 | Elias Loomis - Conic sections - 1857 - 242 pages
...and pyramids generally are to each other as the products of their bases by their altitudes. Cor. 3. Similar pyramids are to each other as the cubes of their homologous edges. Scholium. The solidity of any polyedron may be found by dividing it into pyramids, by planes... | |
 | Elias Loomis - Conic sections - 1858 - 256 pages
...and pyramids generally are to each other as the products of their bases by their altitudes. Cor. 3. Similar pyramids are to each other as the cubes of their homologous edges. Scholium. The solidity of any polyedron may be found by dividing it into pyramids, by planes... | |
 | George Roberts Perkins - Geometry - 1860 - 472 pages
...of the pyramid S- ABCDE, and abcde, x £So is that of the pyramid S— abcde (T. XVII., C.) ; hence two similar pyramids are to each other as the cubes of their homologous sides. SEVENTH BOOK. THE THREE ROUND BODIES. DEFIN1TIONS. I. A cylinder is a solid, which may be produced... | |
 | Benjamin Greenleaf - Geometry - 1862 - 532 pages
...solidity of the pyramid А В С - R, and DEFX è SP that of the pyramid DEF — S (Prop. XX.) ; hence two similar pyramids are to each other as the cubes of their homologous edges. PRO POSITION XXIII. — Tn EOREM . 495. There can be no more than fire regular polyedrous. For,... | |
 | Benjamin Greenleaf - Geometry - 1862 - 518 pages
...represents the solidity of the pyramid ABC-S, and DEFX i SP that of the pyramid DEF-S (Prop. XX.) ; hence two similar pyramids are to each other as the cubes of their homologous edges. DE2. PROPOSITION XXIII. — THEOREM. 495. There can be no more than five regular polyedrons.... | |
 | Benjamin Greenleaf - Geometry - 1863 - 504 pages
...represents the solidity of the pyramid ABC-S, and DEFX i SP that of the pyramid DEF-S (Prop. XX.) ; hence two similar pyramids are to each other as the cubes of their homologous edges. PROPOSITION XXIII. — THEOREM. 495. There can be no more than five regular polyedrons. For,... | |
 | Adrien Marie Legendre - Geometry - 1863 - 464 pages
...the cubes of their altitudes, or as the cubes of any other homologous lines. PROPOSITION XX. THEOKEM. Similar pyramids are to each other as the cubes of their homologous edges. Let &-ABCDE, and S-abcde^ be two similar pyramids, so placed that their homologous angles at... | |
 | Benjamin Greenleaf - Geometry - 1866 - 328 pages
...3.4641016 0.4714045 Dodecaedron, 12 20.6457288 7.6631189 Icosaedron, 20 8.6602540 2.181G950 Also, since similar pyramids are to each other as the cubes of their homologous edges (Prop. XXII. Bk. VIII.), two polyedrons containing the same number of similar pyramids are to... | |
 | Edward Brooks - Geometry - 1868 - 284 pages
...found by dividing it into triangular pyramids, by passing planes through its vertices. THEOREM XIV. Similar pyramids are to each other as the cubes of their homologous edges. - Let S—ABCDE and S—FGH1K be two similar pyramids ; then will they be to each other as the... | |
 | Benjamin Greenleaf - Geometry - 1868 - 340 pages
...1.7320508 6.0000000 3.4641016 8.6602540 0.1178511 1.0000000 0.4714045 7.6631189 2.1816950 Also, since similar pyramids are to each other as the cubes of their homologous edges (Prop. XX II. Ek. VIII.), two polyedrons containing the same number of similar pyramids are to... | |
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XVII.) ; hence two similar pyramids are to each other as the cubes of their homologous sides. 
