140 Insect Architecture. 



have been constructed without unnecessary vacancies ; but 

 these forms would have both required more material, and have 

 been very unsuitable to the shape of a bee's body. The six- 

 sided form of the cells obviates every objection ; and while 

 it fulfils the conditions of the problem, it is equally adapted 

 with a cylinder to the shape of the bee. 



M. Keaumur further remarks, that the base of each cell, 

 instead of forming a plane, is usually composed of three 

 pieces in the shape of the diamonds on playing cards, and 

 placed in such a manner as to form a hollow pyramid. This 

 structure, it may be observed, imparts a greater degree of 

 strength, and, still keeping the solution of the problem in 

 view, gives a great capacity with the smallest expenditure of 

 material. This has actually, indeed, been ascertained by 

 mathematical measurement and calculation. Maraldi, the 

 inventor of glass hives, determined, by minutely measuring 

 these angles, that the greater were 109 28', and the smaller 

 70 32' ; and M. Keaumur, being desirous to know why these 

 particular angles are selected, requested M. Koenig, a skilful 

 mathematician (without informing him of his design, or tell- 

 ing him of Maraldi's researches), to determine by calculation 

 what ought to be the angle of a six-sided cell, with a concave 

 pyramidal base, formed of three similar and equal rhomboid 

 plates, so that the least possible matter should enter into its 

 construction. By employing what geometricians denominate 

 the infinitesimal calculus, M. Koenig found that the angles 

 should be 109 26' for the greater, and 70 34' for the 

 smaller, or about two-sixtieths of a degree, more or less, than 

 the actual angles made choice of by bees. The equality of 

 inclination in the angles has also been said to facilitate the 

 construction of the cells. 



M. Huber adds to these remarks, that the cells of the first 

 row, by which the whole comb is attached to the roof of a 

 hive, are not like the rest ; for, instead of six sides, they have 

 only five, of which the roof forms one. The base, also, is in 

 these different, consisting of three pieces on the face of the 

 comb, and on the other side of two: one of these only is 

 diamond-shaped, while the other two are of an irregular four- 



