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This question is puzzling me since I learnt about the gravitation law in school. Why did Newton guess/assume that gravitational force is inversely proportional to the square of distance?

Did he verify that experimentally? (I remember hearing that the first experimental verification of the law of gravitation was after Newton's death.)

If the answer to the above question is no, is it for example more plausible to suppose that $F\propto1/r^2$ than to suppose that $F\propto1/r^4$? Did Newton carry out a thought experiment that makes $F\propto1/r^2$ a plausible guess?

So in summary: Why did Newton choose exponent of $-2$ instead of any other exponent? Was it a guess that depended on pure luck or an educated guess?

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  • $\begingroup$ In addition to the observations Newton had to verify his guess mentioned in the answers, $\propto 1/r^2$ is an easy guess since so many things follow an inverse square law. Newton very well may have guessed, then used observational evidence to verify, rather than starting from evidence and deriving. $\endgroup$
    – Phil Frost
    Jul 27, 2014 at 12:09

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For a uniform circular orbit of radius $r$, the acceleration is

$$\tag{A} a~=~ \omega^2r, \qquad \omega~=~\frac{2\pi}{T},$$

where $T$ is the orbital period. Comparing eq. (A) with Kepler's third law

$$\tag{B} T^2 ~\propto~ r^3,$$

we conclude that the gravitational acceleration

$$\tag{C} a~\propto~ r^{-2} $$

is proportional to the inverse square distance $r$.

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  • $\begingroup$ So Newton chose an inverse square field because it fits Kepler's law, right ? $\endgroup$
    – Amr
    Jul 28, 2014 at 3:07
  • $\begingroup$ $\uparrow$ Yes. $\endgroup$
    – Qmechanic
    Jul 28, 2014 at 8:00
  • $\begingroup$ In that case, how did Kepler formulate his laws of planetary motion? Was it based on astronomical evidence, or was that a guess too? $\endgroup$
    – Abhigyan
    Jan 17, 2018 at 16:17
  • $\begingroup$ Johannes Kepler famously used the astronomical data of Tycho Brahe. $\endgroup$
    – Qmechanic
    Jan 17, 2018 at 19:11
  • $\begingroup$ @Qmechanic hello! May I ask why he thought that a circular approximation was good enough even though they knew that the path of planets was an ellipse? $\endgroup$
    – GDGDJKJ
    May 1, 2020 at 19:19
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An inverse square law for gravity was already being considered in several places prior to Newton taking it up, and Newton was probably at least partly inspired to consider it by Hooke and Halley (exactly to what extent would be the subject of one of Newton's several priority disputes).

The basic reason for choosing an inverse square as opposed to some other function was picturing gravity arising from some sort of constant physical emanation. Since the surface at a given distance goes as $1/r^2$, presumably whatever caused gravity would fall in the same proportion.

Newton went further then his contemporaries, however, by showing that under such a force an orbiting object would follow Kepler's laws.

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Well, for one, if $F$ goes like $r^{-4}$, all orbits except the unstable circular orbit will fly out or will collapse into the center! So that's a no-no. (I know that because Newton's laws in a central force field give conservation of angular momentum, and in that case, if $U$ is your potential, you can get a differential equation for $r$, the radius, as $\ddot{r}=-\frac{\partial}{\partial r}(U(r)+\frac{M}{2r^2})$. With $U=-\frac{1}{r}$ you have a nice stable dip in which $r$ can oscillate. With $U=\frac{1}{r^3}$ a particle can always lose potential energy by moving in the same direction.)

I haven't read the history in detail, but I believe that the order of events was something like this:

First, Kepler shows that "The orbit of every planet is an ellipse with the Sun at one of the two foci." from wikipedia

Then, Newton shows that an elliptical orbit together with the methods of Newtonian mechanics, implies that the force must act as an inverse square. (A proof of this can be found in A P French's Newtonian Mechanics, though I forget how much history is given there and how geometric his methods are).

Disclaimer: You can see my sources are shaky. There may be more to the story.

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  • $\begingroup$ Not to mention that it's an obvious consequence of living in 3 orthogonal spacial dimensions under some basic assumptions and geometric reasoning. $\endgroup$ Jul 27, 2014 at 3:58
  • $\begingroup$ @BrandonEnright which basic assumptions are you talking about? $\endgroup$
    – user12029
    Jul 27, 2014 at 4:14
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    $\begingroup$ Assume gravity doesn't dissipate and all and that we live in exactly 3 spatial dimensions. Then all that matters in the density of the lines of force. Taking a sphere with force lines radiating out from the center, the density of the lines on the surface of the sphere drops with the square of the radius. A crude analogy is a red balloon. As you blow it up the density of the dye decreases with the square of r so the color gets spread out and turns pink / transparent. If we lived in something other than 3 dimensions or if gravity didn't travel infinitely the square law would be violated. $\endgroup$ Jul 27, 2014 at 6:25
  • $\begingroup$ Idealism and symmetry has it's beautiful attractions, till it discovers it's nemesis in it's foundations like Euclidean geometry. Newton was very worried that it created a monster - action at a distance. He found it necessary to write some mad religious arguments for it. $\endgroup$
    – Sukii
    May 30, 2023 at 19:34

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