the third almost vanishes, or one of the faces may have any size between this extreme and the normal one.

The fourth face, which has been so often noticed, has generally been spoken of as belonging more especially to those cells which are intermediate between the cells of drones and workers. Although it occurs in these, we have found it quite common in the middle of pieces of comb consisting solely of either worker, drone, or honey cells.* In one piece of worker comb containing about five hundred cells, nearly all had a fourth face.

The causes which lead to the introduction of the fourth face are chiefly two, irregularity in the size of the cells and incorrect alignment of them on the two sides of the comb. Each cell on one side of the comb being normally in contact, by its rhombic faces, with three cells on the other, and these fitting exactly, if a cell is increased, it will project beyond them, and thus come in contact with a fourth, and a new face will be formed. We have seen this happen in a single cell, but very commonly a row of cells increases for four or five cells, and gradually diminishes again to the ordinary size. With this increase and decrease of the cell, the fourth face comes and goes.

Incorrect alignment is the more common cause.f If a given row of

* These were studied either after cutting away the body of the cells, leaving only the basal plate which separates those of opposite sides, or by means of casts obtained by filling the cells with plaster of Paris. After this last has dried, if the mass is heated, the wax is absorbed by the plaster, when the casts of the two sets of cells separate. In old brood-combs, where many successive cocoons have been spun, these form a thick and resisting cast of the base of the cell and may be extracted, giving its precise form. In some instances, fourteen distinct layers of cocoons were counted, showing the number of broods which had occupied the cells.

† This introduction of the fourth face to the basal pyramid, through incorrect alignment, was thoroughly investigated several years since by Mr. Chauncey Wright, of the Nautical Almanac Office, and who, at the same time, constructed models illustrating his views. These models are deposited in the Museum of Comparative Anatomy and Physiology at Cambridge. For a discussion of various points connected with the geometry of the cell, see his article, entitled The Economy and Symmetry of the Honey-Bee's Cell, in the Mathematical Monthly for June, 1860. VOL. VII.

10

Fig. 2.

B

A

cells on one side of the comb ceases to be parallel with those on the other, with which it was connected when the comb was begun, and diverges from them, it is gradually transferred to a new series; as the cells come in contact with those of the new series, the fourth side appears, and, at the same time, one of the original sides, viz. that directly opposite to it, is gradually diminished, and may vanish. This divergence is, however, sometimes insufficient to make the separation of the rows complete, and may gradually diminish again, as they are extended by the construction of new cells, so as to bring them back to the original position, when the irregularity is corrected.

If, however, the separation of the two rows at length becomes complete, so that one of the faces is lost and a new one formed, all the basal portion of the cell becomes reversed, as will be seen by reference to Fig. 1.

Figs. 1 and 2 ; the first repC resenting the cells when the

base is viewed, and the second when looked at perpendicularly to one of the sides. In both figures A indicates the ordinary form of the cell. The whole series of Fig. 1 shows the gradual introduction of the new face, which is seen on

the lower border, and the elimination of one of the original faces, which is seen on the upper border. At B, which is intermediate between the two extremes, the four faces consist of two equal rhombs, – one of which is the outgoing and the other the incoming one, — and two equal hexagons. B, Fig. 2, represents the sides of the same cell, which, instead of forming three trapeziums, as at A, a, b, c, now form two pentagons, a' and c', and a parallelogram, b'. At C, Figs. 1 and 2, the forms are in all respects the reverse of those of A. A and Care symmetrical with each other, and B is symmetrical in itself. No precise number of cells is necessary for the purpose of making this transition, for it may take place in two or three, or extend through a long series, as in Fig. 1.

There is another variation which we have noticed twice, - once in drone, and once in worker comb, involving a large number of cells. If a piece of normal comb be held in the position in which it was built,

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