The investigation of the ends of the cell was a more difficult matter, and came later, than that of its sides. In general terms this arrangement was doubtless often studied and described: as for instance, in the Garden of Cyrus: “And the Combes themselves so regularly contrived that their mutual intersections make three Lozenges at the bottom of every Cell; which severally regarded make three Rows of neat Rhomboidall Figures, connected at the angles, and so continue three several chains throughout the whole comb.” But Maraldi[368] (Cassini’s nephew) was the first to measure the terminal solid angle or determine the form of the rhombs in the pyramidal ending of the cell. He tells us that the angles of the rhomb are 110° and 70°: “Chaque base d’alvéole est formée par trois rhombes presque toujours égaux et semblables, qui, suivant les mesures que nous avons prises, ont les deux angles obtus chacun de 110 degrés, et par conséquent les deux aigus chacun de 70°.” He also stated that the angles of the trapeziums which form the sides of the body of the cell were identical angles, of 110° and 70°; but in the same paper he speaks of the angles as being, respectively, 109° 28′ and 70° 32′. Here a singular confusion at once arose, and has been perpetuated in the books[369]. “Unfortunately Réaumur chose to look upon this second determination 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 {330} appears to have thought of the impossibility of measuring such a thing as the end of a bee’s cell to the nearest minute.” At any rate, it now occurred to Réaumur (as curiously enough, it had not done to Maraldi) that, just as the closely packed hexagons gave the minimal extent of boundary in a plane, so the actual solid figure, as determined by Maraldi, might be that which, for a given solid content, gives the minimum of surface: or which, in other words, would hold the most honey for the least wax. He set this problem before Koenig, and the geometer confirmed his conjecture, the result of his calculations agreeing within two minutes (109° 26′ and 70° 34′) with Maraldi’s determination. But again, Maclaurin[370] and Lhuilier[371], by different methods, obtained a result identical with Maraldi’s; and were able to shew that the discrepancy of 2′ was due
Fig. 132.
to an error in Koenig’s calculation (of tan θ = √2),—that is to say to the imperfection of his logarithmic tables,—not (as the books say[372]) “to a mistake on the part of the Bee.” “Not to a mistake on the part of Maraldi” is, of course, all that we are entitled to say.
The theorem may be proved as follows:
ABCDEF, abcdef, is a right prism upon a regular hexagonal base. The corners BDF are cut off by planes through the lines AC, CE, EA, meeting in a point V on the axis VN of the prism, and intersecting Bb, Dd, Ff, at X, Y, Z. It is evident that the volume of the figure thus formed is the same as that of the original prism with hexagonal ends. For, if the axis cut the hexagon ABCDEF in N, the volumes ACVN, ACBX are equal. {331}
It is required to find the inclination of the faces forming the trihedral angle at V to the axis, such that the surface of the figure may be a minimum.
Let the angle NVX, which is half the solid angle of the prism, = θ; the side of the hexagon, as AB, = a; and the height, as Aa, = h.
Then,
AC = 2a cos 30° = a√3.