CONE, Properties of the. 1. Cones and pyramids having the same bases and alti tudes are equal to each other. It is shewn, that every triangular prism may be di vided into three equal pyramids, and therefore that a triangular 1)2, ramid is one-third of a prism standing on the same base, and having the same altitude. Hence, since every multangular body may be resolved into triangular ones, every pyramid is the third part of a prism standing upon the same base, and having the same altitude ; and as a cone may be esteemed an infinite angular py ramid, and a cylinder an infinite angular prism, a cone is the third part of a cylin der which has the same base and alti tude. Hence we have a method of mea suring the solidity and surface of a cone and pyramid. Thus, End the solidify of a prism or cylinder, having the same base with the cone or pyramid, which found, divide by three, the quotient will be the solidity of the cone or pyramid. Or the solidity' of any cone is equal to the area of the base, multiplied into one third part of its altitude. As for the surfaces, that of a right pone, not taking in the base, is equal to a triangle, whose base is the periphery and altitude•of the side of the cone ; therefore the surfaCe of a right cone is had by multiplying the pe*phery of the base into half of the -ide, adding the product to that of the base.
2. The altitudes of similar cones a* as the radii of the bases. and the axes. like.
are as the radii of the bases, and term the same angle with them.
3. Comes are to one another in a ratio compounded of their bases and altitudes. 1. Similar cones are in a triplicate ratio of their homologous sides, and likewise of their altitudes.
5. Of the cones standing upon the same base, and having the same altitude, the superfices of that which is most oblique is the greatest, and so the super fices of the right cone is the least ; but the proportion of the superfices of an oblique cone to that of a right one, or, which is the same thing, the comparison to a circle, or the conic sections, has not yet been determined.
CoxEs of the higher kinds, those whose bases, and sections parallel to the bases, are circles of the higher kinds. They are generated by supposing a right line fixed in a point on high, but conceived to be capable of being extended more or less on occasion, and moved round the peri phery of a circle.