428 HOMES WITHOUT HANDS. 



it is clear that they would have but little adhesion to each 

 other, and that a large amount of space would be wasted. The 

 simplest plan of obviating these defects is evidently to square oflF 

 the rounded bases, and to fill up the ends of each cell with a 

 hexagonal flat plate, which is actually done by the wasp. If, 

 however, we look at a piece of bee-comb, we shall find that no 

 such arrangement is employed, but that the bottom of each 

 cell is formed into a kind of three-sided cup. Now, if we 

 break away the walls of the cell, so as only to leave the bases, 

 we shall see that each cup consists of three lozenge-shaped 

 plates of wax, all the lozenges being exactly alike. 



These lozenge-shaped plates contain the key to the bee-cell, 

 and their properties will therefore be explained at length. Before 

 doing so, I must acknowledge my thanks to the Kev. Walter 

 Mitchell, Vicar and Hospitaller of St. Bartholomew's Hospital, 

 who has long exercised his well-known mathematical powers on 

 this subject, and has kindly supplied me with the outline of the 

 present history. 



If a single cell be isolated, it will be seen that the sides rise 

 from the outer edges of the three lozenges above-mentioned, so 

 that there are, of course, six sides, the transverse section of 

 which gives a perfect hexagon. Many years ago Maraldi, being 

 struck with the fact that the lozenge-shaped plates always had 

 the same angles, took the trouble to measure them, and found 

 that in each lozenge, the large angles measured 109° 28', and the 

 smaller, 70° 32', the two together making 180°, the equivalent of 

 two right angles. He also noted the fact that the apex of the 

 three-sided cup was formed by the union of three of the greater 

 angles. The three united lozenges are seen at fig. 1. 



Some time afterwards, Reaumur, thinking that this remarkable 

 uniformity of angle might have some connexion with the 

 wonderful economy of space which is observable in the bee- 

 comb, hit upon a very ingenious plan. Without mentioning his 

 reasons for the question, he asked Kcenig, the mathematician, to 

 make the following calculation. Given a hexagonal vessel 

 terminated by three lozenge-shaped plates ; what are the angles 

 which would give the greatest amount of space with the least 

 amount of material ? 



Kcenig made his calculations, and found that the angles were 

 109° 26' and 70° 34', almost precisely agreeing with the measure- 



