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Some sources (modern sources, and Kepler himself) claim that in his Geo-Heliocentric (Tychonic) model, Tycho Brahe saw that the orbs of the Sun and Mars intersect, and this was one of the reasons which led him to reject the Solid Orbs:

When Tycho first devised his system, he was still thinking of celestial bodies being imbedded in solid orbs. However he found that in his geo-heliocentric system, the orb of Mars intersected the orb of the sun. This led Tycho to [reject the solid orbs]. (The Oxford Handbook of Early Modern European History, 1350-1750. p.70 ).

I have two questions (I would appreciate even reply to one of those questions):

  1. Why this could not be shown earlier using the traditional Ptolemy form. Was that Geo-Heliocentric model that was necessary, or merely new numbers(?) that Tycho plugged in?. For on first sight I can't see why; since, if there is a point in the orb of Mars that is closer to the Earth, than some other point in the orb of the Sun, why couldn't this be seen also in Ptolemy system where the epicycle of Mars must also intersect the eccentric of the Sun (If we would plug the same numbers to make the models equivalent with respect to the vision from Earth).

  2. In Kepler's Astronomia Nova he thus writes in chapter 6 (Translation William H. Donahue):

Nevertheless, in the theory of Mars, the planet's eccentric is so small in proportion to the Sun's eccentric that Mars's eccentric and the point O and F are nearer to the Earth C than is the sun S, which was one of the reasons why Brahe denied the solidity of the orbs

I'm not quite sure I understand what Kepler means by "planet's eccentric is so small in proportion to the Sun's eccentric" because how could it be that the eccentric of Mars be smaller than the Sun's?

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I can explain the principle (question 1). In Ptolemy's theory (in Almagest), the sizes of planetary orbits are not specified. They can be arbitrary, since scaling of any single orbit does not change anything from the point of view of an observer at the center. In his later work (Planetary Hypotheses) Ptolemy decided to fit the orbits as tightly as possible, so that they did not intersect. This hypothesis is not justified by any observation. Later people speculated of solid spheres which support planets.

In Brahe (and Copernicus) systems where planets revolve around the Sun, the ratios of the orbits are not arbitrary, they are dictated by the observed motion. So for example Brahe's theory gives a definite ratio of the radius of the orbit of Mars, to the radius of the orbit of Sun (I simplify, assuming that all orbits are circles, and the circles are in the same plane.) This ratio tells us that the orbits intersect. So there cannot be any solid spheres.

Copernicus and Brahe theories are equivalent. And they permit to determine the ratios of sizes of all orbits correctly. Ptolemy theory did not allow this, as exposed in Almagest these ratios can be arbitrary, and his later additional hypothesis is not justified. The main difference is that in Copernicus and Brage theory the observer (on Earth) is at a place different from the planets orbits center (Sun).

Of course, in Brahe's theory Mars never actually collides with Mars, since his theory is geometrically equivalent to the Copernicus, and thus it is essentially correct. But their orbits intersect, and this excludes any physical existence of "supporting celestial spheres".

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    $\begingroup$ Thanks. It is true that in Ptolemy the sizes can be arbitrary, but it worth noting that the ratio between the orbit size of a planet and its epicycle size, is most certainly not arbitrary; if I'm not mistaken it should be equal to the ratio of the orbit size of the planet and the Earth in the Copernican system. My mistake, that I now see thanks to your answer, was to assign the size of Epicycle to be equal to the orbit of the Sun (or more exactly vice versa) - which is indeed not necessary if we are not considering the path of the planet viewed from the Sun. $\endgroup$
    – d_e
    Sep 24 at 7:00

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