108 Mr. J. W. L. Glaisher on the Form of the Cells of Bees. 



sidered deserving of publication in extenso) ; but the matter is 

 noticed at length in pp. 30-35 of the Histoire at the beginning, 

 where, however, only a general account of the matter is given, 

 and no further information about Koenig's solution is added*. 

 The historian of the Academy states that Reaumur also proposed 

 the question to other mathematicians, from whom, however, he 

 did not receive solutions ; he speaks of the bees being only in 

 error to the trifling amount of 2', and adds that the "grande 

 merveille " is their solution of a problem belonging to the higher 

 geometry. 



Maraldi proposed to himself the problem, Supposing the 

 rhombs and trapeziums have equal angles, what are they? and 

 Reaumur proposed to Koenig the problem, What must the 

 angles of the rhombs be, if the surface is a minimum ? The 

 answer to the latter question is, " Equal to the angles of the tra- 

 peziums ; M so that both problems give the same values for the 

 angles ; only Maraldi worked his numerical calculation correctly, 

 Koenig incorrectly. What happened was that one mathematician 

 was right and another wrong, not that the bees were right and 

 the mathematicians wrong. How Kcenig, who had himself just 

 been considering the- mathematical problem of the bee-cell, 

 could on reading Maraldi's memoir have failed to see that the 

 latter's values were the result of a geometrical investigation of 

 the same nature as that which he had just completed, and that 

 they ought to have agreed exactly with his own, is perfectly un- 

 accountable ; but it seems to have been the fact. 



Reaumur (t. v. p. 390) gives, on Kcenig's authority, the 

 amount of wax saved as equal to the whole quantity that would 

 be required for a hexagonal bottom ("M. Kcenig a pourtant 

 demontre que les abeilles oeconomisent la cire, en preferant les 

 fonds pyramidaux aux fonds plats, qu'elles menagent en entier 

 la quantity de cire qui seroit necessaire pour un fond plat"). 

 This is incorrect ; it will appear further on that the saving is 

 between one fifth and one sixth of the amount here stated. 

 Maclaurin's paper, " Of the Bases of the Cells wherein the 

 * Reaumur's words (t. v. p. 390) are : " Si je ne craignois qu'on se 

 lassat de m'entendre parler geometrie, je rapporterois volontiers les demon- 

 strations de M. Koenig ; mais ceux qui sont curieux de les voir, n'y per- 

 dront rien pour ne les pas trouver ici. Le memoire qui les donne, a ete 

 ltt a l'Acade'mie en 1739, il en sera fait mention dans 4'Histoire de cette 

 menie annee ; elles y seront exposees plus nettement, et mises dans un 

 plus grand jour, par notre Celebre Historien, que je ne le pourrois faire." 

 This implies that Kcenig's solution was presented in extenso to the Aca- 

 demy, and that Reaumur expected it would be so published. In point of 

 fact, however, as already mentioned, not one word of the solution itself was 

 printed ; and all that can be inferred about it from Reaumur's and the histo- 

 rian's remarks (whieh latter are not remarkable for their profundity) is, 

 that it depended on the differential calculus. 



