The force of gravitation was discovered by Newton along the following line of reasoning: if the Copernicus' System of the World is true, a fact plainly reflected by the astronomical data, then the motion of the Earth around Sun, or of the Moon around Earth, is a circular motion. One of the main facts of experience is that in a circular motion a body experiences a centrifugal force, tending to remove the body away from the center of orbit. The stability of the orbit of Earth or of the Moon is, according to this observation, indication of the existence of a force acting upon Earth, or Moon, and directed toward the center of their orbits. This is the Newtonian force of gravitation.
Now, Kepler discovered that the astronomic observational facts, while in general confirming the Copernicus' System of the World, indicate that it must be a little bit different. Specifically, he found three general patterns, which came to be later known as the Kepler Laws of the planetary motion. One of them, the First Kepler Law, shows indeed that the orbit of the planets around the Sun are closed plane curves, but not quite circles. They are ellipses, kind of deformed circles. These geometrical figures are symmetrical with respect to two perpendicular directions in their plane, being elongated on one of these directions. Along this direction of elongation, and symmetrically with respect to its center, the ellipse has two fixed points  the foci. The Sun occupies one of these foci.
The Second Kepler Law shows that the radius of position of the planet on its orbit sweeps equal areas in equal times. This means that the motion of the planet along its orbit around the Sun is not a uniform motion, but a periodically accelerated one: when closest to the Sun the planet moves faster, then gradually slows down until it reaches the most distant point of the orbit, after which it starts accelerating again.
The Third of the Kepler Laws gives a relationship between the periods of rotation of the planets of the Solar System and the greater axes of their orbits: the ratio of the square of the periods to the cubes of their greater axes is a constant for the Solar System. This is the only one of the Kepler Laws that can be taken as a global property, so to speak, being specific to this system as a whole.
Following Newton, if we take the Third Law of Kepler and combine it with the expression of the centrifugal force acting on a planet, which force tends to remove the planet from the Sun, we end up with a force whose magnitude is inversely proportional with the square of the distance between Sun and planet. Such a force must exist in order to balance the centrifugal force and keep the planet on its orbit around the Sun. This is the Newtonian force of gravitation.
Inferring the existence of such force can therefore be relegated to the general idea that the centrifugal force is a consequence of the motion of the planets, combined with the idea of equilibrium of the forces acting upon planets. Now, if the Newtonian force exists, it must be somehow responsible for the shape of the Solar System. And Newton showed how this happens. First, if we assume the gravitational force and the Second Principle of the Dynamics, we find that the motion of the planet acted upon by the force of gravitation is indeed done in a plane. Therefore, the motion of planets cannot be but a plane motion as stipulated by Copernicus. Secondly, this motion is always respecting the second of Kepler Laws. More specifically, the rate of the area swept by the radius of orbit is constant. Thirdly, the orbit of this motion is an ellipse, with the attracting center in one of its foci.
Here it is apparent that something is not in order. We infer the existence of the force of gravitation by assuming that it is always directed towards the center of the orbit. Nothing strange up to this point. Once we want to see if the consequences of existence of such a force fit the astronomical facts, we proceed by assuming the force. It acts upon the planet attracting it towards the Sun. When calculating the orbit, we find however that its center does not coincide with the point of attraction, as assumed when we inferred the existence of the force.
Let's repeat: we assume the force acting towards the center of the orbit and deduce this orbit by calculations; it is in accordance with the observed facts, but this doesn't satisfy our starting point: the center of this calculated orbit is not the point toward which the gravitation force pulls! This is, indeed, a problem! And it was a problem from the very beginning!
Newton treated it practically: the mathematics involved in Classical Dynamics takes the elements in their absolute determinations. There is not really such a thing like point or straight line, etc. Fact is that the distance between the center of an orbit in the Solar System and the attracting center is so small that it can be usually neglected. For practical evaluations this is enough. Yet, a mathematical formalism should reflect the absolute: the practical things, like neglecting some quantities, must be left to our convenience, which is not the case here!
That's why this little incident marked the whole Newtonian System over time. For once, this was a good thing: it promoted the progress of Science. However, inasmuch as it is still a subject of debate today, let us remind that it was there from old times, as a general attribute of our speculative thinking. It came with the idea of atoms, and still seems to point out in that direction.
The great Greek philosopher Epicurus, imagined the atoms as falling under the action of gravity. However, in order to account for grouping of the atoms into matter formations, he accepted that there is always an unpredictable and undetectable swerving from the straight line of motion of atoms, a declination of the direction of motion. Marx, in more recent times, in his doctoral dissertation, was the one who attached this speculation to the Hegelian philosophy, and he seems to be right.
Indeed, the old Epicurean idea of motion of the atoms is of the same degree of abstraction as the Newtonian idea of action of gravitation. However, the Newtonian idea, sustained by the modern mathematics, proved by calculations what the old Epicurus just speculated: the center of gravity is never in the point where it is supposed to be! Instead of looking for its correct placement, it seems that we have to accept this indetermination of the center of attraction as a law. In other words, we must be more confident in our mathematics which, as the great physicist Eugene Wigner once said, proved time and again to be "unreasonably effective" for the natural sciences.
Nicolae Mazilu, PhD Physics and Engineering, occasionally (very often lately!) writer regarding unsettled problems. Usually perceiving these in places where everything seems to be settled: Classical Mechanics, Cosmology, Thermodynamics, Heat Transfer, Continuum Mechanics, Astrology, Relativity.
