| Benjamin Greenleaf - 1868 - 338 pages
...other as their bases ; and cylinders of equal bases are to each other as their altitudes. 577. Cor. 2. **Similar cylinders are to each other as the cubes of...their bases. For the bases are as the squares of their** radii (Prop. XIII. Bk. VI.), and the cylinders being similar, the radii of their bases are to each... | |
| Benjamin Greenleaf - 1869 - 516 pages
...other as their bases ; and cylinders of equal bases are to each other as their altitudes. 577. Cor. 2. **Similar cylinders are to each other as the cubes of...their bases. For the bases are as the squares of their** radii (Prop. XIII. Bk. VI.), and the cylinders being similar, the radii of their bases are to each... | |
| Elias Loomis - Geometry - 1871 - 302 pages
...as their bases ; cones of equal bases are to each other as their altitudes ; and similar cones are **as the cubes of their altitudes, or as the cubes of the diameters of their bases.** Cor. 2. If A represent the altitude of a cone, and R the radius of its base, the solidity of the cone... | |
| William Chauvenet - Geometry - 1871 - 380 pages
...similar, we have and hence F v _ . r2 . h ~ r* ' h ~ h3 ~ r3 ' that is, similar cones of revolution are to **each other as the cubes of their altitudes, or as the cubes of the** radii of their bases. PROPOSITION VIII.— THEOREM. ^\ '. 1 \ 31. A frustum of any cone is equivalent... | |
| William Chauvenet - Geometry - 1871 - 380 pages
...have H_R h ~ r' and F. v H h ' IP h3 that is, the volumes of similar cylinders of revolution are to **each other as the cubes of their altitudes, or as the cubes of** their radii. THE CONE. 15. Definition. The area of the convex, or lateral, surface of a cone is called... | |
| Edward Olney - Geometry - 1872 - 472 pages
...second couplet of (1) by these equals, we have, PROPOSITION XТI. 409. Theorem. — The volumes of **similar cylinders are to each other as the cubes of their altitudes, or as the cubes of the** radii of their bases. DEM. — Using the same notation ns in the last demonstration, the student should... | |
| Edward Olney - 1872 - 270 pages
...^bh :: A" : a" : : H " : K\ QED PROPOSITION XII. 531, Theorem.—The volumes of similar cones are to **each other as the cubes of their altitudes, or as the cubes of the** radii of their bases. DEM. R and r being the radii of their bases, and H and h their altitudes, R"... | |
| William Chauvenet - Geometry - 1872 - 382 pages
...H_R ihr and hence v ~^.r 2 . h ~ r 2 ' h ~ h* " r' ' __ that is, similar cones of revolution are to **each other as the cubes of their altitudes, or as the cubes of the** radii of their bases. PROPOSITION VIII.— THEOREM. 31. A frustum of any cone is equivalent to the... | |
| Charles Davies - Geometry - 1872 - 464 pages
...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... | |
| William Chauvenet - Geometry - 1872 - 368 pages
...similar, we have HR and xr'h H h ' IP h" that is, the volumes of similar cylinders of revolution are to **each other as the cubes of their altitudes, or as the cubes of** their radii. THE CONE. 15. Definition. The area of the convex, or lateral, surface of a cone is called... | |
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