Editing Polar coordinate system (section)

==History==
{{See also|History of trigonometry}}
[[File:Head of Hipparchus (cropped).jpg |thumb|upright=0.8|Hipparchus]]
The concepts of angle and radius were already used by ancient peoples of the first millennium [[Before Christ|BC]]. The [[Greek astronomy|Greek astronomer]] and [[astrologer]] [[Hipparchus]] (190–120 BC) created a table of [[chord (geometry)|chord]] functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions.<ref>{{Cite web |last=Friendly |first=Michael |date=August 24, 2009 |title=Milestones in the History of Thematic Cartography, Statistical Graphics, and Data Visualization |url=http://www.math.yorku.ca/SCS/Gallery/milestone/milestone.pdf |url-status=dead |archive-url=https://web.archive.org/web/20180926124138/http://www.math.yorku.ca/SCS/Gallery/milestone/milestone.pdf |archive-date=September 26, 2018 |access-date=July 23, 2016}}</ref> In ''[[On Spirals]]'', [[Archimedes]] describes the [[Archimedean spiral]], a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system.

From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to [[Mecca]] ([[qibla]])—and its distance—from any location on the Earth.<ref>{{Cite book |last=King |first=David A. |title=Mathematics and the Divine: A Historical Study |publisher=Elsevier |year=2005 |isbn=0-444-50328-5 |editor-last=Koetsier |editor-first=Teun |location=Amsterdam |pages=162–78 |chapter=The Sacred Geography of Islam |editor-last2=Luc |editor-first2=Bergmans |chapter-url=https://books.google.com/books?id=AMOQZfrZq-EC&pg=PA161}}</ref> From the 9th century onward they were using [[spherical trigonometry]] and [[map projection]] methods to determine these quantities accurately. The calculation is essentially the conversion of the [[Geodetic coordinates#Coordinates|equatorial polar coordinates]] of Mecca (i.e. its [[longitude]] and [[latitude]]) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the [[great circle]] through the given location and the Earth's poles and whose polar axis is the line through the location and its [[antipodal point]].<ref>King ([[#CITEREFKing2005|2005]], [https://books.google.com/books?id=AMOQZfrZq-EC&pg=PA169 p. 169]). The calculations were as accurate as could be achieved under the limitations imposed by their assumption that the Earth was a perfect sphere.</ref>

There are various accounts of the introduction of polar coordinates as part of a formal coordinate system. The full history of the subject is described in [[Harvard University|Harvard]] professor [[Julian Lowell Coolidge]]'s ''Origin of Polar Coordinates.''<ref name="coolidge">{{Cite journal |last=Coolidge |first=Julian |author-link=Julian Lowell Coolidge |year=1952 |title=The Origin of Polar Coordinates |url=http://www-history.mcs.st-and.ac.uk/Extras/Coolidge_Polars.html |journal=American Mathematical Monthly |publisher=Mathematical Association of America |volume=59 |issue=2 |pages=78–85 |doi=10.2307/2307104 |jstor=2307104}}</ref> Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653. Cavalieri first used polar coordinates to solve a problem relating to the area within an [[Archimedean spiral]]. [[Blaise Pascal]] subsequently used polar coordinates to calculate the length of [[parabola|parabolic arcs]].

In ''[[Method of Fluxions]]'' (written 1671, published 1736), Sir [[Isaac Newton]] examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems.<ref>{{Cite journal |last=Boyer |first=C. B. |year=1949 |title=Newton as an Originator of Polar Coordinates |journal=American Mathematical Monthly |publisher=Mathematical Association of America |volume=56 |issue=2 |pages=73–78 |doi=10.2307/2306162 |jstor=2306162}}</ref> In the journal ''[[Acta Eruditorum]]'' (1691), [[Jacob Bernoulli]] used a system with a point on a line, called the ''pole'' and ''polar axis'' respectively. Coordinates were specified by the distance from the pole and the angle from the ''polar axis''. Bernoulli's work extended to finding the [[Radius of curvature (mathematics)|radius of curvature]] of curves expressed in these coordinates.

The actual term ''polar coordinates'' has been attributed to [[Gregorio Fontana]] and was used by 18th-century Italian writers. The term appeared in [[English language|English]] in [[George Peacock (mathematician)|George Peacock]]'s 1816 translation of [[Sylvestre François Lacroix|Lacroix]]'s ''Differential and Integral Calculus''.<ref>{{Cite web |last=Miller |first=Jeff |title=Earliest Known Uses of Some of the Words of Mathematics |url=http://jeff560.tripod.com/p.html |access-date=2006-09-10}}</ref><ref>{{Cite book |last=Smith |first=David Eugene |title=History of Mathematics, Vol II |publisher=Ginn and Co. |year=1925 |location=Boston |page=324}}</ref> [[Alexis Clairaut]] was the first to think of polar coordinates in three dimensions, and [[Leonhard Euler]] was the first to actually develop them.<ref name="coolidge" />