...thirty. This radius, and that, of the orbit of Uranus, are in the ratio of 1 to 49 ; and the volumes of two spheres are to each other as the cubes of their radii. If, therefore, we adopt the hypothesis of the equal distribution of comets in all the regions of our...

...all similar solids are to each other as the cubes of their homologous lines ; thus, the solidities of two spheres are to each other as the cubes of their radii, as the cubes of their diameters, or as the cubes of' their circumferences, etc., etc. ; the solidities...

...3.1416 = J of 1728 times 3.1416 = 288 times 3.H16 cu. ft. = 904.7808 cu. ft. (m.) The solidities of spheres are to each other as the cubes of their radii or diameters. ILLUSTRATION. — A sphere of 3 ft. radius contains 27 as many cubic feet as a sphere of...

...diameter, or it equals -g- of the cube of the diameter multiplied by 3.1416* ^ (n.) The solidities of spheres are to each other as the cubes of their radii or diameters. * See foot note, page 343. (n ) The solidiiios of similar solids are to each other as the...

...as the volume of the latter is to the volume of the former. But these volumes or solid contents vary as the cubes of their radii, or as the cubes of their distances from the vertex. Therefore the force of gravity varies inversely as the cubes of the distances....

...all similar bodies are to each other as the cubes of their homologous lines ; thus, the volumes of two spheres are to each other as the cubes of their radii, as the cubes of their diameters, or as the cubes of their circumferences, etc., etc. ; the volumes...

...radius and that of the orbit of Neptune are in the ratio of one to seventy-eight. But the volumes of two spheres are to each other as the cubes of their radii. If, then, we adopt the hypothesis of an equal distribution of comets in our system, — and no reason...

...$xR*. Or, if D is the diameter of the sphere, whence D3 = (2R)3 = 8R3, 48. Corollary II. The volumes of two spheres are to each other as the cubes of their radii, or as the cubes of their diameters. PROPOSITION XV.— THEOREM. 49. The solid generated by a circular segment revolving about a diameter...

...= ^rx 47rr", the, volume of a sphere = the surface X 3 the radius. 237- COR. 2. — The volumes of spheres are to each other as the cubes of their radii or their diameters. 258. COR. 3. — The portion of a sphere included between two parallel planes is called...

...Or, if D is the diameter of the sphere, whence D* — (2R)5 = 8R", 48. Corollary II. The volumes of two spheres are to each other as the cubes of their radii, or as the cubes of their diameters. PROPOSITION XV.— THEOREM. 49. The solid generated by a circular segment revolving about a diameter...