the number of triangles, in each set, is the same, it follows that these sums are equal. But in the triedral angle whose vertex is B, we have (P. XIX.),,

ABS+ SBC > ABC;

and the like may be shown at each
of the other vertices, C, D, E, A:
hence, the sum of the angles at the
bases, in the triangles whose common
vertex is S, is greater than the sum
of the angles at the bases, in the set
whose common vertex is 0: therefore,
the sum of the vertical angles
sum of the angles about 0:
angles; which was to be proved.

about S, is less than the that is, less than four right

Scholium. The above demonstration is made on the supposition that the polyedral angle is convex, that is, that the diedral angles of the consecutive faces are each less than two right angles.

If the plane angles formed by the edges of two triedral angles are equal, each to each, the planes of the equal angles are equally inclined to each other.

Let S and T be the vertices of two triedral angles, and let the angle ASC be equal to DTF, ASB to DTE, and BSC to ETF: then will the planes of the equal angles be equally inclined to each other.

For, take any point of SB, as B, and from it draw in the two faces ASB and CSB, the lines BA and BC, respectively perpendicular to SB: then will the angle ABC measure the inclination of these faces. Lay off TE equal

to SB, and from E draw in the faces DTE and FTE, the lines ED and EF, respectively perpendicular to TE then will the angle DEF

DTE; hence, AB is equal to DE, and AS to TD. In like manner, it may be shown that BC is equal to EF, and CS to FT. The triangles ASC and DTF, have the angle ASC equal to DTF, by hypothesis, the side AS equal to DT, and the side CS to FT, from what has just been shown; hence, the triangles are equal in all their parts, and consequently, AC is equal to DF. Now, the triangles ABC and DEF have their sides equal, each to each, and consequently, the corresponding angles are also equal; that is, the angle ABC is equal to DEF: hence, the inclination of the planes ASB and CSB, is equal to the inclination of the planes DTE and FTE. In like manner, it may be shown that the planes of the other equal angles are equally inclined; which was to be proved.

Scholium. If the planes of the equal plane angles are like placed, the triedral angles are equal in all respects, for they may be placed so as to coincide. If the planes of the equal angles are not similarly placed, the triedral angles are equal by symmetry. In this case, they may be placed so that two of the homologous faces shall coincide, the triedral angles lying on opposite sides of the plane, which is then called a plane of symmetry. In this position, for every point on one side of the plane of symmetry, there is a corresponding point on the other side.

BOOK VII.

POLYE DRON 8.

DEFINITIONS.

1. A POLYEDRON is a volume bounded by polygons.

The bounding polygons are called faces of the polyedron; the lines in which the polygons meet, are called edges of the polyedron; the points in which the edges meet, are called vertices of the polyedron.

2. A PRISM is a polyedron, two of whose faces are equal polygons having their homologous sides parallel, the other faces being parallelograms.

The equal polygons are called bases of the prism; one the upper, and the other the lower base; the parallelograms taken together

make up the lateral or convex surface of the prism; the lines in which the lateral faces meet, are called lateral edges of the prism.

3. The ALTITUDE of a prism is the perpendicular dis tance between the planes of its bases.

4. A RIGHT PRISM is one whose lateral edges are perpendicular to the planes of the bases.

In this case, any lateral edge is equal to the altitude.

5. An OBLIQUE PRISM is one whose lateral edges are oblique to the planes of the bases.

In this case, any lateral edge is greater than the altitude.

6. Prisms are named from the number of sides of their bases; a triangular prism is one whose bases are triangles; a quadrangular prism is one whose bases are quadrilaterals; a pentangular prism is one whose bases are pentagons, and

so on.

A PARALLELOPIPEDON is a prism whose bases are parallelograms.

A Rectangular Parallelopipedon is a right parallelopipedon, all of whose faces are rectangles; a cube is a rectangular parallelopipedon, all of whose faces are squares.

8. A PYRAMID is a polyedron bounded by a polygon called the base, and by triangles meeting at a common point, called the vertex of the pyramid.

The triangles taken together make up the lateral or convex surface of the pyramid; the lines in which the lateral faces meet, are called the lateral edges of the pyramid.

9. Pyramids are named from the number of sides of their bases; a triangular pyramid is one whose base is a triangle; a quadrangular pyramid is one whose base is a quadrilateral, and so on.

10. The ALTITUDE of a pyramid is the perpendicular distance from the vertex of the pyramid to the plane of its base.

11. A RIGHT PYRAMID is one whose base is a regular polygon, and in which the perpendicular drawn from the vertex to the plane of the base, passes through the centre of the base.

This perpendicular is called the axis of the pyramid.

12. The SLANT HEIGHT of a right pyramid, is the perpendicular distance from the vertex to any side of the base.

13. A TRUNCATED PYRAMID is that portion of a pyramid included between the base and any plane which cuts the pyramid.

When the cutting plane is parallel to the base, the truncated pyramid is called a FRUSTUM OF A PYRAMID, and the inter

section of the cutting plane with the pyramid, is called the upper base of the frustum; the base of the pyramid is called the lower base of the frustum.

14. The ALTITUDE of a frustum of a pyramid, is the perpendicular distance between the planes of its bases.

15. The SLANT HEIGHT of a frustum of a right pyramid, is that portion of the slant height of the pyramid which lies between the planes of its upper and lower bases.

16. SIMILAR POLYEDRONS are those which are bounded by similar polygons, similarly placed.

Parts which are similarly placed, whether faces, edges, or angles, are called homologous.

17. A DIAGONAL of a polyedron, is a straight line joining the vertices of two polyedral angles not in the same face.