HIVE-BEES. 121 



each other without leavuig a vacant and superfluous 

 space between every three contiguous cells. JH ad the 

 cells, on the other hand, been square or triangular, they 

 might have been constructed without unnecessary 

 vacancies; but these forms would have both required 

 more material and 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 con- 

 ditions of the problem, it is equally adapted with a 

 cylinder to the shape of the bee. 



M. Reaumur further remarks, that the base of 

 each cell, instead of forming a plane, is usually com- 

 posed 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. Ma- 

 raldi, the inventor of glass hives, determined, by 

 minutely measuring these angles, that the greater 

 were 109° 28', and the smaller, 70° 32'; and M. 

 Reaumur, being desirous to know why these parti- 

 cular angles are selected, requested M. Koenig, a 

 skilful mathematician, (without informing him of his 

 design, or telling 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 de- 

 nominate the infinitesimal calculus, M, Koenig found 

 that the angles should be 1 09° 26' for the greater, 

 arjd 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 



VOL. IV. 11 



