is 8 inches long, then its solidity is 8 × 8 × 8=512 cubic inches. Thus 1 cubic foot 12 X 12 X 12 = 1728 cubic inches. = 199. Let us suppose a rectangular and an oblique parallelopiped of equal altitude, to be constructed upon the same base. The rectangular parallelopiped is formed of an infinite number of rectangles, the bases of which compose the base of the parallelopiped; and the oblique parallelopiped is formed of an infinite number of oblique parallelograms, the bases of which compose the base of the parallelopiped. These component rectangles and parallelograms are equivalent, (149;) their sums must be equivalent, that is, the two parallelopipeds are equivalent. The solidity of a rectangular parallelopiped is found by multiplying its base by its altitude; consequently, the solidity of any parallelopiped may be found by multiplying its base by its altitude. 200. Every parallelopiped may be divided into two equal prisms, the bases of which are equal triangles. The solidity of each of these triangular prisms is one half the solidity of the whole parallelopiped; it may therefore be found by multiplying the altitude of the parallelopiped by one half its base. But each of the triangles which are the bases of the triangular prism is one half of the base of the parallelopiped; consequently, the solidity of the triangular prism may be found by multiplying its base by its altitude. 201. Every prism may be divided into as many triangular prisms, as the polygon taken for its base can, by diagonals, be divided into triangles. The solidity of the entire prism is equal to the sum of the products of the base of each triangular prism multiplied by its altitude. But the triangular prisms and the entire prism have an equal altitude, and the sum of all the bases of the triangular prisms is equal to the base of the entire prism; consequently, the solidity of any prism is equal to the product of its base multiplied by its altitude. 202. The cylinder may be considered as a prism whose convex surface is composed of an infinite number of parallelograms; consequently the solidity of the cylinder may be found by multiplying its base by its altitude. By the altitude is to be understood a perpendicular which measures the distance between the two bases. Example. If the altitude of a cylinder is 8 inches, and the diameter of its base 6 inches, what is the solidity of the cylinder? Answer. The circumference of the sequently the solidity of the cylinder is 2813 X 8= 226 cubic inches. The Pyramid and the Cone. 203. Every triangular prism may be divided into 3 triangular pyramids. Experiment will best show how this may be done. In this figure (fig. 110) we have a representation of it. ABC and DEF are the bases of a triangular prism, which is divided by the planes ABF and DFB into 3 triangular pyramids. These pyramids are equivalent, that is, equal in solidity, one to the other. It may be demonstrated that pyramids which have equal bases and equal altitudes are equivalent. It is evident that the bases and altitudes of the two pyramids F-ABC and B-DEF are equal, consequently these two pyramids are equivalent. Let us now compare the pyramid F-ADB with B-DEF, of which we now suppose DEB to be the base. It is at once evident that these two pyramids have equal bases and equal altitudes, and therefore are equivalent, consequently F-ADB=B-DEF=F-ABC. The triangular pyramid is, therefore, of a triangular prism of equal base and altitude. The solidity of the triangular prism is equal to the product of its base by its altitude, consequently, the solidity of a triangular pyramid is equal to of the product of its base by its altitude, or, to the product of its base by of its altitude. 204. Every pyramid may be divided into triangular pyramids; consequently, the solidity of any pyramid is equal to the product of its base multiplied by of its altitude. 205. The Cone may be considered as a pyramid whose convex surface is composed of an infinite number of triangles, consequently, the solidity of a cone is equal to the product of its base multiplied by ↓ of its altitude. Other Solid Bodies. 206. The Tetraedron is a triangular pyramid. The Octaedron may be divided into 8 pyramids having equal bases and altitudes. Each of these pyramids will have a face of the polyedron for its base, and of a face axis for its altitude. The vertices lie together at the middle of the octaedron. The solidity of the octaedron is equal to 8 times the solidity of each pyramid. In a similar manner the dodecaedron may be divided into 12, and the icosaedron into 20 pyramids. In general the solidity of any polyedron may be found by dividing it into pyramids; find the solidity of each pyramid, and the sum of all will be the solidity of the entire solid. The Sphere. 207. Let us suppose the sphere to be converted into a polyedron with an infinite number of faces. We can suppose each of these faces to be the base of a pyramid, with its vertex at the centre of the sphere. Each of these pyramids will have a radius of the sphere for its altitude. Thus the solidity of the sphere is equal to that of a pyramid having a base equivalent to the surface of the sphere, and an altitude equal to the radius of the sphere. Consequently the solidity of the sphere may be found by multiplying its surface by its radius or its diameter. The surface of a sphere is 4 times the area of a great circle of the same sphere; therefore the solidity of a sphere is equal to the product of a great circle of that sphere multiplied by the diameter. Example. If the diameter of a sphere is 4 feet, what is its solidity? Answer. The circumference of a great 314×4 314 circle is feet; therefore the area is X4 square 100 = 100 feet 12 square feet; and the solidity is 12 × 23 3343 cubic feet. = 208. The surface of a sphere is equal to the product of the circumference of a great circle multiplied by the diameter; and the product of this surface multiplied by the diameter is the solidity of the sphere. Thus the solidity of a sphere may be found by multiplying the circumference of a great circle by the square of the diameter. Again, the circumference of a great circle is 314 100 equal to the product of the diameter multiplied by consequently the solidity of the sphere is equal to the product of of the cube of the diameter multiplied by which is equal to the cube of the diameter multi 314 100' plied by 314 or by its equivalent decimal fraction, .523. That is, the solidity of a sphere is equal to the product of the cube of its diameter multiplied by .523. MISCELLANEOUS PROPOSITIONS. 209. To find the distance between two objects, when it cannot be directly measured, as, for example, between A and B, (fig. 111,) which are separated by a sea. Sol. Take any point C, so that the lines CA and CB can be directly measured with a rod or a chain. Produce these lines beyond C, taking CD=CA, and CE=CB. Because CA=CD; CB=CE and angle ACB=DCE, (107;) therefore triangle ACB=DCE, (126;) consequently AB= DE. Measure DE and we have the required distance AB. 210. To find the distance between two objects, only one of which can be reached. Suppose the line AB, (fig. 111,) the point A being accessible. Sol. Draw from A a straight line in the direction of AB. Draw AC perpendicular to such line, and of any convenient length; produce AC, taking CD=AC; upon CD at the point D erect a perpendicular; through C |