newtonian mechanics - How did Newton discover the universal law of gravitation?

I am having trouble comprehending how anyone could come up with this formula:

$$F = \frac{GMm}{d^2}.$$

Could someone walk me through this?

Answer

Well, there are 4 parts to the right-hand side; let's look at each in turn.

The first $M$ is the mass of the gravitating body. You would expect that the force it exerts should be larger if it has a larger mass. Moreover, it's not unreasonable to think the force should be directly proportional to this mass: twice as much mass should grab things with twice as much force.

By symmetry, the dependence on $m$ should be the same as on $M$. This symmetry is really that of Newton's Third Law: for every action (force that $M$ exerts on $m$) there is an equal and opposite reaction (force that $m$ exerts on $M$).

How about the $d^2$ in the denominator? Well, we expect that the force with which one thing attracts another decreases with distance: get far enough away and you should feel something's effect on you diminish to arbitrarily small magnitude. Moreover, the power of 2 is an effect of living in 3D space. Consider a point source of light. Any sphere of radius $r$ centered on the source will capture the same total power, so the power per unit area (how bright the light appears) is proportional to $1/r^2$, since the area of a sphere is proportional to $r^2$.

In fact, this $1/r^2$ dependence was known for certain things and even suggested for gravity. Hooke in particular felt Newton got too much credit because he (Hooke) had suggested an inverse square law earlier (though he didn't really have rigorous math or predictions relating to it). Needless to say, this soured their relations, prompting Hooke to join the Leibniz camp in the calculus debate.

Finally, there's the $G$, which is just a reflection of the fact that our arguments are all about proportions and so they leave an overall proportionality constant undetermined. The value of $G$ is in fact notoriously difficult to measure, and this wasn't done in Newton's time. Instead, one would make do with taking ratios of quantities such that $G$ dropped out.