Four hundred years ago today Johannes Kepler rediscovered his most important contribution to the evolution of astronomy, his third law of planetary motion.

He had originally discovered it two months earlier on 8 March but due to a calculation error rejected it. On 15 May he found it again and this time recognised that it was correct. He immediately added it to his *Harmonices Mundi*:

For when the true distances between the spheres were found, through the observations of Brahe, by continuous toil for a very long time, at last, at last, the genuine proportion of the periodic times to the proportion of the spheres –

Only at long last did she look back at him as she lay motionless,

But she look back and after a long time she came[Vergil,EclogueI, 27 and 29.]And if you want the exact moment in time, it was conceived mentally on the 8

^{th}of March in this year one thousand six hundred and eighteen, but submitted to calculation in an unlucky way, and therefore rejected as false, and finally returning on the 15^{th}of May and adopting a new line of attack, stormed the darkness of my mind. So strong was the support from the combination of my labor of seventeen years on the observations of Brahe and the present study, which conspired together, that at first I believed I was dreaming, and assuming my conclusion among my basic premises.But it is absolutely exact that proportion between the periodic times of any two planets is precisely the sesquialterate, though with this in mind, that[1]proportion of their mean distances, that is of the actual spheresthe arithmetic mean between the two diameters of the elliptical orbit is a little less than the longer diameter. Thus if one takesone third of the proportion from the period, for example, of the Earth, which is one year, and the same fromthe periodof Saturn, thirty years, that is,the cube roots, and one double that proportion,by squaringthe roots,he has in the resulting numbers the exactly correct proportion of the mean distancesof the Earth and Saturnfrom the Sun.[2]

writing a few days later:

*Now, because eighteen months ago the first dawn, three months ago the broad daylight, but a very few days ago the full sun of a most remarkable spectacle has risen, nothing holds me back. Indeed, I give myself up to a sacred frenzy*.

He finished the book on 27 May although the printing would take a year.

In modern terminology:

The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit: i.e. for two planets with P = orbital period and R = semi-major axis P_{1}^{2}/P_{2}^{2}=R_{1}^{3}/R_{2}^{3}

Kepler’s third law is probably the most important discovery on the way to the establishment of a heliocentric astronomy but its importance was initially overlooked and its implications were somehow neglected until Isaac Newton displayed its significance in his *Principia Mathematica*, published in 1687 sixty-eight years after the third law first appeared in print.

What the third law gives us is a direct mathematical relationship between the size of the orbits of the planets and their duration, which only works in a heliocentric system. In fact as we will see later it’s actually equivalent to the law of gravity. There is nothing comparable for either a full geocentric system or for a geo-heliocentric Tychonic or semi-Tychonic system. It should have hit the early seventeenth-century astronomical community like a bomb but it didn’t, which raises the question why it didn’t.

The main answer lies in Kepler’s own writings. Although he viewed its discovery as the crowning glory of his work on the *Harmonices Mundi* Kepler didn’t give it any prominence in that work. The *Harmonices Mundi* is a vast sprawling book explicating Kepler’s version of the Pythagorean theory of the harmony of the spheres in five books. After four introductory books covering plane geometry, music theory and astrology Kepler gets down to harmonic planetary theory in the fifth and final book. Book V, 109 pages in the English translations, contains lots of musical relationships between various aspects of the planetary orbits, with the third law presented as just one amongst the many with no particular emphasis. The third law was buried in what is now regarded as a load of unscientific dross. Or as Carola Baumgardt puts it, somewhat more positively, in her *Johannes Kepler life and letters *(Philosophical Library, 1951, p. 124):

*Kepler’s aspirations, however, go even much higher than those of modern scientific astronomy. As he tried to do in his “Mysterium Cosmographicum” he coupled in his “Harmonice Mundi” the precise mathematical results of his investigations with an enormous wealth of metaphysical, poetical, religious and even historical speculations. *

Although most of Kepler’s contemporaries would have viewed his theories with more sympathy than his modern critics the chances of anybody recognising the significance of the harmony law for heliocentric astronomical theory were fairly minimal.

The third law reappeared in 1620 in the second part of Kepler’s *Epitome Astronomiae Copernicanae*, a textbook of heliocentric astronomy written in the form of a question and answer dialogue between a student and a teacher.

