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Scientific Understanding in Astronomical Models from Eudoxus to Kepler

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Current Debates in Philosophy of Science

Part of the book series: Synthese Library ((SYLI,volume 477))


In the following essay I present a narrative of the development of astronomical models from Eudoxus to Kepler, as a case-study that vindicates an insightful and influential recent account of the concept of scientific understanding. Since this episode in the history of science and the concept of understanding are subjects to which Professor Roberto Torretti has dedicated two wonderful books—De Eudoxo a Newton: modelos matemáticos en la filosofía natural (2007), and Creative Understanding: philosophical reflections on physics (1990), respectively—this essay is my contribution to celebrate his outstanding work and career in this volume. I dedicate this piece to Roberto, dear friend and mentor, in gratitude for all his inspirational work and personal support, which has greatly helped me, and many others, to better understand that human wonder we call scientific knowledge.

Sadly, during the preparation for publication of this volume, Professor Torretti passed away. May this essay be my contribution to honor and cherish his memory as an outstanding scholar, generous mentor, and dear friend.

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  1. 1.

    For a conceptual overview of philosophical stances on scientific explanation, see Woodward (2019).

  2. 2.

    Torretti (1990) is an early approximation to understanding as a central goal of science. For an overview of the different contemporary stances on understanding, see de Regt and Baumberger (2020). See also the articles in de Regt et al. (2009), and in Grimm et al. (2017).

  3. 3.

    In this conception of scientific objectivity, empirical adequacy and internal consistency are also values. That is, although both are basic values that transcend revolutionary changes in the history of science, there are always contexts in which they can be traded for other values. As we will see below, scientific understanding can be obtained from false theories. For a treatment of inconsistency in scientific theories, see Vickers (2013) and Frisch (2014).

  4. 4.

    As we will see below, false models can be successful, and therefore convey UP. However, even in these cases some degree of empirical adequacy is involved.

  5. 5.

    Another fixed point around which the stars are seen to rotate clockwise is observed from the Southern Hemisphere. The main characters in our story were inhabitants of the Northern Hemisphere, so we will adopt their perspective.

  6. 6.

    The sidereal day must be distinguished from the solar day: the time it takes the Sun to return to the same local meridian (the time between consecutive noons). The solar day thus defined is variable along the year. Using the idealized mean Sun (see below for the distinction between the real and the mean Sun), the solar day can be defined to last 24 h.

  7. 7.

    Since the observed motion of the Moon, Mercury, Venus, the Sun, Mars, Jupiter and Saturn is not as patently regular as the motion of the stars, the ancient Greeks named them ‘planetai’, i.e., wanderers.

  8. 8.

    The numerical values for the parameters in Eudoxus’ model are contemporary reconstructions. Neither Aristotle nor Simplicius included precise values in their mostly qualitative explanations of the model.

  9. 9.

    Eudoxus included a third sphere, with its axis inclined a small angle with respect to the second sphere’s axis. It is generally affirmed that he wrongly attributed the Sun a small latitudinal motion with respect to the ecliptic. However, Linton (2004, 28) affirms that in Eudoxus’ times the ecliptic was vaguely defined as some great circle within the zodiac in the celestial sphere. The definition of the ecliptic in terms of the motion of the Sun, Linton states, was introduced about two centuries later by Hipparchus. If this is correct, Eudoxus’ third sphere was justified. Support for this interpretation (see Neugebauer, 1975, 633) lies on the fact that an observational determination of the precise path of the Sun with respect to the background stars was very difficult, so an estimation of a 1/15 part of a circumference, i.e., 24°, naturally suggests. Once established this pseudo-ecliptic plane, more precise observations of the path of the Sun would lead to the introduction of some latitudinal motion with respect to the pseudo-ecliptic.

  10. 10.

    Or perhaps 24°, see footnotes 9 and 10.

  11. 11.

    Elongation at opposition is not exactly 180°, and elongation at conjunction is not exactly 0°, due to the fact that along their orbits of planets show a small latitudinal distance from the ecliptic (although they always stay within the zodiac).

  12. 12.

    From a heliocentric perspective, for interior planets a distinction between superior and inferior conjunction can be traced. An interior planet is in inferior conjunction when it lies between the Earth and the Sun, and in superior conjunction when the Sun lies between the planet and the Earth. The retrograde motion of Mercury and Venus is observed near inferior conjunction. From a geocentric perspective, as in Eudoxus’ model, this distinction cannot be traced, for the Sun cannot lie between an inferior planet and the Earth.

