face and its opposite diedral are either both acute, both right, or both obtuse.

POLYEDRALS.

605. A POLYEDRAL is the figure formed by several planes which meet at one point. Thus, a polyedral is composed of several angles having their vertices at a common point, every edge being a side of two of the angular faces. The triedral is a polyedral of three

faces.

606. Problem. - Any polyedral of more than three faces may be divided into triedrals

For a plane may pass through any two edges which are not adjacent. Thus, a polyedral of four faces may be divided into two triedrals; one of five faces, into three; and so on.

607. This is like the division of a polygon into triangles. The plane passing through two edges not adjacent is called a diag

onal plane.

A polyedral is

called convex, when every possible diagonal plane lies within the figure; otherwise

it is called concave.

608. Corollary.-If the plane of one face of a convex polyedral be produced, it can not cut the polyedral. 609. Corollary.-A plane may pass through the vertex of a convex polyedral, without cutting any face of the polyedral.

610. Corollary.-A plane may cut all the edges of a convex polyedral. The section is a convex polygon.

Geom.-18

611. When any figure is cut by a plane, the figure that is defined on the plane by the limits of the figure so cut, is called a plane section.

Several properties of triedrals are common to other polyedrals.

612. Theorem.-The sum of all the angles of a convex polyedral is less than four right angles.

For, suppose the polyedral to be cut by a plane, then the section is a polygon of as many sides as the polyedral has faces. Let n represent the number of sides of the polygon. The plane cuts off a triangle on each face of the polyedral, making n triangles. Now, the sum of the angles of this polygon is 2n—4 right angles (424), and the sum of the angles of all these triangles is 2n right angles. Let v right angles represent the sum of the angles at the vertex of the polyedral; then, 2n right angles being the sum of all the angles of the triangles, 2n v is the sum of the angles at their bases.

Now, at each vertex of the polygon is a triedral having an angle of the polygon for one face, and angles at the bases of the triangles for the other two faces. Then, since two faces of a triedral are greater than the third, the sum of all the angles at the bases of the triangles is greater than the sum of the angles of the polygon. That is,

2n-v2n4.

Adding to both members of this inequality, v + 4, and subtracting 2n, we have 4>v. That is, the sum of the angles at the vertex is less than four right angles.

This demonstration is a generalization of that of Article 587. The student should make a diagram and special demonstration for a polyedral of five or six faces.

613. Theorem. In any convex polyedral, the sum of the diedrals is greater than the sum of the angles of a polygon having the same number of sides that the polyedral has faces.

Let the given polyedral be divided by diagonal planes into triedrals. Then this theorem may be demonstrated like the analogous proposition on polygons (423). The remark made in Article 346 is also applicable here.

DESCRIPTIVE GEOMETRY.

614. In the former part of this work, we have found problems in drawing to be the best exercises on the principles of Plane Geometry. At first it appears impossible to adapt such problems to the Geometry of Space; for a drawing is made on a plane surface, while the figures here investigated are not plane figures.

This object, however, has been accomplished by the most ingenious methods, invented, in great part, by Monge, one of the founders of the Polytechnic School at Paris, the first who reduced to a system the elements. of this science, called Descriptive Geometry.

DESCRIPTIVE GEOMETRY is that branch of mathematics which teaches how to represent and determine, by means of drawings on a plane surface, the absolute or relative position of points or magnitudes in space. It is beyond the design of the present work to do more than allude to this interesting and very useful science.

EXERCISES.

615.-1. What is the locus of those points in space, each of which is equally distant from three given points?

2. What is the locus of those points in space, each of which is equally distant from two given planes?

3. What is the locus of those points in space, each of which is equally distant from three given planes?

4. What is the locus of those points in space, each of which is equally distant from two given straight lines which lie in the same plane?

5. What is the locus of those points in space, each of which is equally distant from three given straight lines which lie in the same plane?

6. What is the locus of those points in space, such that the sum of the distances of each from two given planes is equal to a given straight line?

7. If each diedral of a triedral be bisected, the three planes have one common intersection.

8. If a straight line is perpendicular to a plane, every plane parallel to the given line is perpendicular to the given plane.

9. Given any two straight lines in space; either one plane may pass through both, or two parallel planes may pass through them respectively.

10. In the second case of the preceding exercise, a line which is perpendicular to both the given lines is also perpendicular to the two planes.

11. If one face of a triedral is rectangular, then an adjacent diedral angle and its opposite face are either both acute, both right, or both obtuse.

12. Apply to planes, diedrals, and triedrals, respectively, such properties of straight lines, angles, and triangles, as have not already been stated in this chapter, determining, in each case, whether the principle is true when so applied.

CHAPTER X.

POLYEDRONS.

616. A POLYEDRON is a solid, or portion of space, bounded by plane surfaces. Each of these surfaces is a face, their several intersections are edges, and the points of meeting of the edges are vertices of the polyedron.

617. Corollary.-The edges being intersections of planes, must be straight lines. It follows that the faces of a polyedron are polygons.

618. A DIAGONAL of a polyedron is a straight line joining two vertices which are not in the same face. A DIAGONAL PLANE is a plane passing through three vertices which are not in the same face.

TETRAEDRONS.

619. We have seen that three planes can not inclose a space (581). But if any

point be taken on each edge of a triedral, a plane passing through these three points would, with the three faces of the triedral, cut off a portion of space, which would be inclosed by four triangular faces.

A TETRAEDRON is a polyedron having four faces.