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



the bee-cell ; and when the base is square the roof is formed of 

 four rhombs, which happen to be of the same form as in the bee- 

 cell (viz. ratio of diagonals = V2), with the only difference that 

 when the base is square they are placed with the shorter dia- 

 gonals horizontal, and when the base is hexagonal the longer. 

 He thence concludes that with twelve such rhombs a regular 

 figure of twelve sides could be made exhibiting both the roofs of 

 the 4- and 6-angled buildings, and therefore also of the bee-cell ; 

 after w T hich he adds that this last fact was noted by Kepler in 

 his Harmonice, so that the question Reaumur proposed to Koenig 

 was not so new as he thought (nicht mehr so ganz neu, als es 

 Reaumur geglaubt zu haben scheint). Lhuillier says Lambert's 

 assertion about the bee-cell is contrary to the contents of his 

 (Lhuillier's) paper (meaning that certain other forms would be 

 preferable for the house) ; and he is right. In fact Lambert has 

 not at all considered the general problem, what would be the 

 best form for the hexagonal house ? but merely the particular 

 case which arises in connexion with the bee-cell, viz., if the base 

 and capacity are given and the roof is to be rhomboidal, what 

 must the angles of the rhombs be ? 



The next discussion of the form of the bee's cell that' I have 

 to notice is by Lord Brougham ; and it occurs in vol. i. of his 

 * Dissertations on subjects of Science connected with Natural 

 Theology, being the concluding volumes of the new edition of 

 Paley's Works/ two vols. 1839*. The essay in question occu- 

 pies pp. 218-368; but it is only with the parts of it that relate 

 to the matters already noticed in this communication that we are 

 here concerned, and it must be understood that it is only to these 

 portions that any remarks that may be made have reference, 

 unless otherwise stated. 



In 1858 Lord Brougham extracted the more salient points 

 from his earlier essay and communicated them to the French 

 Academy f- I cannot find that his memoir has yet been printed 

 by the Academy ; but an abstract appears on p. 1024 et seq. of 

 vol. xlvi. (1858) of the Comptes Rendus ; and it is published in 

 extenso by its author as Tract vi. (pp. 103-121) of 'Tracts, Ma- 

 thematical and Physical' (London and Glasgow, 1860). All the 

 statements that will be noticed in this paper appear also in the 

 French memoir, although the wording &c. is sometimes modified 



* The essay appears in vol. vi. of the collected works of Lord Brougham, 

 by A. and C. Black, Edinburgh ; but the Appendix of Demonstrations, con- 

 taining the solutions of the mathematical questions referred to in the text, 

 is left out ; and this is done without even an accompanying statement of so 

 important an omission having been made. 



•f* It is not, however, stated in the memoir (at all events as printed in 

 the ' Tracts ' . . . . 1860) that the facts contained in it had been previously 

 published. 



