  Section 9 includes another often overlooked result that is pivotal to the evidential reasoning for universal gravity in Book 3. Proposition 45 applies the result on precessing orbits mentioned earlier to the special case of nearly circular orbits, that is, orbits like those of the then known planets and their satellites. This proposition establishes that such orbits, under purely centripetal forces, are stationary — that is, do not precess — if and only if the centripetal force governing them is exactly inverse-square. It does this by deriving a formula relating the exponent n in the force law to the angle θ between the point where the orbiting body is furthest from the force center to the point where it is nearest, that is, the apsidal angle: n = (180/θ)2-3. (To illustrate, if the apsidal angle is 180 degrees, as in a Keplerian ellipse, then the exponent in the force law is -2, and if the apsidal angle is 90 degrees, as in an ellipse for which the force center is in the center, the exponent is +1.) This result is striking in three ways. First, insofar as the cumulative effect of even a very small precession is detectable after several revolutions, this formula turns the rate of precession (2θ per revolution) into a sensitive measure of the exponent in the force law. Second, it yields a conditional beyond “If the orbit is stationary, then the centripetal force is inverse-square,” namely, “If the orbit is nearly stationary, then the centripetal force is nearly inverse-square.” Using Newton's preferred phrasing, quam proxime (literally, “most nearly as possible”), this latter conditional has an “If…quam proxime, then…quam proxime” form. Newton illustrates this by taking the mean precession rate of the lunar orbit, 3 degrees per revolution, to conclude that the exponent for the net centrifugal force acting on the Moon is -2 and 4/243. Third, even when an orbit does precess, once such a fractional departure of the exponent from -2 is shown to result from the perturbing effect of outside bodies, then one can still conclude that the force toward the central body is exactly -2. This is precisely the strategy Newton follows in concluding that the centripetal force on the Moon, once a correction is made for the perturbing effects of the Sun, is inverse-square.