Again, it has been demonstrated that, by making the bottoms of the cells to consist of three planes meeting in a point, there is a saving of material and labour in no way inconsiderable. The bees, as if acquainted with these principles of solid geometry, follow them most accurately. It is a curious mathematical problem, at what precise angle the three planes which compose the bottom of a cell ought to meet, in order to make the greatest possible saving, or the least expense of material and labour. This is one of the problems which belong to the higher parts of mathematics. It has accordingly been resolved by some mathematicians, particularly by the ingenious Maclaurin, by a fluctionary calculation, which is to be found in the Transactions of the Royal Society of London. He has determined precisely the angle required, and he found, by the most exact mensuration the subject would admit, that it is the very angle in which the three planes in the bottom of the cell of a honeycomb do actually meet.[58] Marvellous as these facts undoubtedly are, they may now be regarded as having been satisfactorily explained. Long ago Buffon sought to account for the hexagonal form of the cells by an hypothesis of mutual pressure. Supposing the bees to have a tendency to build tubular cells, if a greater number of bees were to build in a given space than could admit of all the parallel tubes being completed, tubes with flat sides and sharp angles might result, and if the mutual pressure were exactly equal in all directions, these sides and angles would assume the form of hexagons. This hypothesis of Buffon was sustained by such physical analogies as the blowing of a crowd of soap-bubbles in a cup, the swelling of moistened peas in a confined space, &c. The hypothesis, however, as thus presented was clearly inadequate; for no reason is assigned why the mutual pressure, even if conceded to exist, should always be so exactly equal in all directions as to convert all the cylinders into perfect hexagons--even the analogy of the soap-bubbles and the moistened peas failing, as pointed out by Brougham and others, to sustain it, seeing that as a matter of fact bubbles and peas under circumstances of mutual pressure do not assume the form of hexagons, but, on the contrary, forms which are conspicuously irregular. Moreover, the hypothesis fails to account for the particular prismatic shape presented by the cell base. Therefore it is not surprising that this hypothesis should have gained but small acceptance. Kirby and Spence dispose of it thus:--'He (Buffon) gravely tells us that the boasted hexagonal cells of the bee are produced by the reciprocal pressure of the cylindrical bodies of these insects against each other!!'[59] The double note of admiration here may be taken to express the feelings with