Mr. J. W. L. Glaisher on the Form of the Cells of Bees. 119 



cells" by "thebees do not manufacture the wax for the walls of their 

 cells," and regards the two assertions as identical. The fact that 

 the depth of the cells is variable is exactly analogous to the fact 

 that the angles of the rhombs vary : none of the cells are per- 

 fect mathematical figures ; but we consider merely the figure 

 which we believe to represent the mean or typical cell. 



In reality Lhuilliers calculation of the saving is under the 

 mark ; for if, as Lord Brougham so strongly insists in reference 

 to the minimum minimorum, the hexagonal plate that closes the 

 mouth of the cell be taken into account also, then the amount 

 of wax in the real cell is 26*370 against 26*847 in the hypothe- 

 tical cell, and the saving is only A- of the wax used, or even less 

 if we consider a comb, as each cell has its own hexagonal plate, 

 while the sides and rhombs are, except in the outside cells, 

 common to two contiguous cells. 



With regard to the argument drawn from the increased thick- 

 ness of the rhombs and triangles, it is only necessary to remark 

 that, unless we know the reason for this thickening (which we 

 do not), its recognition removes the question of the bee-cell from 

 the province of mathematics (or exact reasoning) entirely. We 

 are comparing the real cell that the bees do make with a hypo- 

 thetical cell which, a priori might have appeared more simple 

 and suitable ; and unless we are sure why the bottom (using 

 the word as above denned) is thicker than the walls, how can we 

 know how much of the hypothetical cell would require to have 

 been thickened if it had been adopted ? If the change of thick- 

 ness destroys Lhuillier's argument, it at the same time destroys 

 all mathematical reasoning as applied to the question. Lord 

 Brougham assumes that what I have called the bottom of the 

 hypothetical cell would have needed thickening had that form 

 been adopted; but this assumption is purely arbitrary, and how 

 much (if any) of the walls would have required to have been 

 thickened is merely matter of opinion ; and the question is 

 thereby removed out of the reach of mathematics, where it is 

 not my object to follow it. 



I purposely abstain from any remarks on Lhuillier's minimum 

 minimorum and Lord Brougham's objections, partly because I 

 may possibly make the problem there treated of the subject of a 

 separate communication at some future time, and partly because 

 so shallow a cell (whether the hexagonal plate be excluded or not) 

 is clearly unsuitable for the other purposes to which the bees apply 

 their cells besides storing honey. It is unnecessary here to do 

 more than allude to Lord Brougham's remarks on Lhuillier and 

 Castillon in reply to the argument marked thirdly in his essay, as 

 any one who has read in this paper what Maraldi, Boscovich* &c. 



* Leslie Ellis corrects Lord Brougham for representing that Bosco- 



