  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 out in equal times. Again, after establishing that Kepler's 3/2 power rule holds exactly for concentric uniform circular motions if and only if an exact inverse-square centripetal force holds across all the orbits, he adds the generalization, “And universally, if the periodic time is as any power Rn of the radius R, … the centripetal force will be inversely as the power R2n-1 of the radius, and  conversely.”[37] This result holds for non-integer values of n, and hence it yields the further result that the 3/2 power rule holds quam proxime for uniform circular orbits if and only if the centripetal force is quam proxime inverse-square. These propositions— which Newton has taken the trouble to show still hold in a quam proxime form — are the very ones he invokes in Book 3 to conclude that the forces retaining bodies in their orbits in our planetary system are all centripetal and inverse-square. (By contrast, as noted earlier, while the proposition, “if a Keplerian ellipse exactly, then inverse-square exactly,” is true, the proposition, “if a Keplerian ellipse quam proxime, then inverse-square quam proxime,” is not true when the eccentricity of the ellipse is not large, as explained in Smith, 2002.) A failure to notice these quam proxime forms in Book 1 blinds one to the subtlety of the approximative reasoning Newton employs in Book 3.