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according already analysis apply arrangement assigned assuming base becomes branches Chap circle closed figures co-ordinates common comparing condition considered contain corresponding cosine curve deduced denoted determine direction distance divergent lines drawn elements employed equal equation examine example exist expressed follow four geometry give given greater idea inclination included inquiry intersection known latter less measured meet merely method nature necessary number of points object observe obtained origin parallel parameters pass perpendicular plane position preceding present principles problem properties proposition pyramid quantity question ratios reduced referred regard relations remaining remark represent respect result right angles rule Sect sides similar simple sine solid angle space sphere straight line substituting suppose surfaces taken tangent theory tion triangle unit values whence whilst
Page 415 - Therefore all the interior angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Page 395 - The sum of any two sides of a spherical triangle is greater than the third side, and their difference is less than the third side. DEM.— Let ABC be any spherical triangle; then l3 BO' < BA + AC, and BC - AC < BA ; and the same is true of the sides in any order.
Page 129 - In every triangle the sum of the three angles is equal to two right angles.
Page 290 - A . sin b = sin a . sin B sin A . sin c — sin a . sin C sin B . sin c = sin b . sin C...
Page xxi - ... set of prime numbers cannot be finite — since the product of any set of finite numbers plus one gives a new prime number — is as aesthetically neat in our times as it was in Euclid's. But a problem takes on extra luster if, in addition to its logical elegance, it provides useful knowledge. That the shortest distance between two points on a sphere is the arc of a great circle is an agreeable curiosity ; that ships on earth actually follow such paths enhances its interest.
Page 310 - In practice however, there will generally be some circumstances which will determine whether the angle required is acute or obtuse. If the side opposite the given angle be longer than the other given side...
Page 123 - ... are identical with angles of the triangle, and the third, b, which forms a space indefinitely extended, differs from the opening we call the angle C merely by the small space included in the triangle. "This last, by bringing the triangle nearer to C, may be rendered as small as we please ; and thus a triangle can always be assigned whose angles shall differ from a...
Page 330 - A — cos B cos C — sin B sin C cos a ; and changing the signs of the terms, we obtain, cos A = sin B sin C cos a — cos B cos C.