The Elements of Solid Geometry |
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Page 55
... angle of the polygon , corresponds to the diedral angle B - AO - D ; to which angle it is equal in value ; i.e. , BAD and B - AO - D contain the same number of degrees . 189. A spherical triangle is a spherical polygon of three sides ...
... angle of the polygon , corresponds to the diedral angle B - AO - D ; to which angle it is equal in value ; i.e. , BAD and B - AO - D contain the same number of degrees . 189. A spherical triangle is a spherical polygon of three sides ...
Page 56
... spherical angle is the arc of a great circle described from its vertex as a pole , in- cluded between its sides produced if necessary . B ' B C Let BB ' be the arc of a great circle described from the vertex of the spherical angle BAB ...
... spherical angle is the arc of a great circle described from its vertex as a pole , in- cluded between its sides produced if necessary . B ' B C Let BB ' be the arc of a great circle described from the vertex of the spherical angle BAB ...
Page 57
... spherical angle between AB and AB ' ( 190 ) . Q. E. D. 197 . COROLLARY . Tangents to the sides of a spherical angle , at its vertex , include a plane angle equal in value to the spherical angle . For such a plane angle measures the diedral ...
... spherical angle between AB and AB ' ( 190 ) . Q. E. D. 197 . COROLLARY . Tangents to the sides of a spherical angle , at its vertex , include a plane angle equal in value to the spherical angle . For such a plane angle measures the diedral ...
Page 60
... angle A ( 196 ) ; A180 ° — B'C ' . Q. E. D. 206 SCHOLIUM . If in two spherical triangles all the angles of the one are equal to all the angles of the other , each to each , the triangles are said to be mutually equiangu- lar ; if in the ...
... angle A ( 196 ) ; A180 ° — B'C ' . Q. E. D. 206 SCHOLIUM . If in two spherical triangles all the angles of the one are equal to all the angles of the other , each to each , the triangles are said to be mutually equiangu- lar ; if in the ...
Page 61
... spherical polygon is less than four right angles . Use the diagram of ( 209 ) , and complete the demonstra- tion by ( 45 ) . PROPOSITION XXXIV . 211. THEOREM . The sum of the angles of a spherical triangle is greater than 180 ° and less ...
... spherical polygon is less than four right angles . Use the diagram of ( 209 ) , and complete the demonstra- tion by ( 45 ) . PROPOSITION XXXIV . 211. THEOREM . The sum of the angles of a spherical triangle is greater than 180 ° and less ...
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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.