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 21
... whole circumference ACE , the part which remains after substracting the triangle ABC from the hemisphere , is a new triangle also designated by ABC , and having AB , BC , AEDC for its sides . Here , it is plain , the side AEDC is ...
... whole circumference ACE , the part which remains after substracting the triangle ABC from the hemisphere , is a new triangle also designated by ABC , and having AB , BC , AEDC for its sides . Here , it is plain , the side AEDC is ...
Page 22
... whole sphere must contain 96 of those partial triangles , the lune AMBNA will contain 10 of them ; hence the lune is to the sphere as 10 is to 96 , or as 5 to 48 , in other words , as the arc MN is to the circumference . 1 If the arc MN ...
... whole sphere must contain 96 of those partial triangles , the lune AMBNA will contain 10 of them ; hence the lune is to the sphere as 10 is to 96 , or as 5 to 48 , in other words , as the arc MN is to the circumference . 1 If the arc MN ...
Page 23
... whole sphere will be expressed by 8T , and the surface of the lune whose angle is A , will be expressed by 2AXT . for 4 : A : 8T : 2AXT in which expression , A represents such a part of unity , as the angle of the lune is of one right ...
... whole sphere will be expressed by 8T , and the surface of the lune whose angle is A , will be expressed by 2AXT . for 4 : A : 8T : 2AXT in which expression , A represents such a part of unity , as the angle of the lune is of one right ...
Page 25
... whole surface of the sphere . Scholium . While the spherical triangle ABC is compared with the tri - rectangular triangle , the spherical pyramid , which has ABC for its base , is compared with the tri - rectangular pyramid , and a ...
... whole surface of the sphere . Scholium . While the spherical triangle ABC is compared with the tri - rectangular triangle , the spherical pyramid , which has ABC for its base , is compared with the tri - rectangular pyramid , and a ...
Page 28
... whole circumference of any circle , for it is evident that the 360th part of the circumference of a large circle is greater than the same part of a smaller one , but the number of de- grees in the small circumference is the same as in ...
... whole circumference of any circle , for it is evident that the 360th part of the circumference of a large circle is greater than the same part of a smaller one , but the number of de- grees in the small circumference is the same as in ...
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Common terms and phrases
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 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 trian triangle ABC trigonometrical 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.