The surface area of a sphere is $4\pi r^2$ and when you increase the distance to a point charge the force diminishes like the $r^{-2}$. Who was the first person to realize this?

  • $\begingroup$ Corrected some small mistakes: Hope you don't mind. $\endgroup$
    – Danu
    Commented Feb 15, 2015 at 21:55
  • $\begingroup$ Related question. $\endgroup$
    – HDE 226868
    Commented Feb 15, 2015 at 22:31

2 Answers 2


Curiously, Kepler thought that gravity falls as $1/r$, and he had a peculiar ether vortex theory borrowed from Gilbert's work on magnetism, to support it. But he argued that the intensity of light falls as $1/r^2$ along the same lines that others later applied to gravity: "there is as much light in the narrower spherical surface, as in the wider, thus it is as much more compressed and dense here than there". French astronomer Bullialdus seems to be the first to do so in 1645:

"... it turns with the body of the Sun; now, seeing that it is corporeal, it becomes weaker and attenuated at a greater distance or interval, and the ratio of its decrease in strength is the same as in the case of light, namely, the duplicate proportion, but inversely, of the distances."

But... he did not believe that such a force existed:

"I say that no kind of motion presses upon the remaining planets … indeed [I say] that the individual planets are driven round by individual forms with which they were provided".

It is unclear if Hooke knew of Bullialdus when he gave a similar argument, which he did believe, in his Micrographia of 1666. However, Bullialdus was elected to the Royal Society in 1667, and Newton credits him in Principia.

After Huygens's De vi centrifuga of 1659 the inverse square law could be inferred from combining Huygens's formula $a=v^2/r$ for the centrifugal acceleration with the Kepler's third law. However, De vi centrifuga was not published until 1703, Newton independently rederived the relation in 1669, also privately, and it only became public with Huygens's Horologium Oscillatorum (1673), see Who first derived $a=v^2/r$?

Assuming that the planets moved around the Sun in circles, to get the Kepler's relation between the radii and periods the balancing attraction force had to fall as $1/r^2$. Indeed, if $a=v^2/r\propto1/r^n$ then $v^2\propto1/r^{n-1}$, and since $T=2\pi r/v$ we get $T\propto r^{(n+1)/2}$. For squares of periods to be in the same ratio as cubes of radii we must have $n+1=3$ or $n=2$. The general concern was if it still holds for elliptic orbits, and that was only resolved by Newton in Principia.

Yaglom in Felix Klein and_Sophus Lie (p.184) quotes Kant's 1747 paper Thoughts on the True Estimation of Living Forces which takes Bullialdus' dimensional reasoning further:

"substances in our universe interact with each other so that the acting force is inversely proportional to the square of the distance... If the number of dimensions were different, the forces of attraction would have different properties and dimensions."

  • 2
    $\begingroup$ Perhaps even more curiously, there are modern theories claiming that gravity falls as 1/r, with some varying coefficients. See arXiv:0908.3842. $\endgroup$ Commented Feb 26, 2015 at 17:56
  • $\begingroup$ Interesting answer, I just wonder why you refer for the Bullialdus quote to a blog site that essentially just copies content from Wikipedia, rather than link to the Wikipedia page directly, also considering that private blogs are more likely to disappear. $\endgroup$
    – uUnwY
    Commented Nov 19, 2021 at 10:48
  • $\begingroup$ @StephanMatthiesen Good point, I changed the link. $\endgroup$
    – Conifold
    Commented Nov 19, 2021 at 11:06
  • $\begingroup$ Is this how $a=v^2/r$ is combined with Kepler third law? (The derivation in your paragraph starting "assuming that the planets moved around...") This is related to my other question too, can you clarify? Thanks. $\endgroup$
    – zeynel
    Commented Aug 24, 2022 at 13:04

One possibility is Johannes Kepler, however one could argue that John Dumbleton was homing in on the law as well perhaps about 250 years before Kepler. It's amazing it was being realized that long ago.


I'm interested in how the inverse square law can be thought of as a point source that is emanating a known quantity of "whatever" which propagates outward and causes effects on a per unit area basis.

For example;

$$F = \frac{GM_1M_2}{r^2}$$

Could be rewritten as;

$$F = \frac{KM_1M_2}{4{\pi}r^2}$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.