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ABED axis base becomes bisect c.sin Calculation called centre circle common section complement computed cone consequently construction contained cosine cotan cutting describe diameter difference distance draw drawn ellipse equal equation evidently extremes follows Form given greater half Hence hyperbola intersection join known less logarithms meet oblique opposite original circle parallel pass perpendicular plane of projection pole positive primitive circle produced projecting point proposed Q. E. D. Art quadrant radius ratio represented respectively Result right angles right line secant semitangent sides similar sine sphere spherical triangle suppose surface tangent tion touch triangle whence wherefore
Page 32 - The circumference of every circle is supposed to be divided into 360 equal parts, called degrees ; each degree into 60 equal parts, called minutes ; and each minute into 60 equal parts, called seconds.
Page 39 - 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 95 - TO THEIR DIFFERENCE ; So IS THE TANGENT OF HALF THE SUM OF THE OPPOSITE ANGLES', To THE TANGENT OF HALF THEIR DIFFERENCE.
Page 98 - In a right angled spherical triangle, the rectangle under the radius and the sine of the middle part, is equal to the rectangle under the tangents of the adjacent parts ; or', to the rectangle under the cosines of the opposite parts.
Page 40 - Def. 10. 1.) If then CE is made radius, GE is the tangent of GCE, (Art. 84.) that is, the tangent of half the sum of the angles opposite to AB and AC. If from the greater of the two angles ACB and ABC, there be taken ACD their half sum ; the remaining angle ECB will be their half difference.
Page 36 - The side of a regular hexagon inscribed in a circle is equal to the radius of the circle.
Page 115 - The straight line joining the vertex and the centre of the base is called the axis of the cone.
Page 97 - The cotangent of half the sum of the angles at the base, Is to the tangent of half their difference...
Page 82 - If two angles of a spherical triangle are equal, the sides opposite these angles are equal and the triangle is isosceles. In the spherical triangle ABC, let the angle B equal the angle C. To prove that AC = AB. Proof. Let the A A'B'C