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kept most of his ideas on the subject to himself. Sometime around 1595 he constructed an argument for the cause of tides based on the daily rotation of the Earth, but he did not put it into print until 1616. He argued that, since the

1 There is a wealth of literature concerning Galileo. A good starting point is Machamer (1998). See Dugas (1988). That Galileo was not in total command of the dynamics of free fall at this time is clear from the fact that he also believed that the speed was proportional to the distance travelled rather than the time.

Galileo's progression from rejection to acceptance of Copernicanism is described in Drake (1987). During his early years as a teacher, Galileo composed a Treatise on the Sphere, which basically followed Sacrobosco's thirteenth-century work. He was still teaching this elementary Ptolemaic material after he had converted to heliocentrism (see Drake (1978)). The Discourse on the Tides was written in the form of a letter to Cardinal Orsini (translated in Finocchiaro 1989). Prior to Galileo, most scholars (e.g. Kepler) attributed tidal motion correctly to the influence of the Moon on the oceans, based on the easily observed fact that certain tidal phenomena are related to the Moon's phases. The first detailed treatment of the relationship between the Moon and the tides is due to Posidonius (first century BC (see Darwin (1962),

Fig. 7.1. Galileo's theory of the tides.

motion of a point on the surface of the Earth is made up of two components -one due to the daily rotation of the Earth and the other to the annual motion of the Earth around the Sun - the linear velocity of the point P1 in Figure 7.1 (where it is midnight) is the sum of the linear velocity of the Earth in its orbit around the Sun and the linear velocity due to the diurnal rotation. However, at P2 (where it is noon) the linear velocity is the difference between these quantities. Galileo put it like this:

Thus I believe that it is clear how, though each part of the earth's surface moves with two very uniform motions, nevertheless within a period of twenty-four hours it moves sometimes very fast, sometimes slowly, and twice at intermediate speeds; and this change results from the combination of these two uniform motions, diurnal and annual.

This constant speeding-up and slowing-down of a point on the Earth's surface caused the oceans to slosh back and forth, resulting in the tides. This theory is flawed,6 but it is clear from his work on the subject that he recognized the essentially localized nature of tidal phenomena. There is a periodic tide-generating force (which Galileo got wrong) but it is the geometry of the ocean basin that determines the actual behaviour of the fluid within it.

When Kepler sent him a copy of the Secret of the Universe in 1597, Galileo thanked him and told him that he had been convinced of the Copernican hypothesis for some time but added that he was not prepared to publish anything to that effect:

Many years ago I came to agree with Copernicus, and from this position the causes of many natural effects have been found by me which doubtless cannot be pp. 81-5). Galileo's theory is discussed at length in Palmieri (1998). The influence of Copernicanism on Galileo's early investigations in dynamics is discussed in Naylor (2003). Quoted from Finocchiaro (1989).

The argument confuses two different frames of reference. The motion of the Earth is considered-relative to the Sun, but it is the motion of the water relative to the Earth that Galileo was trying to explain.

explained by the ordinary supposition. I wrote down many reasons and arguments, and also refutations of opposite arguments, which, however, I did not venture until now to divulge, deterred by the fate of Copernicus himself, our master, who, although having won immortal fame with some few, to countless others appears ... as an object of derision and contumely. Truly, I would venture to publish my views if more like you existed; since this is not so, I will abstain.

Kepler encouraged Galileo to be less cautious. In his reply he wrote:

Be confident, Galilei, and proceed! If I am right, only a few of the chief mathematicians of Europe will keep aloof from us; such is the power of truth.8

In 1604, Galileo lectured on the supernova of that year and showed that the absence of diurnal parallax indicated that the new star occupied the region of the heavens above the Moon, supposedly unchanging according to Aristotle. By 1609, Galileo was what might be described as a cautious Copernican; in private he was prepared to advocate the motion of the Earth, but in public, for fear of ridicule, he was not. One of Galileo's reasons for not recoiling from the concept of a moving Earth was that his researches into the nature of motion had convinced him that Aristotle's theory was false. For example, Aristotle said that bodies fall with speeds proportional to their weights, but Galileo's experiments revealed that the speed of fall was independent of weight. More fundamental was Galileo's realization that motion could persist without any applied force. He did not quite formulate the principle of inertia that Newton arrived at a century or so later, because in Galileo's mind it was circular motion that was the natural state; thus, no force was necessary for the Earth to spin continually and, similarly, no force was required for the planets to orbit the Sun. This circular inertia was consistent with the ancient Pythagorean doctrine of uniform circular motion, but Galileo also applied his principle to terrestrial physics. In Galileo's famous work on projectiles, he assumed that the horizontal part of the motion (which was, in the absence of air resistance, not subject to change) actually was part of a circular motion, the radius of which was that of the Earth, and thus it was only approximately rectilinear.

While Galileo was Professor of Mathematics at Padua - perhaps the leading Italian university of the time - he learned of an invention that had the effect of making objects appear to be closer than they really were. Galileo was interested,

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