How is the ratio of the periodic times, which you have assigned to the mobile bodies, related to the aforesaid ratio of the spheres wherein, those bodies are borne?The ration of the times is not equal to the ratio of the spheres, but greater than it, and in the primary planets exactly the ratio of the 3/2th powers. That is to say, if you take the cube roots of the 30 years of Saturn and the 12 years of Jupiter and square them, the true ration of the spheres of Saturn and Jupiter will exist in those squares. This is the case even if you compare spheres that are not next to each other. For example, Saturn takes 30 years; the Earth takes one year. The cube root of 30 is approximately 3.11. But the cube root of 1 is 1. The squares of these roots are 9.672 and 1. Therefore the sphere of Saturn is to the sphere of the Earth as 9.672 is to 1,000. And a more accurate number will be produced, if you take the times more accurately.[3]

Here the third law is not buried in a heap of irrelevance but it is not emphasised in the way it should be. If Kepler had presented the third law as a table of the values of the orbit radiuses and the orbital times and their mathematical relationship, as below[4], or as a graph maybe people would have recognised its significance. However he never did and so it was a long time before the full impact of the third law was felt in astronomical community.

The real revelation of the significance of the third law came first with Newton’s *Principia Mathematica*. By the time Newton wrote his great work the empirical truth of Kepler’s third law had been accepted and Newton uses this to establish the empirical truth of the law of gravity.

In Book I of Principia, the mathematics and physics section, Newton first shows, in Proposition 11[5], that for a body revolving on an ellipse the law of the centripetal force tending towards a focus of the ellipse is inversely as the square of the distance: i.e. the law of gravity but Newton is not calling it that at this point. In Proposition 14[6] he then shows that, *If several bodies revolve about a common center and the centripetal force is inversely as the square of the distance of places from the center, I say that the principal latera recta of the orbits are as the squares of the areas which bodies describe in the same time by radii drawn to the center.* And Proposition 15[7]: *Under the same supposition as in prop. 14, I say the square of the periodic times in ellipses are as the cubes of the major axes*. Thus Newton shows that his law of gravity and Kepler’s third law are equivalent, although in this whole section where he deals mathematically with Kepler’s three laws of planetary motion he never once mentions Kepler by name.

Having established the equivalence, in Book III of *The Principia*: *The System of the World* Newton now uses the empirical proof of Kepler’s third law to establish the empirical truth of the law of gravity[8]. Phenomena 1: *The circumjovial planets, by radii drawn to the center of Jupiter, describe areas proportional to the times, and their periodic times—the fixed stars being et rest—are as 3/2 powers of their distances from that center*. Phenomena 2: *The circumsaturnian planets, by radii drawn to the center of Saturn, describe areas proportional to the times, and their periodic times—the fixed stars being et rest—are as 3/2 powers of their distances from that center*. Phenomena 3: *The orbits of the five primary planets—Mercury, Venus, Mars, Jupiter, and Saturn—encircle the sun*. Phenomena 4: *The periodic times of the five primary planets and of either the sun about the earth or the earth about the sun—the fixed stars being at rest—are as the 3/2 powers of their mean distances from the sun*. “This proportion, which was found by Kepler, is accepted by everyone.”

Proposition 1: *The forces by which the circumjovial planets are continually drawn away from rectilinear motions and are maintained in their respective orbits are directed to the center of Jupiter and are inversely as the squares of the distances of their places from that center*. “The same is to be understood for the planets that are Saturn’s companions.” As proof he references the respective phenomena from Book I. Proposition 2: *The forces by which the primary planets are continually drawn away from rectilinear motions and are maintained in their respective orbits are directed to the sun and are inversely as the squares of the distances of their places from its center*. As proof he references the respective phenomenon from Book I:

One of the ironies of the history of astronomy is that the general acceptance of a heliocentric system by the time Newton wrote his *Principia *was largely as a consequence of Kepler’s *Tabulae Rudolphinae* the accuracy of which convinced people of the correctness of Kepler’s heliocentric system and not the much more important third taw of planetary motion.

