Elements of geometry and trigonometry . d then lie in one .straight line,and an infinite number of great circles might be made to passthrough the two given [)oints. 170 GEOMETRY. PROPOSITION VIII. THEOREM. Kveiij plane. pcrpemUcidar to a induis at its extremity is tangentto the sphere. Let FAG he a piano perpendicularto the radius OA. at its extremity point M in this plane being as-sumed, and OM, AjM, being drawn,the angle OAM will l)e a right angle,and henee the distance OM Mill be^n-eater than OA. Hence the pointiM lies without the sphere ; and as thesame can be shown lor every otherpo
Elements of geometry and trigonometry . d then lie in one .straight line,and an infinite number of great circles might be made to passthrough the two given [)oints. 170 GEOMETRY. PROPOSITION VIII. THEOREM. Kveiij plane. pcrpemUcidar to a induis at its extremity is tangentto the sphere. Let FAG he a piano perpendicularto the radius OA. at its extremity point M in this plane being as-sumed, and OM, AjM, being drawn,the angle OAM will l)e a right angle,and henee the distance OM Mill be^n-eater than OA. Hence the pointiM lies without the sphere ; and as thesame can be shown lor every otherpoint of the j)lane FAG, this plane canhave no [)oint btit A common to it and the surface of the sphere ;hence it is a tangent plane (Def. 12.) Scholium, In the same w^ay it may be shown, that twospheres have but one point in common, and therefore toucheach other, when the distance between their centres is equal tothe sum, or the difterence of their r^dii ; in which case, the(centres and the point of contact lie in the same straight line. 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
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