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 38
... describe the arc DE , measur- ing the angle A ; through E and D , 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 ...
... describe the arc DE , measur- ing the angle A ; through E and D , 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 ...
Page 39
... describe the circle DCEF ; meeting AB , produced in D and E ; and CB , produced in F ; join AF , DC ; and through E 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 ...
... describe the circle DCEF ; meeting AB , produced in D and E ; and CB , produced in F ; join AF , DC ; and through E 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 ...
Page 64
... describe the equilateral triangle ABD ; and about that triangle describe a circle ; join DC , cutting the circle in E ; then is E the point required . Join AE , BE ; then , since the angles of the triangle ABD are all equal ( cor . to ...
... describe the equilateral triangle ABD ; and about that triangle describe a circle ; join DC , cutting the circle in E ; then is E the point required . Join AE , BE ; then , since the angles of the triangle ABD are all equal ( cor . to ...
Page 66
... describe a circle passing through A , and cutting FEG in G ; join GD , and produce it to cut the circle in C ; join CA , CB ; and ACB will be the triangle required . = Since AE BE , and the angles at E are right ones , the line AF BF ...
... describe a circle passing through A , and cutting FEG in G ; join GD , and produce it to cut the circle in C ; join CA , CB ; and ACB will be the triangle required . = Since AE BE , and the angles at E are right ones , the line AF BF ...
Page 67
... describe the segment of a circle containing an angle equal 90 ° + } the vertical angle ( 33.3 ) ; place in this circle the right line AD = the difference of the sides : Pro- duce AD ; join DB ; and make the angle SECTION II . 67.
... describe the segment of a circle containing an angle equal 90 ° + } the vertical angle ( 33.3 ) ; place in this circle the right line AD = the difference of the sides : Pro- duce AD ; join DB ; and make the angle SECTION II . 67.
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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 EC² ecliptic ED² 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.