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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.

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  • $\begingroup$ Certainly not. Kepler was familiar with ellipses from his classical education. In fact he spent many years trying to fit planetary orbits with "ovals", and only after failing at that did he try ellipses. $\endgroup$ Apr 27 at 0:24
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No. Conic sections were studied in Greece since the IVth century BC. And Apollonius of Perga (c. 240 BC – c. 190 BC), who coined the term ellipse, wrote a treatise on conic sections (Conics) in eight books.

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