alternately all round the base of the pyramid, there being three of each kind and six in all. The axis of each cell coincides not with the axis of the cell on the opposite surface, but with one of its angles ; so that each of the three obtuse angles at the base of the terminal pyramid cor responds to the central parts of three of the cells on the opposite side, and each of the sides of the pyramid which closes a cell on one side contributes in part to the enclos ing of three of the cells on the opposite side. We may easily satisfy ourselves that such is the case by piercing the centres of each of the three planes which close the bottom of a cell with a pin, when on turning the comb the three pins will be found to have passed into three different
cells on the opposite side.
A structure of this kind is obviously the one of all others calculated to afford the greatest space for each cell with the same quantity of materials. It is easy to perceive, in the first place, that in a plane surface, when a number of small spaces are to be divided by partitions, the hexagonal form is the one which comprehends the largest space com patible with the extent of the lines which enclose them ; for the equilateral triangle, the square, and the regular hexagon, are the only regular forms that admit of being joined together in the same plane without leaving interstices ; and the proportion of the area to the periphery iu every polygon increases as the figure consists of a greater number of sides, and is, therefore, greater in the hexagon than in either of the other two. The truth of this proposition was perceived by Pappus ; and even its application to the subject of the honeycomb was made by that ancient geometrician. But the determination of the form and inclination that should be given to the partitions that close the bottom of the cells, and which may, of course, belong equally to those on both sides of the comb, is a problem much more complicated and difficult of solution. It has exercised the skill of several modern mathematicians of great eminence. Reaumur proposed to Konig, pupil of the celebrated Bernouilli, and an expert analyst, the solution of the problem : To find the construction of a hexagonal prism terminated by a pyramid composed of three equal and similar rhombs (and the whole of given capacity), such that the solid may be made with the least possible quantity of materials ; which in other words was asking him to determine the angles of the rhombs that should cut the hexagonal prism so as to form with it the figure of the least possible surface, since the hexagon being given, this decided both their dimensions and their intersections with the sides of the cell. Maraldi had previously measured the angles of the rhombus and found them to be 109 28 and 70 32 respectively ; but Konig was not aware of this until after he had solved the problem, and assigned 109 26 and 70 34 as the angles, when he had sent him the Memoirs of the Academy of Science for 1712, containing Maraldi s paper; and Konig was equally surprised and pleased to find how nearly the actual measurement agreed with the result of his investiga tion. The measurement of Maraldi is correct, and the bees have, with rigorous accuracy, solved the problem, for the error turns out to be in Konig s solution. The construc tion of cells, then, is demonstrated to be such that no other that could be conceived would take so little material and labour to afford the same room.
Boscovich, who has also given a solution of the same problem, supposes that the equality of inclination of the planes gives greater facility to the construction of the comb, and might, therefore, be a motive of preference, indepen dently of the greater economy of wax. Maclaurin has offered a solution of this problem, and has demonstrated by simple geometry, that the most advantageous form is that which results from the supposed equality of the three plane angles forming the solid angles at the base. He estimates the saving of wax by partition so constructed, above what would be required for a flat partition, at one- fourth of the wax which would be wanted to complete the truncated sides of the cells, so as to form them into rectangles. L Huillier, in the Mtmoirs of the Berlin Academy, has given a demonstration which is remarkable for its simplicity, and for its involving none but elementary propositions ; he values the economy of wax at -^ of the whole wax employed. Le Sage, as appears from the life of that philosopher by Professor Prevost, has shown that this celebrated problem reduces itself to the finding of the angle at which two planes with a given inclination (such as 120) can be cut by a third plane, so as to make all the angles resulting from the section equal to one another.
But a more essential advantage than even the economy of wax results from this structure, namely, that the whole fabric has much greater strength than if it were composed of planes at right angles to one another ; and when we consider the weight they have to support when stored with honey, pollen, and the young brood, besides that of the bees themselves, it is evident that strength is a material requisite in the work.
It has often been a subject of wonder how such diminutive insects could have adopted and adhered to so regular a plan of architecture, and what principles can actuate so great a multitude to co-operate, by the most effectual and systematic mode, in its completion. Buffon has endeavoured to explain the hexagonal form by the uniform pressure of a great number of bees all working at the same time, exerted equally iu all directions in a limited space ; and illustrates his theory by supposing a number of similar cylinders compressed together, and taking the form of hexagonal prisms by the uniform expansion of each. The analogy of the forms produced by the law of crystal lization, of the figures assumed by various organs in the animal and vegetable world, such as the skin of the bat, and the inner coat of the second stomach of ruminant quadrupeds, is also adduced by this captivating but superficial writer in support of his argument. But however plausible this theory may at first sight appear, it will not stand the test of a serious examination. The explana tion he has attempted applies no further than to the inclination of the sides of the cells ; but he did not take into account, perhaps from not having studied the subject mathematically, the inclinations and forms of the planes which close each cell, and so curiously conspire on both sides to serve a similar office, while they at the same time accurately fulfil a refined geometrical condition. But it is sufficient confutation of the whole theory to show, that it is directly at variance with the actual process employed by the insects in the construction of their combs.
nature with instruments for building of a form somewhat analogous to the angles of the cells; but in no part, either of the teeth, antenna?, or feet, can any such correspondence be traced. Their shape in no respect answers to that of the rhombs, which are constructed by their means, any more than the chisel of the sculptor resembles the statue which it has carved. The shape of the head is indeed triangular, but its three angles are acute, and are different from that of the planes of the cells. The form of the plates of wax, as they are moulded in the pouches in which this substance is secreted, is an irregular pentagon, in no respect affording a model for any of the parts which compose the honeycomb. Hunter, observing that the thickness of the partition was nearly equal to that of the scale of wax, thought that the bees apply these scales immediately to the formation of the partition, by merely cementing them together. Reaumur, notwithstanding the use of glass hives,
had not been able to discover the mystery of their process