1 1 6 Mr. J. W. L. Glaisher on the Form of the Cells of Bees. 



used*. Now to both of these suggestions, and to the second espe- 

 cially, there are great objections. Koenig gave a wrong result 

 as the answer to an arithmetical question ; and all that can be 

 said is that he somewhere must have made a mistake in his 

 work : if he neglected too many decimal places, that is of course 

 a blunder as much as if he had written a 5 for a 3 ; and, as far as 

 I can see, there is no reason to assume that he made this error 

 rather than any other. The second suggestion, so far being 

 most probable, is possible, but highly improbable. Given that a 

 wrong numerical result is published and nothing further, it is 50 

 to 1 that the blunder was in the computations, not in the tables ; 

 besides, Koenig calculated the saving of waxf (which required 

 no tables or numerical computations) wrongly, so that there is 

 prima facie evidence that he did not use sufficient care. Lord 

 Brougham was especially afraid of any error in the tables, and 

 therefore obtained two solutions of Kcenig's problem., in one of 

 which the dihedral angle between two of the rhombs was the quse- 

 situm ; and he also desired a friend to investigate the question 

 independently. But all this was quite superfluous; if be 

 the smaller angle of the rhombs, the analysis readily gives 

 cos = 1= '3333333, so that 6 can be obtained at once from a 

 table of natural sines ; and any one who knows any thing at all 

 about mathematical tables can see at once, by taking a few dif- 

 ferences, whether the table contains an error (particularly so 

 large an error as to produce an alteration of 2'); but even if 

 this is not done, an obvious transformation gives 



tan 10=^/2 = 1-4142136, 



and \9 (0 being here the larger angle) can be found directly 

 either from a natural or a logarithmic canon at once ; or any 

 number of similar transformations can be made. Lord Brougham 

 uses (p. 246) the singular phrase "difference introduced by the 

 logarithmic approximations ; " perhaps the last two words are 

 merely a euphemism for " erroneous table," as it is needless to 

 say that 7-figure logarithms generally give results true to a tenth 

 of a second at least, so that an error of two minutes is out of 

 the question. 



Whence Mr. Wood or Mr. Mitchell derived the statement 

 that Koenig was not to blame, but that the error really was in 

 in the logarithms, I am unable to say; but 1 think it is a fair 

 hypothesis to suggest that Lord Brougham's, or some similar 

 statement to the effect that the error was most likely in the 



* This is intensified in the French memoir, " M'etant assure .... que 

 Koenig etait tombe dans l'erreur par les tables de sinus ou des logarithmes 

 &c.,"p. 112. 



t Lord Brougham does not mention the fact of Koenig having errone- 

 ously calculated the saving. 



