Elements of geometry and trigonometry . lidity of a cylinder is equal to the productof its base by its altitude. Cor, 1. Cylinders of the same altitude are to each other astheir bases ; and cylinders of the same base are to each otheras their altitudes. Cor, 2. Similar cylinders are to each other as the cubes oftheir altitudes, or as the cubes of the diameters of their the bases are as the squares of their diameters ; and thecylinders being similar, the diameters of their bases are toeach other as the altitudes (Def. 4.) ; hence the bases areas the squares of the altitudes ; hence th
Elements of geometry and trigonometry . lidity of a cylinder is equal to the productof its base by its altitude. Cor, 1. Cylinders of the same altitude are to each other astheir bases ; and cylinders of the same base are to each otheras their altitudes. Cor, 2. Similar cylinders are to each other as the cubes oftheir altitudes, or as the cubes of the diameters of their the bases are as the squares of their diameters ; and thecylinders being similar, the diameters of their bases are toeach other as the altitudes (Def. 4.) ; hence the bases areas the squares of the altitudes ; hence the bases, multipliedby the altitudes, or the cylinders themselves, are as the cube»of the altitudes. Scholium, Let R be the radius of a cylinders base ; H thealtitude : the surface of the base will be ^ (Book V. ^or. 2.) ; and the solidity of the cylinder will be tiR^x H;or TriftlH. BOOK VIII. m PROPOSITION III. THEOREM. Ttic cqtivcj: surfdcp of a cone is equal to iJie circumference of its has?, multiplied by ha fits Let tlic circle ABCD be tlicbase of a cone, S the vertex,SO tije altitude, ami SA tlicside : then will its convex sur-lace becqnal x ^SA. For. in.îeribe in the base ofihe cone any regular polygonABCD. and on this polygon asa base conceive a pyramid tobe constructed havuig S tor its _^, vertex : this pyramid will be aregular pyramid, and will be inscribed in the cone. From S, draw SG perpendicular to one of the sides of thepolygon. Tlve convex surlhce of the inscribed pyramid is equalto the perimeter of the polygon w hich forms its base, multipliedby half the slant heigiât *SG (Book VII. Prop. IV.). Let nowthe number of sides of the inscribed polyLon be indefinltelvrincreased : the perimetv^r of the inscribod polygon will thenl>ecomc equ?l to ci re, , the slant height 8G will becomeequal to the side SA of the cone, and the convex surface ofthe pyramiil to the convex surface of the cone. But whateverbe tlie nunjber of sides <
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