[W]hat sharpness of mind was employed by John Kepler... when, from there being just five regular solids... he inferred that the number of the planets… - David Gregory

But, since the law of centripetal force employed by nature is to be discovered from its symptoms, the indisputably elliptical orbit and the sesquialteral ratio of the periodic times and the distances from the centre of forces, the same great Newton solved not only the universal problem of determining the trajectory and the motion in it for any given centripetal force, but also its converse. After this universal problem had been solved the sequel was to find other [quantities] in the geometric figure that are measures of physical qualities; for example, that the periodic times in ellipses are in the sesquiplicate ratio of the transverse axes [the squares of the times are as the cubes of the axes], and as many other things similar to these as possible. Also, for instance, to compare this force, which we experience in the planets, with another given force near to us, namely gravity. But also the new philosophy was to concern itself with movable elliptical orbits, in which the line of apsides either advances or retires. Also, for instance, a more exact [theory] of rectilinear descent and of the motion of pendulous bodies than the Huygenian one, since that supposes the centre to be infinitely removed. Therefore also, other s different from the common one and variously devised according as the pendulum oscillates inside or outside the surface of the Earth. And let that suffice for this problem. But also on account of the mutual actions of bodies moving around a centre the orbits usually turn out to be deformed, and also an investigation of these actions and of the deformity arising from them, whence arise many minor inequalities of the planets, such as the motion of the nodes, the variation of maximum latitude, and other things in the moon.

[T]he Opinion of the Ancients concerning Gravity... they were perswaded that Gravity was not an affection of Terrestrial Bodies only, but of the Celestial also, that all Bodies gravitate towards one another; and that the Planets are retained in their Orbits by the force of Gravity, and lastly, that the Gravity of the Planets towards the Sun are reciprocally as the Squares of their Distances from it. What the industry and skill of the Moderns have added to these inventions of the Ancients, the following Pages do declare at large.

Pythagoras... applied the proportion he had thus found by experiments, to the Heavens, and from thence learn'd the Harmony of the Spheres. And, by comparing these Weights with the Weights of the Planets, and the intervals of the Tones, produced by the Weights, with the interval of the Spheres; and lastly, the lengths of Strings with the Distances of the Planets from the Center of the Orbs; he understood, as it were by the Harmony of the Heavens, that the Gravity of the Planets towards the Sun (according to whose measures the Planets move) were reciprocally as the Squares of their Distances from the Sun.