Elements of geometry and trigonometry . 146 Cor, 1. Let S-ABCDE,S-XYZ be two pyramids, hav-ing a common vertex and thesame altitude,, or having theirbases situated in the sameplane ; if these pyramids arecut by a plane parallel to theplane of their bases, giving thesections ahcde^ xyz. then willthe sections abode, xyz, he to eachother as the bases ABCDE,XYZ. For, the polygons ABCDE, ahcde, being similar, their sur-faces are as the squares of ihe homologous sides AB, ab ; butAB : ab^i : SA I Sa; hence ABCDE : abode : : SA^ : S^^,For the same reason, XYZ : xyz : : SX^ : Sx^. But since

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Elements of geometry and trigonometry . 146 Cor, 1. Let S-ABCDE,S-XYZ be two pyramids, hav-ing a common vertex and thesame altitude,, or having theirbases situated in the sameplane ; if these pyramids arecut by a plane parallel to theplane of their bases, giving thesections ahcde^ xyz. then willthe sections abode, xyz, he to eachother as the bases ABCDE,XYZ. For, the polygons ABCDE, ahcde, being similar, their sur-faces are as the squares of ihe homologous sides AB, ab ; butAB : ab^i : SA I Sa; hence ABCDE : abode : : SA^ : S^^,For the same reason, XYZ : xyz : : SX^ : Sx^. But sinceabc and xyz are in one plane, we have likewise SA : Sa : :SX : Sx (Book VI. Prop. XV.) ; hence ABCDE : abcde : :XYZ : xyz ; hence the sections abode, xyz, are to each otheras the bases ABCDE, XYZ. Cor. 2. If the bases ABCDE, XYZ, are equivalent, any sec-tions abcde, xyz, made at equal distances from the bases, willbe equivalent likewise. PROPOSITION IV. THEOREM. The convex surface of a regular pyramid is equal to the périmeter of its base rnultipUed

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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry