Some Proofs in Elementary Geometry

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. ; draw the chords ab, bc . . . ; and draw a^b^, b^C'' . . .
parallel to ab, bo . . ., tangent to the arc ak, and cut ofE by the radii oa, ob, oc . . ; then the isosceles triangles oab, obc . . . are all equal, and so are oa^b'', q-r'c' . . .
Draw the perpendiculars ax, by, cz . . . Ku, A^x^, k'u^ to PQ, and revolve the whole figure about pq ; then three surfaces of revolution are generated : the zone by the arc ak, the inner surface by the broken line of chords, the outer surface by the broke
...n line of tangents ; and three solids of revolution : the spherical sector bounded by the zone and the two cones generated by the radii OA, OK, 14 the inner solid bounded by the inner surface and the cones, the outer solid bounded by the outer surface and the cones.
(1) The surface of the zone ak is the product f its alti- tude xiT and the 2Krimeter of a great circle.
For •.• surf. AB = XY- jDer.oJi, [lem. 1.
surf . BC = iz • per.OM, .-. surf. A. . . K = xu-per.OM, i.e. the inner surface = prod.


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