It is well known the Isaac Newton was an aggressively querulous man ready to start a scientific dispute at the drop of a bodkin. One of his favourite sparring partners was the equally cantankerous Robert Hooke with whom he disputed on more than one occasion. One of their most famous spats concerned the inverse square law of gravity with Hooke aggressively claiming priority for its discovery and demanding that Newton acknowledge this in the *Principia*, which he was in the process of writing. Newton of course reacted belligerently to Hooke’s claims and threatened to withdraw his masterpiece from publication. It took all of Halley’s diplomatic skill, he was at the time midwifing the book through the press, to advert disaster and persuade Newton to publish but only after Newton had removed all references to his rival from the text. This incident led to the famous correspondence between Halley and Newton that outlined the probably fictionalised account of the genesis of Newton’s book.

The story goes that Halley, Hooke and Christopher Wren were chewing the fat in a London coffee house one day (wouldn’t you just have liked to be sitting on a neighbouring table) when the discussion turned to the newest astronomical theories. The question raised was whether the assumption of an inverse square law of gravity would lead to Keplerian elliptical orbits. Wren, who was himself an excellent mathematician and an early Keplerian, offered a prize of a valuable book-token if either of the others could provide a mathematical proof that this would be the case. Hooke claimed that he could but would not reveal his solution until Halley and Wren had had time to try the problem for themselves. Halley, also a good mathematician, tried to solve the problem but failed and whilst on a visit to Cambridge posed the question to Newton who already had a reputation as a master mathematician. Newton supposedly immediately answered that under the assumption of an inverse square law of gravity the planetary orbits would be ellipses. Halley asked him how he could be so certain to which the good Isaac replied, “because I have calculated it”. Unfortunately Newton had misplaced his calculations but he promised Halley that he would supply him with the necessary proof. The end result of that promise after several intermediary stages and three years hard work was the *Principia*.

Now it should be clear from this story that neither Newton not Hooke could lay claim to being the father of the inverse square law of gravity as it was obvious from the original coffee house discussion that this hypothesis was common currency in the then astronomical community. This was indeed the case and the true originator of the hypothesis, as acknowledged by Newton in the *Principia*, was the French mathematician and astronomer Ismael Boulliau who was born on 28th September 1605.

Boulliau who was born a Calvinist, converted to Catholicism and was eventually ordained a priest. Like his friend Fermat he studied law and became a public notary. Later he became a librarian in Paris where he was part of the mathematical scene counting Pascal, Huygens, Mersenne and Gassendi amongst his friends and correspondents. He corresponded with scholars throughout Europe and was one of the earliest foreign members of the Royal Society. He wrote several important mathematical works of which the most significant was his *Astronomia philolaica* published in 1645. In this work Boulliau, like Wren an early Keplerian, presented his version of the elliptical astronomy. It is here that we can find the earliest statement of the inverse square law.

In his own work Kepler had departed from tradition in not only describing mathematically the planetary orbits but in also suggesting a cause i.e. a driving force for those orbits. This had been previously the terrain of the philosophers and Kepler earned himself much criticism for poaching on their territory. His suggestion was that the sun rotated on its axis and sent out a force, he thought a magnetic one, that swept the planets round their orbits. In Kepler’s theory this force diminished in inverse ratio to its distance from the sun. In his version of the Keplerian astronomy Boulliau argued that if this force existed it would not diminish in direct inverse ratio but in the inverse squared ratio and thus the law of gravity first saw the light of day. It should be pointed out that Boulliau did not accept the existence of such a force but he had introduced it into the contemporary astronomical discussion and we have already seen where it led.

It is interesting to ask why Boulliau thought that the force must diminish according to the inverse square of the distance and the answer takes us back to Kepler and an important piece of methodology in the process of scientific discovery. Kepler was anything but a one trick pony making significant contributions to several areas of astronomy, physics and mathematics, one of his major areas of work being physical optics. In his work in optics Kepler formulated what is considered to be the first modern mathematical law of physics his law of light propagation. This law states that the intensity of a light ray diminishes in strength inversely to the square of the distance travelled. Boulliau argued by analogy that if Kepler’s planetary driving force existed then it too would diminish according to the inverse square of the distance travelled. History and Newton would prove him right against his own better judgement.

