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
... 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 straight line meet BOOK I. 23 *
... 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 straight line meet BOOK I. 23 *
Page 27
... , we have , BD + DC < BA + AC . But it was shown that BO + OC is less than BD + DC ; still more , then , is BO + OC less than BA + AC ; which was to be proved . PROPOSITION IX . THEOREM . If two triangles have two BOOK I. 27.
... , we have , BD + DC < BA + AC . But it was shown that BO + OC is less than BD + DC ; still more , then , is BO + OC less than BA + AC ; which was to be proved . PROPOSITION IX . THEOREM . If two triangles have two BOOK I. 27.
Page 35
... shown , as in the first case , that AD is equal to DF Then , because the point C lies within the triangle ADF , the sum of the lines AD and DF will be greater than the sum of the lines AC and CF ( P. VIII . ) : hence , AD , the half of ...
... shown , as in the first case , that AD is equal to DF Then , because the point C lies within the triangle ADF , the sum of the lines AD and DF will be greater than the sum of the lines AC and CF ( P. VIII . ) : hence , AD , the half of ...
Page 36
... shown , DA and DB will be equal ; but IB is less than the sum of ID and DB ( P. VII . ) ; and because the sum of ID and DB is equal to the sum of ID and DA , or IA , we have IB less than IA : hence , I is unequally distant from A and B ...
... shown , DA and DB will be equal ; but IB is less than the sum of ID and DB ( P. VII . ) ; and because the sum of ID and DB is equal to the sum of ID and DA , or IA , we have IB less than IA : hence , I is unequally distant from A and B ...
Page 39
... shown , AB and CD are parallel . Cor . 2. If two lines are cut by a third , making the oppo- site exterior and interior angles equal , the two lines will be - parallel . Let the angles EGB and GHD be equal : Now , EGB and AGII are equal ...
... shown , AB and CD are parallel . Cor . 2. If two lines are cut by a third , making the oppo- site exterior and interior angles equal , the two lines will be - parallel . Let the angles EGB and GHD be equal : Now , EGB and AGII are equal ...
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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