Around 1670, following Renee DesCartes' mid-century introduction of the Cartesian Coordinate system, Isaac Newton is supposed to have identified ten different coordinate systems. I am seeking to identify these ten coordinate systems and whether those were all those known to mathematicians of that era, or what others were known.
In Carl Boyer's "A History of Analytic Geometry" (QA551 .B684), Chapter VII , page 142 reads "The contributions of Newton to coordinate geometry are not well known, and one aspect in particular has been completely overlooked by historians - his use, in the Geometria Analytica, of polar coordinates. Having shown how to illustrate the method of fluxions... in Cartesian coordinates - oblique as well as rectagular ..." (some excerpts from Newton) "...to illustrate this point, he suggested eight further types of coordinate systems. One of these, the 'Third Manner' of determining a curve analytically, is what would now be called bipolar coordinates." So what I see in this history on page 142 are Cartesian, polar, and bipolar coordinates. On page 146, the Geometria analytica or Method of Fluxions is told to have not finally been published until 1736. The history reference can be viewed at this link, with page 142 appearing on the electronic page 156: https://babel.hathitrust.org/cgi/pt?id=mdp.39015015621322&seq=156&q1=newton
I found The Method of Fluxions available for purchase online in a Kindle format, with 666 pages. {N.B. - fluxions were introduced by Isaac Newton to describe his form of a derivative with respect to time.} I was able to use the search function in Kindle to find "First Manner" on page 113, "Second Manner" on page 122, "Third Manner" on page 126, and then the shorter fourth through eighth are contained in pages 128 to 130, where the last two are labeled "Seventh Manner: for Spirals" and "Eighth Manner: for Quadrices." The seventh manner seems to refer to polar coordinates as was mentioned in the history book excerpts and as also stated on Wikipedia Polar Coordinates with the reference Boyer, C. B. (1949). "Newton as an Originator of Polar Coordinates". American Mathematical Monthly. Mathematical Association of America. 56 (2): 73–78. doi:10.2307/2306162. JSTOR 2306162
A number of coordinate systems appear on Wikipedia including bipolar coordinates.
What I am trying to sort out, is which coordinate systems were known to mathematicians in the mid-1600's to mid-1700's? Do the ten manners of finding tangent curves in the Method of Fluxions by Newton describe meaningful coordinate systems (given that several resources refer to these methods as coordinate systems) and encompass all of those that were known at that time?