Mr. J. W. L. Glaisher on the Form of the Cells of Bees. 107 



" Chaque base <F Alveole est formee par trois rombes presque 

 toujours egaux et seinblables, qui suivant les mesures que 

 nous avons prises, out les deux angles obtus chacun de 110 

 degres, et par consequent les deux aigus chacun de 70 degres;" 

 and on p. 312 the result of the equality of the angles of the 

 rhombs and trapeziums is given : " Outre ces avantages qui 

 viennent du cote de la figure de la base, il y en a encore qui 

 dependent de la quantite des angles des rombes ; c'est de leur 

 grandeur que depend celle des angles dez trapezes, qui forment 

 les six cotez de 1' Alveole ; or on trouve que les angles aigus des 

 rombes, etant de 70 degres 32 minutes et les obtus de 109° 28 

 minutes, ceux des trapezes qui leur sont contigus, doivent etre 

 aussi de la meme grandeur." The etant, as Ellis remarks, being 

 the prothesis of a hypothetical proposition, if they are &c. then so 

 and so ; and in this way the mathematicians Maclaurin, Bosco- 

 vich, and Lhuillier have read it. Maraldi was an able mathe- 

 matician ; and no doubt, after his previous statement of what his 

 measurements gave, it no more occurred to him that a non- 

 mathematical reader might imagine these angles were the result 

 of more accurate measurements, than it did that there was any 

 necessity to give the details of the solution of the geometrical 

 problem involved. 



Reaumur's researches are to be found on pp. 389-391 of t. v. of 

 his Memoir es pour servir a Vhistoire des Insectes (Paris, 1740). 

 The following is the account of the proposition of the question 

 to Koenig: — " Convaincu que les abeilles employent le fond 

 pyramidal qui merite d'etre prefere, j ; ai soupconne que la raison, 

 ou une des raisons qui les avoit decidees, etoit Pepargne de la 

 cire ; qu'entre les cellules de meme capacite et a fond pyramidal, 

 celle qui pouvoit etre faite avec moius de matiere ou de cire, 

 etoit celle dont chaque rhombe avoit deux angles, chacun d' en- 

 viron 1 10 degres, et deux chacun d'environ 70. Sans parler de 

 la grandeur de ces angles, apres avoir fait admirer la disposition 

 des rhombes a M. Koenig . . . . je lui proposal de resoudre le 

 probleme suivant. Entre toutes les cellules exagonales a fond 

 pyramidal, compose de trois rhombes semblables et egaux, de- 

 terminer celle qui peut etre construite avec le moins de matiere/* 

 Koenig gave 109° 26' and 70° 34' as the angles of the rhombs ; 

 and then Reaumur sent him the ' Memoirs of the Academy ' for 

 1712, and he (Koenig) " was agreeably surprised" to find there- 

 from that his results only differed by 2' from Maraldi's 109° 28' 

 and 70° 32', which Reaumur speaks of as " les mesures les plus 

 precises de ces angles." Reaumur unfortunately does not give 

 Koenig' s investigation ; but he mentions that it was presented to 

 ths Academy in 1739. The volume for this year contains no 

 memoir by Koenig (so that, if he presented one, it was not con- 



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