| William Chauvenet - Geometry - 1871 - 368 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 to the sum of three... | |
| William Chauvenet - Geometry - 1871 - 368 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... | |
| Elias Loomis - Geometry - 1871 - 58 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,... | |
| Edward Olney - Geometry - 1872 - 239 pages
...the terms of the second couplet of (1) by these equals, we have, PROPOSITION XТI. 409. Theorem. — **The volumes of similar cylinders are to each other...altitudes, or as the cubes of the radii of their bases.** DEM. — Using the same notation ns in the last demonstration, the student should be able to give the... | |
| William Chauvenet - Geometry - 1872 - 368 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 sum of three cones whose... | |
| Edward Olney - 1872 - 201 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" : r " :: H' : A' (?),... | |
| Edward Olney - Geometry - 1872
...: : A1 : a" : : H1 : h3. QED PROPOSITION XIL 831. Theorem. — Tlie volumes of similar cones are to **each other as the cubes of their altitudes, or as the cubes of the radii of their** DEM. R and r being the radii of their bases, and H and A their altitudes, R* : r* :: H« : A« (?),... | |
| Charles Davies - Geometry - 1872 - 455 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... | |
| Adrien Marie Legendre - Geometry - 1874 - 455 pages
...pyramid S.AB CDE : pyramid S.abcde : : AB3 • ab" ; wluch was to be proved. Cor. Similar pyramids are to **each other as the cubes of their altitudes, or as the cubes of** any other homologous lines. GENERAL FORMULAS. If we denote the volume of any prism by V, its base by... | |
| |