THE HIVE-BEE. 449 



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 off the 

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

 onal 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 ar- 

 rangement 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. Be- 

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

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

 who has long exercised his well-known mathematical powers on 

 this subject, and fias 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 afterward, Eeaumur, thinking that this remarkable 

 uniformity of angle might have some connection with the won- 

 ; derful 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 ma- 

 terial ? 



Kcenig made his calculations, and found that the angles were 

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

 urements of Maraldi. The reader is requested to remember these 



Fr 



