# What Kepler and Newton really did.

This has been a good week[1] for people getting the history of astronomy in the seventeenth century wrong. Darin Hayton drew my attention to what is basically a rather good article by John O’Neil in the New York Times on the equation of time and the difference between local time measured by the sun and standard time measured by the clock. The article is just fine except for one sentence that instantly awoke the Histsci Hulk in me.  The author wrote:

The changes in the solar time follow a different cycle. In the early 1600s Kepler discovered that planets move faster at the part of their orbit that is closest to the sun, the perihelion. For Earth, perihelion comes a little after the winter solstice, so from November on, Earth is accelerating. [my emphasis]

Kepler didn’t discover that planets (actually in the case under discussion the sun) move faster in some part of their orbits he suggested a new, and as it turned out correct, solution to explain the differences in speed in the various segments of planetary orbits; a phenomenon that had been know about for at least two thousand years.

In what follows I will mostly talk about the sun. In ancient Greek astronomy explanations of the suns apparent progression throughout the year were governed by the so-called Platonic axioms. All celestial motion including planetary motion, and the sun was considered to be one of the seven planets, was circular and uniform. These axioms as the name suggest go back at least to Plato in the fourth century BCE and probably to Empedocles in the fifth century BCE. The only problem was that planetary motion was obviously neither circular nor uniform. The major problem of Greek astronomical thinking was therefore to fit the observed facts to the a priori theory rather than to develop a theory to fit the facts. This has become know as saving the phenomena. Already in the second century BCE Hipparchus knew, and he probably wasn’t the first to do so, that the seasons measured from equinox to solstice and from solstice to equinox differed in length, whereas if the sun’s orbit were truly circular they should be equally long thus demonstrating that if the segments were equally long as the circular orbit demanded the speed of the sun during its orbit must vary i.e. it was not uniform. Over the centuries various Greek astronomers came up with various geometric models to explain away this anomaly peaking in the epicycle deferent model of Ptolemaeus in the second century CE. This model being further modified, that is improved, by various Islamic astronomers in the Middle Ages. Then along came Johannes!

Using the new more accurate data of Tycho Brahe Kepler, after much calculation and even more soul searching, abandoned the Platonic Axioms and determined the planetary orbits to be ellipses and not circles and the speed of the heavenly bodies to be non-uniform but to follow his second law of planetary motion. As stated above didn’t make the discovery that the speed of planets vary during their orbits he just found the correct explanation for it.

What exactly Kepler did or didn’t do cropped up in another post this time on Chad Orzel’s Uncertain Principles blog. Chad wrote a nice post about the relative merits of theoretical and experimental physics, basically complaining correctly that people tend to underrate experimental physics. The misconceptions about seventeenth century astronomy turned up in the comments column. Commentator Peter Morgan wrote:

I suppose there is a difference between Ptolemy, Copernicus and Kepler that is not much connected to experiment, which are largely different models for more-or-less the same experimental data (that is, all models depend on there being data *to* model, but sometimes there are different models for more-or-less the same data). Even Newton didn’t have that much more experimental data to work with; new data mostly came after him, when it was partly his theories that gave physicists the tools to imagine and construct new apparatus.

This is of course fundamentally wrong as was pointed out to him by Steinn Sigurdsson:

Kepler had Brahe’s data, which was a qualtitative improvement on previous quantifications. Newton most certainly had access to new experimental data, notably that of Kepler and Galileo, but Halley! Newton also did his own experiments, eg in optics.

To add to the fun Eric Lund decide to have his tuppence worth:

@Peter: Your argument about Ptolemy vs. Copernicus might be granted, but by the time Kepler started looking at things, Galileo had disproven Ptolemy’s model–Galileo, using the recently invented telescope, observed phases of Venus that Ptolemy’s model predicted would never occur. And as Steinn points out, Kepler had new data that were not available to Ptolemy or Copernicus–data which in fact disproved the notion (adhered to based on philosophical arguments) that the orbits of celestial bodies were necessarily circular. Without Kepler’s data analysis, Newton probably would not have come up with the inverse square law.

Taken together the three statements contain quite a few errors, which I will now attempt to correct.

Firstly seventeenth century astronomical theory isn’t based on experimental data but on observational data, which isn’t really the same thing. Steinn is perfectly correct to point out that Kepler had a completely new set of observational data, supplied by Tycho Brahe, on which to base his theories; a fact that separates him from Ptolemaeus and Copernicus. The data however does not, as Eric claims, disprove the notion of circular orbits. If Kepler had kept to the Platonic Axioms he could have, using Tycho’s data, provided circular models to save the phenomena. His obsession with accuracy led him to abandon the Platonic Axioms when he realised he could get a better fit with ellipses, a move that cost him an immense amount of soul searching. If he had had a more advanced set of mathematical tools (Fourier Analysis!) he could have obtained exactly the same level of accuracy with an epicycle deferent model.

Turning to Newton we have from all three commentators a lot of confusion concerning the data and theories available to him when he wrote his Principia.

