CH. VIII] BEES AND THEIR CELLS 163 



pointed and formed by three equal rhombs, the angles at the 

 apex being obtuse. So much probably has been familiar as long 

 as bees have been kept and their habits noted. 



As a result of investigations* made in the eighteenth cen- 

 tury, we may go further, and say that the cells are built on a 

 uniform plan, though they vary slightly among themselves. In 

 a normal comb the height of the longest side of one of the 

 trapeziums is a little more than four times the radius of the in- 

 circle of the hexagonal base ; the angle between every pair of 

 adjacent planes is 120°; and each acute angle of every rhomb 

 and trapezium is about 70°, the obtuse angles being of course 

 supplementary to this. Some writers on natural history are 

 more definite, and say that each such acute angle is 70° 31' 44". 

 We may take this as the typical value, though the edges of a 

 cell are not sharply denned, and these angles cannot be measured 

 to this accuracy. 



It happens that for these angular values a cell of this form 

 with a given volume and base has the smallest surface, or in 

 other words gives the maximum storage space which can be 

 provided with a given amount of wax. As soon as this was dis- 

 covered, it was only natural that, taking into account the con- 

 ditions under which bees live and work, it should be conjectured 

 that they use the available wax in the most economical manner, 

 and whether by reasoning, instinct, or some other process have 

 arrived at a result which certainly is far from obvious. 



The story of this "economy conjecture" starts with a memoir 

 presented to the French Academy in 1712 by G. F. Maraldi. 

 In this he said that, as nearly as he could measure them, each 

 of the smaller angles of the rhombs and trapeziums was 70°, each 

 of the larger angles being of course the supplement of this. 

 Naturally it occurred to him to investigate the problem of the 

 values of these angles on the supposition that those of the rhombs 

 and trapeziums were exactly equal and, on this hypothesis, he 

 found that their values were 70° 32' and its supplement: a result 

 consistent with his measurements. The imposition of the con- 



* For references, see J. W. L. Glaisher, Philosophical Magazine, London, 

 1873, fourth series, vol. xlvi, pp. 103-122. 



11—2 



