made by various Wasps and by the Honey -Bee. 419 



more remarkable, viz. that, of plane figures, which are equi- 

 lateral and equiangular, and have equal perimeters, that is 

 always the greatest which consists of most angles, and the circle, 

 is the greatest of all, provided it be included in a perimeter 

 equal to theirs." — Pappus. 



In 1712, Maraldi published, in the ' Memoires de F Academic 

 des Sciences, Paris/ 1712, p. 299, a remarkable paper, in which 

 is investigated, for the first time, the terminal pianes of the 

 bees' cell, which are now well known to be formed of the faces 

 of the rhombic dodecahedron. He appears to have believed 

 that the object of having lozenges of the same form, as termi- 

 nating planes, was to enable the bees to carry in their mind the 

 idea of one geometrical form only, in addition to their original 

 idea of the hexagon. The angles of the lozenge are found by 

 him to be 110° and 70°, by observation; and 109° 28' and 

 70° 32' by calculation. He gives, also, the following mean 

 measurements of the cells : — In a foot long of comb there are 

 from 60 to 66 cells, about two lines for each cell, and the depth 

 of the cell is five lines. 



Reaumur appears to have been the first who introduced the 

 fantastic idea of economy of wax, as the motive cause of the 

 peculiar shape of the terminating planes, and, not being a geo- 

 meter, he obtained the assistance of Konig to calculate the 

 angle of the lozenge which should give the least surface with a 

 given volume. Konig determined this angle at 109° 26', agree- 

 ing with Maraldi within two minutes. 



MacLaurin published, in the 'Philosophical Transactions/ 

 1743, p. 565, an elaborate geometrical paper on the subject, in 

 which he proves that the tangent of the angle in question is the 

 square root of 2, and that it is therefore equal to 109° 28' 16"; 

 and he computes the saving of wax as " almost one-fourth part 

 of the pains and expense of wax they bestow, above what was 

 necessary for completing the parallelogram side of the cells/'' 



I/Hullier, in 1781, published, in the ' Berlin Memoirs/ p. 277, 

 an elaborate discussion of the entire problem, in which he 

 arrived at the following results, already found by MacLaurin's 

 geometrical method: — 



a. That the economy of wax is less than one-fifth of what 

 would make a flat base. 



b. That the economy of wax, referred to the total expenditure, 

 is in-st, so that the bees can make fifty-one cells, instead of fifty, 

 by the adoption of the rhombic dodecahedron. 



He does not share, however, in the enthusiasm of the natu- 

 ralists, but maintains and proves that mathematicians could 

 make cells, of the same form as those of the bees, which, instead 

 of using only a minimum of wax, would use the minimum mini- 



