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CHAPTER VIII. 



BEES AND THEIR CELLS. 



The general form of the honey cells of bees has long been 

 known. Early in the eighteenth century it was suggested that 

 these cells were shaped by the bees so as to use their available 

 wax in the most economical manner, and Samuel Koenig, a young 

 Swiss mathematician of some repute, was asked to look into the 

 problem. He said that the suggestion was true, and added that 

 the bees had thus solved a question beyond the power of mathe- 

 maticians unacquainted with the calculus. In fact the bees do 

 not use their wax in the most economical way, and the problem 

 of determining rhomboidal ends of a hexagonal prismatic cell of 

 given base and volume so that the surface is a minimum is 

 easily soluble by classical geometry; but Koenig's ponderous 

 joke about the ability of the bees has been repeated so often 

 that it seems worth while to give the history of the subject. 



A honey comb, taken from a bee hive under normal con- 

 ditions, is in the shape of a thick slab with two parallel faces 

 containing between them cells made of wax and filled with 

 honey. The open bases of these cells are on the faces of the 

 slabs and are regular hexagons. If the faces of the slab are 

 carefully drawn apart, not an easy task, the cells will be found to 

 be arranged in two systems, each system resting on one of the 

 faces, and so placed that the top of each cell fits into the space 

 formed by the tops of three adjacent cells of the other system, 

 its axis being the continuation of the line of junction of these 

 three cells. Each cell is a prism, whose six sides (each perpen- 

 dicular to the hexagonal base) are trapeziums, and whose top is 



