48 NATURAL HISTORY. [OH. in. 



Hence it appears that both the body and the base of 

 the tube are the best fitted for their purposes ; that 

 the greatest strength and the greatest -capacity are 

 obtained with the least expenditure of wax, in an 

 hexagonal tube with a pyramidal base. 



As the bottom of each cell is formed of three 

 lozenge-shaped pieces, it is obvious that their junc- 

 tion might have been formed at any imaginable 

 angle. Like the slated roofs of our houses, it might 

 have been of any inclination. Reaumur suspected 

 that, as the bottom of the cells had a uniform in- 

 clination, this particular direction was the one which 

 caused the least expenditure of wax. He therefore 

 asked Kcenig, an able analyst, to solve the follow- 

 ing question : among all the hexagonal tubes with 

 pyramidal bases, composed of three similar and 

 equal rhombs, to determine that which can be con- 

 structed with the least possible quantity of matter'? 

 Kcenig, not at all aware of the object which Reau- 

 mur had in view when he proposed this problem, 

 worked it out, and found, that if three rhombs or 

 lozenges were so inclined to each other, that the 

 great angles measured 109 26', and the little angles 

 70 34', this construction would require the least 

 quantity of matter. Maraldi measured the angles 

 formed at the bottom of a cell, and found that the 

 great angles gave 109 28', the little one 70 32'! 

 Such an agreement between the solution and the 

 actual measurement is, it must be acknowledged, 

 sufficiently surprising. It is impossible to look at a 

 cell without fancying that some profound geometri- 

 cian had not only furnished the general plan, but 

 also assisted in its execution. The bees appear 

 says Reaumur, to have had a problem to solve 

 which w r ould puzzle many a mathematician. "A 

 quantity of matter being given, it is required to form 

 out of it cells which shall be equal and similar, and 

 of a determinate size, but the largest possible with 

 relation to the quantity of matter employed, while 



