1 12 Mr. J. W. L. Glaisher on the borm of the Cells of Bees. 



clined to one another at this angle, and so is each trapezium to 

 the planes it cuts. This leads to the belief that the bees have 

 some means (instrumenta) of constructing planes inclined to one 

 another at this angle. Ellis (who had not seen what Boscovich 

 had written) practically remarked the same thing, and made a 

 guess at what the instrumenta were a century later (' Writings/ 

 p. 356). It thus appears that Boscovich discussed the whole 

 subject with completeness, penetration, and (but for the one trivial 

 slip) accuracy. Had his remarks been published in a work 

 better known and more accessible to naturalists, a detailed refu- 

 tation of Reaumur and Koenig a hundred and thirteen years 

 later would have been rendered superfluous. 



The next memoir* that I know of in which the form of the 

 bee-cell is considered is by Lhuillier^ and occupies pp. 277-300 

 of the Berlin Memoirs for 1781 (published 1783). The in- 

 troduction (of two pages) and conclusion were written by Cas- 

 tillon, who presented the memoir to the Academy. Lhuillier 

 gives a short account of the history of the subject without ad- 

 verting particularly to Reaumur's misreading of Maraldi. He 

 argues with Boscovich "que Pegalite supposee des angles des 

 rhombes du fond et de ceux des trapezes des faces d'un alveole, 

 est le principe qui a guide M. Maraldi dans ^estimation de ces 

 angles;" and adds a new fact, viz. that Cramer, his fellow- 

 citizen (of Geneva) had given some developments on the subject 

 to Koenig which had not been published. After stating that 

 one of Boscovich's methods is very similar to Maclaurin's, he 

 begins the next paragraph: "Tous les mathematiciens ont re- 

 garde cette matiere comme passant les forces de la Geometrie 

 elementaire," which is untrue, as the solutions in question are 

 geometrical : the error is a curious one for Lhuillier to have 

 made, as he had certainly read Boscovich's remarks, and in all 

 probability seen Maclaurin's paper. He very properly regards 

 the difference of method as of very slight importance in a mathe- 

 matical point of view (and thereby he shows, I think, a truer 

 appreciation of the relations between the modern and ancient 

 methods than either Koenig or the historian of the French Aca- 

 demy) ; and he justifies his own geometrical solution that follows 

 on the ground that it is thus made intelligible to naturalists ; 

 for although it is unlikely that the same person should be both 

 an able mathematician and an able naturalist, still a particular 

 study of the one does not prevent a knowledge of the elements 

 of the other. After his geometrical solution he proceeds to cal- 

 culate the saving of wax; and he finds that the amount used by 

 the bee in the real cell is to the amount that would be required for 



* " Memoire sur le minimum de cire des alveoles des Abeilles et en 

 particulier sur un minimum minimorum relatif a cette matiere." 



