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
... Hence no point of the shortest line from A to B can lie out of the arc ADB ; hence this arc is itself the shortest distance be- tween its two extremities . PROPOSITION IV . THEOREM . The sum of the three sides of a spherical triangle is ...
... Hence no point of the shortest line from A to B can lie out of the arc ADB ; hence this arc is itself the shortest distance be- tween its two extremities . PROPOSITION IV . THEOREM . The sum of the three sides of a spherical triangle is ...
Page 12
... hence the points D and E are each equally distant from all the points of the cir- cumference AMB ; hence ( Def . 5. ) they are the poles of that circumference . Again , the radius DC , perpendicular to the plane AMB , is ...
... hence the points D and E are each equally distant from all the points of the cir- cumference AMB ; hence ( Def . 5. ) they are the poles of that circumference . Again , the radius DC , perpendicular to the plane AMB , is ...
Page 13
... hence it is perpendicular to their plane ( Prop . V. B. I. El . S. Geom . ) ; hence the point D is the pole of the arc AM ; and consequently the angles DAM , AMD are right . Scholium . The properties of these poles enable us to de ...
... hence it is perpendicular to their plane ( Prop . V. B. I. El . S. Geom . ) ; hence the point D is the pole of the arc AM ; and consequently the angles DAM , AMD are right . Scholium . The properties of these poles enable us to de ...
Page 14
... hence the distance OM will be greater than OA . Hence the point M lies without the sphere ; and as the same can be shown for every other point of the plane FAG , this plane can have no point but A common to it and the sur- face of ...
... hence the distance OM will be greater than OA . Hence the point M lies without the sphere ; and as the same can be shown for every other point of the plane FAG , this plane can have no point but A common to it and the sur- face of ...
Page 15
... hence the point E is removed the length of a quadrant from each of the points A and C ; hence ( Prop VI . Cor . 3. ) is the pole of the arc AC . It might be shown , by the same method , that D is the pole of the arc BC , and F that of ...
... hence the point E is removed the length of a quadrant from each of the points A and C ; hence ( Prop VI . Cor . 3. ) is the pole of the arc AC . It might be shown , by the same method , that D is the pole of the arc BC , and F that of ...
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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.