Elements of geometry and trigonometry . the circle anyregular polygon, BDEFGA, andconstruct on this polygon a rightprism having its altitude equal to II, the altitude of the cylin-der : this prism will be inscribed in the cylinder. The convexsurface of the prism is equal to the perimeter of the polygon,multiplied by the altitude II (Book Vll. Prop. I.). Let nowthe arcs which subtend the sides of the polygon be continuallybisected, and the number of sides of the polygon indefinitelyincreased : the {)erimeterof the polygon will then become equalto circ. CA (Book V. Prop. VIII. Cor. 2.), and the
Elements of geometry and trigonometry . the circle anyregular polygon, BDEFGA, andconstruct on this polygon a rightprism having its altitude equal to II, the altitude of the cylin-der : this prism will be inscribed in the cylinder. The convexsurface of the prism is equal to the perimeter of the polygon,multiplied by the altitude II (Book Vll. Prop. I.). Let nowthe arcs which subtend the sides of the polygon be continuallybisected, and the number of sides of the polygon indefinitelyincreased : the {)erimeterof the polygon will then become equalto circ. CA (Book V. Prop. VIII. Cor. 2.), and the convex sur-face of the prism will coincide witii the convex surface of thecylindei*. But the convex surface of the prism is equal to theperimeter of its base multiplied by II, whatever be the numberof sides : hence, the convex surface of the cylinder is equal tothe circumference of its base multiplied by its altitude. PROPOSITION II. THEOREM. The solidity of a cylinder is equal to the product of its base by its altitude. 170 Let CA be the radius of thebase of* tlie cyhnder, and IIihe altitude. Let the circlewhose radius is CA be repre-sented by area CA, it is to beproved that the solidity of thecylinder is equal io areaCK x in the circle any regu-lar polygon BDEFGA, and con-struct on this polygon a rightprism having its altitude equalto H, the altitude of the cylinder : this prism will be inscribedin the cylinder. The solidity of the prism will be equal to of the polygon multiplied by the altitude II (Book XIV.). Let now the number of sides of thn polygon beindefinitely increased : the solidity of the new prism will stillbe equal to its base multiplied by its altitude. But when the number of sides of the polygon is indefinitelyincreased, its area becomes equal to the area CA, and its pe-rimeter coincides with circ. CA (Book V. Prop. Vlll. Cor. 2.) ; the inscribed prism then coincides with the cylinder,since their altitudes are equal,
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