Elements of Spherical Trigonometry1833 - Spherical trigonometry - 32 pages |
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A+B+C Acos arcs B₁ centre CHAPTER chordal triangle circles cos.a cos.b cos.c cosines ctan deduce determine dihedral angles equation equiangular equiangular polygon FRENCH LANGUAGE geometrical given gives greater Hence hypothenuse less than 90 less than unity limit logarithms Lune to angle number of edges number of faces number of sides O₁ oblique-angled triangles opposite angles pentagons perpendicular plane angles plane triangle PO.OQ positive preceding formulæ price 38 quadrilaterals R₁ radius rectilinear angles rectilinear triangle regular solids remaining angles right angles right-angled triangles sides and angles sin.b sin.c sine solid angle spherical excess spherical polygons SPHERICAL TRIGONOMETRY ß² Substitute subtended supplemental triangle three angles three planes triangle A B C vers whence whole number α β αβ
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Page 13 - C denote angles , a, b, c, opposite sides, _~*V^r_— «• and 2 sin A sin B sin C sin a sin 6 sin c...
Page 25 - FOR PLANE AND SPHERICAL TRIANGLES. (47.) IF a rectilinear triangle be drawn through three points A, B, C, fixed in space, and if from the point where the perpendiculars bisecting the sides meet one another, a straight line be drawn perpendicular to the plane of the triangle, it may easily be proved that any point in the lastmentioned line is the centre of a sphere which passes through the three given points. Let us call this line the axis of the triangle.
Page 4 - A. (7.) A spherical triangle is made only by those circles which pass through the centre of the sphere, or by great circles. All triangles made by other circles are not considered. (8.) When we talk of the side of a spherical triangle, we mean, therefore, the angle which that side subtends, at the centre of the sphere; and, as it is an angle which we are speaking of in reality, we are liable to the apparent confusion of calling a line an angle.
Page 13 - IT — 6, and ir — c. (G. vi. 7.) These two triangles are called polar or supplemental, and the geometrical discussion of them is contained in the reference just made. (32.) The formula: given are amply sufficient for the direct solution of triangles : we now proceed to give an example of all the functions of sides and angles used, with the logarithms of their sines, cosines and tangents. The table, as well as the idea, is taken from Dclambre's Astronomy.
Page 29 - ... a number of polygons, the whole number of sides being that of the edges of the solid, and so on. If a solid be inscribed inside the sphere as in (57), we shall have a second solid inscribed in a sphere, having the same number of faces, edges, and solid angles as the first. (58.) Hence any relation which is found to exist between the number...
Page 29 - By proceeding in this way for the other angles, we find the following very remarkable theorem. If a spherical triangle of very small curvature be flattened without altering the length of its sides, its angles are diminished by quantities which are very nearly equal to one another, and to the third of the spherical excess in the spherical triangle. CHAPTER Vlf.
Page 5 - These formulae will be readily seen from the definitions (Tr. 13.), and from what has just been stated, to be identical relations existing among the ratios of the sides of the pyramid PQOR. Each of the first three has another like it, similarly deduced from the other side : we shall range them as follows; in which a side and ils opposite angle are in the same type, whether Roman or Italic, in the same formulae.
Page 21 - Let a, 6, and A, be the data in case 5. Then in all cases sin. 6 sin. A must be < or = sin. a. And when b < 90° A > 90° «... must be . . . > b b > 90°...
Page 26 - ... formula which remains is true of the. chordal triangle, when its sides are substituted for arcs of the spherical triangle, and its angles for angles of the spherical triangle. (52.) To obtain what is called the remaining formula in the preceding theorem, it will be necessary to substitute for the sines, cosines, and tangents, their developments derived from tlie following theorems*, in which 0 is an angle measured by the ratio of its arc to the radius.