BOOK X.

POLYHEDRONS.

740. A Polyhedron is a solid bounded by planes.

741. The bounding planes, by their intersections, determine the Faces of the polyhedron, which are polygons.

742. The Edges of a polyhedron are the sects in which its faces meet.

743. The Summits of a polyhedron are the points in which its edges meet.

744. A Plane Section of a polyhedron is the polygon in which a plane passing through it cuts its faces.

745. A Pyramid is a polyhedron of which all the faces, except one, meet in a point.

746. The point of meeting is called the Apex, and the face not passing through the apex is taken as the Base.

747. The faces and edges which meet at the apex are called Lateral Faces and Edges.

748. Two polygons are said to be parallel when each side of the one is parallel to a corresponding side of the other.

749. A Prism is a polyhedron two of whose faces are congruent parallel polygons, and the other faces are parallelograms.

750. The Bases of a prism are the congruent parallel polygons.

751. The Lateral Faces of a prism are all except its bases. 752. The Lateral Edges are the intersections of the lateral faces.

753. A Right Section of a prism is a section by a plane perpendicular to its lateral edges.

754. The Altitude of a Prism is any sect perpendicular to both bases.

755. The Altitude of a Pyramid is the perpendicular from its vertex to the plane of its base.

756. A Right Prism is one whose lateral edges are perpendicular to its bases.

757. Prisms not right are oblique.

758. A Parallelopiped is a prism whose bases are parallelograms.

759. A Quader is a parallelopiped whose six faces are rectangles.

760. A Cube is a quader whose six faces are squares.

THEOREM I

761. All the summits of any polyhedron may be joined by one closed line breaking only in them, and lying wholly on the surface.

B

A

For, starting from one face, ABC..., neighboring polygon.

each side belongs also to a

Therefore, to join A and B, we may omit AB, and use the remainder of the perimeter of the neighboring polygon a. In the same way, to join B and C, we may omit BC, and use the remainder of the perimeter of the neighboring polygon b, unless the polygons a and b have in common an edge from B. In such a case, draw from B in b the diagonal nearest the edge common to a and b; take this diagonal and the perimeter of b beyond it around to C, as continuing the broken line; and proceed in the same way from C around the neighboring

polygon c.

When this procedure has taken in all summits in faces having an edge in common with ABC..

we may, by proceeding from the closed broken line so obtained, in the same way take in the summits

on the next series of contiguous faces, etc.

So continue until the single closed broken line goes once, and only

once, through every summit.