Editing Spherical trigonometry (section)

{{Short description|Geometry of figures on the surface of a sphere}}
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{{Use British English|date=March 2018}}
[[File:Triangle trirectangle.png|thumb|The [[octant of a sphere]] is a spherical triangle with three right angles.]]

'''Spherical trigonometry''' is the branch of [[spherical geometry]] that deals with the metrical relationships between the [[edge (geometry)|sides]] and [[angle]]s of '''spherical triangles''', traditionally expressed using [[trigonometric function]]s. On the [[sphere]], [[geodesics]] are [[great circle]]s. 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]], [[Jean Baptiste Joseph Delambre|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''.<ref name=todhunter>{{cite book
|last = Todhunter
|first = I.
|author-link = Isaac Todhunter
|title = Spherical Trigonometry
|year = 1886
|publisher = MacMillan
|edition = 5th
|url = http://www.gutenberg.org/ebooks/19770
|access-date = 2013-07-28
|archive-date = 2020-04-14
|archive-url = https://web.archive.org/web/20200414233849/http://www.gutenberg.org/ebooks/19770
|url-status = live
}}</ref>
Since then, significant developments have been the application of vector methods, quaternion methods, and the use of numerical methods.
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[[Tycho Brahe]] remarks<ref>http://renæssancesprog.dk/tekstbase/Tycho_Brahe_De_nova_stella_1573/9/view?query_id=None {{Bare URL inline|date=October 2021}}</ref> that the nature of understanding spherical triangles is so divine and elevated that it is not appropriate to extend its mysteries to everyone. (''Diuinior et excellentior sit Triangulorum sphæricorum cognitio, quam fas sit eius mysteria omnibus propalare.'')
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