Proposition 2 establishes that purely centripetal force governance implies equal areas swept in equal times

This is not the only place in Book 1 where Newton takes the trouble to derive an “If…quam proxime, then…quam proxime” version of an exact “If…, then…” proposition. Propositions 1 and 2 establish that a motion is governed purely by centripetal forces if and only if equal areas are swept out in equal times. The second and third corollaries of Proposition 3 then yield the conclusion that a motion is quam proxime governed purely by centripetal forces if and only if equal areas are quam proxime swept