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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?

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  • $\begingroup$ Corrected some small mistakes: Hope you don't mind. $\endgroup$ – Danu Feb 15 '15 at 21:55
  • $\begingroup$ Related question. $\endgroup$ – HDE 226868 Feb 15 '15 at 22:31
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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 On Centrifugal Force of 1659 the inverse square law was usually inferred from combining Huygens's formula $a=v^2/r$ for the centrifugal acceleration with the Kepler's third law. Assuming 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 (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."

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  • $\begingroup$ Perhaps even more curiously, there are modern theories claiming that gravity falls as 1/r, with some varying coefficients. See arXiv:0908.3842. $\endgroup$ – Pasha Zusmanovich Feb 26 '15 at 17:56
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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.

http://en.wikipedia.org/wiki/Inverse-square_law

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;

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

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