The geometric principles brought into play in the construction of honey-comb have been a favourite study with mathematicians of all ages, and especially this rhombiform method adopted by the bee in flooring her cells. The rhomb is best described as a plane-figure whose four sides are equal, like those of a square, but whose angles are not right angles. In such a figure there are necessarily two greater angles and two smaller, facing each other in pairs. The three rhombs composing the base of the honey-cell lean together, as has been seen, in the form of a blunt pyramid; and—treating all angles as negligible factors—the bluntness of this pyramid is found to coincide very aptly with the shape of the full-grown larvæ. But this is not the only reason for the particular inclination given by the bee to the rhombs forming the base of each cell. Economy rules here, as in everything else she undertakes; and the truth that she has chosen the one and only form of cell-base which takes the least possible material to construct has received very striking confirmation. The story is an old and famous one, but it will bear repeating. A great naturalist once put himself to an infinity of trouble in measuring the angles formed by the rhombs in a vast number of comb-cell bases, and he found that these showed remarkable uniformity. It will be clear that the hollow pyramid of the cell-bottom will be either deep or shallow, according to the shape of the three rhombs composing it. The apex of the pyramid is formed by the meeting of three equal angles, one from each rhomb; and it is plain that this apex will be sharp or blunt, according to whether the meeting angles are wide or narrow. It was, of course, impossible to ascertain the dimensions of these angles with absolutely microscopical nicety; but, dealing only with the most perfect comb, the naturalist found that the two greater angles in the rhombs measured very nearly 110°, and the two lesser angles 70°. He also found that the angles formed by the conjunction of the cell-sides with the bases had the same dimensions as those of the rhombs. Assuming therefore that, mathematically, the angles of the rhombs and cell-sides should be equal, he was able to calculate exactly the angles for which the bees were evidently striving in the construction of the rhombs—109° 28′ and 70° 32′. Another bee-lover scientist, ruminating over these figures, was much impressed by them, and determined to find out the reason why the bee made such constant choice of this particular shape of rhomb. He therefore conceived the idea of submitting the bee’s judgment on this cell-base question to an independent authority. Without disclosing his object, he propounded the following problem to one of the greatest mathematicians of the day.