The Elements of Solid Geometry |
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Page 1
... however far extended . 6. A line and a plane are parallel if they do not meet , however far extended . 1 PROPOSITION I. 7. THEOREM . Through a straight line an SECTION LINES AND PLANES IN SPACE PAGE DEFINITIONS: LINES AND PLANES TO XII 2- ...
... however far extended . 6. A line and a plane are parallel if they do not meet , however far extended . 1 PROPOSITION I. 7. THEOREM . Through a straight line an SECTION LINES AND PLANES IN SPACE PAGE DEFINITIONS: LINES AND PLANES TO XII 2- ...
Page 2
... planes may be passed . B Let AB be a straight line lying in a plane ( 1 ) . Now , we may conceive of the plane as ... parallel lines . 12. COROLLARY 4. The intersection of two planes is a straight line . For if through two points of the ...
... planes may be passed . B Let AB be a straight line lying in a plane ( 1 ) . Now , we may conceive of the plane as ... parallel lines . 12. COROLLARY 4. The intersection of two planes is a straight line . For if through two points of the ...
Page 5
... plane , i.e. , the plane of the parallel lines AA ′ and CC ' ( 11 ) ; and they cannot meet , since they lie in the parallel planes MN and PQ ( 5 ) ; hence they are parallel . AC ' is therefore a parallelogram ( 371 ) ; and AA ' is equal ...
... plane , i.e. , the plane of the parallel lines AA ′ and CC ' ( 11 ) ; and they cannot meet , since they lie in the parallel planes MN and PQ ( 5 ) ; hence they are parallel . AC ' is therefore a parallelogram ( 371 ) ; and AA ' is equal ...
Page 6
... parallel , each to each , and lying in different planes . · 20. COROLLARY 3. Lines parallel to the same line are parallel to each other . For , from the parallelograms AC ' and AB ' , we derive that , the lines CC ' and BB ' are each ...
... parallel , each to each , and lying in different planes . · 20. COROLLARY 3. Lines parallel to the same line are parallel to each other . For , from the parallelograms AC ' and AB ' , we derive that , the lines CC ' and BB ' are each ...
Page 7
... parallel to each other . Let the two planes MN and PQ be perpendicular to the straight line AD at the points A and D ; then will these planes be parallel to each other . For , pass through A a plane which is parallel to PQ , and it will ...
... parallel to each other . Let the two planes MN and PQ be perpendicular to the straight line AD at the points A and D ; then will these planes be parallel to each other . For , pass through A a plane which is parallel to PQ , and it will ...
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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.