nal angles. Therefore (53) the triangles are equal, and the angle H G E the angle G E F. Then these must be alternate internal angles, and H G is parallel to E F (37).

131. Two straight lines comprehended between three. parallel planes are divided proportionally. Let the three planes be A B, C D, E F (fig. 89). 1. Suppose the two F 89 lines to meet as G H and G I. By the preceding proposition the plane G H I will make the intersections K L and H I parallel. Then (70) GK: KH:: GL:LI. 2. Suppose the two lines do not meet as G H and M I. Still we shall have G K K H:: MN: N I. For by drawing G I, the plane G I M makes the intersections L N and GM parallel. Then (70) GL:LI:: MN: NI. But we had GK: KH:: GLLI. Hence, leaving out the common ratio GL: LI, we have G K÷KH:: M N NI.

:

SECTION THIRD.

Solids

132. —A solid is that magnitude which has the three dimensions of extension, namely, length, breadth, and thickness ; and we may conceive it to be generated by the motion of a surface in any direction but that of its length or breadth Thus we have the origin of the three dimensions: for the moving surface has two, length and breadth, and the motion produces a third, namely thickness. We have seen that points are the boundaries of lines, and lines the boundaries of surfaces. In like manner surfaces are the boundaries of solids. These surfaces may be either plane or curved, and the solids enclosed by them will have different denominations and properties accordingly.

133. The general name for solids bounded by planes is polyedron. The planes are called faces, and their lines of intersection edges or sides. The least number of planes which can enclose a space or bound a solid, is four. Three

"

-A

planes meeting each other, would make an opening called a solid angle, and a fourth is necessary to close up this opening. Thus the three planes B A C, B A D, CA D F90 (fig. 90), which meet in A, form an opening or solid angle at A, and a fourth plane B C D is necessary to close up this opening. The points A, B, C, D are called vertices. The solid A B C D is called, from the number of its faces, a tetraedron. For the same reason a solid of six faces is called a hexaedron, one of eight, an octaedron, and so on. But other denominations, depending upon the form and relative positions of the faces, are more important. prism is a solid comprehended under several parallelograms which terminate in two equal and parallel polygons. Thus F 91 if the polygon A B C D E (fig. 91) is equal and parallel to the polygon FGHIK, and if all the other faces are parallelograms, as they evidently must be (129, 130), then the solid A H is a prism. The two equal and parallel polygons are called the bases of the prism, and the sum of the parallelograms A F G B, BGH C, &c. are called the convex surface of the prism. If the faces are perpendicular to the bases the prism is called a right prism. The altitude of a prism is a perpendicular let fall from one base to the other. If the bases of a prism be triangles, the prism is said to be triangular; if quadrilaterals, quadrangular, and so on. If the bases of a prism be parallelograms, then all the faces will be parallelograms, and the prism is called a F92 parallelopiped. Thus AG (fig. 92) is a parallelopiped. If the bases be right parallelograms, and if the other faces be perpendicular to the bases, the prism is called a right parallelopiped. Among right parallelopipeds the cube is most remarkable, being comprehended under six equal squares. The only remaining polyedron to be mentioned is the pyramid. A pyramid is a solid comprehended under several triangles proceeding from the same point and terminating in the sides of a polygon-. Thus A B C D E F F93 (fig. 93) is a pyramid. The point A is called the vertex, and the polygon B C D E F the base. The altitude of a pyramid is a perpendicular let fall from the vertex to the base. The sum of the triangles form the convex surface of the pyramid. If a plane as G H I K L pass through the pyramid parallel to the base, the part cut off below is called a frustum of a pyramid. If the base of a pyramid is a regular polygon and if the altitude passes through the cen

tre of the base, the pyramid is said to be regular, and the altitude is called the axis of the pyramid.

