HABITATIONS OF INSECTS. 273 



The interior of a bee-hive consists of several of these combs fixed to its 

 upper part and sides, arranged vertically at a small distance from each 

 other, so that the cells composing them are placed in a horizontal position, 

 and have their openings in opposite directions not the best position one 

 would have thought for retaining a fluid like honey, yet the bees find no 

 inconvenience on this score. The distance of the combs from each other 

 is about half an inch, that is, sufficient to allow two bees busied upon the 

 opposite cells to pass each other with facility. Besides these vacancies, 

 which form the high roads of their community, the combs are here and 

 there pierced with holes which serve as posterns for easy communication 

 from one to the other without losing time by going round. 



The arrangement of the combs is well adapted for its purpose, but it is 

 the construction of the cells which is most admirable and astonishing. As 

 these are formed of wax, a substance secreted by the bees in no great 

 abundance, it is important that as little as possible of such a precious ma- 

 terial should be consumed. Bees, therefore, in the formation of their cells, 

 have to solve a problem which would puzzle some geometers, namely, a 

 quantity of wax being given, to form of it similar and equal cells of a de- 

 terminate capacity, but of the largest size in proportion to the quantity of 

 matter employed, and disposed in such a manner as to occupy in the hive 

 the least possible space. Every part of this problem is practically solved 

 by bees. If their cells had been cylindrical, which form seems best adapted 

 to the shape of a bee, they could not have been applied to each other with- 

 out leaving numberless superfluous vacuities. If the cells were made square 

 or triangular, this last objection, indeed, would be removed : but besides 

 that a greater quantity of wax would have been required, the shape would 

 have been inconvenient to a cylindrical-bodied animal. All these difficulties 

 are obviated by the adoption of hexagonal cells, which are admirably fitted 

 to the form of the insect, at the same time that their sides apply to each 

 other without the smallest vacant intervals. Another important saving 

 in materials is gained by making a common base serve for two strata of 

 cells. Much more wax as well as room would have been required, had 

 the combs consisted of a single stratum only. But this is not all. The 

 base of each cell is not an exact plane, but is usually composed of three 

 rhomboidal or lozenge-shaped pieces, placed so as to form a pyramidal con- 

 cavity. From this form it follows that the base of a cell on one side or 

 stratum of the comb is composed of portions of the bases of three cells on 

 the other. You will inquire, Where is the advantage of this arrangement ? 

 First, a greater degree of strength ; and, secondly, precisely the same as 

 results from the hexagonal sides a greater capacity with less expenditure 

 of wax. Not only has this been indisputably ascertained, but that the 

 angles of the base of the cell are exactly those which require the smallest 

 quantity of wax. It is obvious that these angles might vary infinitely ; 

 but, by a very accurate admeasurement, Maraldi found that the great 

 angles were in general 109 28', the smaller ones 70 32'. Reaumur, in- 

 geniously suspecting that the object of choosing these angles from amongst 

 so many was to spare wax, proposed to M. Konig, a skilful geometrician, 

 who was ignorant of Maraldi's experiments, to determine by calculations 

 what ought to be the angle of a hexagonal cell, with a pyramidal bottom 

 formed of three similar and equal rhomboid plates, so that the least matter 

 possible might enter into its construction. For the solution of this pro- 

 blem the geometrician had recourse to the infinitesimal calculus, and found 



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