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 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 will coincide throughout their whole extent , and form one and the same line . Let A ...
... 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 will coincide throughout their whole extent , and form one and the same line . Let A ...
Page 26
... hence , the triangles coincide throughout , therefore equal in all their parts ( I. , D. 14 ) ; which was to be proved . 1 PROPOSITION VII . THEOREM . The sum of any two sides of a triangle is greater than the third side . Let ABC be a ...
... hence , the triangles coincide throughout , therefore equal in all their parts ( I. , D. 14 ) ; which was to be proved . 1 PROPOSITION VII . THEOREM . The sum of any two sides of a triangle is greater than the third side . Let ABC be a ...
Page 29
... Hence , in each case , BC is greater than EF ; which was to be 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 ...
... Hence , in each case , BC is greater than EF ; which was to be 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 ...
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 D 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 D Cor . 1. An ...
Page 32
... hence , the hypothesis that AB and AC are unequal , is false . must , therefore , be equal ; which was to be proved . Cor . An equiangular triangle is equilateral . They PROPOSITION XIII . THEOREM . In any triangle , the greater side is ...
... hence , the hypothesis that AB and AC are unequal , is false . must , therefore , be equal ; which was to be proved . Cor . An equiangular triangle is equilateral . They PROPOSITION XIII . THEOREM . In any triangle , the greater side is ...
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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