If

134. Of the solids terminated by curved surfaces, only three are considered in the elements of geometry. These are the cylinder, the cone, and the sphere, which are usually denominated the three round bodies, or the three solids of revolution. If a right parallelogram be supposed to revolve about one of its sides as a fixed axis, the solid thus generated will be a cylinder-. Thus if the right parallelogram A B GH (fig. 94) be supposed to revolve about A B, the solid F 94 E G is a cylinder. The two equal and parallel circles described by the radii A H and B G, are called the bases of the cylinder, the axis A B the altitude, and the path described by H G, the convex surface. If a right triangle be supposed to revolve about one of its sides which include the right angle, the solid thus generated will be a cone-. Thus if the right triangle S A D (fig. 95) revolve about S A as F95 an axis, the solid S-B D C E is a cone. The circle described by the revolution of A D is called the base, the point S the vertex, and the path described by the hypothenuse S D, the convex surface. The axis S A is the altitude, and any line S B drawn from the vertex to the circumference of the base, is called the side of the cone. a plane as F G H I pass through the cone parallel to the base, the part cut off below is called a frustum of a cone. -If a semicircle be supposed to revolve about its diameter, the solid thus generated will be a sphere. Thus if the semicircle M AP (fig. 96) revolve about M P, the solid F 96 thus generated will be a sphere. MP, the diameter of the generating circle, is the diameter of the sphere, and C P the radius. From the manner in which the sphere is generated, it follows that-every point in the surface of a sphere is equally distant from the centre-. Also-if a plane be made to pass through the sphere in any direction, the section will be a circle. If the plane pass through the centre as MD P, this is evident, since every point in the curve M DP is equally distant from the centre C. In this case the circle is called a great circle. If the plane does not pass through the centre as E H G, still the curve EH G Fis a circle. Suppose the plane in question to be perpendicular to the diameter of the generating circle. It is immaterial whether this diameter be considered as M P or A B. Let it be A B. Then the curve EH GI may be con

ceived to be traced by the motion of the point G. But G remains always at the same distance from H. Therefore it describes a circle of which H is the centre. Now in whatever direction we suppose a plane to pass, it is evident that a diameter may be drawn perpendicular to it, and that this may be considered as the diameter of the generating circle. Then, from the reasoning just made use of, the section will be a circle. Hence the proposition is universally true. In this case, when the plane does not pass through the centre of the sphere, the circle is called a small circle. If two parallel planes pass through a sphere, or if one be a tangent to the sphere, that is, if it touch the sphere only in one point, while the other passes through it, in either case the portion of the surface comprehended between the two parallel planes is called a zone. Thus the portions of the surface A-E H GI and E H G I—M DP F are zones, and the circular planes are called their bases. Also the portion of the sphere comprehended between two parallel planes is called a spherical segment. Thus the solids A-E H GI and E H G I-M DPF are spherical segments, and the circular planes are their bases. The altitude of a zone or segment is the perpendicular drawn between its bases. While the semicircle A P B. generates the sphere, the sector B C K generates a solid which is called a spherical sector.

Surface of Polyedrons.

135. With regard to the solids defined in the two preceding articles, two questions present themselves. First, how shall we measure their surfaces? Secondly, how shall we measure their volume or solidity? We shall consider all the above solids in these two points of view. We begin with the surface of a prism. To find the surface of a prism, take double the area of the base, and add it to the sum of the areas of the parallelogram which form the convex surface. This is evident from the definition (133). —If the prism be a right prism, the convex surface is equal to the perimeter of the base multiplied by the altitude of the prism-. For the convex surface is made up of right parallelograms, of which the altitude is that of the prism, and the sum of the bases the perimeter of the base of the prism. If the prism is a circle, its surface is six times the area of one of its

faces, or six times the square of one of its sides. This is evident from the definition.

136. To find the surface of a pyramid, add the area of the base to the sum of the areas of the triangles which form its convex surface-. This is evident from the definition

(133). 137. To find the surface of the frustum of a pyramid, add the sum of the areas of the upper and lower bases to the sum of the areas of the trapezoids which form the convex surface. The convex surface of the frustum of a pyramid is made up of trapezoids, because H I (fig. 93) is parallel F 93 to C D, I K to D E, &c. (130).

Solidity of Polyedrons.

138. We now proceed to find the solidity of polyedrons. For this purpose we must fix upon some known solid as a unit of solidity, and see how many times it is contained in the solid to be measured. Of all solids the cube is most regular and simple; and accordingly the same reasons which induced geometers to adopt the square as the unit of surface, have also induced them to adopt the cube as the unit of solidity. The cube is a solid comprehended under six equal squares, and consequently has all its three dimensions the same; in other words its length, breadth, and thickness are expressed by the same linear unit, and each of its faces is the square of that linear unit. Thus a cubic inch is an inch long, an inch broad and an inch thick, and so of a cubic foot, a cubic yard, &c. The unit of solidity, as well as the unit of surface, depends upon the linear unit. It is a cubic inch, when the length, breadth, and thickness are expressed in inches, a cubic foot, when they are expressed in feet, and so on.

139. The solidity of a right parallelopiped is equal to the area of its base multiplied by its altitude. Let the right parallelopiped be E C (fig. 97), having the right F 97 parallelogram EH G F for its base and F B for its altitude. Suppose E H to contain a given number of inches as 9, and E F a given number as 5. Then (100) E H G F will contain 45 squares. Now each of these squares may be made the base of a cube, whose three dimensions are an inch. The the first layer will contain 45 of

these cubes. And it is evident that there will be as ma