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



they differ from one another will agree with Boscovich in his 

 surprise that Maraldi should have been able to obtain by mea- 

 surement the typical angle correctly to within one ninetieth of 

 a right angle. Of course Boscovich does not impute the slight- 

 est insincerity to Maraldi ; he merely points out that the know- 

 ledge of the solution of the theoretical question may have exer- 

 cised au unconscious bias over his mind, so as to make him 

 announce 110° with more confidence than he might otherwise 

 have felt. 



Boscovich gives two solutions of Kosnig's problems, the one 

 by pure geometry, the other by differential calculus. The latter 

 is obtained in exactly the way any one to whom the question was 

 proposed would probably now proceed (at all events it is nearly 

 identical with my own solution undertaken before I had seen 

 any of the investigations on the matter) : the correct result is 

 obtained ; and it is suggested that Koenig's error was produced 

 by his having adopted some complicated method leading to a 

 formula which had to be solved by approximation. With regard 

 to the saving of the wax the process is quite correct ; only there 

 is a slip in the course of the work which renders the result inac- 

 curate : the ratio of the amount saved to that required to form 



a hexagonal base is given as - — ~ — , whereas it should be 

 — ^ . lne ratio in question is truly tound to be . ; 



but in the next line this is by a mistake written j- , and 



the error is retained. This upsets an ingenious suggestion of 

 Boscovich's, who thought that perhaps Koenig's error was intro- 

 duced by the accidental substitution of \^6— 2 for \/6 — 2 

 in the numerator of the fraction quoted, which would change 

 the ratio into one of equality*. 



One point of very considerable importance in the form of the 

 cell is first noticed by Boscovich, who observes that by the 

 arrangement adopted every plane cuts every other plane at an 

 angle of 120° : thus the three rhombs forming the apex are in- 



* It is scarcely worth speculating how it was that Kcenig did make the 



error in determining the ratio ; and, of course, the facts can never be known 



with certainty ; but the following is a guess. Taking the length of a side of 



• , • • 3^2 , 3V3 9^/2 



the hexagonal base as unity, the amount or the saving is — 4 — -r — q— — — j — 



(the first term being the area of the six triangles, the second of the hexa- 

 gonal base, and the third of the three rhombs) ; if, then, Kcenig forgot to 



multiply the area of a single rhomb, viz. — ^— , by 3, the expression would 

 reduce to the middle term, viz. the area of the hexagonal base. 



