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 9
... PROPOSITION I. THEOREM . Every section of a sphere , made by a plane , is a circle . Let AMB be a section , made by a plane , in the sphere , whose centre is C. From the point C , draw CO perpendicu- A lar to the plane AMB ; and ...
... PROPOSITION I. THEOREM . Every section of a sphere , made by a plane , is a circle . Let AMB be a section , made by a plane , in the sphere , whose centre is C. From the point C , draw CO perpendicu- A lar to the plane AMB ; and ...
Page 10
... PROPOSITION II . THEOREM . In every spherical triangle , any side is less than the sum of the other two . A C B Let O be the centre of the sphere , and ACB the triangle : draw the radii OA , OB , OC . Imagine the planes AOB , AOC , COB ...
... PROPOSITION II . THEOREM . In every spherical triangle , any side is less than the sum of the other two . A C B Let O be the centre of the sphere , and ACB the triangle : draw the radii OA , OB , OC . Imagine the planes AOB , AOC , COB ...
Page 11
... PROPOSITION IV . THEOREM . The sum of the three 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 ...
... PROPOSITION IV . THEOREM . The sum of the three 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 ...
Page 12
... proposition we have been engaged with . The demon- stration here given is different from that of Prop . XXII . B. I. El . S. Geom .; both , however , suppose that the polygon ABCDE is convex , or that no side produced will cut the ...
... proposition we have been engaged with . The demon- stration here given is different from that of Prop . XXII . B. I. El . S. Geom .; both , however , suppose that the polygon ABCDE is convex , or that no side produced will cut the ...
Page 13
... to a quadrant ; then , from the pole S , and with the same distance , describe the arc PM , which will be the perpendicular required . PROPOSITION VII . THEOREM . Every plane perpendicular to a 2 * SPHERICAL GEOMETRY . 13.
... to a quadrant ; then , from the pole S , and with the same distance , describe the arc PM , which will be the perpendicular required . PROPOSITION VII . THEOREM . Every plane perpendicular to a 2 * SPHERICAL GEOMETRY . 13.
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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.