The Elements of Solid Geometry |
From inside the book
Results 1-5 of 5
Page 55
... triangle as poles , arcs of great circles be de- scribed , a second spherical triangle I will be formed which is called the polar triangle of the first . If A , B , and C are poles of B'C ' , C'A ' , and A'B ' , respectively , then will ...
... triangle as poles , arcs of great circles be de- scribed , a second spherical triangle I will be formed which is called the polar triangle of the first . If A , B , and C are poles of B'C ' , C'A ' , and A'B ' , respectively , then will ...
Page 60
... triangle , and if B'C ' is the side opposite A , in the polar tri- angle , then will A be equivalent to 180 ° B'C ' . - For , produce the arcs AB and AC until they meet B'C ' at some points D and E. or , But , ( a ) Now , arc B'E = 90 ...
... triangle , and if B'C ' is the side opposite A , in the polar tri- angle , then will A be equivalent to 180 ° B'C ' . - For , produce the arcs AB and AC until they meet B'C ' at some points D and E. or , But , ( a ) Now , arc B'E = 90 ...
Page 61
... triangle is greater than 180 ° and less than 540 ° . Α ' B B C ' αν Let A , B , and C represent the angle of any spherical triangle ABC ; then will A + B + C > 180 ° and < 540 ° . Construct A'B'C ' , the polar triangle of ABC . Now , A ...
... triangle is greater than 180 ° and less than 540 ° . Α ' B B C ' αν Let A , B , and C represent the angle of any spherical triangle ABC ; then will A + B + C > 180 ° and < 540 ° . Construct A'B'C ' , the polar triangle of ABC . Now , A ...
Page 64
... polar triangle are supplements of the opposite parts in the primitive triangle ( 205 ) . But , if A'B'C ' and D'E'F ' are mutually equal with respect to two sides and their included angle , they are mutually equal in all their parts ...
... polar triangle are supplements of the opposite parts in the primitive triangle ( 205 ) . But , if A'B'C ' and D'E'F ' are mutually equal with respect to two sides and their included angle , they are mutually equal in all their parts ...
Page 84
... triangle are 40 ° 10 ' , 80 ° 20 ' , and 140 ° 30 ′ , respectively . What are the values of the sides of the polar triangles ? See 205 . 281. On a sphere whose diameter is 2000 feet , the angle of a lune is 30 degrees . What is the ...
... triangle are 40 ° 10 ' , 80 ° 20 ' , and 140 ° 30 ′ , respectively . What are the values of the sides of the polar triangles ? See 205 . 281. On a sphere whose diameter is 2000 feet , the angle of a lune is 30 degrees . What is the ...
Other editions - View all
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.