Copernican Revolution in the Complex Plane an Algebraic Way to Show the “Chief Point” of Copernican Innovation

GENERAL ¨ ARTICLE Copernican Revolution in the Complex Plane An Algebraic Way to Show the “Chief Point” of Copernican Innovation Giorgio Goldoni Starting from a simpli¯ed model of the Ptole- maic system in the complex plane, we show that Copernicus' innovation did not merely consist of choosing a reference frame in which theplan- etary motions were simpler, but in ¯nding the size of the planetary orbits expressed in what we now call astronomical units. In modern times a misleading appealtorelativity and a comparison Giorgio Goldoni is a high only on the grounds of precision hasledsome school teacher of math- to consider the Ptolemaic and the Copernican ematics and a lecturer at the Civic Planetarium of systems asbasically equivalent. This erroneous Modena, Italy.Heis point of view hasresultedintheneglectof the mainly interested in main scienti¯c content of the Copernicanthe- astronomy and mathemat- ory and hasleftCopernicusonlytohistorians ics education. and philosophers of science. It is time to restore Copernicus to the teachers of physics as an in- comparable opportunity to show the formidable power of theoreticalinvestigation. Introduction It is a widespread and ¯rmly established opinion, clearly expressed by some of the well-known scienti¯c ¯gures of the last century (see Box 1), thatthe Copernican system maybebasically obtained from the Ptolemaic one through a change in the reference frame. This view which had a dramatic impact on the psycheof medieval manbecame quite trivial in the eyes of the modern Keywords physicist accustomed to the ideasof Einstein's general Ptolemaic system, Copernican system, Tychonic system, ret- theory of relativity. Thus, in modern times, Coperni- rograde motion, complex plane, cus's achievement has been belittled to such anextent superior and Inferior planets, thatthename of Copernicus, the founder of modern sci- ecliptic, equant point, epicycle– ence, is rarely mentioned in most modern textbooks of deferent. RESONANCE ¨ November 2012 1065 GENERAL ¨ ARTICLE Box 1. “Relatively, not considering the unknown medium of space, the motions of the universe are the same whether we adopt the Ptolemaic or the Copernican mode of view. Both views are, indeed, equally correct; only the latter is more simple and more practical.” Ernst Mach ... from Einstein’s point of view Ptolemy and Copernicus are equally right. What point of view is chosen is a matter of expediency.” Max Born “It is meaningless to speak of a difference in truth claims of the theories of Copernicus and Ptolemy; the two conceptions are equivalent descriptions.” Hans Reichenbach “Today we cannot say that the Copernican theory is “right” and the Ptolemaic one is “wrong” in a meaningful physical sense … So the passion loosed on the world by the publication of Copernicus’ book, De revolutionibus orbium coelestium libri VI, were logically irrelevant.” Fred Hoyle “Copernicus’s theory was only a modified form of the Ptolemaic system – assuming the same celestial machinery, but with one or two of the wheels interchanged through the transposition of the roles of the Earth and the Sun.” Herbert Butterfield “Astronomy is easier if we take the Sun as fixed than if we take the Earth, just as accounts are easier in decimal coinage. But to say more for Copernicus is to assume absolute motion, which is a fiction.” Bertrand Russell physics. At best,we can ¯nd a hint thatCopernicus only chose a reference frame in which themotionsof the planets were simpler and that he persisted in the error of considering circular orbits. So, the Copernicantheory, 1066 RESONANCE ¨November 2012 GENERAL ¨ ARTICLE belittled, aswesaid, in its originalscienti¯c content, Copernicus, like hasinevitably become a subject only for philosophers most of the great and historians of science. scientists, was a In this article, we demonstrate thattheassumed equiv- unifier. alence between the `two chief world systems' is com- pletely erroneous and thattheidea underlying Coper- nican innovation goes beyond a mere transformation of reference frame. Starting from a simpli¯ed Ptolemaic system in the complex plane, this article demonstrates thattheCopernican theory consisted of much more than a change in the reference frame. Indeed, theCopernican theory allowed the determination of the relative sizes of the planetary orbits starting from essentially the same observationaldata asthatusedbyPtolemy.Coperni- cus himself considered this and not the motionof the Earth, the chief point of his theory, aswecaninfer from a careful reading of some passages in his De Revolution- ibus and in NarratioPrima written by his only disciple, Rethicus. A Simpli¯ed Ptolemaic System in the Complex Plane Every great scienti¯c theory consists of a basic idea,of- ten very