What me know what Im talking about just commiting a bit of original research and shouting history Anyway I did a bit of digging in  google which turned up Gregory St Vincent Grgoire de SaintVincent with  The Greeks described a spiral using an angle and a radius vector but it was St Vincent and Cavalieri who simultaneously and independently introduced them as a separate coordinate system In an article The Origin of polar coordinates J J Coolidge refers to the priority dispute between Cavalieri and St Vincent over their discovery St Vincent wrote about this new coordinate system in a letter to Grienberger in  and published the process in  On the other hand Cavalieris publication appeared in  and the corrected version in  and Milestones in the History of Thematic Cartography Development of the use of polar coordinates for the representation of functions Newtons Method of Fluxions was written about  but not published until  Jacob Bernoulli published a derivation of the idea in  p  attributes the development of polar coordinates to Fontana with no date Isaac Newton   England and Gregorio Fontana   and Jacob Bernoulli   p   is probably too early  would probably be more appropriate There are earlier references to Hipparchus BC regarding the use of polar coordinates in establishing stellar positions and Abu Arrayhan Muhammad ibn Ahmad alBiruni  regarding the use of three rectangular coordinates to establish a point in space  Use of polar coordinates in a graphfrequency of wind directions Lon Lalanne  France  La Habra High School Math History Timeline Jacob Bernoulli invents polar coordinates a method of describing the location of points in space using angles and distances Sherlock Holmes in Babylon and Other Tales of Mathematical History has a section Newton as an originator of polar coordinates Jacob Amsler httpwwwhistorymcsstandacukhistoryPrintonlyAmslerhtml Amsler invented the polar planimeter a device for measuring areas enclosed by plane curves It was based on polar coordinates whereas earlier instruments were based on cartesian coordinates In  Amsler published a paper ber das Planimeter in which he gave details of his idea As Mahoney writes in  Amslers planimeter  adapted easily to the determination of static and inertial moments and to the coefficients of Fourier series it proved especially useful to shipbuilders and railway engineers  So it seems there is considerable different accounts of their introduction and quite a few notable applications