The Elements of Solid Geometry |
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Page 5
... MN and PQ ; then will AA ' , BB ' , and CC ' be equal . Draw the lines AC and A'C ' . Now these lines are paral- lel . For , they lie in the same plane , i.e. , the plane of the parallel lines AA ′ and CC ' ( 11 ) ; and they cannot meet ...
... MN and PQ ; then will AA ' , BB ' , and CC ' be equal . Draw the lines AC and A'C ' . Now these lines are paral- lel . For , they lie in the same plane , i.e. , the plane of the parallel lines AA ′ and CC ' ( 11 ) ; and they cannot meet ...
Page 6
... MN and PQ be parallel planes , and let AD be per- pendicular to PQ ; then will AD also be perpendicular to MN . Through AD pass two planes whose intersections with MN and PQ are the lines DF , AC , and DE , AB . Now DF and AC are ...
... MN and PQ be parallel planes , and let AD be per- pendicular to PQ ; then will AD also be perpendicular to MN . Through AD pass two planes whose intersections with MN and PQ are the lines DF , AC , and DE , AB . Now DF and AC are ...
Page 7
... MN must coincide with this plane which has been passed through A , parallel to PQ ; i.e. , MN is parallel to PQ . PROPOSITION VII . THEOREM . If one of two parallel lines is perpendicu lar to a plane , the other is also perpendicular to ...
... MN must coincide with this plane which has been passed through A , parallel to PQ ; i.e. , MN is parallel to PQ . PROPOSITION VII . THEOREM . If one of two parallel lines is perpendicu lar to a plane , the other is also perpendicular to ...
Page 8
... MN and PQ , of PROP . V. Let AA ' and CC ' be any two perpendiculars to the plane PQ between the parallel planes ; then will AA ' and CC ' be equal . For , AA ' is parallel to CC ' ( 24 ) , Hence , AA ' is equal to CC ' ( 17 ) ...
... MN and PQ , of PROP . V. Let AA ' and CC ' be any two perpendiculars to the plane PQ between the parallel planes ; then will AA ' and CC ' be equal . For , AA ' is parallel to CC ' ( 24 ) , Hence , AA ' is equal to CC ' ( 17 ) ...
Page 10
... MN ; and let the plane PQ be passed through BA ; then will PQ be perpendicular to MN . In MN draw the line BS perpendicular to BQ , the inter- section of PQ and MN . Now since AB and BS lying in their respective planes PQ and MN , are ...
... MN ; and let the plane PQ be passed through BA ; then will PQ be perpendicular to MN . In MN draw the line BS perpendicular to BQ , the inter- section of PQ and MN . Now since AB and BS lying in their respective planes PQ and MN , are ...
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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.