simple, obscured by a lot of technicaldetails. For the expert, technicaldetails are far frombeing irrelevant and represent the very subject of his work. This is not the case for the teacher, whose primary purpose is to communicate to his students, with the greatest clarity and simplicity, the idea that allowed us toencapsulate in a single conceptual framework, phenomena that pre- viously seemed unrelated to each other. Copernicus, like most of the great scientists, was a uni¯er, and in order to understand very clearly his brilliant andincompara- ble contribution, let us begin by analysing a simpli¯ed version of Ptolemaic system. It maybethatPtolemy'smain purpose wastoprovide a means to compile horoscopes, but the really important RESONANCE ¨ November 2012 1067 GENERAL ¨ ARTICLE Figure 1. (left) Most of the time the planets move eastward (direct motion) in front of the distant stars, but periodically they reverse their motion (retrograde motion) for a few weeks and trace a loop in the starry sky. Here is the last loop of Mars in Cancer on January 2010. Mars reached the maximum retrograde speed and the maximum brightness on January 29th at the top of the loop. Figure 2. (right) In the model deferent–epicycle by Apollonius, a planet moves on a minor circle (epicycle) whose centre moves on a major circle (deferent) centred at Earth. By adjusting the angular speeds and the ratio of the radii, it is possible to explain the retrograde motion of the planet. thing is thathedeveloped a powerful mathematicaltool to predict the positions of the planets against the back- ground of the ¯xed stars. Unlike the Sun, that always moves eastward (direct motion) on the CelestialSphere completing the circle of the ecliptic in oneyear, the planets periodically reversetheir motion for a few weeks (retrograde motion) describing characteristic loops among the stars (Figure 1). In the fourth century B C Apollo- nius explained themotion of the planets by means of the composition of two uniform circularmotions. Following theevidenceof the senses, theEarth was considered still, while each planetmovedon a minor circle (epicycle) whose centre in turn moved on a major circle (deferent) centredintheEarth. With an appropriate choice of the radii and of the angularspeedsitwaspossible to account for the retrogrademotions of the planets (Figure 2). To reproduce the deviations in latitude it was su±cient to place the epicycle and the deferent on slightly di®erent planes. Ptolemy improved the epicycle{deferent device, 1068 RESONANCE ¨November 2012 GENERAL ¨ ARTICLE introducing the so-called equant point. The deferent was notcentredontheEarth and the centre of the epicycle movedwithuniform angular speed around a point di- rectly opposite the Earth from the centre of the deferent, the equant point (Figure 3). In the case of the Sun there wasnoneedfor an epicycle, but only an eccentric circle with anequant point. By means of these math- ematicaltoolsPtolemyobtained a good approximation for the ¯rst two of Kepler's laws1.Incidentally, Coperni- 1 See Resonance, Vol.14, No.12, cus obtained the same result by adding a small epicycle 2009. (epicyclette) to his circular orbits centred in the Sun (Figure 4). Therefore it is not true that intheCoper- nican system the planetary orbits were circles, nor that they were centred on the Sun. They were the composition of two uniform circularmotionsand the resulting Figure 3. (left) In the Ptolemaic system the deferent of a planet is not centred at Earth and the centre of the epicycle moves at constant angular speed around a point opposite the Earth with respect to the centre of the deferent, called equant point or simply equant. This expedient gives a good approximation for Kepler’s law of areas. In fact for an ellipse the difference between the semiaxes goes with the square of the eccentricity and the orbit of a planet with small eccentricity can be approximated by means of a circle eccentric with respect to the Sun. Moreover, with respect to the focus of the ellipse, the planet moves at constant angular speed. Figure 4. (right) Copernicus resorted to a device similar to Ptolemy’s equant, consisting of a little epicycle (epicyclette) revolving in the same direction as the deferent but with double the angular speed and with a diameter equal to the eccentricity e. To a first approximation the planet describes a circle (actually it is an algebraic curve of 5th degree) whose centre is shifted by e/2 with respect to the centre of the deferent. The Sun stands at a distance 3e/2 from the centre of the deferent, while the point on the opposite side at distance 2e from the Sun plays the role of Ptolemy’s equant. RESONANCE ¨ November 2012 1069 GENERAL ¨ ARTICLE orbits were not circles but an approximation of Kepler's ellipses. ClaimingthattheCopernican

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