Elements of geometry and trigonometry . PROPOSITION IX. LEMMA. If a rerrular semi-pohjgon he revolved about a line passingthrough the centre and the vertices of two x)pposite angles^ thesurface described by its perimeter rtnllbe equal to the axis mul-tiplied by the circumference of the inscribed circle. Let the regular semi-polygon ABCDEF,be revolved about tlie line AF as an axis :then will the surface flescribed by its pe-rimeter be equal to AF multiplied by tJîecircumference of the inscribed circle. From E and D, the extremities of one oftlie equal sides, let fall the perpendicularsEH, DI, o
Elements of geometry and trigonometry . PROPOSITION IX. LEMMA. If a rerrular semi-pohjgon he revolved about a line passingthrough the centre and the vertices of two x)pposite angles^ thesurface described by its perimeter rtnllbe equal to the axis mul-tiplied by the circumference of the inscribed circle. Let the regular semi-polygon ABCDEF,be revolved about tlie line AF as an axis :then will the surface flescribed by its pe-rimeter be equal to AF multiplied by tJîecircumference of the inscribed circle. From E and D, the extremities of one oftlie equal sides, let fall the perpendicularsEH, DI, on the axis AF, and from the cen-tre O draw ON perpendicular to the sideDE : ON will be the radius of the inscribedcircle (Book V. Prop. II.). Now, the sur-face described in the revolution by any oneside of the regular polygon, as I>F. has. BOOK VIII. 177 been shown to be equal to DEx circ. NM (Prop. IV. Sch.).But since the triangles EDK, 0\M, are similar (Book l\.Prop. XXL), ED : EK or 111 : : ON : NM, or as circ. ON :circ. NM ; hence ED X circ. NM=:II1 X circ. ON ;and since the same may l)e shown lor each of the other sides,it is plain that the surface described by the entire perimeter isequal to (F1Ï + HI + IP+ PQ + QA)x czVc. ON-:AFx r/rc. ON. Cor. The surface described by any portion of the perime-ter, as EDC, is equal to the distance between the two perpen-diculars let fall from its extremities on the axis, multi[)lied bythe circumference of the inscribed circle. For, the surfacedescribed by DE is equal to III x circ. ON, and the surfacedescribed by DC is equal to IPx circ. ON : hence the surfacedescribed by ED-f DC, is equal to (III + IP) x tiVc. ON, orequal to IIP X circ. ON. PROPOSITION X. THEOREM. The surface of a sphere is equal to the product of its diameter hyii/ie circumference of a great circU^ Let ABODE
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