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 14
... respectively . Denote the new value of y by y ' ; so that wherefore y ' = x + h + z + k ; y ' — y = h + k ; or the increment of y Now , this being true , whether the increments are taken in their vanishing state or any other , we ...
... respectively . Denote the new value of y by y ' ; so that wherefore y ' = x + h + z + k ; y ' — y = h + k ; or the increment of y Now , this being true , whether the increments are taken in their vanishing state or any other , we ...
Page 22
... respectively of the corresponding terms , q- - n - 1 = 0 , p - 1 = n , q— 1 = p , r − 1 = q , & c . na △ ―m = 0 , paB + nA = 0 , qaC + pB = 0 , raD + qC = 0 , & c . n = 1 , p = 2 , q Hence , = 3 , & c .; m m m m and A = = , B = C = D ...
... respectively of the corresponding terms , q- - n - 1 = 0 , p - 1 = n , q— 1 = p , r − 1 = q , & c . na △ ―m = 0 , paB + nA = 0 , qaC + pB = 0 , raD + qC = 0 , & c . n = 1 , p = 2 , q Hence , = 3 , & c .; m m m m and A = = , B = C = D ...
Page 23
... respectively , we have x2 y = 1 + x + x4 + & c . x3 + + 2 2.3 2.3.4 ART . 17. From the formulæ contained in the last article , the logarithms of small numbers are readily computed ; and those of large ones are easily deduced from the ...
... respectively , we have x2 y = 1 + x + x4 + & c . x3 + + 2 2.3 2.3.4 ART . 17. From the formulæ contained in the last article , the logarithms of small numbers are readily computed ; and those of large ones are easily deduced from the ...
Page 33
... respectively at right angles to CA and CH ( 18.3 ) ; hence CAG and KHC are right angled triangles . Now the lines AG and CH , being at right angles to AC , are pa- rallel to each other ( 28.1 ) ; consequently , the alternate angles HCK ...
... respectively at right angles to CA and CH ( 18.3 ) ; hence CAG and KHC are right angled triangles . Now the lines AG and CH , being at right angles to AC , are pa- rallel to each other ( 28.1 ) ; consequently , the alternate angles HCK ...
Page 36
... respectively , are also the sines of the angle at A. The lines BL , FO , KP , which are the tangents of the same arcs , are likewise the tangents of the angle at A. ART . 26. It appears from cor . to 15.4 , that the side of a regular ...
... respectively , are also the sines of the angle at A. The lines BL , FO , KP , which are the tangents of the same arcs , are likewise the tangents of the angle at A. ART . 26. It appears from cor . to 15.4 , that the side of a regular ...
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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.