VII] OF THE BEE'S CELL 529 



de r Alveole." In short, Maraldi takes the two principles of sim- 

 phcity and mathematical beauty as his sure and sufficient guides. 



The next step was that which had been foreshadowed long before 

 by Pappus. Though Euler had not yet published his famous 

 dissertation on curves maximi minimive proprietate gaudentes, the 

 idea of maxima and minima was in the air as a guiding postulate, 

 an heuristic method, to be used as Maraldi had used his principle 

 of simphcity. So it occurred to Reaumur, as apparently it had not 

 done to Maraldi, that a minimal configuration, and consequent 

 economy of material in the waxen walls of the cell, might be at 

 the root of the matter: and that, just as the close-packed hexagons 

 gave the minimal extent of boundary in a plane, so the figure deter- 

 mined by Maraldi, namely the rhombic dodecahedron, might be 

 that which employs the minimum of surface for a given content: 

 or which, in other words, should hold the most honey for' the least 

 wax. "Convaincu que les abeilles employent le fond pyramidal qui 

 merite d'etre prefere, j'ai soup9onne que la raison, ou une des 

 raisons, qui les avoit decidees etait I'epargne de la cire; qu'entre 

 les cellules de meme capacite et a fond pyramidal, celle qui pouvait 

 etre faite avec moins de matiere ou de cire etoit celle dont chaque 

 rhombe avoit deux angles chacun d'environ 110 degres, et deux 

 chacun d'environ 70°." He set the problem to Samuel Koenig, 

 a young Swiss mathematician: Given an hexagonal cell terminated 

 by three similar and equal rhombs, what is the configuration which 

 requires the least quantity of material for its construction? Koenig 

 confirmed Reaumur's conjecture, and gave 109° 26' and 70° 34' as 

 the angles which should fulfil the condition; and Reaumur then 

 sent him the Memoires de I'Academie for 1712, where Koenig was 

 "agreeably surprised" to find: "que les rombes que sa solution 

 avait determine, avait a deux minutes pres* les angles que 

 M. Maraldi avait trouves par des mesures actuelles a chaque rhombe 

 des cellules d'abeilles. . . . Un tel accord entre la solution et les 

 mesures actuelles a assurement de quoi surprendre." Koenig 

 asserted that the bees had solved a problem beyond the reach of 



* The discrepancy was due to a mistake of Koenig's, doubtless misled by his 

 tables, in the determination of V2; but Koenig's own paper, sent to Reaumur, 

 remained unpublished and his method of working is unknown. An abridged 

 notice appears in the Mdm. de VAcad. 1739, pp. 30-35. 



