"When one has well examined," says Réaumur,[84] "the true shape of each cell, when one has studied their arrangement, geometry seems to have guided the design for the whole work, and to have presided over its execution. One finds that all the advantages which could have been desired are here combined. The bees seem to have had to solve a problem containing conditions which would have made the solution appear to be difficult to many geometricians. This problem may be thus enunciated:--Given a quantity of matter, say of wax, it is required to form cells which shall be equal and similar to each other, of a determined capacity, but as large as possible in proportion to the quantity of matter which is employed, and the cells to be so placed that they may occupy the least possible space in the hive. To satisfy this last condition, the cells should touch each other in such a manner that there may remain no angular space between them, no gap to fill up. The bees have satisfied these conditions, and at the same time they have satisfied the first conditions of the problem in making cells which are tubes having six equal sides, or in other words, hexagonal tubes.... We see still further that the best thing the bees could do to economise their space and materials, was to compose their honeycombs of two rows of cells turned in opposite directions." [84] "Mémoires pour servir à l'Histoire des Insectes," tome v., p. 379. [Illustration: Fig. 317.--Cells constructed by Bees.] This arrangement, it will be seen, enables them to economise the half of the wax intended for making the bases of the cells. They economise it still more by making the bases and the sides of the tubes extremely thin; the borders only of the comb being fortified by an excess of wax. These two-sided combs descend from the roof of the hive in parallel series, their thickness being about half an inch. They are fixed to the top by a sort of wax foot, and fastened to the sides by numerous bands. The bees pass between the rows, besides excavating circular openings, which serve as doors of communication. The form and the general arrangement of these buildings are otherwise very varied, according to circumstances. The bees always accommodate themselves to the nature of the hive. In all these operations they exhibit great judgment. It is impossible, when one has once seen them at work, to look on them as mere organised machines, whose instinct is their spring of action; we are forced to concede to them intelligence. [Illustration: Fig. 318.--The cells of a Bee-hive. A, large cell intended for the larvæ of the _queens_. B, middling-sized cells intended for the larvæ of the _males_. C, small cells intended for the larvæ of the _workers_.] The cells are of three dimensions: the small ones intended for the larvæ of the workers, the middling-sized ones for the larvæ of the males, and the large ones for the larvæ of the queens.