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Elements of Geometry and Trigonometry from the Works of A.M. Legendre; Adapted to the Course of Mathematical Instruction in the United States - Softcover
Synopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1885 Excerpt: ...cV. But, base CBD x AH is equal to the volume of the prism CDB-A, and base cbdxah is equal to the volume of the prism cbd-p: hence, prism CDB-P: prism cbd-j:: CB which was to be proved. Cor. 1. Any two similar primus are to each other as the cubes of their homologous edges. For, since the prisms are similar, their bases are similar polygons (D. 16); and these similar polygons may each be divided into the same number of similar triangles, similarly placed (B. IV., P. XXVI.); therefore, each prism may be divided into the same number of triangular prisms, having their faces similar and like placed; consequently, the triangular prisms are similar (D. 16). But these triangular prisms are to each other as the cubes of their homologous edges, and being like parts of the polygonal prisms, the polygonal prisms themselves are to each other as the cubes of their homologous edges. Cor. 2. Similar prisms are to each other as the cubes of their altitudes, or as the cubes of any other homologous lines. PROPOSITION XX. THEOREM. Similar pyramids are to each other as the cubes of their homologous edges. Let S-ABCDE, and S-abcde, be two similar pyramids, so placed that their homologous angles at the vertex shall coincide, and let AB and ab be any two homologous edges: then are the pyramids to each other as the cubes of AB and ab. For, the face SAB, being similar to Sab, the edge AB is parallel to the edge ab, and the face SBC being similar to Sbc, the edge BC is parallel to be; hence, the planes of the bases are parallel (B. vi., P. xni.). Draw SO perpendicular to the base ABCDE; it will also be perpendicular to the base abcde. Let it pierce that plane at the point o; then SO is to So, as SA is to So (P. DX), or as AB is to ab; hence, JSO: $So:: AB: ab. But the bases being si...
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