Peter is of course wrong as Newton did have substantially new data, which I will explain in a minute. Steinn is wrong as he did not have any experimental data from Kepler but he did have Galileo’s theory of parabolic motion, his laws of fall and the law of inertia, which he falsely believed came from Galileo. These are theories and laws derived from experimental data but not in themselves experimental data. The data that Newton had and which were central to his theories was that on the orbits of the moons of Jupiter and Saturn. Although Galileo and Marius had supplied the original data on the moons of Jupiter, Newton’s source on both sets of moons was the much more up to date and accurate data of Cassini. Newton of course also had access to the new and considerably more accurate observational data of John Flamsteed.

The data on the orbits of the moons of Jupiter and Saturn provided empirical proof of Kepler’s third law of planetary motion, something that Newton explicitly states in the Principia, this law playing a central role in Newton’s argumentation for his theory of universal gravity, of which more in a minute.

Eric Lund muddies the water with his claim that Galileo’s observations of the phases of Venus (Ptolemaeus’ theory also predicts phases for Venus but they are different to the ones observed) were carried out before “Kepler started looking at things”. Kepler did the work on his Astronomia Nova containing his first two laws of planetary motion between 1600 and 1606 although the book itself was first published in 1609. Galileo, Harriot and Marius first started astronomical telescopic observations in 1609 and the first publication of such observations was Galileo’s Sidereus Nuncius in 1610. It’s difficult to date exactly but the first observations of the phases of Venus are later than the publication of the Sidereus Nuncius.

Also as should be well known to diligent readers of this blog Newton didn’t “come up with the inverse square law”, as claimed by Eric Lund. That honour goes to Ismael Boulliau. I think I’ve probably said this before but its worth repeating Newton’s great achievement was in showing that under the assumption of his three laws of motion the inverse square law of gravity implies Kepler’s third law and under the same assumption Kepler’s third law implies the inverse square law of gravity, i.e. the inverse square law of gravity and Kepler’s third law are, under this assumption, equivalent. As Kepler’s third law had been proved to be empirically valid, remember those Jupiter and Saturn moons, it follows that the law of gravity is also (empirically) valid.

Given that the story of the so-called astronomical revolution is probably the most often told and repeated piece of the history of science I find it sad that even educated people mostly have a very vague and largely inaccurate idea of what actually took place.

[1] I actually wrote this post a couple of weeks ago and have only now got round to posting it. There are a couple of others in the pipeline, too.

### 14 responses to “What Kepler and Newton really did.”

1. “The data on the orbits of the moons of Jupiter and Saturn provided empirical proof of Kepler’s third law of planetary motion, something that Newton explicitly states in the Principia”

Do you have the reference for this in the Principia? I am interested to read what exactly Newton wrote about Kepler.

By the way, a while back I also asked about reference to Boulliau. Do you have that reference too?
https://thonyc.wordpress.com/2011/09/28/the-man-who-inverted-and-squared-gravity/#comment-2244

Very interesting articles, thanks.

2. Peter L. Griffiths

Kepler obtained his laws of planetary motion from recorded observations both of the velocity at a moment of time as well as of the average velocity of the whole orbit round the Sun. Newton’s inverse square law is consistently wrong because in attempting to combine Kepler’s distance law with Galileo’s law of falling bodies, Newton fails to recognise that Kepler’s definition of distance is the inverse of Galileo’s definition of distance.

3. I hope you don’t me being persnickety; it seems in keeping with the spirit of this post to pick nits.

Kepler didn’t discover that planets (actually in the case under discussion the sun) move faster in some part of their orbits … Already in the second century BCE Hipparchus knew … that the seasons measured from equinox to solstice and from solstice to equinox differed in length, whereas if the sun’s orbit were truly circular they should be equally long

Only if the earth were at the center of the sun’s orbit. Ptolemy’s model for the sun’s orbit was the “simple eccentric”, that is, the sun moved at a uniform speed in a perfect circle (no epicycle), with the center of the orbit displaced (“eccentric”) from the center of the earth. (Ptolemy did show the geometrical equivalence of this to a special kind of deferent-epicycle model.)

For all the other planets, Ptolemy had (at least) an eccentric deferent, an epicycle, and an equant. (The moon and Mercury required even more machinery.) The equant is the part that makes motion along the deferent non-uniform. Ptolemy’s omitted the equant only for the sun.

All astronomers before Kepler adopted essentially the same solar model — or model of the earth’s motion, in the heliocentric version. Kepler bridled at the idea that the earth should be special — if the other planets required equants, so should the earth. (Kepler started off by using equants, only later switching to what ultimately became his area law.)

So Kepler did discover that the sun moved faster in parts of its orbit (speaking geocentrically). On the other hand, it is quite correct to say that this non-uniformity for the “non-sun” planets was old news.

His obsession with accuracy led him to abandon the Platonic Axioms when he realised he could get a better fit with ellipses, a move that cost him an immense amount of soul searching.

As both Stephenson and Voelkel point out, Kepler was convinced in the non-uniformity of the planets’ motion well before he met Tycho — convinced by a combination of theological and physical arguments. He never had any commitment to the Platonic Axioms. Other astronomers praised how Copernicus had replaced the equant with another epicycle (a trick discovered earlier by Ibn al-Shatir), but Kepler rejected it as physically meaningless. Kepler from the start directed his attention to the actual path of the planet, which was clearly neither circular nor traversed uniformly.