[1] *Sesquialterate* means one and a half times or 3/2

[2] *The Harmony of the World* *by Johannes* *Kepler*, Translated into English with an Introduction and Notes by E.J. Aiton, A.M. Duncan & J.V. Field, Memoirs of the American Philosophical Society Held at Philadelphia for Promoting Useful Knowledge, Volume 209, 1997 pp. 411-412

[3] Johannes Kepler, *Epitome of Copernican Astronomy & Harmonies of the World*, Translated by Charles Glenn Wallis, Prometheus Books, New York, 1995 p. 48

[4] Table taken from C.M. Linton, *From Eudoxus to Einstein: A History of Mathematical Astronomy*, CUP, Cambridge etc., 2004 p. 198

[5] Isaac Newton, *The Principia*: *Mathematical Principles of Natural Philosophy*, A New 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, Berkley, Los Angeles, London, 1999 p. 462

[6] Newton, Principia, 1999 p. 467

[7] Newton, Principia, 1999 p. 468

[8] Newton, Principia, 1999 pp. 797–802

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One of the ironies of the history of astronomy…Ironic indeed, since the Tabulae Rudolphinae would make the same observational predictions under a geoheliocentric version of Kepler’s first two laws. It’s unclear to me why their accuracy helped turn the tide of opinion. From a strictly logical standpoint, it should have had no impact. The only explanation that comes to mind is prestige rubbing off.

Also ironic—or maybe just a curiosity—Kepler gave a physical justification for his first two laws. The physics was custom-built and Aristotelian at heart (forces proportional to speeds, not to accelerations), but still… But (afaik) he never suggested any physical basis for the third law, which has, as you point out, major implications for physics.

The tables turned the tide of opinion because up to and including Kepler that was the principle function of astronomical systems. Astronomical systems were required to provide a method of accurately predicting the positions of the celestial bodies at any given time and ephemerides or planetary tables fulfilled this function. In Kepler’s case the acceptance of his version of heliocentricity was initially almost certainly purely an instrumentalist one but gradually over time became a realist one.

The rest of Book IV of

Epitome Astronomiae Copernicanis devoted to a long explanation of the physical explication of the third law. He starts with an explanation and rejection of all previous explanation of what moves the planets and then goes on to explain his own theory. I am not going to try and reproduce it here in a comment but it starts with the sun rotating and a force going out from the sun, which sweeps the planets around their orbits.So it seems it was prestige rubbing off. “Instrumentalist”, i.e., it gave the most accurate predictions. “[G]radually over time became realist”, i.e., the credibility gained from the predictions spilled over into acceptance of Kepler’s heliocentric convictions, even though logically speaking the one doesn’t entail the other. In particular, the parallax objection, or Tycho’s stellar disk objection, lost their persuasiveness. Not that anyone had devised a more compelling rebuttal to them in the interim.

Physical explanations: my goof! Too long since I’ve read Stephenson’s

Kepler’s Physical Astronomy.There’s a tension between the 2nd law and the 3rd: they give

differentrelations between distance and speed. In Newtonian mechanics, this is resolved by the different total energies (potential + kinetic) for different orbits.Kepler also needed to explain away the discrepancy. His “sweeping” force (Kepler’s “species”) gave the 2nd law. To resolve the difficulty with the 3rd law, Kepler assumed that the densities of the planets decreased as the square root of the semi-major axis of the orbit.

When I said “different energies for different orbits”, I should have said “different energies per unit mass”.

My Comment: I always enjoy reading these articles about the history of mathematics. I confess to being a bit confused about the latest one, however, concerning the statement about the equivalence of Kepler’s Third Law with Newton’s Universal Law of Gravity.

400 Years of The Third Law: “What the third law gives us is a direct mathematical relationship between the size of the orbits of the planets and their duration, which only works in a heliocentric system. In fact as we will see later it’s actually equivalent to the law of gravity.”

My Comment: I will have to see how the article justifies this, since the standard statement (based on Toeplitz, Calculus A Genetic Approach, 1963) is that given Newton’s three Laws of Motion, then Kepler’s Three Laws are equivalent to Newton’s Universal Law of Gravity. That is, it is not Kepler’s Third Law that is equivalent to Newton’s Law of Gravity, but all three (assuming Newton’s Laws of Motion). Actually, Kepler’s First Two Laws are equivalent to Newton’s Law (massless, acceleration version: d^2r/dt^2 = –k/r^2, k > 0, along the line between sun and planet), and assuming Kepler’s first two Laws (equivalently Newton’s Law), then Kepler’s Third Law is equivalent to k being constant for all planets. (k may still depend on the sun.) Newton’s Laws of Motion are needed when masses are involved.

