Nicolaus Copernicus (1473-1543) received a sinecure as canon at Frauenburg through the efforts of his influential uncel, the Bishob of Ernland. As a consequence, Copernicus was able to spend several years studying at Italian universities , and to pursue his project of reforming mathematical planetary astronomy. In the De revolutionibus (1543), Copernicus revised Ptolemy’s mathematical moddels by eliminating equant points and by taking the sun to be (roughly) the centre of planetary motions.
    Johannes Kepler (1571-1630) was born in the Swabian city of Weil. Was of delicate constitution, and passed an unhappy childood. Kepler found relief in his studies and his Protestant faith. At the University of Tubingen, Michael Maestlin interested him in the Copernican astronomy. The sun-centred system applead to Kepler on aesthetic and theological grounds, and he devote his life to the discovery of the mathematical harmony according to which God must have created the universe.
    In 1594 he accepted a position as teacher of mathematics in a Lutheran school at Graz. Two years later he published the Mysterium Cosmographicum, in which he stated his “nest of regular solids” theory of planetary distance. This work, like all his writings, displayed a Pythagorean commitment informed by Christian fervour. In 1600, partly to escape pressure from Catholics in Graz, Kepler went to Prague as assistant to Tycho’s observations, and for the most part tempered his enthusiasm for mathematical correlations with respect for the accuracy of Tycho’s data. Kepler published the first two laws of planetary motion in Astronomia Nova (1609), and the third law in the Harmonice Mundi (1619).



Osiander on Mathematical Models and Physical Truth

The question of proper method in astronomy was still debated in the sixteenth century. The Lutheran theologian Andreas Osiander affirmed the tradition of saving the appearances in his preface to Copernicus’s De revolutinbus. Osiander argued that Copernicus was working in the tradition of those astronomers who freely invent mathematical models in order to predict the positions of the planets. Osiander declared that it does not matter whether the planets  really do revolve around the sun. What counts is that Copernicus has been able to save the appearances on this on this assumption. In a letter to Copernicus, Osiander tried to persuade him to present his sun-centred system as a mere hypothesis for which only mathematical truth was claimed.   

Copernicus’s Pythaagorean Commitment

Copernicus, however, didi not subcribe to this approach to astronomy. As a committed Pythagorean, he sought mathematical harmonies in phenomena because he believed they were “really there” . copernicus believed that his sun-centred system was more than a computational device.
    Copernicus recognized that the observed planetary motions could be deduced with about the same degree of accuracy from his system, or from Ptolemy’s system. Hence he acknowledged that selection of one of these competing models was based on considerations other than successful fit. Copernicus arguedfor the superiority of his own system by appealing to “conceptual integration” as a criterion of acceptability. He contrasted his own unified model of the solar system with Ptolemy’s collection of separate models, one for each planet. He noted, moreover, that the sun-centred system explains the magnitudes and frequencies of the retrograde motions of the planets. The sun-centred system implies, for instance, that jupiter’s retrograde motions is more pronounced than that of saturn, and that the frequency with which retrogression  occurs is greater for Saturn than for jupiter. By contrast, Ptolemy’s Earth-centred system provides no explanation of these facts.
    Copernicus died before having a chance to respon to Osiander’s Preface to his book. Consequently, the sixteenth-century confrontation of the two methodological orientations—Phytagoreanism and the concern to save appearances – was not as sharp as it might have been.  
 Kepler was able to achieve a rough agrrement between the observed ratios of the radii of the planets and rations calculated from the geometry of the nest of regular solids. However, he took values of planetary radii from data of Copernicus, which referred planetary distances to the centre of the Earth’s orbit. Kepler hoped to improve the rough correlation achieved by his theory by referring planetary distances to the sun, thereby taking account of the eccentricity of the Earth’s orbit. He recomputed the ratios of the planetary radii on this basis, using Tycho Brahe’s more accurate data, and found that these ratios differed substantially from the ratios calculated from the regular-solid theory. Kepler accepted this as a refutation of his theory, but his pythagorean faith was unshaken. He was convinced that the discrepancies between observation and theory themselves mest be a manifeststion of yet-to-be-discovered mathematical harmonies.
Kepler’s persevered in the search for mathematical regularities in the solar system, and eventually succeeded in formulating three laws of planetary motion:
1). The orbitb of planet is an ellipse with the sun at one focus
2). The radius vector from the sun to a planet sweeps over equal areas in  equal times
3). The ratio of the square of the periods of any two planets is directly proportional to the      ratio of the cubes of their mean distances from the sun.
Kepler’s discovery of the Third Law is a striking application of Phytagorean principles. He was convinced that there must be a mathematical correlation between planetary distances and orbital velocities. He discovered the Third Law only after having tried a number of possible algebraic relations.
The commited Phytagorean believes that if a mathematical relation fits phenomena, this can hardly be a coincidence. But Kepler, in particular, formulated a number of
 Mathematical correlations whose status is suspect. For example, he correlate planetary distance and their “densities”. He suggested that the densities of planets are inversely proportional to the square roots of their distances from the sun. Kepler had no way to determine independently the desities of the planets. In spite of this, he noted that the densities calculated from this mathematical relation could be correlated with the densities of well-known terrestrial substances (p.44, ‘Kepler’s Distance-Density Relation’).
Kepler noted with satisfaction that it would be appropriate to correlate the sun with gold, the density of which is greater than that of quicksilver. Of course, Kepler did not believe that the Earth was composed of silver and Venus of lead, but he did believe it important thaty his calculated planetary densities correspond to the densities of these terrestrial substance.
 

