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Page 11 - 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 2 - ... 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 2 - 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 12 - 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 6 - 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�...