354 ON THE FORM OF BEES' CELLS. 



equality was absolute, and hence the determination of the angles 

 became, as I need hardly point out, a matter of spherical trigo- 

 nometry, namely to determine the sides of the equilateral spheri- 

 cal triangle whose angles are each equal to 120, that is, to the 

 angle between the two adjacent sides of a hexagon. This 

 angle has for its cosine J, and is, I believe, to the nearest second 

 109 28 V 16 V \ 



Unfortunately Eeaumur chose to look upon this second deter- 

 mination of Maraldi's as being, as well as the first, a direct 

 result of measurement, whereas it is in reality theoretical. 

 He speaks of it as Maraldi's more precise measurement, and this 

 error has been repeated in spite of its absurdity, to the present 

 day: nobody appears to have thought of the impossibility of 

 measuring such a thing as the end of a bee's cell to the nearest 

 minute. One can only suppose that Maraldi made so many ob- 

 servations, varying between 109 and 110, that the arithmetical 

 mean came out with this excessive amount of accuracy, a suppo- 

 sition in itself highly improbable, and at variance with Maraldi's 

 distinct statement. The subsequent history of the matter is 

 this, Eeaumur employed Koenig to determine the form the 

 rhombs ought to have in order to give the greatest volume to 

 the cell with the least expenditure of wax. Koenig, for a reason 

 which will be mentioned by and by, gave as his solution the 

 following values to the angles of the rhomb, 109 26' and 70 34\ 

 He was agreeably surprised, says Eeaumur, to find that his 

 result agreed within two minutes with Maraldi's measurements, 

 whereas in reality they only agreed with Maraldi's theoretical 

 determination. With this determination they ought to have 

 been absolutely coincident : for it is easy to show that when the 

 surface of the cell is made a minimum, the angles of the rhombs 

 are equal to those of the trapeziums. It was soon afterwards 

 shown that Kcenig's results were wrong, and thus we have been 

 pleasantly told, that the bees proved to be right and the mathe- 

 matician wrong, that there was no mistake on the part of the 

 bees, and so on. Maclaurin was, I believe, the first to correct 

 Kcenig : he has not always had credit for this priority. Thus Dr 

 Carpenter in his Physiology, informs us that Lord Brougham, 

 not satisfied with Kcenig's determination, took into account cer- 

 tain small quantities previously neglected, and showed that the 

 coincidence between theory and observation was absolute. 



