The Elements of Solid Geometry |
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... LUNE , ETC. EXERCISES 54 70 74 SECTION V REGULAR POLYEDRONS . DEFINITIONS PROPOSITION XLIII .. 76 CONSTRUCTION : - REGULAR POLYEDRONS 25 76 77 SECTION VI . MENSURATION : SOLIDS AND THEIR SURFACES . PRISMS AND CYLINDERS PYRAMIDS AND ...
... LUNE , ETC. EXERCISES 54 70 74 SECTION V REGULAR POLYEDRONS . DEFINITIONS PROPOSITION XLIII .. 76 CONSTRUCTION : - REGULAR POLYEDRONS 25 76 77 SECTION VI . MENSURATION : SOLIDS AND THEIR SURFACES . PRISMS AND CYLINDERS PYRAMIDS AND ...
Page 70
... lune is a spherical surface bounded by two semi- circumferences of great circles . A lune then has two equal angles and two equal sides . A'B'CB is a lune . See ( 226 ) . PROPOSITION XLI . 228. THEOREM . A lune is to the entire surface ...
... lune is a spherical surface bounded by two semi- circumferences of great circles . A lune then has two equal angles and two equal sides . A'B'CB is a lune . See ( 226 ) . PROPOSITION XLI . 228. THEOREM . A lune is to the entire surface ...
Page 71
... lune by A : then LS : A : 4 rt . angles . Let abcda be the circumference of a great circle whose poles are A and V , and let ac be an arc of it intercepted by the sides of the lune ; then ac is the measure of A ( 196 ) . Let ab , a ...
... lune by A : then LS : A : 4 rt . angles . Let abcda be the circumference of a great circle whose poles are A and V , and let ac be an arc of it intercepted by the sides of the lune ; then ac is the measure of A ( 196 ) . Let ab , a ...
Page 72
... equilateral are mutually equal in all their parts ( 215 ) , and must be equal or symmetrical ( 217 ) and equivalent ( 220 ) ; and , the lune which is made up of the triangles ABC and A'BC 72 THE ELEMENTS OF SOLID GEOMETRY .
... equilateral are mutually equal in all their parts ( 215 ) , and must be equal or symmetrical ( 217 ) and equivalent ( 220 ) ; and , the lune which is made up of the triangles ABC and A'BC 72 THE ELEMENTS OF SOLID GEOMETRY .
Page 73
... lune BACA ' ABC + AB'C ' . ( 2 ) . But lune ABCB ' ABC + ACB ' . ( 3 ) . And , lune ACBC ' ACB + ABC ' . Equations ( 1 ) , ( 2 ) , and ( 3 ) give by addition , ( 4 ) . BACA ' + ABCB ' + ACBC ' 2 ABC + ACB + AB'C ' + ACB ' + ABC ' ( 5 ) ...
... lune BACA ' ABC + AB'C ' . ( 2 ) . But lune ABCB ' ABC + ACB ' . ( 3 ) . And , lune ACBC ' ACB + ABC ' . Equations ( 1 ) , ( 2 ) , and ( 3 ) give by addition , ( 4 ) . BACA ' + ABCB ' + ACBC ' 2 ABC + ACB + AB'C ' + ACB ' + ABC ' ( 5 ) ...
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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.