Isaac Newton Derives Kepler's Ellipses from the Law of Universal Gravitation

In my post, Johannes Kepler and Inverse-Square Laws, I wrote that Johannes Kepler (1571-1630) was the first to propose an inverse-square law, for the propagation of light, but that he missed this opportunity for gravity; this task would fall on the shoulders of Isaac Newton (1643-1727). But the story of how this came about is not straightforward.

I came across a little book by the great Soviet mathematician, Vladimir Igorevich Arnol’d (Владимир Игоревич Арнольд, 1937-2010), Huygens and Barrow, Newton and Hooke. Pioneers in mathematical analysis and catastrophe theory from evolvents to quasi crystals.¹ Therein, Arnol’d writes how the mathematical work undertaken by Isaac Barrow (1630-1677), Christiaan Huygens (1629-1695) and Newton laid the foundations for a number of branches of mathematics, some only developed in the 20th century.

In the same book, Arnol’d also writes about the interactions between Robert Hooke (1635-1703) and Newton that led to the latter’s universal gravitation and his publishing in 1687 of the Philosophiæ Naturalis Principia Mathematica, or Principia for short.

Robert Hooke

Arnol’d begins with an introduction to Hooke, explaining his rôle as Curator to the Royal Society:

Robert Hooke was an older contemporary of Newton, though much less well known. He was born in 1635 and died in 1703. Hooke was a poor man and began work as an assistant to Boyle (who is now well known thanks to the Boyle-Mariotte law discovered by Hooke…). Subsequently Hooke began working in the recently established Royal Society (that is, the English Academy of Sciences) as Curator. The duties of Curator of the Royal Society were very onerous. According to his contract, at every session of the Society (and they occurred every week except for the summer vacation) he had to demonstrate three or four experiments proving the new laws of nature.

Hooke held the post of Curator for forty years, and all that time he carried out his duties thoroughly. Of course, there was no condition in the contract that all the laws to be demonstrated had to be devised by him. He was allowed to read books, correspond with other scientists, and to be interested in their discoveries. He was only required to verify whether their statements were true and to convince the members of the Royal Society that some law was reliably established. For this it was necessary to prove this law experimentally and demonstrate the appropriate experiment. This was Hooke’s official activity. [pp.11-12]

It turns out that Hooke was remarkably productive!

In the line of duty Hooke was interested in all discoveries in natural science by others, but it also fell to him to make discoveries. Towards the end of his life he counted 500 laws that he had discovered. It needs to be said that the numerous discoveries of Hooke form the basis of modern science. Very many of them were discovered more or less in parallel with other scientists, hence very often now laws discovered by Hooke are known, but attributed to other people. As a result the law of elasticity (the force is proportional to the extension) bears the name of Hooke, but his other discoveries bear other people’s names. For example, Hooke discovered the cellular structure of plants. He improved the microscope and was the first to observe that plants consist of cells. He scrutinized various objects in the microscope and everything he saw he sketched. It is clear that as he saw new things in the microscope he quickly made new discoveries. Hooke personally engraved pictures which he saw in the microscope, and even published on the basis of this a book “Micrography”, which later led Leeuwenhoek to his famous biological discoveries. [p.12]

At that time it was easy to carry out fundamental discoveries, and large numbers of them were carried out. Huygens, for example, improved the telescope, looked at Saturn and discovered its ring, and Hooke discovered the red spot on Jupiter. At that time discoveries were not unusual events, they were not registered, not patented, as they are now, they were quite an everyday occurrence. (This was the case not only in the natural sciences. Mathematical discoveries at that time also poured forth as if from a horn of plenty….) [p.12]

Because he was so busy, Hooke did not always have the time to fully develop his ideas. Nevertheless, he did work on the question of gravity. In an exchange between the two in late 1679, initiated by Hooke, they discussed what should happen to falling objects under the Copernican hypothesis of diurnal rotation of the Earth. In a letter dated January 6th, 1680, Hooke wrote that he had conducted some experiments demonstrating that an object should fall slightly east and south of where it was dropped, as a result of the different speeds of rotation at altitudes and latitudes. In this letter, Hooke wrote:

My supposition is that the attraction always is in duplicate proportion to the distance from the centre reciprocal, and consequently that the velocity will be in a subduplicate proportion to the attraction, and consequently as Kepler supposes reciprocal to the distance ... Mr. Halley, when he returned from St. Helena, told me that his pendulum at the top of the hill went slower than at the bottom, which he was much surprised at, and could not imagine a reason. But I presently told him that he had solved me a query I had long desired to be answered... and that was to know whether the gravity did actually decrease at a greater height from the centre ... what I mentioned concerning the descent within the body of the Earth ... not that I really believe there is such [inverse proportional to the squared distance] an attraction to the very centre of the Earth, but on the contrary I rather conceive that the more the body approaches the centre, the less will it be urged by the attraction, possibly somewhat like the gravitation on a pendulum or a body moved in a concave sphere where the power continually decreases the nearer the body inclines to a horizontal motion ... But in the celestial motions the Sun, Earth or central body are the cause of the attraction, and though they cannot be supposed mathematical points yet they may be conceived as physical, and the attraction at a considerable distance may be computed accordingly to the former proportion [inverse square] as from the very centre.... [p.22]

So in that letter, Hooke had proposed to Newton that an inverse square law for gravity would lead to Kepler’s elliptic orbits. This is what Arnol’d wrote:

Most likely the situation was as follows. Hooke, not having the necessary mathematical technique, was unable to solve exactly the equations of motion obtained from the inverse square law and, in order to find the orbits, he integrated these equations numerically, graphically or on an analogue machine like the concave surface he mentioned. It is known that Hooke had such a machine: he investigated the nature of motion under various laws of attraction, modelling the attraction by the action of a surface on a weight sliding over it. (We observe that all this happened six years before Newton wrote his book and stated the general laws of mechanics. According to our modern ideas, at that time there was no mechanics. Nevertheless, in these pre-mechanics times Hooke found approximate solutions of the equations of motion under the inverse square law, and Huygens stated the law of conservation of energy. It is true that Huygens did not give it in its most general form, but in his formulation the law was applicable in our case, and made it possible to realize that in the absence of air resistance the orbits of a stone inside the Earth must be closed.) Having integrated the equations of motion, Hooke drew the orbits and saw that they were similar to ellipses. This is how the word elliptoid arose. His scientific honesty did not allow him to call them ellipses, since he could not prove that they were elliptic. Hooke suggested to Newton that he do this, saying that he did not doubt that Newton with his superior methods could cope with this problem and also check that Kepler’s first law (which asserts that the planets move in ellipses) also follows from the inverse square law. [p.23]

So how did Newton respond? He didn’t. He got to work:

In sending Newton a letter with this suggestion, Hooke was turning to later discoveries, since he had no time for the mathematical details. Newton was silent and never wrote any more to Hooke (except for one case, when he sent Hooke a request from an Italian doctor who wished to collaborate with the Royal Society and took the opportunity to thank him for his information about his experiments with falling balls), he never referred to the correspondence (although he kept the letters) and he did not speak about the fact that Hooke had posed him the problem of gravitation.

But Newton took up this problem, investigated the law of motion, checked that elliptic orbits had actually been obtained, and proved conversely that the inverse square law follows from Kepler’s law on ellipticity of orbits. In order to put this properly into shape and present it in accessible form, he needed to state the basic principles, referring to such general concepts as mass, force, acceleration. This is how the famous “three laws of Newton” appeared, to which Newton himself did not pretend (the first law, Galileo’s law of inertia, had been well known for a long time, and the other two could not have been discovered later than, say, Hooke’s law of elasticity or Huygens’ formula for centrifugal force). But in connection with the law of universal gravitation Newton behaved less scrupulously. [my emphasis, p.24]

Further to the question of primacy, Arnol’d wrote in a note:

Later, in 1694, Newton wrote that he had discovered the law of universal gravitation in 1665 or 1666. Still later, in 1714, Newton dated his derivation of the ellipticity of orbits from the inverse square law as 1676 or 1677. However, neither in correspondence with Hooke in 1679 nor earlier did Newton recall his discoveries in this field: he did not publish them and did not speak about them. Newton explained this by the fact that because of the false value of the radius of the Earth he accepted that the calculated accelerations of stones and the Moon do not fit the inverse square law with sufficient accuracy. Hooke’s first publication on the force of gravitation as a possible reason for the ellipticity of orbits was his report read to the Royal Society in 1666, and published in 1674 as part of a 1670 Kutlerian lecture. [note 5, pp.107-108]

André Assis sent me a copy of a Scientific American article on Newton’s discovery², written by I. Bernard Cohen, co-translator of the Principia³ and author of The Newtonian Revolution⁴. Therein, Cohen’s chronology is not identical with that of Arnol’d, but he also concurs that Newton’s discovery was the result of the exchange with Hooke.

When Newton heard the question, he responded immediately: an ellipse. Halley asked him how he knew and Newton replied: “I have calculated it.” Newton apparently could not find the calculations, but at Halley’s urging he wrote them up for the Royal Society in the small tract De Motu (Concerning Motion). In De Motu Newton described his work on terrestrial and celestial dynamics, including his ideas on motion in free space and in a resistive medium. Newton must have finished De Motu by December 10, 1684, because Halley told the Royal Society then that Newton had recently shown him the curious treatise.

The exact progression of Newton's ideas in the time between his correspondence with Hooke and his completion of the first draft of De Motu is not documented. Nevertheless, I am certain it was Hooke’s method of analyzing curved motion that set Newton on the right track. Although not all historians would agree with me, I believe the approach Newton takes to terrestrial and celestial dynamics in De Motu, which he further developed the following spring in the first book of the Philosophiae Naturalis Principia Mathematica, represents his thinking on planetary dynamics inspired by his correspondence with Hooke. In a few autobiographical manuscripts Newton said the correspondence either preceded or coincided with his demonstration published first in De Motu and then in the Principia that an object that has an inertial motion and is subject to an inverse-square centripetal force moves in an elliptical orbit. [p.169]

Putting aside questions of primacy, Newton put together into a single work, the Principia, a number of concepts, including mass, acceleration and inertia, and the principle of universal gravitation, based on an inverse square law, and derived Kepler’s first law, that orbits are elliptic. This complete package turned out to be highly productive, and became the foundation for all of the research in mechanics that was undertaken by the great physicists of the 18th century.

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1. V.I. Arnol’d. Huygens and Barrow, Newton and Hooke. Pioneers in mathematical analysis and catastrophe theory from evolvents to quasi crystals. Translated from the Russian by Eric J.F. Primrose. Basel: Birkhauser Verlag, 1990.

2. I. Bernard Cohen. Newton’s Discovery of Gravity. Scientific American. March 1, 1981, pp.168-179.

3. Isaac Newton. The Principia: Mathematical Principles of Natural Philosophy. The Authoritative Translation by I. Bernard Cohen and Anne Whitman. Assisted by Julia Budenz. Preceded by A Guide to Newton’s Principia, by I. Bernard Cohen. University of California Press, 1999.

4. I. Bernard Cohen. The Newtonian Revolution. With illustrations of the transformation of scientific ideas. Cambridge University Press, 1980.