The process of scientific discovery is a complex and oft irrational process that nobody has, in my opinion, really succeeded in explaining. However analogy has in the history of science often played an important role in the formulation of new hypotheses and sometimes as in this case it proves fruitful.

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Interesting. I’ve only vaguely heard of Boulliau before. That makes Newton and Hooke’s work seem comparatively less impressive if the basic idea of an inverse square law was already floating around to that extent.

In case anyone is interested in the decisive passage:

http://diglib.hab.de/drucke/2-1-4-astron-2f-1/start.htm?image=00053 (the paragraph starting with “Virtus autem illa…”)

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Nice article but I thought it was unlikely that Newton would give credit to Boulliau for inverse square law so I checked the Principia in Google Books and all I see is that Newton gives credit to Boulliau for determining “the distances of the planets from the sun…”

http://books.google.com/books?id=ySYULc7VEwsC&pg=PA630&dq=newton+principia&hl=en&ei=4fvnToDFDcrf0QG3msH7CQ&sa=X&oi=book_result&ct=result&resnum=3&ved=0CEEQ6AEwAg#v=onepage&q=Boulliau&f=false

Do you know where Newton gives credit to Boulliau for discovering the inverse square law?

As far as I can tell, not in the

Principia, but in the Halley-Newton correspondence that thony mentions:…so Bullialdus [i.e., Boulliau] wrote that all force respectingye Sun as its center & depending on matter must be reciprocally in a duplicate ratio of ye distance from ye center, & used that very argument for it by wch you, Sr, in the last Transactions have proved this ratio in gravity. Now if Mr Hook from this general Proposition in Bullialdus might

learn ye proportion in gravity, why must this proportion here go

for his invention?

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Kepler’s inverse force was not attractive, but (as stated) swept the planets around the sun. Thus it wasn’t really a gravitational force at all. Was Boulliau’s force attractive?

Kepler’s inverse force was an attractive force in fact he states quite explicitly that it is a magnetic force emanating from a magnetic mono-pole sun that hold the planets in their orbits.

It is not a gravitational force because, unlike Newton, he doesn’t equate it with the gravitas of Aristotelian physics responsible for causing things to fall on the surface of the earth.

However it was Kepler’s force that inspired Boulliau to hypotheses that such a force must be squared and it was Boulliau’s suggestion that informed the discussion on an attractive planetary force in the 1660s leading to Newton’s discovery of the law of universal gravity.

The discussion between Boulliau and Seth Ward on the Keplerian Laws is thought to have been the source from which Newton drew his knowledge of Keplerian astronomy rather than Kepler’s own writings.

I’m not sure we’re talking about the same part of Kepler’s work. I am relying on secondary sources (Koestler and the MacTutor archive), which do add up to a coherent picture. But if these are incorrect, I am glad to learn something new. Let me lay out the argument in detail.

In the introduction to Astronomia Nova, Kepler discusses an attractive force, which he calls gravity. (Quoted extensively by Koestler, and also at some length in Wikipedia.) But in III Cap. 33 (according to Koestler, p326) Kepler introduces a tangential force, inversely proportional to the distance, caused by the sun’s rotation on its axis.

This makes perfect sense in the context of Kepler’s physics, since his notion of inertia (he coined the term, didn’t he?) is a “laziness” notion; he doesn’t make use of Newtonian momentum. I suppose he might have used medieval impetus theory (though Koestler says he doesn’t), but even that was held to decay over time by most medieval philophers. So you need a tangential force to preserve (or “hold”) a planet in its orbit.

