Elements of Geometry and Trigonometry: With Notes |
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Page 241
... tang . x = - x2 3 x2 5 - - 1. tang . x , we shall have 7- & c . This formula will serve as the basis of our demonstration NOTE IV . 241.
... tang . x = - x2 3 x2 5 - - 1. tang . x , we shall have 7- & c . This formula will serve as the basis of our demonstration NOTE IV . 241.
Page 245
... tang . ma n n m2 3n ma 5n 7 . - & c . Now this continued fraction falls under Lemma II ; for since the de- nominators 3n , 5n , 7n , & c . increase continually , whilst the numerator m2 continues of the same magnitude , the component ...
... tang . ma n n m2 3n ma 5n 7 . - & c . Now this continued fraction falls under Lemma II ; for since the de- nominators 3n , 5n , 7n , & c . increase continually , whilst the numerator m2 continues of the same magnitude , the component ...
Page 248
... tang . 2B = 1+ cos B ( a + c + d - b ) ( b + c + d - a ) ( a + b + c - d ) ( a + b + d - c ) ( a + b ) ( cd ) & = Hence tang . B = √ ( 2.4 ) . p - a.p- p - c.p - d PROBLEM III . In the quadrilateral ABCDE , of which the opposite angles ...
... tang . 2B = 1+ cos B ( a + c + d - b ) ( b + c + d - a ) ( a + b + c - d ) ( a + b + d - c ) ( a + b ) ( cd ) & = Hence tang . B = √ ( 2.4 ) . p - a.p- p - c.p - d PROBLEM III . In the quadrilateral ABCDE , of which the opposite angles ...
Page 259
... tang . C = — 2 . In the icosaedron , we must make c = C'B'D ' = 3 * , a = b = C'B ' A ' = T ; and we shall have Cos C Cos - cos2 7 sin ᅲ ( 1 — √5 ) — 1 2 hence sin C } . H G 5 - cos a 1 + cos a No5 √5 ' E ; but cos r T = √5 ' sin C ...
... tang . C = — 2 . In the icosaedron , we must make c = C'B'D ' = 3 * , a = b = C'B ' A ' = T ; and we shall have Cos C Cos - cos2 7 sin ᅲ ( 1 — √5 ) — 1 2 hence sin C } . H G 5 - cos a 1 + cos a No5 √5 ' E ; but cos r T = √5 ' sin C ...
Page 260
... tang Ja tang ; also , making AB = a , we have CA = m , and conse- sin queatly R2 = re + sin2 1a2 n These two equations will give , for each polyedron , the values of R and r , the radii of the circumscribed and of the inscribed sphere ...
... tang Ja tang ; also , making AB = a , we have CA = m , and conse- sin queatly R2 = re + sin2 1a2 n These two equations will give , for each polyedron , the values of R and r , the radii of the circumscribed and of the inscribed sphere ...
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Elements of Geometry and Trigonometry from the Works of A. M. Legendre A. M. Legendre No preview available - 2017 |
Common terms and phrases
AC² adjacent adjacent angles altitude angle ACB angle BAC centre chord circ circle circular sector circumference circumscribed common cone consequently construction continued fraction convex surface cos² cosine cylinder demonstration determined diagonal diameter draw drawn equal angles equation equivalent faces figure formulas frustum greater homologous sides hypotenuse inclination inscribed intersection isosceles join less likewise manner measure multiplied number of sides opposite parallel parallelepipedon parallelogram perpendicular plane MN polyedron prism PROBLEM Prop PROPOSITION quadrilateral quantities radii radius ratio rectangle rectilineal triangle regular polygon right angles right-angled triangle SABC Scholium sector segment shew shewn side BC similar sin² sines solid angle sphere spherical polygon spherical triangle square straight line suppose tang tangent THEOREM third side three angles three plane angles triangle ABC triangular pyramids vertex vertices
Popular passages
Page 152 - AMB be a section, made by a plane, in the sphere, whose centre is C. From the...
Page 24 - THEOREM. In the same circle, or in equal circles, equal arcs are subtended by equal chords ; and, conversely, equal chords subtend equal arcs.
Page 22 - CIRCLE is a plane figure bounded by a curved line, all the points of which are equally distant from a point within called the centre; as the figure ADB E.
Page 62 - Similar triangles are to each other as the squares of their homologous sides.
Page 211 - If two angles of one triangle are equal to two angles of another triangle, the third angles are equal, and the triangles are mutually equiangular.
Page 187 - Similar cylinders are to each other as the cubes of their altitudes, or as the cubes of the diameters of their bases.
Page 140 - AT into equal parts .Ax, xy, yz, &c., each less than Aa, and let k be one of those parts : through the points of division pass planes parallel to the plane of the bases : the corresponding sections formed by these planes in the two pyramids will be respectively equivalent, namely, DEF to def, GHI to ghi, &c.
Page 150 - The radius of a sphere is a straight line, drawn from the centre to any point...
Page 168 - THEOREM. The surface of a spherical triangle is measured by the excess of the sum of its three angles above two right angles, multiplied by the tri-rectangular triangle.
Page 135 - XII.) ; in like manner, the two solids AQ, AK, having the same base, AOLE, are to each other as their altitudes AD, A M.