6. The acute angle which a line makes with its own projection on a plane, is the least angle which

it makes with any line on the plane.

See Def. of projection, and use the last Ex. and (I. 23).

7. If a plane be passed through a diagonal of a parallelogram, the perpen

diculars to this plane from the extremities of the other diagonal are equal.

Pass a plane through the perpendiculars and the other diagonal, and work in equal triangles.

8. If each of the projections of a line on two intersecting planes be straight, the line itself is a straight line.

9. If, from a point A, a perpendicular be drawn to a plane meeting it at B, and from B a perpendicular be drawn on a straight line DE in the plane, meeting it in C, then AC is perpendicular to DE.

D

E

A

B

10. Within a given triangle is inscribed another triangle; then the sum of the angles subtended by the sides of the interior triangle at a point not in the plane of the triangle, is less than the sum of the angles subtended at the same point by the sides of the exterior triangles (VII. 19).

11. A straight line is drawn within a triedral angle; the angles which this line makes with the lines which form the sides of the angle, are greater than the half sum of the three angles forming the triedral angle (VII. 19).

15

BOOK VIII.

POLYEDRONS, CYLINDERS, CONES.

DEFINITIONS.

1. A solid is a figure having length, breadth and thickness. 2. A polyedron is a solid bounded by plane surfaces.

3. A prism is a polyedron of which two opposite faces, called bases, are equal, similar, and parallel polygons, and the other faces are parallelograms.

If the lateral edges stand at right angles to the base, it is said to be a right prism. If the base be a triangle, it is called a triangular prism; if a pentagon, a pentangular prism, etc., etc.

4. A parallelopiped is a prism whose base is a parallelogram.

Corollary.-Hence, all its faces are parallelograms.

If all its faces be rectangles, it is said to be rectangular.

If all its faces be squares, the solid is called a cube.

170

5. A pyramid is a polyedron, of which one of its faces, called the base, is any polygon, and the other faces are triangles having a common vertex, which is not in the plane of the base. If the base be a regular polygon, and the line from the vertex to the middle of the base be at right angles to its plane, the pyramid is a right pyramid.

If the base be a triangle, it is called a triangular pyramid, etc. etc.

6. A cylinder is a solid formed by the revolution of a rectangle about one of its sides.

7. This side is the axis of the cylinder; the circular ends are the bases.

8. A cone is a solid formed by the revolution of a right-angled triangle about one of the sides adjacent the right angle.

9. This side is the axis of the cone; the base is the circle formed by the other side, and the other extremity of the axis is the vertex.

10. The slant height of a cone is the distance from any point of the circumference of the base to the vertex. The slant height of a right pyramid is the distance of any side of the perimeter of the base from the vertex.

11. The frustum of a cone or pyramid is the portion cut off toward the base by a plane parallel to the base.

12. Similar solids are such as are made up of the same number of similar planes, similarly placed. Similar cylin

ders and cones are such as have their altitudes and diameters of their bases proportional to each other.

13. The terms cylinder and cone have often a wider meaning than that just given. If a straight line be moved on any curve, so as to be always parallel to its original position, the surface generated is a cylindrical surface. If moved so that all the different positions of the line pass through a point not in its plane, the surface is a conical surface. The different positions of the generating line are called elements of the surface. The elements of a cone may be continued beyond the vertex when another cone is formed similar to the first. If the base be a circle, and the elements of the cylinder be perpendicular to the base, the ordinary cylinder is formed, which is therefore said to be a right cylinder with a circular base. If the line from the vertex of the cone to the centre of the circular base be perpendicular to the plane of the base, the cone is a right cone.

14. A prism is inscribed in a cylinder when all the lateral edges of the prism are elements of the cylinder. A pyramid is inscribed in a cone when all the lateral edges of the pyramid are elements of the cone.

Proposition 1.

Theorem.-If two prisms have the three faces which bound any solid angle in each, equal, similar and similarly placed, the prisms will be equal.

Let the two prisms A-BDEFC, a-bdefc have the three faces which bound the solid angles B, b, equal, similar

and similarly placed. The prisms will be equal to each other.

Apply the prism ACE to the prism ace, so that B shall be on b, and the bases coincident.

And because the three angles

which contain the angle B are respectively equal to the three which contain the angle b, their faces are equally inclined to each other (VII. 21). Therefore the faces AD, AC will coincide with

ad, ac; and because they are

H

G

B D

b

d

equal and similar, the lines AG, AH will coincide with ag, ah. Therefore the upper bases, being equal and similar to the lower bases and to each other, will coincide altogether, and hence the other faces will coincide, and the prisms will coincide altogether, and are therefore equal to each other.

Proposition 2.

Theorem.-The opposite faces of a parallelopiped are equal and similar figures.

Let A-BCED be a parallelopiped; its opposite faces are equal and similar.

The two bases GF, DC are equal and similar by (Def. 3). Because AC is a parallelogram, AB is equal and parallel to FC; for the same reason, the other sides of the face AD are equal and parallel to the other sides of the face FE.

There

fore the angles of one face are equal

D

G

H

B

E

to those of the other (VII. 9). Therefore the faces are equal