  The distinct possibility of the ellipse being only an approximation to the true trajectory explains the appropriateness of the question Hooke put to Newton in 1679 and Halley put to him again in 1684 — what trajectory does a body describe when moving under an inverse-square force directed toward a central body? The inverse-square part of this question came from combining the mathematical theory of uniform circular motion, which Huygens had published in his Horologium Oscillatorium of 1673, with Kepler's 3/2 power rule: the force in a string retaining a body in a uniform circular orbit varies directly as the radius of the circle and inversely as the square of the period; but the squares of the periods of the planets vary as the cubes of their mean distances; and hence, at least to a first approximation, the forces retaining the planets in their orbits vary inversely with the square of the radii of their nearly circular orbits. But now allow the distance of the orbiting body from the center to vary rather than remaining constant, as in a circle. What trajectory would result if the force toward the center varies as the inverse-square of the distance from the center toward which the force is always directed? The answer in the nine page tract “De Motu Corporum in Gyrum” that Newton sent to Halley in November 1684 is, an ellipse, provided the velocity is not too high (and if it is, then instead a parabola or a hyperbola, depending on the velocity). The key step in developing this answer is a generalization of uniform circular motion to the case of motion under a “centripetal” force — a term Newton coined from Huygens's “centrifugal” force, by which he meant the tension in the string keeping the body in a circle; and a key to this step was the discovery that a body moving under any form of centripetal force always sweeps out equal areas in equal times with respect to that center, so that the appropriate geometrical representation of time for generalizing uniform circular motion is area swept out rather than angle or arc length. The tract also confirms that Kepler's 3/2 power rule continues to hold for bodies orbiting in confocal ellipses governed by inverse-square centripetal forces.