The Elements of Solid Geometry |
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Page 19
... base are to each other as their altitudes . G F E D L H с K b 82 a C A B Let the right prisms AF and AL have the common base AC , and the altitudes CF and CL respectively ; then will AF : AL :: CF : CL . Let us suppose that CF and CL ...
... base are to each other as their altitudes . G F E D L H с K b 82 a C A B Let the right prisms AF and AL have the common base AC , and the altitudes CF and CL respectively ; then will AF : AL :: CF : CL . Let us suppose that CF and CL ...
Page 21
... base we may write , AL : AN :: BM : BO . Regarding BN as a common base we may write , AN : KN : BA : BK ... altitude . PROPOSITION XV . 76. THEOREM . An oblique parallelopiped is SOLIDS : PRISMS AND CYLINDERS . 21.
... base we may write , AL : AN :: BM : BO . Regarding BN as a common base we may write , AN : KN : BA : BK ... altitude . PROPOSITION XV . 76. THEOREM . An oblique parallelopiped is SOLIDS : PRISMS AND CYLINDERS . 21.
Page 22
... base and altitude are equal respectively to a right section and a lateral edge of the oblique parallelopiped ... base , LO is a right parallelopiped whose base JQ equals a right section of the oblique parallelopiped AF ; and the altitude ...
... base and altitude are equal respectively to a right section and a lateral edge of the oblique parallelopiped ... base , LO is a right parallelopiped whose base JQ equals a right section of the oblique parallelopiped AF ; and the altitude ...
Page 23
... base or right section involved , it applies as well to any other prism as to the parallelopiped . Hence , 77. COROLLARY 1. An oblique prism is equivalent to that right prism whose base and altitude are equal respec- tively to a right ...
... base or right section involved , it applies as well to any other prism as to the parallelopiped . Hence , 77. COROLLARY 1. An oblique prism is equivalent to that right prism whose base and altitude are equal respec- tively to a right ...
Page 25
... base and altitude . Let a represent the altitude of ABC - B ' ; then will the volume of ABC - B ' be equal to ( ABC ) ( a ) . ( 1 ) . For , vol . BD ' = ( 2 ) ( ABC - B ' ) . . . ( 80 ) , ( 2 ) . And , area DB = ( 2 ) ( ABC ) ... ( 359 ) ...
... base and altitude . Let a represent the altitude of ABC - B ' ; then will the volume of ABC - B ' be equal to ( ABC ) ( a ) . ( 1 ) . For , vol . BD ' = ( 2 ) ( ABC - B ' ) . . . ( 80 ) , ( 2 ) . And , area DB = ( 2 ) ( ABC ) ... ( 359 ) ...
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Common terms and phrases
ABC and DEF ABC-B altitude are equal axis base and altitude bases are equal bisecting centre circle circumference coincide common altitude common vertex conical surface COROLLARY cube cuboid DEFINITIONS diameter dicular diedral angle EDC-O equal with respect equivalent faces is called feet Find the volume frustum Hence inscribed prism intersection lateral edges lateral faces lateral surface Let the line line BA lune MN and PQ mutually equal O-ABCD oblique parallelopiped parallel planes parallelogram pass perimeter perpen perpendicular to MN plane angle plane MN plane PQ polar triangle polyedral angle prisms whose bases radii radius regular polyedrons regular polygon right angles right circular cone right prism right section SCHOLIUM sides slant height sphere spherical angle spherical excess spherical polygon spherical triangle straight line tetraedron THEOREM triangles ABC triangular prism triangular pyramids triedral
Popular passages
Page 48 - Assuming that the areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles...
Page 46 - Similar cylinders are to each other as the cubes of their altitudes, or as the cubes of the diameters of their bases.
Page 4 - If a straight line is perpendicular to each of two other straight lines at their point of intersection, it is perpendicular to the plane of the two lines.
Page 49 - A sphere is a solid bounded by a curved surface, every point of which is equally distant from a point within called the center.
Page 94 - Two triangles are equal if two sides and the included angle of the one are equal respectively to two sides and the included angle of the other (sas = sas). Hyp. In A ABC and A'B'C', AB = A'B', BC = B'C', and Z B = Z B'.
Page 46 - The lateral areas, or the total -areas, of two similar cones of revolution are to each other as the squares of their altitudes...
Page 6 - Theorem: If a straight line is perpendicular to one of two parallel planes, it is perpendicular, also, to the other plane.
Page 9 - ... meeting the plane at unequal distances from the foot of the perpendicular the more remote is the greater.
Page 56 - A spherical angle is measured by the arc of a great circle described from its vertex as a pole, and included between its sides, produced if necessary.