CH. VIII] BEES AND THEIR CELLS 165 



stated that the saving of wax by using rhomboidal ends instead 

 of a plane hexagonal top was almost one-quarter of the wax 

 required for a plane hexagonal end: the actual fraction is 

 (V3-V2)/V3. 



Boscovich also had his attention called to the problem, and 

 he gave two solutions, published in 1760, one by pure geometry 

 and the other by the calculus. In these he pointed out the 

 numerical slip made by Koenig, the extreme difficulty of making 

 accurate measurements of actual cells, and the closeness to the 

 typical form given by Maraldi. In his calculation of the 

 amount of wax saved there is a misprint, but the error is obvious 

 to any careful reader. He was the first to point out that the 

 angle between every two adjacent planes is 120°. 



In 1781 Lhulier gave another geometrical solution of the 

 problem, and treated the obvious extension of determining the 

 best ratio of the depth of the cell to the width of the base. He 

 also discussed the saving of wax obtained by using cells of 

 different forms, and came to the conclusion, now generally ac- 

 cepted, that economy in using wax cannot have been the main 

 reason for the form of cell adopted. 



Most people are interested in bee life, but the economy con- 

 jecture about cell construction presents no mathematical diffi- 

 culty, and probably the papers by Maclaurin and Boscovich 

 would not have been written had not Koenig stated that the 

 question was insoluble by classical geometry. Their papers with 

 that of Lhulier completely solved the problem as presented to 

 them. Unluckily, Lord Brougham in the nineteenth century 

 described it afresh, and in doing so made certain inaccurate 

 statements. These were pointed out by Leslie Ellis and Glaisher, 

 but Brougham's account had a wide circulation, and his mistakes 

 have been copied by some modern popular writers. 



Brougham, however, brought out the fact, previously noted 

 but perhaps not widely known, that the bees usually make the 

 walls of the rhombs and adjoining parts thicker than the walls of 

 the rest of the sides. This disposes finally of the conjecture that 

 the form of cell adopted is that which is most economical of wax, 

 for that hypothesis rests on the assumption that the walls of 



