The Elements of Solid Geometry |
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Page 13
... ABC , let AOC be the greatest ; then will AOC AOB + BOC . Draw OD making the angle AOD equal to the angle AOB . Take DO , AO , and BO equal to each other . Draw AD pro- duced to meet OC in some point C. Draw AB and BC . The triangles ...
... ABC , let AOC be the greatest ; then will AOC AOB + BOC . Draw OD making the angle AOD equal to the angle AOB . Take DO , AO , and BO equal to each other . Draw AD pro- duced to meet OC in some point C. Draw AB and BC . The triangles ...
Page 15
... abc and a'b'c ' are the measures of the diedral angles a - ob - c and a ' - o'b ' - c ' ( 30 ) . The triangle aob equals the triangle a'o'b ' ( 356 ) ; hence , ab equals a'b ' , and ao equals a'o ' . In like manner we can prove that be ...
... abc and a'b'c ' are the measures of the diedral angles a - ob - c and a ' - o'b ' - c ' ( 30 ) . The triangle aob equals the triangle a'o'b ' ( 356 ) ; hence , ab equals a'b ' , and ao equals a'o ' . In like manner we can prove that be ...
Page 35
... triangles ) ; AB : A'B ' :: PO : P'O ` AB2 : A'B22 :: PO2 : P′O2 . But hence , or , But , hence , AC : A'C ' :: AB2 ... ABC and O ' - A'B'C ' ; and let h be the common distance from the vertices of the sections DEF and D'E'F ' . Let the ...
... triangles ) ; AB : A'B ' :: PO : P'O ` AB2 : A'B22 :: PO2 : P′O2 . But hence , or , But , hence , AC : A'C ' :: AB2 ... ABC and O ' - A'B'C ' ; and let h be the common distance from the vertices of the sections DEF and D'E'F ' . Let the ...
Page 38
... ABC - O , constructed with the same base and altitude as the prism , has a volume one - third as great . Hence , the ... triangle be revolved about one of its sides as an axis it will generate a right circular cone . 125. If two similar ...
... ABC - O , constructed with the same base and altitude as the prism , has a volume one - third as great . Hence , the ... triangle be revolved about one of its sides as an axis it will generate a right circular cone . 125. If two similar ...
Page 50
... ABC be any section not through the centre O ; then will ABC be a circle . Draw a diameter perpendicular to the ... triangles OBO ' and OAO ' , the angle 00'A equals the angle OO'B ( 4 ) ; OB equals AO being radii , and OO ' is common ; hence ...
... ABC be any section not through the centre O ; then will ABC be a circle . Draw a diameter perpendicular to the ... triangles OBO ' and OAO ' , the angle 00'A equals the angle OO'B ( 4 ) ; OB equals AO being radii , and OO ' is common ; hence ...
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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.