The Elements of Solid Geometry |
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Page 3
... perpendicular to the plane MN , drawn from P , a point in the plane ; and let TP be any other line drawn from P ; then TP cannot be perpendicular to MN . For , pass through QP and PT a plane , intersecting MN in the line HK ; then ...
... perpendicular to the plane MN , drawn from P , a point in the plane ; and let TP be any other line drawn from P ; then TP cannot be perpendicular to MN . For , pass through QP and PT a plane , intersecting MN in the line HK ; then ...
Page 4
... perpendicular to the line at the given point . M B A N Let any line AE be drawn perpendicular to the line AB at a given point A ; and let MN be a plane passing through A , and perpendicular to AB ; then will AE lie in MN . For , if AE ...
... perpendicular to the line at the given point . M B A N Let any line AE be drawn perpendicular to the line AB at a given point A ; and let MN be a plane passing through A , and perpendicular to AB ; then will AE lie in MN . For , if AE ...
Page 5
... MN . Hence MN is perpen- dicular to AB ; or , AB is perpendicular to MN . Q. E. D. i.e. , the plane of the lines PROPOSITION V. 17. THEOREM . Parallel lines included between two paral- lel planes are equal . M B A C N P B Let the ...
... MN . Hence MN is perpen- dicular to AB ; or , AB is perpendicular to MN . Q. E. D. i.e. , the plane of the lines PROPOSITION V. 17. THEOREM . Parallel lines included between two paral- lel planes are equal . M B A C N P B Let the ...
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 parallel ...
... 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 parallel ...
Page 7
... parallel to each other . For , pass through A a plane which is parallel to PQ , and it will be perpendicular to AD ( 21 ) . But only one plane can be passed through A which is perpendicular to AD ( 15 ) . Therefore MN must coincide with ...
... parallel to each other . For , pass through A a plane which is parallel to PQ , and it will be perpendicular to AD ( 21 ) . But only one plane can be passed through A which is perpendicular to AD ( 15 ) . Therefore MN must coincide with ...
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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.