Moreover, Kepler has shown at this point that the speed of the planet is inversely proportional to the distance from the sun at aphelion and perihelion, and has made the natural (but incorrect) generalization to all points in the orbit. So an inverse force is just what’s required. And an attractive force makes no sense in this context. It

doesmake sense in the context of the preface, where it invoked to explain the tides.If Kepler really had the idea that his attractive version of gravity held the planets in the “Newton’s apple” sense, why would he write in the preface:

[quote]

If the earth and the moon were not kept in their respective orbits by a

spiritual or some other equivalent force, the earth would ascend towards the moon [and vice versa][end quote]

(Emphasis added by me to Koestler’s translation)

OK, let’s turn to Boulliau. MacTutor quotes him thus:

[quote]

As for the power by which the Sun seizes or holds the planets, and which, being corporeal, functions in the manner of hands, it is emitted in straight lines throughout the whole extent of the world, and like the species of the Sun, 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 that is, 1/d2.[quote]

No mention here of gravity. Again I’ve added emphasis. Clearly Boulliau is referring to Kepler’s tangential force of III:33 and not the gravitational force in the preface.

Is there some place where Kepler invokes the medieval impetus theory, or makes a clear reference to something like Newton’s notion of centripetal force? Without that, it seems to me that the “sweeping force” interpretation makes Kepler a much more rational thinker than the alternative, which turns him into some kind of mystic seer.

One more thought, totally speculative.

The geometry here — spokes originating from the sun, in all directions, but turning with the sun — in a crucial sense really resembles a dipolar field more than a monopole. At the poles of rotation (of the sun) we have no sweeping effect, and we have maximum sweeping effect at the equator.

So the magnitude of the tangential force might well die off in an inverse linear fashion instead of inverse square.

Following your advide, I’ve perused the key parts of the Astronomia Nova[1], also Stephenson’s book[2] and Aiton’s 2nd law paper[3]. For Boulliau, besides the MacTutor bio[4], there’s a web-site by Robert A. Hatch[5].

In brief, crediting Boulliau with the inverse-square law for gravity strikes me as odd, to say the least, and based on a conflation of Kepler’s gravity with his orbital forces.

Kepler’s gravity is attractive, but it does not act between the sun and the planets, nor does he give a distance-dependence law for it (just a mass-dependence). He invokes his gravitational force only to refute some standard anti-Copernican arguments, and to explain the tides.

Kepler’s sun-planet forces play a different role. First, there is the “species immateriata”. As you say yourself in your earlier post, this sweeps the planets around the sun. (Kepler compares it to a whirlpool, carrying the planets round in its gyre.) Kepler bestows a distance-dependence law

onlyon this force, namely inverse-linear.To explain the elliptical orbit, Kepler proposes a magnetic sun-planet force, varying sinusoidally around the orbit between attraction and repulsion, but without a distance dependence.

Boulliau clearly refers to the tangential force: the only one Kepler says is inverse-linear, and the only one associated with the solar rotation. Kepler devotes an extended argument explaining why, unlike light, this force is not inverse-square. (With the modern example of dipolar radiation in mind, it is easy to see that Kepler got it right.) Boulliau either missed this discussion, or failed to understand it, or disbelieved it.

In fact, after tossing off the inverse-square remark, Boulliau rejects the very idea of solar forces moving the planets[4]. Boulliau had his own, purely geometrical, conical model to explain the elliptical orbits. (Hatch has some neat animations on his web-page.)

I know that Newton mentions Boulliau in connection with the inverse-square law (in a letter to Halley, not in the

Principia). To be precise, Newton does not credit Boulliau with an inverse-square law for an attractive force, but for any sun-centered force. Anyway, the entire letter is an extended rant against Hooke’s priority claims, with Newton grabbing any ammunition he can lay hand to.I suppose it is possible that Hooke, or Wren, or Halley, got the idea of inverse-square laws (of any ilk) from Boulliau’s remark, rather than directly from Kepler’s Optica. Perhaps they mistakenly assumed Boulliau If so, drunken hotel guest!

