# Robert Hooke and the Inverse Square Law

In

borel, emile, 1922, l'espace et le temps, paris: alcan

there is a claim that Robert Hooke discovered an inverse square relation of gravitating bodied prior Isaac Newton, who later applied it more successfully in astronomy (Hooke apparently studied bodies on earth).

This is said to result in attribution of the principle to Newton, rather than Hooke, since the astronomical example were much more obvious than the small measurements with bodies on earth.

Is this true? What primary sources exist? Writing, books or letters, by Hooke, prior 1682, as opposed to later texts about him discovering the principle, if he really did discover it as Borel claims.

• See my answer to this post : the two books have detailed discussion of Hooke's letter and subsequent development by Newton. Nov 20 '14 at 14:48

Hooke did propose the inverse square law to Newton, and even the program of ”compounding the celestiall motions of the planetts of a direct motion by the tangent & an attractive motion towards the central body.” These suggestions from Hooke ”occasioned my findings” on planetary motion, Newton admitted. For more details, see my summary of Ofer Gal’s book on this.

Added: Hooke stated the law in print in his 1665 Micrographia. See The Archaeology of the Inverse Square Law.

• I suppose Gal's book has all the citations to the primary sources. Thank you for the reference. I'll check it out. I'm trying to nail down the date of the conjecture by Hooke. He wrote Newton in 1680 with the result, which is the last communication. From your review with quotes of the book it seems the conjecture was made earlier. Did Hooke ever publish it, in a book, outside earlier letters, as explicitly as he later stated it? Nov 20 '14 at 11:53
• The passage from Micrographia only applies the ISL to atmosphere. Apparently, he never claimed it might apply to celestial bodies until 1674, and by then the idea "was rather common and had been advanced by a number of different people for different reasons" (Ofer Gal). Hooke himself based his plagiarism claim on the 1679 letter to Newton rather than on Micrographia of 1665. en.wikipedia.org/wiki/… Nov 25 '14 at 20:03

The following stories show that the inverse square law was widely discussed at the time of Newton.

First story is about Hooke. He wrote to Newton proposing to determine "how a point will be moving under the inverse square law". Specifically he discussed the example of an object with some initial velocity (NOT directed towards the center) how would it move if the Earth did not offer any resitanace. Newton's answer was INCORRECT (he wrote it will move on a spiral winding towards the center of the Earth). In the next letter Hooke wrote the correct answer: it will move on an ellipse. (Hook did not have a mathematical proof of this: he probally EXPERIMENTED!)

To this last letter Newton did not reply. Several years later Sir Christopher Wren in a conversation with Hooke and Halley (in a pub:-) proposed to prove that the inverse square law of attraction would imply Kepler orbits, and offered a reward for a proof. Halley passed this to Newton, and Newton replied that he has a proof. But he could not find it among his papers. Some time later he sent to Halley a manuscript with a "proof". This is what later become Principia. (It is still discussed whether Newton's proof was a valid one. I share the majority opinion that it was. However, WHEN did Newton obtain it remains a mystery. Certainly he did not know the result, not speaking of the proof at the time when Hooke wrote to him. Later Newton spread rumors that he knew it almost from his childhood:-).

Halley proposed to mention Hooke in Principia. But Newton stubbornly resisted, and his angry answer to Halley is well known and documented.

The whole story is told in many places, one of them is Arnold's book, Barrow and Huygens, Newton and Hooke. Unlike some other stories told by Arnold this story is true, and is well documented: I read the corresponding letters of Newton and Hooke myself. And everyone can check this in the published collection of Newton's correspondence.

Conclusion. The Inverse square law was discussed at the time of Newton as a plausible CONJECTURE. Newton proved it.

• I found the letter you refer to :) : see p.147 below books.google.com/… This is Hooke's letter in January 1680. Nov 20 '14 at 11:44

The idea that gravity acted with an inverse square relation was not a "done deal" because Newton or Hooke said so. On Nov 15, 1747 "Clairaut, at a public session of the [French] Academy, announced in rather pompous phrases that the Newtonian Theory of gravity was false!”
One of the boldest attempts to reconcile the observed and theoretical descriptions of the moon's motion was made not by Euler, but Clairaut, who announced in at a public session in the French Academy of Sciences that Newton's theory of gravity was wrong. Euler and d’Alembert simultaneously came to the same conclusion as both had been working on the motion of the moon as a special case of the three body problem.
Clairaut suggested that the strength of gravity was proportional not to $1/r^2$, but the more complicated $1/r^2 + c/r^4$ for some constant $c$. Over large distances, the $c/r^4$ term would effectively disappear, accounting for the utility of the inverse square law over large distances. He then began trying to find a value of c which could account for the moon's motion. He would continue to pursue this idea until May 17, 1749, when he made an equally dramatic announcement in which he claimed that Newton was right after all.

• Is "Clairaut, at a public session of the [French] Academy, announced in rather pompous phrases that the Newtonian Theory of gravity was false!” a citation from somewhere? Dec 17 '14 at 12:40