Higher Geometry and Trigonometry: Being the Third Part of a Series on Elementary and Higher Geometry, Trigonomentary and Mensuration : Containing Many Valuable Discoveries and Improvements in Mathematical Science, Especially in Relation to the Quadrature of the Circle, and Some Other Curves, as Well as the Cubature of Certain Curvilinear Solids : Designed as a Text-book for Collegiate and Academic Instruction, and as a Practical Compendium of Mensuration |
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Page 11
... sides of a spherical triangle is less than the circumference of a great circle . Let ABC be any spherical tri- angle ; produce the sides AB , AC , till they meet again in D. The arcs ABD , ACD , will be semicircumferences , since ( Prop ...
... sides of a spherical triangle is less than the circumference of a great circle . Let ABC be any spherical tri- angle ; produce the sides AB , AC , till they meet again in D. The arcs ABD , ACD , will be semicircumferences , since ( Prop ...
Page 15
... sides : hence it is easy to make an angle of this kind equal to a given angle . Scholium . Vertical angles , such as ... sides of the first . From the vertices A , B , C , as poles , let the arcs EF , FD , ED be described , forming ...
... sides : hence it is easy to make an angle of this kind equal to a given angle . Scholium . Vertical angles , such as ... sides of the first . From the vertices A , B , C , as poles , let the arcs EF , FD , ED be described , forming ...
Page 16
... side of BC , the two B and E on the same side of AC , and the two C and Fon the same side of AB . D F E d Various names have been given to the triangles ABC , DEF ; we shall call them polar triangles . PROPOSITION XI . THEOREM . If ...
... side of BC , the two B and E on the same side of AC , and the two C and Fon the same side of AB . D F E d Various names have been given to the triangles ABC , DEF ; we shall call them polar triangles . PROPOSITION XI . THEOREM . If ...
Page 17
... side AD = AC , DB = BC , and AB is common ; hence those two triangles have their sides equal , each to each . We are now to show , that the angles opposite these equal sides are also equal . A ۱ 0 1 F If the centre of the sphere is ...
... side AD = AC , DB = BC , and AB is common ; hence those two triangles have their sides equal , each to each . We are now to show , that the angles opposite these equal sides are also equal . A ۱ 0 1 F If the centre of the sphere is ...
Page 18
... sides . This truth is evident from Proposition XI , where it is shown that , with three given sides AB , AC , BC , ( see the diagram , ) there can only be two triangles ACB , ABD , differing as to the position of their parts , and ...
... sides . This truth is evident from Proposition XI , where it is shown that , with three given sides AB , AC , BC , ( see the diagram , ) there can only be two triangles ACB , ABD , differing as to the position of their parts , and ...
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abscissa altitude arithmetical progression axes base bisected chord circle circular circular segment circumference cone conjugate construction convex surface corresponding cosec cosine cylinder described diameter distance divided draw drawn ellipse equal to half equation expression feet formed formula frustum Geom geometrical given height hence hyperbola inches infinite series latus rectum length logarithm major axis multiplied opposite ordinates parabola parallel parallelogram passing perpendicular plane portion prism Prop PROPOSITION pyramid quadrant quadrature quantity radii radius ratio rectangle represent revoloidal surface right angles Scholium sector segment sides similar similar triangles sine solidity specific gravity sphere spherical triangle spheroid spindle square straight line tangent THEOREM tion triangle ABC Trigonometry ungula versed sine vertex vertical virtual centre whence
Popular passages
Page 81 - The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.
Page 81 - N .-. by definition, x — x" is the logarithm of ^ ; that is to say, The logarithm of a fraction, or of the quotient of two numbers, is equal to the logarithm of the numerator minus the logarithm of the denominator. III. Raise both members of equation (1) to the nth power. N"=a".
Page 68 - In the same way it may be proved that a : b : : sin. A : sin. B, and these two proportions may be written a : 6 : c : : sin. A : sin. B : sin. C. THEOREM III. t8. In any plane triangle, the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference. By Theorem II. we have a : b : : sin. A : sin. B.
Page 7 - The radius of a sphere is a straight line, drawn from the centre to any point of the surface ; the diameter, or axis, is a line passing through this centre, and terminated on both sides by the surface.
Page 138 - B . sin c = sin b . sin C cos a = cos b . cos c + sin b . sin c cos b = cos a . cos c + sin a . sin c cos A cos B cos c = cos a . cos b + sin a . sin b . cos C ..2), cotg b . sin c = cos G.
Page 8 - A spherical triangle is a portion of the surface of a sphere, bounded by three arcs of great circles.
Page 27 - The circumference of every circle is supposed to be divided into 360 equal parts, called degrees...
Page 78 - In a system of logarithms all numbers are considered as the powers of some one number, arbitrarily chosen, which is called the base of the system, and the exponent of that power of the base which is equal to any given number, is called the logarithm of that number. Thus, if a be the base of a system of logarithms, N any number, and x such that N = a* then x is called the logarithm of N in the system whose base is a.