[1]

Selections from Kepler’s Astronomia Nova, selected, translated, and annotated by William H. Donahue (2008). The acknowledgements state that the selections focus on Kepler’s physics.[2]

Kepler’s Physical Astronomy, Bruce Stephenson (1987).[3] “Kepler’s Second Law of Planetary Motion”, E. J. Aiton,

Isis60:1 (1969) p.75-90.[4] http://www-history.mcs.st-andrews.ac.uk/Biographies/Boulliau.html

[5] http://web.clas.ufl.edu/users/ufhatch/pages/11-ResearchProjects/boulliau/index.htm

[Since it’s awaiting moderation, I thought I would correct the typos in this version.]

Following your advice, I’ve perused the key parts of the Astronomia Nova[1], also Stephenson’s book[2] and Aiton’s 2nd law paper[3]. For Boulliau, besides the MacTutor bio[4], there’s a web-site by Robert A. Hatch[5].

In brief, crediting Boulliau with the inverse-square law for gravity strikes me as odd, to say the least, and based on a conflation of Kepler’s gravity with his orbital forces.

Kepler’s gravity is attractive, but it does not act between the sun and the planets, nor does he give a distance-dependence law for it (just a mass-dependence). He invokes his gravitational force only to refute some standard anti-Copernican arguments, and to explain the tides.

Kepler’s sun-planet forces play a different role. First, there is the “species immateriata”. As you say yourself in this post, this sweeps the planets around the sun. (Kepler compares it to a whirlpool, carrying the planets round in its gyre.) Kepler bestows a distance-dependence law

onlyon this force, namely inverse-linear.To explain the elliptical orbit, Kepler proposes a magnetic sun-planet force, varying sinusoidally around the orbit between attraction and repulsion, but without a distance dependence.

Boulliau clearly refers to the tangential force: the only one Kepler says is inverse-linear, and the only one associated with the solar rotation. Kepler devotes an extended argument explaining why, unlike light, this force is not inverse-square. (With the modern example of dipolar radiation in mind, it is easy to see that Kepler got it right.) Boulliau either missed this discussion, or failed to understand it, or disbelieved it.

In fact, after tossing off the inverse-square remark, Boulliau rejects the very idea of solar forces moving the planets[4]. Boulliau had his own, purely geometrical, conical model to explain the elliptical orbits. (Hatch has some neat animations on his web-page.)

I know that Newton mentions Boulliau in connection with the inverse-square law (in a letter to Halley, not in the

Principia). To be precise, Newton does not credit Boulliau with an inverse-square law for an attractive force, but for any sun-centered force. Anyway, the entire letter is an extended rant against Hooke’s priority claims, with Newton grabbing any ammunition he can lay hand to.I suppose it is possible that Hooke, or Wren, or Halley, got the idea of inverse-square laws (of any ilk) from Boulliau’s remark, rather than directly from Kepler’s Optica. If so, drunken hotel guest!

[1]

Selections from Kepler’s Astronomia Nova, selected, translated, and annotated by William H. Donahue (2008). The acknowledgements state that the selections focus on Kepler’s physics.[2]

Kepler’s Physical Astronomy, Bruce Stephenson (1987).[3] “Kepler’s Second Law of Planetary Motion”, E. J. Aiton,

Isis60:1 (1969) p.75-90.[4] http://www-history.mcs.st-andrews.ac.uk/Biographies/Boulliau.html

[5] http://web.clas.ufl.edu/users/ufhatch/pages/11-ResearchProjects/boulliau/index.htm

I just ran across two not-so-recent papers that have more to say about Boulliau’s

Astronomia philolaica:[1] “Kepler’s Laws of Planetary Motion: 1609-1666”, J. L. Russell,

BJHS, 2:1 (June 1964) p.1-24.[2] “From Kepler’s Laws, So-called, to Universal Gravitation: Empirical Factors”, Curtis Wilson,

Archive for History of Exact Sciences, 6:2 (1970), p. 89-170.Russell writes:

[Bouilliau’s] view, vigorously asserted in the introductionto this book, was that we must look not for physical but for geometrical

causes of planetary motion. By this he meant that the ultimate reason

for the shape of an orbit was simply the exemplification of a geometrical

form. Kepler had explained the orbit as arising from a physical interaction

between planet and sun; Boulliau denied any such interaction.

