Were Kepler's Laws of Planetary Motion the first formal definition of an ellipse? It seems to me that Kepler's Laws necessitate some definition of an ellipse in terms of a coordinate system. I am wondering whether Kepler's Laws mathematically defined what an ellipse is, or if he used an already defined shape in his laws.
The reason I got curious about this in the first place was that I realized that Descartes may not have invented the Cartesian coordinate system by the time that Kepler discovered these laws and I have only seen proofs of Kepler's Laws in a Cartesian system.
 A: Due to the geometric formulation, Kepler's Astronomia Nova is full of elaborate illustrations of ellipses, epicircles and whatnot.  It is not necessary to spell out the position of points in Cartesian coordinates, because the relation between points are specified in angles, lengths, and geometric constructions.
Kepler presents the use of ellipses for planetary motion in Astronomia Nova (1609, originally in Latin; translated by William H. Donahue, 1992, Cambridge University Press), when René Descartes was thirteen years old.  Paging through introductory words by Kepler himself, translators, and commenters (Max Caspar, Kepler, 1993, Dover Publications) is highly recommended because, oh, do they throw shade.
The definition of an ellipse as a conic section, as known to the Greeks, already is a formal definition.  Kepler actually cites the Greek philosophers for many propositions.  For an in depth historical view, see chapter 2.1 of The Ellipse: A Historical and Mathematical Journey by Arthur Mazer (2010, Wiley).
