Higher Geometry and Trigonometry: Being the Third Part of a Series on Elementary and Higher Geometry, Trigonometry, and Mensuration, Containing Many Valuable Discoveries and Improvements in Mathematical Science, Especially in Relation to the Quatrature 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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1+cos a+b+c abscissę AKXG altitude angle opposite arcs or angles base bisected called centre circle circumference conjugate hyperbolas construction cosec cosine curve determine distance draw drawn ellipse equal equation expression find the side formula frustum Geom geometrical given angle gles hence hyperbolas hypothenuse intersection Join known latus rectum logarithm major axis manner mean proportional multiplied ordinate parallel parallelogram perpendicular plane triangle PM² pole principal vertex Prop PROPOSITION pyramid quadrant quantities R+log radius rectangle right angles right-angled triangle Scholium segment side opposite similar triangles sine sines and cosines slant height solid angle sphere spherical polygon spherical triangle square straight line surface tables tangent THEOREM three angles tri-rectangular trian triangle ABC trigonometrical vertex vertices whence
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 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 27 - The circumference of every circle is supposed to be divided into 360 equal parts...
Page 7 - The radius of a sphere is a straight line, drawn from the centre to any point of the...
Page 8 - A spherical triangle is a portion of the surface of a sphere, bounded by three arcs of great circles.
Page 14 - ... point of intersection, as a pole, and limited by the sides, produced if necessary. Let the angle BAC be formed by the two arcs AB, AC ; then...
Page 18 - For, if the arc AD be drawn from the vertex A to the middle point D of the base, the two triangles ABD, ACD will have all the sides of the one respectively equal to the corresponding sides of the other, namely, AD common, BD=DC, and AB= AC : hence by the last Proposition, their angles will be equal ; therefore B=C.