Elements of geometry and trigonometry . o four of its great circles : that is, equal ^ (Book V. Prop. XII. Cor. 2.). Scholium 1. The surface of a zone is equal to its altitude mul-tiplied by the circumference (fa great circle. For, the surface described by any portionof the perimeter of the inscribed polygon, asBC + CD, is ecjual to Elixc/rc. OF ( Cor.). But when the number of sidesof the polygon is indefinitely increased, BC+ CD, becomes the arc BCD, OF becomesequal to OA, and the surface described byBO-fCD, becomes the surface of the Konedescribed l)y the arc BCD; hence the s
Elements of geometry and trigonometry . o four of its great circles : that is, equal ^ (Book V. Prop. XII. Cor. 2.). Scholium 1. The surface of a zone is equal to its altitude mul-tiplied by the circumference (fa great circle. For, the surface described by any portionof the perimeter of the inscribed polygon, asBC + CD, is ecjual to Elixc/rc. OF ( Cor.). But when the number of sidesof the polygon is indefinitely increased, BC+ CD, becomes the arc BCD, OF becomesequal to OA, and the surface described byBO-fCD, becomes the surface of the Konedescribed l)y the arc BCD; hence the sur-face of the zone is equal to EH x circ, OA. Scholium 2. When the zone has but onebase, as the zone described by the arc ABCD, its surface willstill be equal to the altitude AE multiplied by Uie circumferenceof a great circle. SchoUum 3. Two zones, taken in the same sphere or in equalspheres, are to each other as their altitudes ; and any zone is tothe surface of the sphere as the altitude of the zone is to thediameter of the PROPOSITION XI. LEMMA. If a triangle and a rectangley havivg the same hase and the samealtitude^turn together about the common base^ the solid describedtry the triangle will be a third of the cylinder described by therectangle. Let ACB be the triangle, and BE the rectangle. On the axis, let fall the perpen-licular AI): tlie cone described bythe triangle ABD is the third part ofthe cylinder <lescribed bv the rectan-gle ÀFBD (In^p. V. CoV.) ; also thecone described by th<r triangle ADCis the third part of the cylinder de-scribed by the rectaiigle* ADCE ; henco the sum of the twocones, or the solid described by ABC\ is the third pail of thetwo cylinders taken t«*gethcr, or of the cylinder described bythe rectangle BCEF.
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