Russell has a bit more to say about the

Astronomia philolaica, but Wilson’s paper has 17 pages on it. He quotes from Boulliau’s preface:From physical conjectures we draw back, for they do not suffice, not is there great force in themAs Wilson notes, this is in keeping with Aristotle’s distinction between formal and efficient causes. Wilson traces “Boulliau’s fixation on uniform circular motion”, and how it lead to his development of an elaborate alternative to Kepler’s 2nd law. (Boulliau justified the 1st law in these terms: the ellipse is a conic section, and a cone is made up of circles.)

Embarrassingly for Bouilliau, Seth Ward showed that Bouilliau’s new speed law was equivalent Ptolemy’s old equant, adapted to the ellipse. (Bouilliau had explicitly, and incorrectly, denied this possiblility.) Although Bouilliau’s law was conspicuously less accurate than Kepler’s area law, it proved more popular in England for a while, simply because it made the computations easier (if you used Ward’s equivalence result). Bouilliau later modified his law to improve its accuracy.

All told, Bouilliau was no fan of celestial forces of any stripe, certainly not of an inverse-squared gravity.

An intelligent set of exchanges; I suggest that readers consider doing a Google of “Boulliau” to find primary sources [in translation] about his planetary model. It is easy to do and there are images available. There is also a printed precisely on Boulliau and Borelli et al on the i.s.l. The article aims to focus on creativity and imagination not simply calculation and demonstration.

The Andrews site is sometimes derivative and not always generous with scholarly citations.

Kepler first assumed that orbits are naturally circular; second, he argued that each orbit is corrupted by something like a magnetic attraction and repulsion; third, Kepler did not accept Galileo’s notion of circular inertia, and hence he imagined that a power or virtue came from the sun [in the plane of the planets] as it rotated and pushed the planets around, hence the analogy of the lever and the simple inverse distance rule. Boulliau aimed at a simple and elegant theory with light as the obvious analogy expanding radially. Minimal assumptions. B argued planets are moved by an internal form or tendency and can be imagined at any instant to be moving on a circle with uniform motion around central points and at any instant to also move on an ellipse, both defined by the surface of a 3-dimensional cone, by construction [conic sections key here]. The Cause of the orbit shapes and the changing speeds is Explained by Reason of geometry. With Kepler, Boulliau agreed that God is a geometer.

Great to have

theBoulliau expert chiming in. Readers should go to href=”http://web.clas.ufl.edu/users/ufhatch/pages/11-ResearchProjects/boulliau/index.htm”>your website — faster than googling.Rereading what I wrote, I think I may have been a little hard on Boulliau, mostly as a reaction to the original post. The post came perilously close to anointing him The Father of the Inverse Square Law of Gravity, or so it seemed to me. Whereas actually the law’s discovery was a process that played out over 70 years or more. (Do we start the clock with Nicolas of Cusa’s early thoughts on inverse square laws?)

What do you think about Wilson’s comment on efficient vs. formal causes?

Translating into modern terms, it would seem that physicists nowadays prefer formal causes: just find the right equations, don’t worry about forces.

Oops, I messed up the html code for your website.

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28.IX.2016: Happy BD Izzy Boulliau … some readers may wish to see an article I wrote on this issue with a broader nod to questions of creativity not simply calculation. ‘Nature’s Profoundest Secret: First Inklings, Second Guesses, Second Thoughts’ in Astronomy as a Model for the Sciences in Early Modern Times, (Ed. Menso Folkerts & Andreas Kuhne) Algorismus (Verlag, Augsburg, 2006): 307-331.

For Boulliau (see above) planets are moved by their form and they are explained (not merely described) not by physics but by reason of geometry. There are no forces or attractions and repulsions and importantly, as with Kepler, the motions of bodies are not perfect given the finally flawed character of their composition.

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