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Kepler apparently arrived at his first two laws based on the Tycho's data for Mars. But Mars has the largest eccentricity except for Mercury, so it is easier to tell the difference between a circle and an ellipse. What about other planets, like Venus? It seems difficult, its eccentricity is only 0.006772. Wouldn't it be very hard to tell a circle from an ellipse with Tycho's data? Did Kepler really establish the two laws based on a single orbit of Mars?

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  • $\begingroup$ First and Second Laws was discovered when studying Mars. For the Third Law he needed all planets. $\endgroup$ – Alexandre Eremenko Mar 14 '17 at 23:57
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This is a natural question given Kepler's own account of his discoveries in his most famous astronomical work, Astronomia Nova. It is a fascinating story of of how he arrived at the idea of the first two planetary laws (orbits are ellipses with the Sun at a focus, radius vector sweeps equal areas in equal times) by trying to fit Tycho's data on Mars first to epicycles, then to ovals, and finally to ellipses. The specific difficulty in the OP would not have been an issue: even if it was impossible to tell a circle from an ellipse for Venus and Mercury circle is a particular case of ellipse, so both laws would hold for a planet uniformly moving along a circle with the sun at its center. However, anomalies of the inner planets were known already at the time of Ptolemy, let alone Tycho, see Duke's The Ancient Values of the Planetary Parameters of Venus and Mercury. More importantly, modern scholarship indicates that Astronomia Nova is not a realistic description of how Kepler made his discoveries, it is a carefully crafted rhetorical fiction.

Kepler was a devoted heliocentrist long before 1609 when Astronomia Nova was published, and metaphysical and mystical speculations played as much (if not more) of a role in his conviction than empirical data. He was intimately familiar with the mechanics of the Copernican system and its relation to the geocentric ones. Copernicus had circular orbits, and therefore still required epicycles, although they were much smaller than Ptolemy's (and sometimes referred to as epicyclets), see The Astronomy and Cosmology of Copernicus. Kepler was discussing the fine points of this mechanics along with his own improvements for various planets with his like-minded teacher from Tübingen, Maestlin, before even working for Tycho. See Michael Maestlin's Account of Copernican Planetary Theory by Grafton for his correspondence with Kepler on various planets at the time Kepler's Mysterium Cosmographicum was published (1596).

The area law is implicitly present in the epicyclic astronomy (to the first order of approximation) through the artifice of Ptolemy's equant, placing the point from which a planet's motion looks uniform off-center. Copernicus replaced the equant with a pair of epicyclets, and Kepler realized early on that it closely approximates "equal areas in equal times" when plotting the data for various planets, long before he came up with ellipses. In other words, his basis for the two laws was not just one orbit of Mars, although that alone provided quite a bit of data points to explain by two simple principles, but the entirety of epicyclic astronomy, including Copernicus, with all the known anomalies, and technical devices used to deal with them. As Voelkel writes in The Composition of Kepler's Astronomia Nova:

"Kepler named as one of the works he would produce his Commentaries on Mars--that is, the Astronomia nova. He was thus forced to conceive the book as a preliminary announcement of the fruits of his physical astronomy as applied to the orbit of Mars. It would contain his important finding regarding the bisection of the earth's eccentricity, which vindicated his physical account of the cause of the equant as well as clearing up certain problems in the orbit of Mars (and the orbits of Mercury and Venus as well). At that time, however, he had no clear idea of what the eventual solution to Mars's orbit would be. Although he was employing a form of the area law, the discovery of Mars's elliptical orbit was still two and a half years away."

Although the orbit of Mars was a catalyst for many developments they were accompanied with looking at the data for other planets, if for no other reason than to answer to Kepler's various correspondents and critics. Nor was empirical data his sole guide, his physical, metaphysical and mystical arguments for heliocentrism were deliberately pushed into the background by the circumstances at the time of the writing. Here is Voelkel again:

"Recent research, especially that of William H. Donahue, has shown that the account Kepler offers his readers is not a true history of the course of his research--something Kepler never claimed--but is rather a didactic or rhetorical pseudohistory... Astronomia Nova is only accidentally modern--that is, that the particular context in which the book was composed forced Kepler to rein in his broader arguments for heliocentrism, leaving only a subset of his physical reasoning that appears distinctly modern in retrospect.

[...] the argument of the book was a response to the various criticisms he had encountered during the course of his research. To the charge that his physical astronomy was an unjustified aberration, he responded by constructing his argument to make it appear as though he resorted to a physical approach to planetary theory only after a comprehensive failure of the most general kind of model in the classical form (which he presented in part 2, even though he actually completed the research only after parts of the research presented in part 3). He countered the charge that his radical innovations were themselves the source of the difficulties he had encountered by repeating many of the demonstrations in the book (as with the repeated demonstrations involving the true and the mean sun)... Thus, many features of the Astronomia Nova become comprehensible only when they are viewed in the context of Kepler's experience in writing the book as elements of an elaborate and purposefully-constructed rhetorical argument."

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Let me add a couple of remarks to Conifold's response, based on Voelkel's book (mentioned by Conifold), and the earlier book Kepler's Physical Astronomy by Bruce Stephenson.

Kepler did indeed come up with his first two laws based just on the orbit of Mars. The discovery was convoluted and not purely empirical, and Stephenson and Voelkel give a detailed account. However, once Kepler had his first two laws, he immediately generalized them to all the planets, first in his Epitome of Copernican Astronomy, and then in his Rudolphine Tables.

Although the empirical case for elliptical orbits for the other planets was weak, Kepler belief in harmony and uniformity led him to extend his laws to all the planets. In addition, the Astronomia Nova presents physical explanations for the first two laws; naturally, he assumed that the same physics would lead to the same results. (His physical explanations convinced almost nobody and rapidly disappeard from history.)

Even for Mars, the data was not that strong. As Curtis Wilson says in "Predictive astronomy in the century after Kepler" (in The General History of Astronomy (vol. 2): Planetary astronomy from the Renaissance to the rise of astrophysics, ed. René Taton and Curtis Wilson):

Because of the level of unavoidable error in observations of position, and the near circularity of the orbits, the departure from circularity could be detected observationally only in the orbits of Mars and Mercury, where the eccentricity was exceptionally large; and even in these cases 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. Ironically for Kepler, it was the ellipse that caught on most quickly and widely, as a simple curve that could be produced by a combination of circular motions, and as able to account for the bisection of the eccentricity and for the observably oval paths of Mars and Mercury; while Kepler's rationale of the planetary motions was all but universally rejected. The Keplerian ellipse became emblematic of the new astronomy.

As Wilson notes, several other Keplerian innovations (e.g., his treatment of the "mean sun") made a much bigger impact purely in terms of accuracy, although conceptually they were less important.

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