VII] OF THE BEE'S CELL 537 



and that of the flat-bottomed hexagonal prism 



= (iii) + (iv) = 24-249; 



, 24-249 102 ,.^^. 



23772 "" 100 ' ^^ ^ ^^^^ ' 



so that the saving in the former case amounts to about 2 per cent., 

 as Lhuiller had found it to be. 



Glaisher sums up the matter as follows: "As the result of a 

 tolerably careful examination of the whole question, I may be 

 permitted to say that I agree with Lhuiller in beheving that the 

 economy of wax has played a very subordinate part in the deter- 

 mination of the form of the cell ; in fact I should not be surprised 

 if it were acknowledged hereafter that the form of the cell had 

 been determined by other considerations, into which the saving of 

 wax did not enter (that is to say did not enter sensibly; of course 

 I do not mean that the amount of wax required was a matter of 

 absolute indifference to the bees). The fact of all the dihedral 

 angles being 120° is, it is not unlikely, the cause that determined 

 the form of the cell." This last fact, that in such a cell every plane 

 cuts every other plane at an angle of 120°, was known both to Kliigel 

 and to Boscovich; it is no mere corollary, but the root of the 

 matter. It is, as Glaisher indicates, the fundamental physical 

 principle of construction from which the apical angles of 109° follow 

 as a geometrical corollary. And it is curious indeed to see how 

 the obtuse angle of the rhomb, and its cosine —J, drew attention 

 all the while; but the dihedral angle of 120° of the rhombohedron, 

 and the inclination of its three short diagonals at 90° to one another, 

 got rare and scanty notice. 



Darwin had hstened too closely to Brougham and the rest when 

 he spoke of the bee's architecture as "the most wonderful of known 

 instincts"; and when he declared that "beyond this stage of per- 

 fection in architecture natural selection could not lead ; for the 

 comb of the hive-bee, as far as we can see, is absolutely perfect in 

 economising labour and wax." 



The minimal properties of the cell and all the geometrical reasoning 

 in the case postulate cell -walls of uniform tenuity and edges which 

 are mathematically straight. But the walls, and still more their 

 edges, are always thickened ; the edges are never accurately straight, 