400 Years of The Third Law: “In Book I of Principia, the mathematics and physics section, Newton first shows, in Proposition 11[5], that for a body revolving on an ellipse the law of the centripetal force tending towards a focus of the ellipse is inversely as the square of the distance: i.e. the law of gravity but Newton is not calling it that at this point. In Proposition 14[6] he then shows that, If several bodies revolve about a common center and the centripetal force is inversely as the square of the distance of places from the center, I say that the principal latera recta of the orbits are as the squares of the areas which bodies describe in the same time by radii drawn to the center. And Proposition 15[7]: Under the same supposition as in prop. 14, I say the square of the periodic times in ellipses are as the cubes of the major axes. Thus Newton shows that his law of gravity and Kepler’s third law are equivalent, although in this whole section where he deals mathematically with Kepler’s three laws of planetary motion he never once mentions Kepler by name.”

My Comment: Prop 14 involves k, where the constant time rate of change of area dA/dt = ab pi/T means k = 4(dA/dt)^2/p = 4pi^2 a^3/T^2 where p = b^2/a is the semi-latus rectum, b is the semi-minor axis, a is the semi-major axis, and T is the period of revolution around the ellipse. The assumptions in Prop 14 actually involve the equivalence of Kepler’s first two laws with the acceleration version of Newton’s law of gravity, which in turn yield the above equations for a single planet, where the constant factor k depends a priori on both the sun and the planet (but not the distance between them). Prop 14 is then saying that k actually does not depend on the particular planet and is the same for all planets around the sun (though it may still depend on the sun). Prop 15 says under the same assumptions as Prop 14 that Kepler’s third law holds. But this is not quite correct (which seems to be Newton’s fault), since these Props as stated do not say that given Kepler’s first two laws (or equivalently, Newton’s law, acceleration version), then Kepler’s third law is equivalent to k being constant for all planets, which is the actual situation as shown in the equations above (again see Toeplitz, Calculus A Genetic Approach, 1963). And they certainly do not say that Kepler’s third law is equivalent to the entire acceleration version (or even the full mass version) of Newton’s law of gravity.

As Quine and others have pointed out, scientific theories (especially in physics) are webs of interconnected statements, so these equivalences are all suspect in a purely logical sense.

But let’s take a more relaxed attitude. Even in pure mathematics, one says that two forms of a theorem are “equivalent” if the deductions of one from the other, in both directions, are much shorter and simpler than the proofs from scratch. From the standpoint of mathematical logic, all theorems of a theory are equivalent.

It’s pretty common to say that Kepler’s second law is just the conservation of angular momentum, plus the assumption that gravity is a central force. Now, technically, angular momentum conservation follows from Newton’s second and third laws (and his 1st law is just a special case of his 2nd), but I think this traditional statement is quite illuminating.

For the special case of circular orbits, the deduction of the k/r^2 dependence (with k the same for all planets, as you emphasize) from the formula for centripetal force, plus K’s 3rd law, is just a couple of lines of algebra. And the planetary orbits are nearly circular. The centripetal force law was already known, thanks to Huygens (and independently Newton, though as usual, he didn’t publish till later). So as far as intellectual history goes, I think it’s legitimate to describe K’s 3rd law and N’s law of gravity as “basically the same”.

Now for Kepler’s 1st law vis-a-vis Newton law of gravity. Pretty much all you need to derive “inverse square law” from “particle in an elliptical orbit in a central force field, with the center of field at one focus (*)”, is the conservation of angular momentum, in the form of K’s 2nd law. And in principle it’s easy to see why something like this must be true. You have the path of the orbit; the 2nd law thus gives position as a function of time; all you need to do is take a second derivative! (Of course the details are not so simple, and that’s not what Newton’s derivation looks like.) So I think saying that Newton showed that the inverse square law, for a single planet, follows from K’s first law, is a permissible shorthand.

(*) This second clause is essential; we also get elliptical orbits with a linear force law, but the center of the force field is the center of the ellipse.

Finally, John Baez has a nice post about Newton’s work on the inverse

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