hemisphere, by a solid less than S; but by supposition it differs by the solid S, which is absurd. Therefore the hemisphere ADB and the cone ECF together are not unequal to the circumscribing cylinder ABFE, that is, they are equal to that cylinder. And because the cone ECF is the third part of the cylinder ABFE, the hemisphere ADB is two-thirds of the circumscribing cylinder ABFE; and consequently the whole sphere is two-thirds of the cylinder described by twice the rectangle DB, that is, two-thirds of its circumscribing cylinder. Which was to be proved. Cor. 1. A cone, a hemisphere and a cylinder, all having the same base and the same altitude, are to one another as the numbers 1, 2 and 3. Cor. 2. Spheres are to one another as the cubes of their diameters; for they are to one another as their circumscribing cylinders (V. C), and these being similar are to one another as the cubes of their altitudes, that is, as the cubes of the diameters of the spheres. Cor. 3. From the corollaries to propositions 76 and 77, it follows, as in the above demonstration, that the segment of the sphere described by the revolution of DIG about DC, and the frustum of the cone described by DFHG, are together equal to the cylinder described by DFKG; and that the zone of the sphere described by GIBC and the cone described by GHC, are together equal to the cylinder described by GKBC. THEOREMS TO BE DEMONSTRATED. 1. Three straight lines, meeting in a point, but which are not all in one plane, being given, a parallelepiped three of whose edges are equal to these straight lines may be constructed. 2. Three straight lines, not parallel to a plane, and no two of which are in the same plane, being given, a parallelepiped three of whose edges are in these straight lines may always be constructed. 3. If planes pass through each of the diagonals of two opposite parallelograms of a parallelepiped, they will divide it into four equal prisms. 4. If two opposite solid angles of a parallelepiped be joined by a straight line, and any other two opposite solid angles be likewise joined, these two straight lines will intersect. 5. The intersections of planes perpendicular to the edges of any triangular pyramid at their middle points all intersect in one point; and this point is the centre of a sphere the surface of which passes through the vertices of the four solid angles of the pyramid. 6. If a cylinder be cut by a plane parallel to the base, the section is a circle equal to the base. 7. If a cone be cut by a plane passing through its axis, perpendicular to the base, and planes touch the cone in the opposite lines of section, then if the cone be cut by a plane making the same angle with the tangent plane on one side, which the base makes with the tangent plane on the other, the section (which is called subcontrary) will be a circle; and this section will be to the base in the duplicate ratio of its distance from the vertex, to the altitude of the cone. 8. Show that a sphere may be described about any cone. END OF PART III. PRINTED BY RICHARD AND JOHN E. TAYLOR, |