The “obsession with accuracy” was only one factor leading to the elliptical orbit, and arguably not the most important. Curtis Wilson wrote:

the choice between ellipse and other oval shapes was, as far as the observations could show, a matter for conjecture. Kepler, of course, had reasons for his choice: a causal [i.e., physical] account which led both to the elliptical orbit and to the planet’s motion on that ellipse, with close agreement between the theoretical prediction and observation in the particular case of Mars.

As for Bouilliau, I’ve picked this nit before, and you’ve pushed back. I refer the interested reader to my comments in the linked post, and more recently to my comment to your post “Discovery is a process not an act”.

• You are confusing the knowledge that the sun moves at different speeds at different times of the year (when observed from the earth), a fact that was well known to Greek and later astronomers since at least the time of Hipparchus, as already stated, with mathematical models, in this case that of Ptolomaeus, created to explain this phenomenon away in terms of the so-called Platonic axioms, i.e. saving the phenomenon.

• I’m not confused on this point. Perhaps I did not interpret what you wrote as you intended.

When the New York Times author writes, “Kepler discovered that planets move faster at the part of their orbit that is closest to the sun”, his statement is entirely correct for the actual motion of the earth.

True enough, the reporter doesn’t make it clear that the ancient Greek astronomers had an observationally adequate and even partially correct explanation for the non-uniform apparent motion of the sun. Maybe that’s what woke HistSci Hulk?

Still, the reporter was not wrong. I would guess that he (or more likely the two sources he mentions) didn’t want to get into the distinction between the optical and physical equations, i.e., apparent and actual motion.

When I read your paragraph on the Platonic axioms, I found this same distinction even more unclear. You write,

if the sun’s orbit were truly circular [the seasons] should be equally long thus demonstrating that if the segments were equally long as the circular orbit demanded the speed of the sun during its orbit must vary i.e. it was not uniform.

This didn’t sound like a correct description of the model of the sun’s (*) motion used by all astronomers from Hipparchus up to Kepler: the sun moves in a perfect circle at a perfectly uniform speed, and all apparent non-uniformity can be explained by the displacement of the earth from the center of the circle (i.e., the eccentricity). Did you mean “apparent speed”?

(*) (Or earth’s, speaking heliocentrically.)

Kepler’s innovation in making the sun’s/earth’s actual motion non-uniform met with more resistance than his ellipse. Heilbron devotes over 10 pages to this topic in The Sun in the Church (p.102-112). The issue was finally settled in 1655 by Giovanni Domenico Cassini and his collaborators, using the heliometer in the Cathedral of San Petronio in Bologna.

4. “…and all apparent non-uniformity can be explained…” Observed not apparent and explained mathematically but not in real accord with the Platonic axioms as was criticised by many later Islamic and European astronomers.

Put bluntly the sun, as observed, does not move in uniform circles a fact well known in European and Islamic astronomy for two thousand years before Kepler. Kepler’s great achievement was no longer to attempt to make it look as if it did so.

• I have to insist here: according to Hipparchus, Ptolemy, the Islamic astronomers, Tycho, and (switching earth with sun) Copernicus, the actual path of the sun was a circle traversed uniformly. The apparent motion (i.e., the motion as observed from Earth) was not uniform. (What do you mean by “Observed not apparent”?)

The criticism you refer to (“not in real accord with the Platonic axioms”) concerned Ptolemy’s equant, not the solar theory. The equant was, I repeat, not used in the solar theory. I’ll give a few references for this at the bottom of this comment.

Followers of Aristotle disliked the very idea of non-homocentric spheres, but it doesn’t sound like you have that in mind here. I’d be very interested to see quotes from astronomers criticizing the Hipparchus/Ptolemy solar theory as violating the Platonic axioms.

Olaf Pedersen, A Survey of the Almagest, chap 5.

James R. Voelkel, The Composition of Kepler’s Astronomia nova, p.6. (Note that “bisecting the eccentricity” was jargon for “using an equant”.)

James Evans, “On the function and the probable origin of Ptolemy’s equant.” Am.J. Physics 52 (1984) p.1081 (available from his website). Or see his book The History and Practice of Ancient Astronomy, OUP 1998.

J.L. Heilbron, The Sun in the Church p.103

5. Peter L. Griffiths

In 1604 Kepler did discover the pins and thread method of locating the foci of a cylindric section. This method can also be applied slightly less accurately to an ellipse by applying the formula f= a – [a^2 – b^2]^0.5, where f is the focus, a is the major axis and b is the minor axis.

• Peter L. Griffiths

A slight amendment to my comment of 20.12.16. a should be half the major axis, and b should be half the minor axis. Also Kepler’s definition of
ellipse bears no resemblance to Andrew Wiles’s in his title to his proof of Fermat’s Last Theorem.

6. Peter L. Griffiths

Further to my comments of 20 Dec 2016 and 30 December 2016, I find it extraordinary that modern astronomers fail to mention the foci in their detailed calculations of black holes.