What is "spherical trigonometry"

Spherical \Spher"ic*al\, Spheric \Spher"ic\, a. [L. sphaericus, Gr. ???: cf. F. sph['e]rique.]

1. Having the form of a sphere; like a sphere; globular; orbicular; as, a spherical body.

2. Of or pertaining to a sphere.

3. Of or pertaining to the heavenly orbs, or to the sphere or spheres in which, according to ancient astronomy and astrology, they were set.

   Knaves, thieves, and treachers by spherical predominance.
   --Shak.

   Though the stars were suns, and overburned Their spheric limitations.
   --Mrs. Browning.

   Spherical angle, Spherical co["o]rdinate, Spherical excess, etc. See under Angle, Coordinate, etc.

   Spherical geometry, that branch of geometry which treats of spherical magnitudes; the doctrine of the sphere, especially of the circles described on its surface.

   Spherical harmonic analysis. See under Harmonic, a.

   Spherical lune,portion of the surface of a sphere included between two great semicircles having a common diameter.

   Spherical opening, the magnitude of a solid angle. It is measured by the portion within the solid angle of the surface of any sphere whose center is the angular point.

   Spherical polygon,portion of the surface of a sphere bounded by the arcs of three or more great circles.

   Spherical projection, the projection of the circles of the sphere upon a plane. See Projection.

   Spherical sector. See under Sector.

   Spherical segment, the segment of a sphere. See under Segment.

   Spherical triangle,re on the surface of a sphere, bounded by the arcs of three great circles which intersect each other.

   Spherical trigonometry. See Trigonometry. [1913 Webster] -- Spher"ic*al*ly, adv. -- Spher"ic*al*ness, n.

Trigonometry \Trig`o*nom"e*try\, n.; pl. -tries. [Gr. ? a triangle + -metry: cf. F. trigonom['e]trie. See Trigon.]

1. That branch of mathematics which treats of the relations of the sides and angles of triangles, which the methods of deducing from certain given parts other required parts, and also of the general relations which exist between the trigonometrical functions of arcs or angles.

2. A treatise in this science.

   Analytical trigonometry, that branch of trigonometry which treats of the relations and properties of the trigonometrical functions.

   Plane trigonometry, and Spherical trigonometry, those branches of trigonometry in which its principles are applied to plane triangles and spherical triangles respectively.

n. the trigonometry of spherical triangles

Spherical trigonometry is the branch of spherical geometry that deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons (especially spherical triangles) defined by a number of intersecting great circles on the sphere. Spherical trigonometry is of great importance for calculations in astronomy, geodesy and navigation.

The origins of spherical trigonometry in Greek mathematics and the major developments in Islamic mathematics are discussed fully in History of trigonometry and Mathematics in medieval Islam. The subject came to fruition in Early Modern times with important developments by John Napier, Delambre and others, and attained an essentially complete form by the end of the nineteenth century with the publication of Todhunter's textbook Spherical trigonometry for the use of colleges and Schools. This book is now readily available on the web. The only significant developments since then have been the application of vector methods for the derivation of the theorems and the use of computers to carry through lengthy calculations.