Elements of geometry and trigonometry . om,the two bases, Scholium. If a line AD, lying wholly on one side of the lineDC, and in the same plane, make a revolution around OC,the surface described by AD will have for its measure AD x/amAO-t-<^itmDC\ ^j.^j) ^ ^^v^^ ^K; the lines AO, DC, /K, being perpcodieulara. let fall from the extremities and fromthe middle point of AD. on the axis OC. ^ ., For if AD a«4 OC are produced till they meet in ^, tiiesurface described by AD i» (evidently the frustum of a cone BOOK VIII. 173 having AG and DC lor ihc radii of its base?, the vertex ofthe whole cone
Elements of geometry and trigonometry . om,the two bases, Scholium. If a line AD, lying wholly on one side of the lineDC, and in the same plane, make a revolution around OC,the surface described by AD will have for its measure AD x/amAO-t-<^itmDC\ ^j.^j) ^ ^^v^^ ^K; the lines AO, DC, /K, being perpcodieulara. let fall from the extremities and fromthe middle point of AD. on the axis OC. ^ ., For if AD a«4 OC are produced till they meet in ^, tiiesurface described by AD i» (evidently the frustum of a cone BOOK VIII. 173 having AG and DC lor ihc radii of its base?, the vertex ofthe whole cone being 8. Hence this surface ^vill be measuredas wo have saiil. This measure will always hold good, even when the point1) falls on 8, ami thus ibrms a whole cone ; and also when theline AD is parallel to tlie axis, and thus forms a cylinder. Inthe first case DC would be nothing; in the second, DC wouldbe ecjual to A() and to IK. PROPOSITION V. THEOREM. The soUditj/ of a cone is equal lo its hase multiplied hy a third ofits Let SO be the altitude of a cone,OA t!ie radius of its base, and letthe area of the base be designatedby area OA : it is to be j)roved thatthe soliditv of the cone is equal toarea OAx^SO. Inscribe in the base of the coneany regular polyijon AHDl^F, andjoin the vertices A, B, C, A:c. withliie vertex 8 of the cone : then willthere be inscril)e(l in the cone aregular pyramid having the same vertex as the cone, and hav-mg fjj- its base tJie j)olygon ABDEF. The solidity of thispyrami<J is ^ual to its base Fimltiplied by one third of its alti-tude (Book VJI. Vrnp. XVil.). Let now the number of sidesof the polygon be indefiaitely increased : the polygon will thenbecome equal to th<: circle, and tht; pyramid and cone willcoincide and become xfpial. But the solidity of the pyramidis C(iual to its base riuiliiplied by one third (jf its altitude, what-ever be the number of sides of the u(jlygon which forms itsbase : hence the soJidity of the coue lij e
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
