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 146
... regular polygons are to each other as the radii of their circumscribed or inscribed circles ; and their areas are to each other as the squares of those radii . 1o . Let ABC and KLM be similar regular polygons . Let OA and QK be the ...
... regular polygons are to each other as the radii of their circumscribed or inscribed circles ; and their areas are to each other as the squares of those radii . 1o . Let ABC and KLM be similar regular polygons . Let OA and QK be the ...
Page 147
... regular polygons of the same number of sides can be constructed , the one circumscribed about a circle and the other inscribed in it , which shall differ from each other by less than any given surface . Let ABCE be a circle , O its ...
... regular polygons of the same number of sides can be constructed , the one circumscribed about a circle and the other inscribed in it , which shall differ from each other by less than any given surface . Let ABCE be a circle , O its ...
Page 148
... greater than their differ- ence , at the same time , which is impossible ; therefore , the two quantities are ultimately equal . * Newton's Principia , Book I. , Lemma I. Cor . If we take any two similar regular polygons 148 GEOMETRY .
... greater than their differ- ence , at the same time , which is impossible ; therefore , the two quantities are ultimately equal . * Newton's Principia , Book I. , Lemma I. Cor . If we take any two similar regular polygons 148 GEOMETRY .
Page 149
... regular polygons , the one cir- cumscribed about , and the other inscribed within the circle , and bisect the arcs , and then circumscribe and inscribe two regular polygons having double the number of sides , it is plain that by ...
... regular polygons , the one cir- cumscribed about , and the other inscribed within the circle , and bisect the arcs , and then circumscribe and inscribe two regular polygons having double the number of sides , it is plain that by ...
Page 153
... regular polygons MNPST and EFGKL be inscribed in the circles : then will the perimeters of these polygons be to each other as their apothems , and the areas will be to each other as the squares of their apothems , what- ever may be the ...
... regular polygons MNPST and EFGKL be inscribed in the circles : then will the perimeters of these polygons be to each other as their apothems , and the areas will be to each other as the squares of their apothems , what- ever may be the ...
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Common terms and phrases
AB² AC² adjacent angles altitude angle ACB apothem Applying logarithms base and altitude bisect centre chord circle circumference circumscribed coincide cone consequently convex surface corresponding cosec Cosine Cotang cylinder denote diagonals diameter distance divided draw drawn edges equally distant feet find the area Formula frustum given angle given straight line greater hence homologous hypothenuse included angle interior angles intersection less Let ABC log sin lower base mantissa mean proportional measured by half number of sides opposite parallel parallelogram parallelopipedon perimeter perpendicular plane MN polyedral angle polyedron prism PROPOSITION proved pyramid quadrant radii radius rectangle regular polygons right angles right-angled triangle Scholium secant segment semi-circumference side BC similar sine six right slant height sphere spherical polygon spherical triangle square Tang tangent THEOREM triangle ABC triangular prisms triedral angle upper base vertex vertices whence