 

Kepler’s Nest of Regular Solids

1596, he announced with some ppride that he had succeeded in gainig insight into God’s plan of creation. Kepler showed that the distances of the planets can be cerrelated with the radii off spherical shells, wich are inscribed within, and circumsribed around, a nest of the five regular  solids.


 Kepler’s arrangement was:
                Sphere of Saturn
                             Cube
                Sphere of Jupiter
                    Tetrahedron
                Sphere of Mars
                     Dodecahedron
                Sphere of Earth
                      Icosahedron
                Sphere of Venus
                      Octahedron
                Sphere of Mercury
 

Kepler’s Distance – Density Relation2

Density = 1 √distance        Terrestrial substabce
Planet            (Earth = 1,000)
Saturn             324                       The hardest precious stones
Jupiter            438                        The lodestone
Mars              810                        Iron
Earth              1,000                     Silver
Venus            1,175                      Lead
Mercury        1,605                      Quicksilver



From the phytagorean standpoint, the adequqncy of a mathematical correlation is determined by appeal to the criteria of “successful fit” and “simplicity”. Provide that a relation is not unduly complex mathematically, if it  the phenomena under consideration, it must be important. But a person who does not share the phytagorean faith doubtless would judge Kepler’s distance-density correlation to be a coincidence. Such a person might appeal to criteria other than successful fit and simplicity, on the grounds thats application of these criteria alone is not sifficient to distinguinsh genuine correlations from coincidental correlations.

Bode’s Law
The evalution of mathematical correlations has been a continuing problem in the history of science. In 1772, for example, Johann Itius suggested a correlation that was in the Phytagorean tradition. He noted that bthe “suitably adjusted” terms of the geometrical series 3, 6, 12, 24. . . , viz,:
         
to the Pythagorean orietation.
Then, in 1781, William Herschel discovered a planet  beyond Saturnus. Astronomers on the continent calculated the distace of Uranus from the sun and found it to be in excellent agreement with the term in Bode Law (196). Eyebrows were raised. The sceptics no longer could dismiss correlation as an “after the fact” numerical coincidence. An increasing number of astronomers began to take Bode Law seriosly. A search was undertaken for “missing planet” between Mars and Jupiter, and the stan asteroids Ceres and Pallas were discovered in 1801 and 1802. Although the asteroids were much smaller than Mercury, their distances were such that astronomers who believed in Bode Law were statisfied that the missing term in the series had been filled.
After it became apparent that the motion of Uranus was being affected by a still more distant planet, J. C. Adams and U. J. J.Leverrier independently calculated the position of this new planet. One ingredient in their calculations was the assumption that the mean distance of the new planet would be given by the next trm in Bode’s Law (388). The planet Neptune was discovered by Galle in the region pedicted by Lt everrier. Howere, continued observation of  the planet revealed that its mean distance from the sun (relative to Earth = 10) is about 300, which is not in good agreement with Bode’s Law.
With the inclusion of Neptune, Bode’s Law no longer stisfied the criterion of succesful fit. Hence one may be a Pythagorean today without being impressed by  Bode’s Law. On the other hand, since Pluto’s distance is very cllosee to the Bode’s Law value for the next planet beyoun Uranus, a person with aPythagorean bent might be temted to explain away the anolamous case of Neptune by insisting that Neptune is a lately captured  acquisition of the solar system, and not one of the original planets at all.

Notes
    Copernicus, On the Revolutions of the Heavenly Spheres,
    Kepler,  Epitome of Copernican Astrronomy, trans. C. G. Wallis, in Ptolemy, Copernicus.