Sections made in any two pyramids at the same distance from these summits are in a constant ratio. Parallelopipedon.--Its diagonals. Any polyhedron can always be divided into triangular pyramids.--Two bodies composed of the same number of equal and similarly disposed triangular pyramids, are equal. _Similar polyhedrons._ The homologous edges of similar polyhedrons are proportional; as are also the diagonals of the homologous faces and the interior diagonals of the polyhedrons.--The areas of similar polyhedrons are as the squares of the homologous edges. _Measure of volumes._ Two parallelopipedons of the same base and of the same height are equivalent in volume. If a parallelogram be constructed on the base of a triangular prism, and on that parallelogram, taken as a base, there be constructed a parallelopipedon of the same height as the triangular prism, the volume of this prism will be half of the volume of the parallelopipedon.--Two triangular prisms of the same base and the same height are equivalent. Two tetrahedrons of the same base and the same height are equivalent. A tetrahedron is equivalent to the third of the triangular prism of the same base and the same height. The volume of any parallelopipedon is equal to the product of its base by its height.--What must be understood by that enunciation.--The volume of any prism is equal to the product of its base by its height. The volume of a tetrahedron and that of any pyramid are measured by the third of the product of the base by the height. Volume of the truncated oblique triangular prism. The volumes of two similar polyhedrons are to each other as the cubes of the homologous edges. 3. OF ROUND BODIES. _Of the right cone with circular base._ Sections parallel to the base.--Having the dimensions of the trunk of a cone with parallel bases, to find the height of the entire cone. The area of a right cone is measured by half of the product of the circumference of its circular base by its side.--Area of a trunk of a right cone with parallel bases. Volume of a pyramid inscribed in the cone.--The volume of a cone is measured by the third of the product of the area of its base by its height.[2] [Footnote 2: The volume of the cone is derived from that of the pyramid; and it is to be noted that the demonstration applies to the cone with closed base, whatever the figure of that base.] Which of the preceding properties belong to the cone of any base whatever? _Of the right cylinder with circular base._ Sections parallel to the base. The area of the convex surface of the right cylinder is measured by the product of the circumference of its base by its height.--This is also true of the right cylinder of any base.