Reaumur suggested the following problem to the celebrated mathematician Koenig:  “Of all possible hexagonal cells with pyramidal base composed of three equal and similar rhombs, to find the one whose construction would need the least material.”  Koenig’s answer was, the cell that had for its base three rhombs whose large angle was 109 deg 26’, and the small 70 deg 34’.  Another savant, Maraldi, had measured as exactly as possible the angles of the rhombs constructed by the bees, and discovered the larger to be 109 deg 28’, and the other 70 deg 32’.  Between the two solutions there was a difference, therefore, of only 2’.  It is probable that the error, if error there be, should be attributed to Maraldi rather than to the bees; for it is impossible for any instrument to measure the angles of the cells, which are not very clearly defined, with infallible precision.

The problem suggested to Koenig was put to another mathematician, Cramer, whose solution came even closer to that of the bees, viz., 109 deg 28 1/2’ for the large angle, and 70 deg 31 1/2’ for the small.

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I myself do not believe that the bees indulge in these abstruse calculations; but, on the other hand, it seems equally impossible to me that such astounding results can be due to chance alone, or to the mere force of circumstance.  The wasps, for instance, also build combs with hexagonal cells, so that for them the problem was identical, and they have solved it in a far less ingenious fashion.  Their combs have only one layer of cells, thus lacking the common base that serves the bees for their two opposite layers.  The wasps’ comb, therefore, is not only less regular, but also less substantial; and so wastefully constructed that, besides loss of material, they must sacrifice about a third of the available space and a quarter of the energy they put forth.  Again, we find that the trigonae and meliponae, which are veritable and domesticated bees, though of less advanced civilisation, erect only one row of rearing-cells, and support their horizontal, superposed combs on shapeless and costly columns of wax.  Their provision-cells are merely great pots, gathered together without any order; and, at the point between the spheres where these might have intersected and induced a profitable economy of space and material, the meliponae clumsily insert a section of cells with flat walls.  Indeed, to compare one of their nests with the mathematical cities of our own honey-flies, is like imagining a hamlet composed of primitive huts side by side with a modern town; whose ruthless regularity is the logical, though perhaps somewhat charmless, result of the genius of man, that to-day, more fiercely than ever before, seeks to conquer space, matter, and time.

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