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So, I came here from Mathematics StackExchange where I posted this question.

So, I want to know why polar coordinates came into existence. Why exactly did the mathematician who introduced them...introduce them? Was it because he encountered a problem where defining the position of a point in terms of the distance of that point from the origin and it's "direction" gave rise to easier calculations? If so, what was the problem that he/she encountered and what type of calculations are we talking about here? If not, how did the idea of polar coordinates originate?

Thank you!

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    $\begingroup$ Well, Cartesian, cylindrical, and polar coordinates certainly help solving various problems that exhibit such symmetries. $\endgroup$
    – Jon Custer
    Jan 7, 2021 at 20:31

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The following information is found HERE

According to Daniel L. Klaasen in Historical Topics for the Mathematical Classroom:

Isaac Newton was the first to think of using polar coordinates. In a treatise Method of Fluxions (written about 1671), which dealt with curves defined analytically, Newton showed ten types of coordinate systems that could be used; one of these ten was the system of polar coordinates. However, this work by Newton was not published until 1736; in 1691 Jakob Bernoulli derived and made public the concept of polar coordinates in the Acta eruditorum. The polar system used for reference a point on a line rather than two intersecting lines. The line was called the "polar axis," and the point on the line was called the "pole." The position of any point in a plane was then described first by the length of a vector from the pole to the point and second by the angle the vector made with the polar axis.

According to Smith (vol. 2, page 324), "The idea of polar coordinates seems due to Gregorio Fontana (1735-1803), and the name was used by various Italian writers of the 18th century."

"Polar co-ordinates" is found in English in 1816 in a translation of Lacroix’s Differential and Integral Calculus: "The variables in this equation are what Geometers have called polar co-ordinates" (OED2).

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    $\begingroup$ Is there somewhere one can read what the ten were? $\endgroup$
    – J.G.
    Jan 8, 2021 at 7:26
  • $\begingroup$ @J.G.: Is there somewhere one can read what the ten were? --- Googling, the third hit I got was a Library of (U.S.) Congress listing, where surprisingly it is digitized and freely available (at least where I'm at) here (171 MB, so rather long down-load time needed). There are two table of contents, one on .pdf page 28 and the other on .pdf pages 171-172, but nothing grabbed my attention, so I looked at a few possibly promising chapter beginnings (continued) $\endgroup$
    – Dave L Renfro
    Jan 9, 2021 at 15:08
  • $\begingroup$ and I didn't see anything that I recognized as "different coordinate systems". So I looked in Carl B. Boyer's History of Analytic Geometry (my copy is the 1988 The Scholar's Bookshelf edition, but probably the paging is the same as with the Dover edition and the original 1956 edition) under the index listings for both "Coordinates -- Polar coordinates" and "Newton" for matching pages, which led me to p. 142 where a couple of quotes from Newton's 1736 book are given, with a footnote saying where they're at in the 1736 book. (continued) $\endgroup$
    – Dave L Renfro
    Jan 9, 2021 at 15:13
  • $\begingroup$ In Newton's 1736 book, see Article 29 at the bottom of p. 51 (= .pdf page 79) and the second through ninth "manners" on pp. 52-59 (= .pdf pages 80-87). In trying to follow this I'm reminded a bit of why I tend not to delve into the history of mathematics much earlier than the 1870s or so --- the notation and manners of exposition (plus often in German or Latin, neither of which I can read any of) are just too foreign for me to feel I'm really understanding what's going on. I'm much more comfortable with reading Hilbert, Cantor, du Bois-Reymond, Baire, etc. than Riemann and earlier! $\endgroup$
    – Dave L Renfro
    Jan 9, 2021 at 15:26
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It depends what you consider by "introducing polar coordinates". Polar (and spherical) coordinates were used (without explicitly naming them) long before Cartesian one. Ptolemy gave spherical coordinates of celestial bodies (two angles and distance from Earth). It is the most natural way to describe position of a point relative to an observer. Of course Ptolemy did not described curves analytically in polar coordinates as Newton. So you can say that he did not "introduce" them as in contemporary textbooks. But he certainly used them.

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