For the Sun and Planets are separated from one another by so immense a distance, as renders them incapable of exerting most of those forces whereby a… - David Gregory

[W]e have come into the age where questions that were once cosmographical are being transformed into geometrical problems. For it has now been shown not only that the areas which bodies driven in a circuit describe by the radii drawn to the centre of forces are in immobile planes and proportional to the times, but also conversely that every body moved in this way is impelled by a centripetal force that tends towards the aforesaid point. By this proposition alone the Ptolemaic system is destroyed, for the primary planets by radii drawn to the Earth describe areas in no way proportional to the times, while with the radii drawn to the sun it is established that they run over areas sufficiently proportional to the times.

Mr Issac Newton in addition to the geometric figure in any orbit of a projectile sought also to find the measure of the (tending to a given centre) of the body borne in that orbit, from whatever cause that force may arise, be it from a deeper mechanical one or from a law imposed by the supreme creator of all things. He inquires geometrically into the law of centripetal force of a body moved in the circumference of a circle with the force tending to a given point either on the circumference or anywhere outside it or inside it, or even infinitely removed. By the same method he seeks the law of centripetal force tending to the centre of a plane nautical spiral (that is one that the radii cut in a given angle) which will drive a body in that spiral. Also the law of centripetal force that would make a body rotate in an ellipse when the centre of the ellipse coincides with the centre of forces. If the ellipse is changed into a hyperbola and the centripetal force into a centrifugal one the same things apply to the hyperbola. Also the resolution of the same problem when the centre of forces coincides with either focus of the ellipse shows that the law of centripetal force is reciprocally in the duplicate ratio of the distance [as the inverse square of the distance]; others had long before shown that this was the one and only law that would satisfy the other phenomenon observed by Kepler in the motion of the planets. These results also apply to the hyperbola and the parabola when the centre of forces is situated in a focus of the conic section.

[S]o also they were not unacquainted with the Law and Proportion which the action of Gravity observ'd according to the different Masses and Distances. For that Gravity is proportional to the Quantity of Matter in the heavy Body, Lucretius does sufficiently declare, as also that what we call light Bodies, don't ascend of their own accord, but by the action of a force underneath them, impelling them upwards, just as a piece of Wood is in Water; and further, that all Bodies, as well the heavy as the light, do descend in vacuo, with an equal celerity.