Editing Stereographic projection (section)

==History==
{{see also|Astrolabe#History}}
[[Image:RubensAguilonStereographic.jpg|thumb|Illustration by [[Peter Paul Rubens|Rubens]] for "Opticorum libri sex philosophis juxta ac mathematicis utiles", by [[François d'Aguilon]]. It demonstrates the principle of a general perspective projection, of which the stereographic projection is a special case.]]

The origin of the stereographic projection is not known, but it is believed to have been discovered by [[Ancient Greek astronomy|Ancient Greek astronomers]] and used for projecting the [[celestial sphere]] to the plane so that the motions of stars and planets could be analyzed using [[Euclidean geometry|plane geometry]]. Its earliest extant description is found in [[Ptolemy]]'s [[Planisphaerium|''Planisphere'']] (2nd century AD), but it was ambiguously attributed to [[Hipparchus]] (2nd century BC) by [[Synesius]] ({{circa|400 AD}}),<ref name=dicks>[[Synesius]] wrote in a letter describing an instrument involving the stereographic projection: "Hipparchus long ago hinted at the unfolding of a spherical surface [on a plane], so as to keep a proper proportion between the given ratios in the different figures, and he was in fact the first to apply himself to this subject. I, however (if it is not presumptuous to make so great a claim), have followed it to its uttermost conclusion, and have perfected it, although for most of the intervening time the problem had been neglected; for the great Ptolemy and the divine band of his successors were content to make only such use of it as sufficed for the night-clock by means of the sixteen stars, which were the only ones that Hipparchus rearranged and entered on his instrument." Translation from {{cite book |last=Dicks |first=D.R. |year=1960 |title=The Geographical Fragments of Hipparchus |publisher=University of London, Athlone Press |postscript=none}}, fragment 63 pp. 102–103. {{pb}} Dicks concludes (commentary on fragment 63, pp. 194–207): "Whether Synesius' evidence can be accepted at its face value depends on the view taken as to the strength of the objections raised above. On the whole, it would seem that the value of his testimony has been greatly exaggerated, and its unsatisfactory nature on so many points insufficiently emphasized. At any rate, the 'instrument' he sent to Paeonius was either a modified astrolabic clock of the Vitruvian type or a simple celestial map, and not a planispheric astrolabe. Furthermore, on the evidence available we are not, in my opinion, justified in attributing to Hipparchus a knowledge of either stereographic projection or the planispheric astrolabe."</ref> and [[Apollonius]]'s ''Conics'' ({{circa|200 BC}}) contains a [[theorem]] which is crucial in proving the property that the stereographic projection maps circles to circles. Hipparchus, Apollonius, [[Archimedes]], and even [[Eudoxus of Cnidus|Eudoxus]] (4th century BC) have sometimes been speculatively credited with inventing or knowing of the stereographic projection,<ref name=neugebauer>{{cite journal |last=Neugebauer |first=Otto |author-link=Otto Neugebauer |year=1949 |title=The Early History of the Astrolabe |journal=Isis |volume=40 |number=3 |pages=240–256 |doi=10.1086/349045 |jstor=227240 }}</ref> but some experts consider these attributions unjustified.{{r|dicks}} Ptolemy refers to the use of the stereographic projection in a "horoscopic instrument", perhaps the {{ill|anaphoric clock|fr|Horloge hydraulique antique#L'horloge anaphorique|it|Orologio anaforico}} described by [[Vitruvius]] (1st century BC).<ref>{{cite journal |last1=Sleeswyk |first1=A.W. |last2=Huldén |first2=B. |year=1991 |title=The three waterclocks described by Vitruvius |journal=History and Technology |volume=8 |issue=1 |pages=25–50 |doi=10.1080/07341519108581788 }}</ref><ref>{{cite journal |last=Drachmann |first=A.G . |author-link=A. G. Drachmann |year=1953 |title=The Plane Astrolabe and the Anaphoric Clock |journal=Centaurus |volume=3 |issue=1 |pages=183–189 |doi=10.1111/j.1600-0498.1953.tb00528.x |bibcode=1953Cent....3..183D }}</ref>

By the time of [[Theon of Alexandria]] (4th century), the planisphere had been combined with a [[dioptra]] to form the [[astrolabe|planispheric astrolabe]] ("star taker"),{{r|neugebauer}} a capable portable device which could be used for measuring star positions and performing a wide variety of astronomical calculations. The astrolabe was in continuous use by [[Byzantine Empire|Byzantine]] astronomers, and was significantly further developed by [[Astronomy in the medieval Islamic world|medieval Islamic astronomers]]. It was transmitted to Western Europe during the 11th–12th century, with Arabic texts translated into Latin.

In the 16th and 17th century, the [[equator]]ial aspect of the stereographic projection was commonly used for maps of the [[Eastern Hemisphere|Eastern]] and [[Western Hemisphere]]s. It is believed that already the map created in 1507 by [[Gualterius Lud]]<ref>According to (Snyder 1993), although he acknowledges he did not personally see it</ref> was in stereographic projection, as were later the maps of [[Jean Rotz]] (1542), [[Rumold Mercator]] (1595), and many others.<ref name="Snyder1989">Snyder (1989).</ref> In star charts, even this equatorial aspect had been utilised already by the ancient astronomers like [[Ptolemy]].<ref>Brown, Lloyd Arnold : [https://books.google.com/books?id=Qh7nDfGm7BkC&dq=equator+greek+astronomy&pg=PA40 ''The story of maps'', p.59].</ref>

[[François d'Aguilon]] gave the stereographic projection its current name in his 1613 work ''Opticorum libri sex philosophis juxta ac mathematicis utiles'' (Six Books of Optics, useful for philosophers and mathematicians alike).<ref>According to (Elkins, 1988) who references Eckert, "Die Kartenwissenschaft", Berlin 1921,&nbsp;pp 121&ndash;123</ref>

In the late 16th century, [[Thomas Harriot]] proved that the stereographic projection is [[Conformal map|conformal]]; however, this proof was never published and sat among his papers in a box for more than three centuries.<ref>{{cite journal|author=Lohne, John|title= Essays on Thomas Harriot|journal=Archive for History of Exact Sciences |year=1979|volume=20|issue=3/4|pages=189–312|doi=10.1007/BF00327737|s2cid= 118095486}}</ref> In 1695, [[Edmond Halley]], motivated by his interest in [[star chart]]s, was the first to publish a proof.<ref>Timothy Feeman. 2002. "Portraits of the Earth: A Mathematician Looks at Maps". American Mathematical Society.</ref> He used the recently established tools of [[calculus]], invented by his friend [[Isaac Newton]].