An Elementary Treatise on Plane and Solid Geometry |
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Page xii
... Parallel lines in the same plane ( 326 ) , Line parallel to plane , parallel planes ( 327 ) , 103 105 105 Cases in which a line is parallel to a plane ( 328 , 334 ) , 105 Cases in which planes are parallel ( 329 , 331 xii CONTENTS .
... Parallel lines in the same plane ( 326 ) , Line parallel to plane , parallel planes ( 327 ) , 103 105 105 Cases in which a line is parallel to a plane ( 328 , 334 ) , 105 Cases in which planes are parallel ( 329 , 331 xii CONTENTS .
Page xiii
Benjamin Peirce. Cases in which planes are parallel ( 329 , 331 ) , Parallel lines between parallel planes ( 332 , 335 ) , Intersection of two parallel planes by a third plane ( 333 ) , Solid angle ( 336 ) , Sum of two plane angles of a ...
Benjamin Peirce. Cases in which planes are parallel ( 329 , 331 ) , Parallel lines between parallel planes ( 332 , 335 ) , Intersection of two parallel planes by a third plane ( 333 ) , Solid angle ( 336 ) , Sum of two plane angles of a ...
Page 103
... plane . 319. Theorem . Oblique lines drawn from a point to a plane at equal ... parallel to this perpendicular , is likewise perpendicular to the plane . Perpendiculars to a Plane . Demonstration . Let AB ( CH . XV . $ 321 . ] 103 PLANES ...
... plane . 319. Theorem . Oblique lines drawn from a point to a plane at equal ... parallel to this perpendicular , is likewise perpendicular to the plane . Perpendiculars to a Plane . Demonstration . Let AB ( CH . XV . $ 321 . ] 103 PLANES ...
Page 104
... plane are parallel . 323. Theorem . If two planes are perpendicular to each other , the line , which is drawn in one of the planes perpendicular to their common intersection , must be perpendicular to the other plane . Demonstration . Let ...
... plane are parallel . 323. Theorem . If two planes are perpendicular to each other , the line , which is drawn in one of the planes perpendicular to their common intersection , must be perpendicular to the other plane . Demonstration . Let ...
Page 105
... planes AP and AQ , and must , therefore , be their common intersection . 326. Theorem . Two parallel lines are always in the same plane . Demonstration . Draw a plane MN ( fig . 153 ) perpendicu- lar to one of the parallels AB , it must ...
... planes AP and AQ , and must , therefore , be their common intersection . 326. Theorem . Two parallel lines are always in the same plane . Demonstration . Draw a plane MN ( fig . 153 ) perpendicu- lar to one of the parallels AB , it must ...
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Common terms and phrases
ABCD &c AC AC adjacent angles altitude angle ABC angle BAC arc BC base and altitude bisect CD fig centre chord circumference convex surface Corollary cylinder DEF fig Definitions Demonstration denote diameter divided Draw equal arcs equal distances equilateral equivalent four right angles frustum given angle given circle given line given polygon given sides given square greater half the product Hence homologous sides hypothenuse infinitely small inscribed circle isoperimetrical isosceles Join AC Let ABCD line AB fig lines drawn mean proportional number of sides oblique lines parallel lines parallelogram parallelopiped perimeter perpendicular plane angles plane MN polygon ABCD prism Problem radii radius ratio rectangles regular polygon respectively equal right triangles Scholium secant sector segment side AC similar polygons similar triangles solid angle Solution sphere spherical polygon spherical triangle tangent Theorem triangles ABC triangular prism vertex vertices whence
Popular passages
Page 148 - 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. D c A' D' Hyp. In triangles ABC and A'B'C', ZA = ZA'. To prove AABC = ABxAC. A A'B'C' A'B'xA'C' Proof. Draw the altitudes BD and B'D'.
Page 90 - To construct a parallelogram equivalent to a given square, and having the difference of its base and altitude equal to a given line.
Page 24 - CIRCLE is a plane figure bounded by a curved line, all the points of which are equally distant from a point within called the centre; as the figure ADB E.
Page 40 - One side and two angles of a triangle being given, to construct the triangle. Solution.
Page 79 - Construct, by § 145, a right triangle, of which the hypothenuse BC (fig. 79) is equal to the side of the greater square, and the leg AB is equal to the side of the less square ; and AC is the side of the required square.
Page 142 - THEOREM. If two triangles on the same sphere, or on equal spheres, are mutually equiangular, they will also be mutually equilateral. Let A and B be the two given triangles; P and Q their polar triangles. Since the angles are equal in the triangles A and B, the sides will be equal in. their polar triangles P and Q (Prop.
Page 6 - The preface and commentary to the Antigone are even more creditable to Mr. Woolsey's ability than those to the Alcestis. The sketch of the poem, in the preface, is written with clearness and brevity. The difficulties in this play, that call for a commentator's explanation, are far more numerous than in the Alcestis.
Page 70 - The perimeters of two regular polygons of the same number of sides, are to each other as their homologous sides, and their areas are to each other as the squares of those sides (Prop.
Page 5 - A surface is that which has length and breadth, without thickness. 6. A plane is a surface, in which any two points being taken, the straight line joining those points lies wholly in that surface.
Page 137 - Each side of a spherical triangle is less than the sum of 'the other two sides. 48. The sum of the sides of a spherical polygon is less than 360°.