Elements of geometry and trigonometry . Î64 ab BC : 5c mon base (Book VI. Prop. XVI.). Througha,m the plane BAH,draw ah perpendicular toEH : then will ah also be per-pendicular to the base BDC(Book VI. Prop. XVII.) ; andAH, ak will be the altitudesof the two prisms. Now, because of the similartriangles ABH,«B//, and of thesimilar parallelograms AC, ac,we have AH : ah : : ABBut since the bases are similar, we havehase BCD : base bed : : BC^ : bc^ (Book IV. Prop. XXV.) ;hence, base BCD : base bed : : AH^ : ah\Multiplying the antecedents by AH, and the consequents byahf and we have bas
Elements of geometry and trigonometry . Î64 ab BC : 5c mon base (Book VI. Prop. XVI.). Througha,m the plane BAH,draw ah perpendicular toEH : then will ah also be per-pendicular to the base BDC(Book VI. Prop. XVII.) ; andAH, ak will be the altitudesof the two prisms. Now, because of the similartriangles ABH,«B//, and of thesimilar parallelograms AC, ac,we have AH : ah : : ABBut since the bases are similar, we havehase BCD : base bed : : BC^ : bc^ (Book IV. Prop. XXV.) ;hence, base BCD : base bed : : AH^ : ah\Multiplying the antecedents by AH, and the consequents byahf and we have base BCD X All : base bed x ah : : AH^ ah\But the solidity of a prism is equal to the base multiplied bythe altitude (Prop. XIV.) ; hence, the prism BCD-P : prism bcd-p : : AH^ : ah^ : : BC^ : bc^,or as the cubes of any other of their homologous sides. Cor. Whatever be the bases of similar prisms, the prismswill be to each other as the cubes of their homologous sides. For, since the prisms are similar, their bases will be similaipolygons (Def, 17.) ; a
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