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 parellelograms, F 92 and the prism is called a parallelopiped. Thus AG (fig. 92) is a parallelopiped. If all the plane and linear angles are right angles, 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 (fig. 93) is a pyramid. The point A is F 93 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. 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. The sum of the trapezoids thus formed is the convex surface of the frustum. If the base of a pyramid is a regular polygon and if the altitude passes through the centre of the base, the pyramid is said to be regular, and the altitude is called the axis of the pyramid. If a 134. DEF. Of the solids terminated by curved surfaces, only three are considered in the elements of geometry, namely, 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 BGH F 94 (fig. 94) be supposed to revolve about A B, the solid ÈG 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 AD F 95 (fig. 95) revolve about SA as an axis, the solid S-B D CE 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 crcumference of the base, is called the side of the cone. If 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 F 96 the semicircle M A P (fig. 96) revolve about M P, the solid 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 M D P, this is evident, since every point in the curve MDP 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 EH G, still the curve EH GI is 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 conceived 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 sur face comprehended between the two parallel planes is called a zone. Thus the portions of the surface A-E H GI and E H GI-M D P F are zones, and the circular planes are called their bases. Also the portion of the sphere cut off by a single plane, or comprehended between two parallel planes, is called a spherical segment. Thus the solids A-E H GI and EHGIMDPF 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. THEOREM. The convex surface of a right prism is equal to the product of the perimeter of the base by the altitude. DEM.-By the definition (133) the convex surface is composed of parallelograms. Moreover, the prism being a right prism, the altitude G D of the prism (fig. 91), is the common altitude of F 91 all the parallelograms; and the sum of their bases F G+G H+H I+ &c. is the perimeter of the base of the prism. Therefore, adding together their areas, we have for the convex surface, the measure above enunciated. 136. THEOREM.-The convex surface of a regular pyramid is equal to the product of the perimeter of its base, by half the altitude of one of its triangles. DEM. By the definition (133) all the triangles forming the convex surface of a regular pyramid are equal. For their bases are equal, being sides of a regular polygon; and the other sides are equal, being oblique lines drawn at equal distances from the perpendicular, since the axis passes through the centre. But triangles which are equal and have equal bases, must have the same altitude. Therefore, the sum of their bases multiplied by half the common altitude, is the same as the perimeter of the base of the pyramid multiplied by half this altitude. 137. THEOREM.-The convex surface of the frustum of a regular pyramid is equal to the product of half the perimeters of the two bases, by the altitude of one of the trapezoids. DEM.-By the definitions(133) the trapezoids forming the convex surface are all equal, being remainders after taking equal triangles from equal triangles. Moreover they have their parallel sides equal each to each, since the two polygons are regular. Therefore they must have a common altitude; and this multiplied by half the sum of their parallel sides, is the same as the measure above enunciated. SOLIDITY OF POLYEDRONS. 138. SCHO.-In order to find the solidity of polyedrons, 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 expressed in feet, and so on. 139. THEOREM.-The solidity of a right parallelopiped is equal to the area of its base multiplied by its' F 97 altitude. Let the right parallelopiped be E C (fig. 97), having the right parallelogram E H G F for its base and F B for its altitude. DEM.-Suppose E H to contain a given number of inches as 9, and E F a given number as 5. Then (100) EHGF will contain 45 squares. Now each of these squares may be made the base of a cube, whose three dimensions will be as many such layers as there are inches in the altitude, since this layer only takes up one inch F 1 of the altitude. Let the number of inches in the altitude F B be 8. Then the whole number of cubes contained in the right parallelopiped is 8X45=360. Thus the measure of its solidity is 360 cubic inches. We have here made use of particular numbers, but this is only for the sake of being definite. It is evident that the same reasoning would apply to any other numbers. If the dimensions contained fractions of an inch, the proposition would still be true, as might be shown by reasoning similar to that employed in art. 100. Hence we conclude universally that the solidity of a right parallelopiped is equal to the area of its base multiplied by its altitude, which is the same as the product of its three dimensions. Thus the solidity of E C expressed in lines E HXE FXF B. COR.-The solidity of a cube is found by taking one of its sides three times as a factor. Thus if the right parallelopiped be a cube, then E H-E F=F B, and E HXE FXF B=E H3. This explains the reason why the term cube is used to express the third power of any number. 140. THEOREM.-The solidity of any parallelopiped is equal to the area of its base multiplied by its altitude. This will be evident if we prove that any parallelopiped is equivalent to a right parallelopiped of the same base and altitude. As the demonstration is long, we shall divide it into three distinct propositions. DEM. 1.-If two parallelopipeds have the same inferior base, and their superior bases comprehended between the same parallel lines, they are equivalent. Let the two parallelopipeds be E D and EM (fig. 98) having F 98 the inferior base E F G H common, and their superior bases A B C D and I K L M comprehended between the same parallels A M and B L. The figure thus constructed contains two triangular prisms F B K-E A I and G C L-H D M. This will be true whether I K falls upon D C or upon either side of D C. Now we say that these two prisms are equal. The proof is by superposition. The triangle H D M the triangle EA I, having their three sides respectively equal. Therefore the inferior bases will coincide. Moreover since D C corresponds in length and direction with A B, the point C will fall upon B. For the same rea |