A Treatise on Plane and Spherical Trigonometry: Including the Construction of the Auxiliary Tables; a Concise Tract on the Conic Sections, and the Principles of Spherical Projection |
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Page 31
... drawn from one end of the arc to the other . Thus , BE is the chord of the arc BAE , or BDE . 3. The sine of an arc is a straight line drawn from one ex- P tremity of the arc , at right angles to the diameter , which passes through the ...
... drawn from one end of the arc to the other . Thus , BE is the chord of the arc BAE , or BDE . 3. The sine of an arc is a straight line drawn from one ex- P tremity of the arc , at right angles to the diameter , which passes through the ...
Page 32
... drawn from the centre through the other extremity . Thus AG , which touches the circle at A , is the tangent of AB , 5. The secant of an arc is the right line intercepted be- tween the centre of the circle and the extremity of the ...
... drawn from the centre through the other extremity . Thus AG , which touches the circle at A , is the tangent of AB , 5. The secant of an arc is the right line intercepted be- tween the centre of the circle and the extremity of the ...
Page 34
... negative ; the former being drawn in a direction opposite to AG , and the latter not produced from C through L , the extremity of the arc , but in the oppo- site direction . If we take the arc more than 34 PLANE TRIGONOMETRY .
... negative ; the former being drawn in a direction opposite to AG , and the latter not produced from C through L , the extremity of the arc , but in the oppo- site direction . If we take the arc more than 34 PLANE TRIGONOMETRY .
Page 38
... draw the sine EH and tangent DF . A H D B Then the triangles ABC , ADF and AHE , being similar , As AC BC :: AE : HE ; As AC AB :: AE : AH ; BC : AD : DF ; and that is , and AB As AC : BC :: radius : the sine of A ; As AC AB :: radius ...
... draw the sine EH and tangent DF . A H D B Then the triangles ABC , ADF and AHE , being similar , As AC BC :: AE : HE ; As AC AB :: AE : AH ; BC : AD : DF ; and that is , and AB As AC : BC :: radius : the sine of A ; As AC AB :: radius ...
Page 39
... draw EG parallel to BC , meeting DC produced in G. Then it is evident that DB is the sum , and BE the difference , of AC and AB . The outward angle CAD of the triangle ABC , is equal to the two inward and opposite angles , ABC and ACB ...
... draw EG parallel to BC , meeting DC produced in G. Then it is evident that DB is the sum , and BE the difference , of AC and AB . The outward angle CAD of the triangle ABC , is equal to the two inward and opposite angles , ABC and ACB ...
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Common terms and phrases
ABDP angled spherical triangle base bisect c.cos c.sin centre circle Art common section Comp AC cone conical surface consequently construction cosec cosine cotan directrix distance drawn ECē ecliptic EDē ellipse equal equation given angle greater axis Hence hyperbola hypothenuse join latus rectum less circle Let ABC line of measures logarithms meet opposite ordinate original circle parabola parallel perpendicular plane of projection primitive circle projected circle projected pole projecting point Q. E. D. ART Q. E. D. Cor quadrant radius right angled spherical right ascension right line secant semicircle semitangent sides similar triangles sine sphere spherical angle tangent tangent of half touches the circle triangle ABC vertex vertical angle whence wherefore
Popular passages
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
Page 82 - If two triangles have two angles of the one respectively equal to two angles of the other, the third angles are equal.