PHYSICAL AND MATHEMATICAL SCIENCES. 77

perimetrical polygons. Similar considerations apply tosolids and their surfaces.

The form in which the cells of the honeycomb aremade appears to accord with the deductions of geo-metry, on principles of which this is the foundation, butto which several other considerations must be added.The only figures which, placed together, leave no in-terstices, are equilateral triangles, squares, and hexagons:of these, the last, of course, have the greatest contentwith the same circumference ; in forming contiguouscells, therefore, there is a saving of materials in usingthe hexagonal form. The mode of terminating the cellsis the most curious part of the whole. Geometrically,an extension of the same advantages would be gained ifthey ended in three-sided pyramids, formed by planescutting off each alternate solid angle of the hexagonalcell, and inclined at a certain angle, which is found bycalculation. Precisely such conditions have been foundto be observed by taking the average of a great numberof measurements of the actual cells of honeycombs.

This part of Pappus’s work has come down to us inan imperfect state, but has been restored by Maclaurin .Pappus forcibly expresses his admiration at this verysingular fact in the economy of the bee, exclaiming,—

Kara t iva ■yfw/ASTpnoji/ pVjyjxMmrcu rrpovolav.” (Theywork by a sort of geometrical forethought.)

In his fourth book, Pappus discusses several pointsconnected with the quadrature of the circle, and de-scribes some of the inventions made by preceding geo-meters in their researches having that object in view.One of the most remarkable, perhaps, is the constructionof a curve called the quadratrix, or squaring line, in-vented by Dinostratus and Nicomedes, of which he givesa demonstration; and hy which the length of a circulararc is assigned : but the application of it involves theprinciple of limits, and does not, therefore, materiallyhelp the question. It affords, nevertheless, some verybeautiful geometrical speculations.

The labours of this great geometer, however, extended