Importance of Conic Section "size" in the Derivation of Propositions X-XVI in Newton's Principia Book I

The “direct problem” of the inverse-square law is examined, by retracing the derivations of Propositions X-XVI in Newton’s Principia. It is found that transfers of constants of proportionality are inconsistent between the propositions, leading to errors in the derived relations for multiple orbits. Proposition X has to be corrected for the relations of time period to the force law by including the area constants for conic sections. Propositions XI-XIII assume the expression 2h2 / L to be constant, that in turn implicitly assume a modified version of Kepler’s third law, while Propositions XIV-XVI apply this to multiple orbits leading to further “size” inconsistencies with the original form of Kepler’s third law. It has been shown how this relevance of “size” has been missed in the Newtonian literature, leading to a necessity of revising the existence of an exact inverse-square law.

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