710 



PHILOSOPHICAL TRANSACTIONS. 



[anno 1743. 



de Reaumur, who have found those bases to be of that pyramidal figure, that 

 requires the least wax for containing the same quantity of honey, and which 

 has at the same time a very remarkable regularity and beauty, connected of 

 necessity with its frugality. 



These bases are formed from three equal rhombuses, the obtuse angles of 

 which are found to be the doubles of an angle that often offers itself to mathe- 

 maticians in questions relating to maxima and minima ; that is, the angle, 

 whose tangent is to the radius, as the diagonal is to to the side of the square. 

 By this construction, of the 6 solid angles at the base, that correspond to the 

 angles of the hexagon, 3 are equal as well to each other, as to the solid angle 

 at the apex of the figure, each of which solid angles is respectively formed from 

 3 equal plane obtuse angles: and the other 3 solid angles are also equal to each 

 other, but severally formed each from 4 equal plane acute angles, supplements 

 to the former obtuse ones. 



By this form the utmost improvement is made of their wax, of which they are 

 on all occasions very saving, the greatest regularity is obtained in the construc- 

 tion, and with a particular facility in the execution ; as there is one sort of 

 angle only with its supplement, that is required in the structure of the whole 

 figure. 



^iM. Maraldi* had found by mensuration, that the obtuse angles of the 

 rhombuses were of 110 degrees nearly; on which he observed, that if the 3 

 obtuse angles, which formed the solid angles abovementioned, were supposed 

 equal to each other, they must each be of 109° 28'; whence it has been inferred, 

 that this last was really the true and just measure of them : and lately M. de 

 Reaumur-^ has informed us, that Mr. Koenig having, at his desire, sought 

 what should be the quantity to be given to this angle, in order to employ the 

 least wax possible in a cell of the same capacity, that gentleman had found, by 

 a higher geometry than was known to the ancients, by the method of infinitesi- 

 mals, that the angle in question ought in this case to be of 109° 26'. And we 

 shall now make it appear, from the principles of common geometry, that the 

 most advantageous angle for these rhombuses, is indeed, on that account also, 

 the same which results from the supposed equality of the 3 plane angles that 

 form the abovementioned solid ones. 



Let GN and nm, fig. 1 and 2, pi. 21, represent any two adjoining sides of 

 the hexagon, that is, the section of the cell perpendicular to its length. The 

 sides of the cell are not complete parallelograms as cgnk, bmnk, but trapezia 

 cgne, bmne, to which a rhombus cebc, is fitted at e, and that has the opposite 



* Memoires de I'Acad. Royale des Sciences, 1712. — Orig. 

 t Memoires sur les Insectes, torn. r. — Orig. 



