 | Benjamin Greenleaf - 1873 - 202 pages
...represents the volume of the pyramid ABC-S, and DBFX$SP that of the pyramid DEF-S (Theo. XXII.); hence two similar pyramids are to each other as the cubes of their homologous edges. EXERCISES FOR ORIGINAL THOUGHT, ON REVIEW. 322. 1. A straight line cannot be partly in a plane, and... | |
 | Adrien Marie Legendre - 1874 - 500 pages
...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... | |
 | Benjamin Greenleaf - 1874 - 206 pages
...proportional • between them. Hence the proposition is true of the frustum of any pyramid. THEOREM XXIV. 321. Similar pyramids are to each other as the cubes of their homologous edges. Let ABC-S and S DEF-S be two similar pyramids ; these pyramids are to each other as the cubes of their... | |
 | William Guy Peck - 1876 - 412 pages
...Cor. 1. Any two pyramids are to each other as the product of their bases and altitudes. Cor. 2. Two similar pyramids are to each other as the cubes of their homologous edges. For, their bases are to each other as the squares of any two homologous edges, and their altitudes... | |
 | William Henry Harrison Phillips - 1878 - 236 pages
...altitudes ; and any two p3Tamids are to each other as the products of their bases and altitudes. COR. 2. Similar pyramids are to each other as the cubes of their homologous edges or altitudes (8). COR. 3. A cone is a regular pyramid, whose base has an infinite number of sides :... | |
 | Charles Scott Venable - 1881 - 380 pages
...the volumes of the two polyedrons, are to each other as the cubes of these same edges. COR. — Two similar pyramids are to each other as the cubes of their homologous edges, as also two similar prisms. EXERCISES ON BOOK VI. THEOREMS. I Show that two tetraedrons are equal,... | |
 | Edward Olney - 1883 - 352 pages
...„£ ^ four faces are equiiateral triangles 283^ PROPOSITION XI. 640. Theorem. — The volumes of similar pyramids are to each other as the cubes of their homologous dimensions. SYNOPSIS OF DEMONSTKAHON. Let A and a be homologous sides .of the bases of two similar... | |
 | Charles Davies, Adrien Marie Legendre - 1885 - 538 pages
...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... | |
 | Elias Loomis - 1886 - 436 pages
...be one of the faces of the polyedron ; and since their altitude is the perpendicular from the centre upon one of the faces, the solidity of the polyedron...must be to each other as the cubes of their edges. 135. The solidity of a tetraedron whose edge is unity, may be computed in the following manner : Let... | |
 | 1888 - 758 pages
...radius is i. 5. Two triangular pyramids having equal bases and equal altitudes, are equal in volume. 6. Similar pyramids are to each other as the cubes of their homologous edges. 7. The volume of a sphere is equal to its surface multiplied by one third of the radius. TRIGONOMETRY.... | |
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Also, similar pyramids are to each other as the cubes of their homologous edges (Geom., Prop. 
