An Elementary Treatise on Plane and Solid Geometry |
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Page iii
... demonstration have , by their attacks , induced some mathematicians to waste much time in disguising the principles of the Differential Calculus under a form of words , in which the term " infinitely small " does not occur . The value ...
... demonstration have , by their attacks , induced some mathematicians to waste much time in disguising the principles of the Differential Calculus under a form of words , in which the term " infinitely small " does not occur . The value ...
Page xvi
... demonstration here given requires that the angle SBA be a right angle , or an obtuse angle , and cannot there- fore be extended to the case in which ASB is right or obtuse , so that the following demonstration must be substituted for it ...
... demonstration here given requires that the angle SBA be a right angle , or an obtuse angle , and cannot there- fore be extended to the case in which ASB is right or obtuse , so that the following demonstration must be substituted for it ...
Page 1
... demonstration . A Problem is a question proposed which requires a solution . A Lemma is a subsidiary truth employed in the demonstration of a theorem , or in the solution of a problem . A Corollary is a consequence which follows from ...
... demonstration . A Problem is a question proposed which requires a solution . A Lemma is a subsidiary truth employed in the demonstration of a theorem , or in the solution of a problem . A Corollary is a consequence which follows from ...
Page 8
... Demonstration . For it is equal to the sum of the two right angles APM , MPE , formed by the perpendicular AE . 26. Theorem . The sum of all the successive an- gles APB , BPC , CPD , DPE , and EPA ( fig . 10 ) , formed in a plane about ...
... Demonstration . For it is equal to the sum of the two right angles APM , MPE , formed by the perpendicular AE . 26. Theorem . The sum of all the successive an- gles APB , BPC , CPD , DPE , and EPA ( fig . 10 ) , formed in a plane about ...
Page 9
... Demonstration . Thus the two lines AB and CD ( fig . 11 ) cannot meet at P ; for , if two straight lines are drawn through P , in the same direction , they must coincide and form one and the same straight line . 29. Theorem . Two angles ...
... Demonstration . Thus the two lines AB and CD ( fig . 11 ) cannot meet at P ; for , if two straight lines are drawn through P , in the same direction , they must coincide and form one and the same straight line . 29. Theorem . Two angles ...
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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°.