164 BEES AND THEIR CELLS [CH. VIII 



dition that the acute angle of the rhomb is equal to the acute 

 angle of the trapezium ensures that the surface of the cell is a 

 minimum, but Maraldi did not suspect this. 



Some years later Reaumur took up the problem. Conjec- 

 turing that the shape of the rhombs might arise from the desire 

 of the bees to economise wax, he asked Koenig, who was then 

 staying in Paris, to determine the angles of the rhombs so that 

 for a given capacity the surface of a cell of this form should be a 

 minimum. The probl em presents no difficulty *, and Koenig gave 

 as the result 70° 34' and 109° 26', which are wrong by about 2'. 

 He further stated, also inaccurately, the amount of wax saved 

 by using this form of top instead of a plane one, making the 

 natural assumption that the walls of the cells are of uniform 

 thickness. Koenig used the calculus in his work ; he said this 

 was necessary for the purpose, and jokingly added, as stated 

 above, that the bees had in fact solved a problem beyond the 

 powers of the old geometricians. His answer was not worthy 

 of his reputation ; his numerical mistakes showed carelessness, 

 his statement about the old geometry was untrue, and his joke 

 was at least unlucky, for some writers, whose sense of humour 

 was not highly developed, took it literally. The volume con- 

 taining Reaumur's researches in which Koenig's conclusions are 

 embodied was published in 1740. Reaumur's investigations on 

 bee life are classical and were continued by F. Huber, J. H. 

 Fabre, and M. Maeterlinck, but with this side of the subject I 

 am not here concerned. 



Maclaurin came across Koenig's statements, and in 1743 

 published a geometrical solution of the problem, from which he 

 correctly deduced the values of the angles as 70° 31' 44" and its 

 supplement. He pointed out Koenig's numerical blunders, and 



* Let a be the length of one of the hexagonal edges, which is also the base 

 of one of the trapezium sides, h the longest side of the trapezium, A the area of 

 the hexagonal base, and 8 the inclination of one of the rhombs to the hexagonal 

 base. Then the volume of the cell is equal to hA and is independent of 8. The 

 surface of the cell is 



3a (ih - a tan + a ^3 sec 8)12. 



If this is a minimum, we have sin 2 9 = 1/3. And the acute angle of the rhombus 

 = 28= arc cos (1/3), or to the nearest second 70° 31' 44". 



