Kepler's 1609 Astronomia Nova records his frustration at the oval orbit's quadrature and his explicit rejection of a near-elliptical 'puffy-cheeked' orbit (the via buccosa)

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Before settling on the ellipse, Kepler worked through oval-shaped candidate orbits in Astronomia Nova (1609) and documented both the mathematical difficulty and his own reasoning for rejecting one. In Chapter 47, quoted by Moore, Smith & Mertens, he complains about needing to geometrically square ("quadrature") an oval orbit: "we must call in a God from a tragedy—or rather, a certain reason from a machine—to teach us how to engineer the quadrature of the oval... So now once again, in the oval [ovoid] (or if you prefer, the metopoid), I appeal to geometers and implore their help."

In Chapter 58 he considers and rejects a specific near-elliptical alternative he calls the via buccosa — the "puffy-cheeked" path — on the strength of a precision argument rather than aesthetics: "It is evident, then, that the path is puffy-cheeked, and therefore not an ellipse. And since the ellipse provides the correct equations, it follows that this puffy-cheeked path correctly provides incorrect ones." The orbit was close enough to work, and Kepler threw it out anyway, because the true ellipse's equations were exactly right and this one's were not.

This is the same historical juncture Moore, Smith & Mertens' orbit family revisits from the opposite direction — constructing, centuries later, an exact non-elliptical orbit Kepler himself never found a use for.