Mr. J. W. L. Glaisher on the Form, of the Cells of Bees. 105 



it will be found that the history of the matter is well deserving 

 of atteution, although the account quoted, occurring in a work 

 of a popular character, would not, however glaring its inaccu- 

 racies, per se require refutation in a scientific journal. 



It is convenient to give at the outset, before entering into any 

 details, a brief account of the facts that did occur, for comparison 

 with that quoted above. Maraldi measured the angles of the 

 rhombs and also those of the trapeziums*, and found that, as 

 nearly as he could determine them, the larger angles in both 

 cases were about 110°, and the smaller the supplement of this 

 angle, viz. 70° ; it naturally occurred to him to investigate ma- 

 thematically what the angles must be on the supposition that 

 those of the rhombs and trapeziums were exactly equal, and he 

 found as a consequence of this hypothesis that the values in 

 question were 109° 28' and 70° 32'. These angles were not, 

 therefore, the results of direct measurement, but were derived 

 from the solution of a mathematical problem suggested by the 

 actual measurements. Reaumur, suspecting that economy of 

 wax was the reason for the angles being such as they were, ex- 

 plained the form of the cell to Koenig, and (without mentioning 

 what had been done by Maraldi) asked him to determine ma- 

 thematically what the angles of the rhombs would be if for a given 

 capacity the surface of the cell were made a minimum. Koenig 

 solved the problem, and gave as his result 109° 26' and 70° 34', 

 differing by only 2' from what Reaumur (erroneously) regarded 

 as the angles absolutely measured by Maraldi. Koenig accom- 

 panied his solution by some remarks to the effect that the ques- 

 tion was one that required the methods of Newton and Leibnitz, 

 so that the bees had solved a problem that was beyond the reach 

 of the old geometry. Maclaurin showed that this was inaccurate, 

 by giving a solution depending only on ordinary geometry; and 

 he found the angles to be 109° 28' and 70° 32', agreeing with 

 Maraldi's theoretical values. Maclaurin read Maraldi' s memoir 

 correctly, and understood the angles by measurement to be 110° 

 and 70° ; and he merely just notices the trifling error made by 

 Koenig in his calculation, regarding it as quite an unimportant 

 matter. 



These are all the facts on which Mr. Wood's narrative is 

 founded; and it therefore follows (1) that Maraldi did not find 

 the angles by measurement to be 109° 28' and 70° 32', (2) that 

 Maclaurin was not led to consider the matter by observing the 

 discrepancy between Maraldi* s and Koenig' s values, (3) that he 

 did not make any measurements himself, and (4) that the dis* 



* By the angles of the trapeziums in this paper are invariably meant the 

 two angles where the sides of the cell are intersected by the rhombs; the 

 angles at the mouth of the cell are, of course, right angles. 



Phil. Mag. S. 4. Vol. 46. No. 304. Aug. 1873. I 



