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I am speculating about the value of Tycho Brahe’s - for his time - accurate observations and Kepler’s calculations.

Might it have been possible for Kepler to formulate his laws based on Ptolemy’s system, assuming he adapted the Earth’s position in a heliocentric system?

For example you might expect that the period of the planets might be known relatively accurately also from Ptolemy’s data.

Similarly, if you were ready to accept elliptical orbits, planets such as Mercury and Mars might give some input to the second law. I realize that there is a component of circular reason involved here, in that you must be open to the laws from the beginning in order to want to make such tests using Ptolemy’s system.

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    $\begingroup$ Kepler did adopt the Copernican system long before fitting Tycho's observations, but to be ready to "accept" elliptical orbits one would need to at least consider them. Nothing in pre-Keplerian astronomy suggested that, and even Kepler himself tried ovals first before fortuitously trying ellipses and getting convinced by a close match with Tycho's data. So the first law was not forthcoming, and an epicyclic version of the second was already embodied in Ptolemy's equant, see Did Kepler arrive at his planetary laws based on Mars's orbit alone? $\endgroup$
    – Conifold
    Commented Sep 9, 2020 at 4:40
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    $\begingroup$ Not so much; the fact that planets’orbit were not perfect circkes was wll known since antiquity. This the beeds of devices to account for this: eccentrics, epicycles. The idea of using a non-circular path was totally new and unexpected $\endgroup$
    – Mauro ALLEGRANZA
    Commented Sep 9, 2020 at 10:46

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In process of finding his laws Kepler tried to construct circular motions for planet orbits. The best fit disagree with observations by 8 minutes. Kepler knew that Brahe's observations could not have such errors, so he had to reject his hypothesis. Ptolemy's observation do not have such accuracy. For stars he give coordinates rounded to 20' or 15'. So yes, Brahe's data were crucial

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Brahe's observations were crucial. More than the other things that you mentioned: Copernicus system and knowledge of periods. Copernicus system indeed simplifies calculations for planets a lot, but everyone understood how to transfer in principle calculations in one system into another. Periods+Copernicus system would be enough for the discovery of the Third Law, Copernicus system is needed to find heliocentric distances.)

But the most important discovery of Kepler was the First Law, and for this one really needs observed positions at various times and with high accuracy. Kepler used Mars observations of Brahe. Without these high accuracy observations, one could not guess that the true trajectory is an ellipse rather than a circle (or a combination of circles). The deviation of this ellipse from a circle is quite small. And astronomy, since the time of Aristotle was dominated by the dogma that all motions have to be expressed in terms of uniform motions on circles. This is what everyone from Hipparchus to Copernicus and Galileo accepted without any doubts. It is demolishing of this dogma which constitutes the truly revolutionary discovery of Kepler.

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  • $\begingroup$ Lots of hand-waving and not an iota of the mathematics involved. Scientists since Euclid had a very very good understanding of the conic sections, including the ellipse, so it is not at all clear to me that your observations are worth an iota. $\endgroup$
    – Pieter Geerkens
    Commented Sep 8, 2020 at 18:14
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    $\begingroup$ @Pieter Geerkins: you completely missed the point. Scientists is antiquity knew everything about conic sections. But they missed the crucial fact that planets move on conic sections. $\endgroup$
    – Alexandre Eremenko
    Commented Sep 10, 2020 at 12:22
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Tycho' observations impacted on numerous levels. First, it presented an accurate, detailed calculation of the motion of the heavenly bodies. At the time, 2 theories existed; helio- and earth-centric. The debate, via religion gravitated to the latter. Yet, facts are stubborn things. Given data, Kepler formulated his 3 Laws. The accurate observations where the prodigy of mathematical, astronomical. Theoretical, observational, astrophysical, cosmological,terrestrial. And yes, theological sciences. To answer your question; inestimable.

See: Feynman Lectures on Physics : Law of Gravitation

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