An ellipse makes the algebra heavy. Restrict it to a circle. The shape of the answer will not change when we put the ellipse back.
To keep going round a circle takes an acceleration toward the centre, because something that would travel straight has to be bent continuously. Its size is .
The speed follows from one lap divided by the period: .
Those two are the whole toolkit. The rest is putting in the third law of Kepler.
Eliminate . Substituting into gives .
Now the third law. For a circle the semi-major axis is just the radius, so , with the same for every planet.
Substitute: . An squared has arrived in the denominator.
So the acceleration a planet undergoes falls off as the square of the distance. Granting the third law of Kepler is enough to produce that form.
If the Sun pulls the planet, the planet pulls the Sun. Action and reaction are equal in size and opposite in direction.
That settles the form of the force. Were it proportional to the mass of the planet alone, swapping the Sun and the planet would change its size. To survive the swap it has to go as the product of the two masses.
Together with the inverse square in the distance, this gives .
is the gravitational constant. Cavendish measured it in by weighing the attraction between lead spheres on a torsion balance. It is , and it remains the least precisely known of the physical constants.
A formula has to be checked. What Newton checked it against was the Moon.
The Moon is km from the centre of the Earth. The radius of the Earth is km, so that is times the distance of the ground.
With an inverse square, the acceleration there should be smaller by . Divide the at the surface by and you get .
Now compute the acceleration the Moon needs to hold its circle, from its period and radius. The answer is . A falling apple and a circling Moon, accounted for by a single formula.
So far we went from the three laws to the form of the force. It runs the other way too.
Start from an inverse square force and the paths can only be conic sections: the first law. The force always points at the centre, so the areal speed does not change: the second law. Compute the period and out comes a proportionality to the cube of the semi-major axis: the third law.
The three were not separate rules. They were separate faces of one formula.
That the power is exactly also matters. Depart from it even slightly and orbits stop closing, with the perihelion slowly turning. That such a departure really is found in the solar system is the subject of a much later article.