Elements of Geometry and Trigonometry from the Works of A.M. Legendre: Adapted to the Course of Mathematical Instruction in the United States |
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Page 20
... hence , But , DCA + DCB = ECA ECD + DCB ; ECD + DCB is equal to ECB ( A. 9 ) ; hence , DCA + DCB = ECA + ECB . The sum of the angles ECA and ECB , is equal to two right angles ; consequently , its equal , that is , the sum of the angles ...
... hence , But , DCA + DCB = ECA ECD + DCB ; ECD + DCB is equal to ECB ( A. 9 ) ; hence , DCA + DCB = ECA + ECB . The sum of the angles ECA and ECB , is equal to two right angles ; consequently , its equal , that is , the sum of the angles ...
Page 22
... Hence , the proposition is proved . Cor . 1. If one of the angles about C is a right angle , all of the others are right angles also . For , ( P. I. , C. 1 ) , each of its adjacent angles is a right angle ; and from the proposition just ...
... Hence , the proposition is proved . Cor . 1. If one of the angles about C is a right angle , all of the others are right angles also . For , ( P. I. , C. 1 ) , each of its adjacent angles is a right angle ; and from the proposition just ...
Page 23
... Hence , the sum of the given angles is equal to four right angles . PROPOSITION III . THEOREM . If two straight lines have two points in common , they coincide throughout their whole extent , and form one and the same line . Let A and B ...
... Hence , the sum of the given angles is equal to four right angles . PROPOSITION III . THEOREM . If two straight lines have two points in common , they coincide throughout their whole extent , and form one and the same line . Let A and B ...
Page 31
... hence , the triangles BAD , and DAC , have the three sides of the one equal to those of the other , each to each ; therefore , by the last Proposition , the angle B is equal to the angle C ; which was to be proved . B Cor . 1. An ...
... hence , the triangles BAD , and DAC , have the three sides of the one equal to those of the other , each to each ; therefore , by the last Proposition , the angle B is equal to the angle C ; which was to be proved . B Cor . 1. An ...
Page 32
... hence , the hypothesis that AB and AC are unequal , is false . They must , therefore , be equal ; which was to be proved . Cor . An equiangular triangle is equilateral . PROPOSITION XIII . THEOREM . In any triangle , the greater side is ...
... hence , the hypothesis that AB and AC are unequal , is false . They must , therefore , be equal ; which was to be proved . Cor . An equiangular triangle is equilateral . PROPOSITION XIII . THEOREM . In any triangle , the greater side is ...
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Common terms and phrases
ABē ACē altitude angle is equal apothem Applying logarithms axis base and altitude base multiplied bisects centre chord circumference cone consequently convex surface corresponding cosec cosine cotangent cylinder denote diagonals diameter distance divided draw drawn edges equal altitudes equal in volume equilateral feet find the area formula frustum given line given point greater hence homologous hypothenuse included angle inscribed circle inscribed polygon intersection less Let ABC logarithm lower base mantissa mean proportional measured by half number of sides opposite parallelogram parallelopipedon perimeter perpendicular plane angles plane MN polyedral angle polyedron prism PROBLEM.-To PROPOSITION proved pyramid quadrant radii radius rectangle regular polygons right angles right-angled triangle Scholium segment similar sine slant height sphere spherical angle spherical polygon spherical triangle square straight line tangent THEOREM triangle ABC triangular prisms triedral angle upper base vertex vertices whence
Popular passages
Page 28 - If two triangles have two sides of the one equal to two sides of the...
Page 90 - A cos 6 = cos a cos c + sin a sin c cos B cos c = cos a cos 6 + sin a sin 6 cos C Law of Cosines for Angles cos A = — cos B...
Page 18 - Things which are equal to the same thing, are equal to each other. 2. If equals are added to equals, the sums are equal. 3. If equals are subtracted from equals, the remainders are equal. 4. If equals are added to unequals, the sums are unequal.
Page 4 - The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.
Page 253 - For, the point A being the pole of the arc EF, the distance AE is a 'quadrant ; the point C being the pole of the arc DE, the distance CE is likewise a quadrant : hence the point E is...
Page 61 - A Circle is a plane figure bounded by a curved line every point of which is equally distant from a point within called the center.
Page 94 - A'B'C', and applying the law of cosines, we have cos a' = cos b' cos c' + sin b' sin c' cos A'. Remembering the relations a' = 180° -A, b' = 180° - B, etc. (this expression becomes cos A = — cos B cos C + sin B sin C cos a.
Page 126 - If two polygons are composed of the same number of triangles, similar each to each and similarly placed, the polygons are similar.
Page 46 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Page 125 - THEOREM. Similar triangles are to each other as the squares of their homologous sides.