Elements of Plane and Spherical Trigonometry: With Its Applications to the Principles of Navigation and Nautical Astronomy. With the Logarithmic and Trigonometrical Tables

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John Souter, 1833 - Trigonometry - 264 pages
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Page viii - 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.
Page 22 - In every plane triangle, the sum of two sides is to their difference as the tangent of half the sum of the angles opposite those sides is to the tangent of half their difference.
Page 160 - If the zenith distance and declination be of the same name, that is, both north or both south, their sum will be the latitude ; but, if of different names, their difference will be the latitude, of the same name as the greater.
Page 165 - PS' ; the coaltitudes zs, zs', and the hour angle SPS', which measures the interval between the observations ; and the quantity sought is the colatitude ZP. Now, in the triangle PSS , we have given two sides and the included angle to find the third side ss', and one of the remaining angles, say the angle PSS'. In the triangle zss...
Page vi - MICHAEL O'SHANNESSY, AM 1 vol. 8vo. " The volume before us forms the third of an analytical course, which commences with the * Elements of Analytical Geometry.' More elegant t&xtbooks do not exist in the English language, and we trust they will speedily be adopted in our Mathematical Seminaries. The existence of such auxiliaries will, of itself, we hope, prove an inducement to the cultivation of Analytical Science ; for, to the want of such...
Page 69 - The sum of the three sides of a spherical triangle is less than the circumference of a great circle. Let ABC be any spherical triangle ; produce the sides AB, AC, till they meet again in D. The arcs ABD, ACD, will be semicircumferenc.es, since (Prop.
Page vi - An ELEMENTARY TREATISE ON ALGEBRA, Theoretical and Practical ; with attempts to simplify some of the more difficult parts of the science, particularly the demonstration of the Binomial Theorem, in its most general form ; the Solution of Equations of the higher orders ; the Summation of Infinite Series, &c.
Page 164 - It should also be observed here, that in the preceding examples the celestial object is supposed to be on the meridian above the pole ; that is, to be higher than the elevated pole. But, if a meridian altitude be taken below the pole, which may be done if the object is circumpolar, or so near to the elevated pole as to perform its apparent daily revolution about it without passing below the horizon, then the latitude of the place will be equal to the sum of the true altitude, and the codeclination...
Page 71 - ... that the two angles A and D lie on the same side of BC, the two B and E on the same side of AC, and the two C and F on the same side of AB.
Page 135 - ... the surface of the celestial sphere. The Zenith of an observer is that pole of his horizon which is exactly above his head. Vertical Circles are great circles passing through the zenith of an observer, and perpendicular to his horizon.

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