FORM OF THE CELLS. 



double layer, the pyramidal bases of each layer being placed in 

 contact with each other. 



It might also have been expected that these bases would have 

 received the most simple form of plane surfaces, so that the side 

 of each layer occupied by them would be a uniform plane ; and 

 these planes resting in contact would form the comb ; but to this 

 there would be several objections. In the first place, the capacity 

 of the comb would be less ; the bases of the cells, placed in contact, 

 would be liable to slip one upon the other; and if the cells had a 

 common base, they would have less strength ; but independently 

 of this, the bee itself tapers towards its posterior extremity, and a 

 cell with a flat bottom having no corresponding tapering form 

 would be little adapted to its shape, and would involve a con- 

 sequent waste of space. The bee avoids this disadvantage by 

 giving the bottom of the cell the shape of a hollow angular 

 pyramid, into the depth of which the tapering posterior extremity 

 of the insect enters. 



63. There is another advantage in this arrangement which 

 must not be overlooked. The pyramidal bases of each layer of 

 cells, placed in juxtaposition by reciprocally fitting each other, so 

 that the angular projections of each are received into the angular 

 cavities of the other, are efiiective means of resisting all lateral 

 displacement. 



64. Pyramidal bases, however, might have been given to the 

 cells in a great variety of ways, which would have equally served 

 the purposes here indicated ; but it was essential, on grounds of 

 economy, that that form should be selected which would give 

 the greatest possible capacity with the least possible material. Oa 

 examining curiously the form of the lozenges composing the pyra- 

 midal bases of the cells, Maraldi found by accurate measurement 

 that their acute angle measured 70° 32', and consequently their 

 obtuse angle 109° 28'. Magnitudes so singular as these, invariably 

 reproduced in all the regular cells, could scarcely be imagined to 

 have been adopted by these little engineers without a special pur- 

 pose, and Eeaumur accordingly conjectured that the object must 

 have been the economy of wax. 



Not being himself a mathematician sufficiently profound to 

 solve a problem of this order, he submitted to M. Koenig, an 

 eminent geometer of that day, the general problem to determine 

 the form which ought to be given to the pyramidal bottom of an 

 hexagonal prism, such as those constituting the cones, so that with 

 a given capacity, the least possible material would be necessaiy 

 for the construction. The problem was one requiring for its solu- 

 tion the highest resources to which analytical science had then 

 attained. Its solution, however, was obtained, from which it 



31 



