Arch. Hist. Exact Sci. 53 (1998) 223–375. c Springer-Verlag 1998 Re-examining Galileo’s Theory of Tides PAOLO PALMIERI Communicated by C. WILSON Contents Introduction .............................................................. 224 1. Galileo’s tide-generating acceleration ................................... 227 1.1 The woad-grindstone model .......................................... 227 1.2 Composition of speeds and relativity ................................... 234 1.3 Tide-generating acceleration as a historiographical stumbling block ..................................................... 243 2. Newton’s tide-generating force .......................................... 248 2.1 Newton’s dynamic model ............................................. 248 2.2 Asymmetric tide-generating force and asymmetric tide periods ........... 250 3. Galileo’s oscillatory model .............................................. 255 3.1 A simple oscillating system ........................................... 255 3.2 Galileo’s model: laws of basins and superimposition of waves ............ 263 4. The ‘warping’ of history ................................................ 275 4.1 Galileo’s claim: tides prove Copernicus ................................ 275 4.2 Galileo’s notions on bodies that move around a centre ................... 281 4.3 Tide equations: the quasi-Galilean term ................................ 295 4.4 Tides in a non-Newtonian universe .................................... 302 5. Simulate the winds and the sea .......................................... 305 5.1 Comets and winds ................................................... 305 5.2 Rotating buckets and the terrestrial atmosphere ......................... 314 5.3 Tide experiments: artificial vessels and tide-machines ................... 318 6. Celestial wheel clock ................................................... 326 6.1 Monthly and annual periodicities ...................................... 326 6.2 The celestial balance-stick regulator ................................... 335 7. Moon and waves ....................................................... 338 7.1 Lunar trepidations and tides: a new research programme ................. 338 7.2 A single great wave .................................................. 352 Appendix 1. Newton’s asymmetric tide generating force .................... 356 Appendix 2. The math of warps ........................................... 359 Appendix 3. The repetition of the spinning bucket experiments ............. 362 Appendix 4. The repetition of Galileo’s experiments on tides ................ 365 Appendix 5. The math of the wheel clock .................................. 371 224 P. P ALMIERI Introduction Some fifteen years ago, Stillman Drake pointed out in his article on History of Science and Tide Theories1 that the state of historical research at that time concerning Galileo’s theory of tides was highly unsatisfactory. More specifically, what deserved re-examining was the Italian scientist’s claim to having furnished a physical proof of the Copernican double motion of the Earth (the annual revolution and the diurnal rotation) by means of a model that explained the ebb and flow of the oceans as a consequence of the acceleration and retardation imparted to sea water by the composition of two speeds: that due to the annual motion of the terrestrial globe and that due to its diurnal rotation. Today, Drakes argument still holds true. On the one hand, owing to a lack of mathe- matical knowledge, many professional historians have tended to oversimplify Galileo’s theory, so that what has usually been presented as Galileo’s position is but ‘a parody or caricature of it’.2 On the other hand, the very few working scientists who have paid some desultory attention to the history of their chosen disciplines have drawn overly facile conclusions regarding Galileo’s conception of tide motions, probably without direct con- sultation of all the original sources. As a result, Galileo’s theory of tides has frequently been brushed aside as being an unfortunate episode, unable as it was to explain the flux and reflux of the sea and patently wrong in its claiming to demonstrate the double motion of the Earth, namely, the diurnal rotation about its polar axis and the annual revolution around the Sun. Yet, there is no getting away from the fact that precisely such a ‘bizarre’ theory was uppermost in Galileo’s mind throughout his long scientific career, for it was, in his view, the very ground upon which the Copernican world system was eventually to be founded and physically proved. In this study, I have therefore set myself the twin task of re-proposing the Galilean tide model in all its complexity by bringing to light its completely forgotten vision of the flux and reflux of the sea as a wave-like phenomenon and reassessing its claim to being a physical proof of the double motion which Copernicus had assigned to the Earth in his De Revolutionibus. During the course of my investigations I was on one occasion faced with an impasse. I had long been at a loss as to how to explain how Galileo could possibly have reached at least one of his two most insightful and surprising conclusions regarding the dependence of the reciprocation time – in modern terms, frequency of oscillation – of different sea basins on their different depths and lengths. He clearly formulates two laws: a) that frequency increases with depth (width remaining constant), and b) that frequency decreases with width (depth remaining constant). Amazing as it may seem, he gives us no clue to the specific research strategy which must have brought him to such discoveries, nor does he bother to justify his assertions. Why? I was baffled. The only thing which is clearly asserted, though not directly referring to the laws of oscillation, is that he had been trying in many ways to simulate tide-related phenomena by means of mechanical models. I therefore decided to repeat – re-invent, if you like – his experiments using a very simple parallelepipedal glass tank and, sure enough, found a possible answer 1 S. Drake, “History of Science and Tide Theories”, Physis, XXI, 1979, pp. 61–69. 2 Ibid.,p.63. Re-examining Galileo’s Theory of Tides 225 to my question. As we shall see, Galileo might possibly have observed much more in his experiments than his physics and mathematics were able to cope with. For, he must have found himself face to face with the formidable difficulties of wave motions and undulatory phenomena. At the very heart of this research programme lay his quest for a physical proof of Copernican astronomy. It was the idea of devising a periodic tide-generating cause whose law could be deduced from the kinematic principle of composition of uniform or quasi-uniform motions. The Earth-Moon system orbiting the Sun could be studied as a two-body system giving rise to periodic regularities in the rate of change of motion of the surface of the Earth that, on a first approximation, were comparable with the main periods of tides commonly accepted in Galileo’s time. The ‘crucible’ wherein this periodic cause was to be transformed into tidal ebb and flow was the ocean basin, the geometry of which became, in his theory, the key to the understanding of the striking variety that wave-like tide phenomena manifest, beyond their basic periodicities. This is the very key that has been lost by science historians, a loss that has so far precluded a more balanced assessment of the many aspects of Galileo’s tide theory. Galileo did not wholly succeed in cracking the code of the ‘chemistry’ of tidal waves. This would have required analytical methods much more powerful than the elementary theory of proportions, which was all the daily tools of his mathematical laboratory amounted to. He was nonetheless able to come up with a satisfactorily working ‘principle of superimposition of waves’. According to this principle, different periodic patterns of behaviour could be pre- dicted with some degree of numerical accuracy, and the absolute wave could be thought of as being the result of a wave induced by the ‘external’ cause due to the motions of the Earth-Moon system and a wave due to the ‘internal’ response of the basin. This re- sponse was regulated by the geometric characteristics of each particular sea basin. The global picture that emerges from my study indicates that the two laws of motion within artificial vessels and sea basins must have been regarded by the Pisan scientist as being only a partial success. They represent in all probability just the outcroppings of a more ambitious research programme. This ampler vision aimed to cast light on the multifaceted nature of the relationship between, on the one hand, the motions of sea water on the surface of the moving Earth and of the superficial air strata of the atmosphere girdling a rotating planet and, on the other, the Earth’s Copernican motions. In spite of Galileo’s long labouring at his intuition of the existence of such a complex relationship, his overall programme has hitherto been overlooked and his incomplete achievements dismissed virtually unanimously by science historians as being simply irrelevant and/or patently wrong. In order to illumine the background against which Galileo’s reasoning on the problem of tides evolved, I have also repeated a few extremely interesting experiments with rotating vessels (both closed and open) to test their ability to drag the air contained inside. One of these experiments is clearly described by Galileo himself, while the others are proposed by Galileo’s