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 22
... proved . Cor . 1. If one of the angles about C all of the others will be right angles also . each of its adjacent angles will be a right angle ; and from the proposition just demonstrated , its opposite angle will also be a right angle ...
... proved . Cor . 1. If one of the angles about C all of the others will be right angles also . each of its adjacent angles will be a right angle ; and from the proposition just demonstrated , its opposite angle will also be a right angle ...
Page 23
... proved , and then continuing the reasoning until the assumed hypothesis is shown to be false . Its contradictory is thus proved to be true . This method of demonstration is often used in Geometry . PROPOSITION IV . THEOREM . If a ...
... proved , and then continuing the reasoning until the assumed hypothesis is shown to be false . Its contradictory is thus proved to be true . This method of demonstration is often used in Geometry . PROPOSITION IV . THEOREM . If a ...
Page 26
... proved . 1 PROPOSITION VII . THEOREM . The sum of any two sides of a triangle is greater than the third side . Let ABC be a triangle : then will the sum of any two sides , as AB , BC , be greater than the third side AC . For , the ...
... proved . 1 PROPOSITION VII . THEOREM . The sum of any two sides of a triangle is greater than the third side . Let ABC be a triangle : then will the sum of any two sides , as AB , BC , be greater than the third side AC . For , the ...
Page 29
... proved . Conversely : If in two triangles ABC and DEF , the side AB is equal to the side DE , the side AC to DF , and BC greater than EF , then will the angle BAC be greater than the angle EDF For , if not , BAC must either be equal to ...
... proved . Conversely : If in two triangles ABC and DEF , the side AB is equal to the side DE , the side AC to DF , and BC greater than EF , then will the angle BAC be greater than the angle EDF For , if not , BAC must either be equal to ...
Page 32
... proved . 1 AB > AC ; Conversely : Let AB be greater than AC : then will the angle ACB be greater than the angle ABC . For , if ACB were less than ABC , the side AB would be less than the side AC , from what has just been proved ; if ACB ...
... proved . 1 AB > AC ; Conversely : Let AB be greater than AC : then will the angle ACB be greater than the angle ABC . For , if ACB were less than ABC , the side AB would be less than the side AC , from what has just been proved ; if ACB ...
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Common terms and phrases
AB² AC² altitude angle ACB apothem axis base and altitude base multiplied BC² bisect centre chord circumference coincide cone consequently convex surface corresponding cosec cosine Cotang cylinder denote diameter distance divided draw drawn edges equal bases equal in volume equal to AC equal to half equally distant Formula frustum given angle given line greater hence homologous hypothenuse included angle intersection less Let ABC logarithm lower base mantissa mean proportional measured by half number of sides opposite parallelogram perimeter perpendicular plane MN polyedral angle polyedron prism PROPOSITION XI proved pyramid quadrant radii radius rectangle regular polygons right-angled triangle Scholium segment semi-circumference side BC similar sine slant height sphere spherical angle spherical excess spherical polygon spherical triangle straight line tangent THEOREM triangle ABC triangular prisms triedral angle upper base vertex vertices whence