  13. 13.

    ∠P1OP2 = α + β + 90° and ∠P2OP3 = α – β + 90°, so that ∠P1OP3 = 2α + 180°. The Sun goes from P1 to P2 in 94 ½ = 189/2 days, and from P1 to P3 in 187. Thus, α + β + 90° = 189w/2, and 2α + 180° = 187w, solving the last two equations we get α and β.

  14. 14.

    Interestingly, the distance that Ptolemy obtained for the Sun by this method in the Planetary Hypotheses (1079 Earth radii) is very close to the value he had obtained in the Almagest by a different and independent method (1160 Earth radii). For a treatment of this issue, see Carman (2010).

  15. 15.

    The value in meters of the Greek stadion is a disputed issue, but the proposed estimations range between 150 and 200 meters.

  16. 16.

    As Kuhn (1995, 82) reports, the Arabic astronomer Al-Faraghi (800–870) applied the same method as Ptolemy in the Planetary Hypotheses. Using a value of 3250 Roman miles for the radius of the Earth, he calculated a universe radius equivalent to ca. 120.000.000 kilometers. That Ptolemy himself used this method was discovered only in 1967, when the relevant passage in the Arabic version of the Planetary Hypotheses was found and translated (see Carman, 2010 and the references therein).

  17. 17.

    The obliquity of the ecliptic is indeed variable, and it is due to gravitational perturbations of other planets. However, the real effect is much smaller than the one that Ibn Qurra deduced from the difference between his measurements and Ptolemy’s.

  18. 18.

    What Al-Zarqali discovered was the precession of the apsidal line in the Earth’s orbit. The modern estimation of apsidal precession period for the Earth is 11,6 per year. Apsidal precession is caused by concomitant factors, including gravitational planetary perturbations. A full explanation can only be given using general relativity: the precession of Mercury’s perihelion was a crucial element in Einstein’s formulation of his gravitational theory.

  19. 19.

    For an illustration and a rigorous treatment of the inter-translability between the Ptolemaic and Copernican models of Venus, see Neugebauer (1986, 497) and Barbour (2001, 239).

  20. 20.

    For a comprehensive treatment of Tycho’s work, see (Dreyer, 2014).

  21. 21.

    The figure is a simplification. The actual model includes eccentricities and equants.

  22. 22.

    For the precession of equinoxes, Tycho’s system must go back to an explanation like the one illustrated in Fig. 12.13.

  23. 23.

    Copernicus’ made extensive use of the mean Sun, as we saw above, but never as an equant.

  24. 24.

    This formula for α is obtained using mathematical techniques that were not available to Kepler. The calculations he had to make are tortuous and tedious. Furthermore, the data he had to obtain \( \overline{\alpha } \) the angle were observations for M in kinematic configurations with respect to the mean Sun, rather than to the real Sun, so Kepler had to extrapolate them. (see Linton 2004, 179).

  25. 25.

    A remarkably accurate result, for the true value of the inclination of Mars’ orbit is 1°51′. Kepler determined the nodes of the orbit of Mars from Brahe’s data. He found that the red planet returns to the same node every 687 days, which is exactly its sidereal period, and that the Sun lies on the line joining both nodes. This is evidence for Kepler’s assumption that the Sun lies on the orbital plane of each planet.

  26. 26.

    Kepler’s first law states that the planets describe elliptical orbits, with the Sun in one of the foci. His second law tells us that a straight line from the planet to the Sun sweeps out equal areas in equal times. The first two laws were formulated in the New Astronomy. The third law states that the ratio D3/T2 has the same value for all planets, where D is the planet’s average distance to the Sun, and T is its orbital period. Kepler obtained his third law in The Harmony of the Universe, published in 1619.

  27. 27.

    An interesting contextual difference is that the deferent-epicycle model is intelligible to us in a simpler and easier way because we have modern trigonometry, to which Hipparchus did not have access. His calibration of the solar model was much more tortuous than for us.


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Acuña, P. (2023). Scientific Understanding in Astronomical Models from Eudoxus to Kepler. In: Soto, C. (eds) Current Debates in Philosophy of Science. Synthese Library, vol 477. Springer, Cham.

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