Original form of Kepler's Third Law and its misapplication in Propositions XXXII-XXXVII in Newton's Principia (Book I)

The original form of Kepler's Third Law contains a caveat regarding the requirement of small eccentricities – a fact that has not been incorporated by the traditional Newtonian derivations. This constraint is analyzed, and a re-clarification of the real meaning of “mean distance” in the law is provided, by following up the indications given by Kepler in the Harmonices Mundi. It is shown that the modified expression for the “mean distance” not only clears up conceptual difficulties, but also removes a discrepancy found by Kepler for Mercury. Based on this re-evaluation, the result of ignoring the small-eccentricity constraint is analyzed in the Propositions XXXII-XXXVII in the Principia. It is seen that there are several conceptual and mathematical mistakes that are inevitable with the Newtonian form of Kepler's Third Law.