PROPOSITION XXVIII. THEOREM.

672. The volumes of two similar polyhedrons are to each other as the cubes of any two homologous edges.

Let V, V' denote the volumes, GB, G'B' any two homologous edges, of the polyhedrons P and P.

Proof. Decompose these polyhedrons into tetrahedrons similar, each to each, and similarly placed.

§ 666

Denote the volumes of these tetrahedrons by v, v1, v2,

REGULAR POLYHEDRONS.

673. DEF. A regular polyhedron is a polyhedron whose faces are equal regular polygons, and whose polyhedral angles are equal.

674. To determine the number of regular convex polyhedrons possible.

A convex polyhedral angle must have at least three faces, and the sum of its face angles must be less than 360° (§ 581).

1. Since each angle of an equilateral triangle is 60°, convex polyhedral angles may be formed by combining three, four, or five equilateral triangles. The sum of six such angles is 360°, and hence greater than the sum of the face angles of a convex polyhedral angle. Hence, three regular convex polyhedrons are possible with equilateral triangles for faces.

2. Since each angle of a square is 90°, a convex polyhedral angle may be formed by combining three squares. The sum of four such angles is 360°, and therefore greater than the sum of the face angles of a convex polyhedral angle. Hence, one regular convex polyhedron is possible with squares.

3. Since each angle of a regular pentagon is 108° (§ 206), a convex polyhedral angle may be formed by combining three regular pentagons. The sum of four such angles is 432°, and therefore greater than the sum of the face angles of a convex polyhedral angle. Hence, one regular convex polyhedron is possible with regular pentagons.

4. The sum of three angles of a regular hexagon is 360°, of a regular heptagon is greater than 360°, etc. Hence, only five regular convex polyhedrons are possible.

The five regular polyhedrons are called the tetrahedron, the hexahedron, the octahedron, the dodecahedron, the icosahedron.

Q. E. F.

675. The regular polyhedrons may be constructed as follows: Draw the diagrams given below on stiff paper. Cut through the full lines and paste strips of paper on the edges shown in the diagrams. Fold on the dotted lines, and keep the edges in contact by the pasted strips of paper.

A+

CYLINDERS.

676. DEF. A cylindrical surface is a curved surface generated by a straight line, which moves parallel to a fixed straight line and constantly touches a fixed curve not in the plane of the straight line.

The moving line is called the generatrix, and the fixed curve the directrix,

677. DEF. The genera

trix in any position is

Cylindrical Surface.

called an element of the cylindrical surface.

678. DEF. A cylinder is a solid bounded by a cylindrical surface and two parallel plane surfaces.

679. DEF. The two plane surfaces are called the bases, and the cylindrical surface is called the lateral surface.

680. DEF. The altitude of a cylinder is the perpendicular distance between the planes of its bases. The elements of a cylinder are all equal.

it may be generated by the revolution of a rectangle about one side as an axis.

Inscribed Prism.

684. DEF. Similar cylinders of revolution are cylinders generated by the revolution of similar rectangles about homologous sides.

685. DEF. A tangent line to a cylinder is a straight line, not an element, which touches the lateral surface of the cylinder but does not intersect it.

686. DEF. A tangent plane to a cylinder is a plane which con

tains an element of the cylinder but does not cut the surface. The element contained by the plane is called the element of

contact.

687. DEF. A prism is inscribed in a cylinder when its lateral edges are elements of the cylinder and its bases are inscribed in the bases of the cylinder.

688. DEF. A prism is circumscribed about a cylinder when its lateral edges are parallel to the elements of the cylinder and its bases are circumscribed about the bases of the cylinder.

Circumscribed Prism.

689. DEF. A section of a cylinder is the figure formed by its intersection with a plane passing through it.

A right section of a cylinder is a section made by a plane perpendicular to its elements.