International Catholic University

Galileo: Science and Religion

Galileo the Astronomer

Peter Hodgson

1 The Invention of the Telescope

In 1609 Galileo heard that a Dutch optician, Lippershey, had found that if he put two lenses at either end of a tube and looked though it, distant objects appeared much closer. Many of these telescopes were made, but as their magnification was low and the images rather blurred they were regarded more as interesting toys than as objects of practical value. Galileo, however, immediately realised the importance of this invention and how he could use it to further his career by offering it to the Venetian State. He was alarmed to learn that a Dutchman was already in Venice hoping to sell his telescope to the Doge. He alerted his friend Sarpi, who succeeded in preventing the Dutchman from obtaining an audience with the Doge, and frantically set to work to make a telescope for himself. He fitted two lenses at either end of a lead tube and indeed found that it magnified distant objects. Later on he said that he succeeded in making a telescope in a single day, but this seems very unlikely. It takes a long time to grind a lens, and it is unlikely that he already had commercial lenses available or that they would be of sufficient quality. By a process of trial and error he went on to make a series of telescopes of increasing magnification and technical excellence. As soon has he had made a good telescope he arranged with the help of Sarpi to have an audience with the powerful Doges of Venice, who were impressed by its value to the Venetian Navy. Astutely, Galileo presented his best telescope to the Doge as a gift and, not to be outdone, the Doge's Senate soon after voted to double his salary, to reappoint him for life, and to give him a large bonus. Subsequently he made many more telescopes and presented them to many eminent friends and powerful princes. This story illustrates very well Galileo's ruthless opportunism and technical genius. Certainly his best telescopes were far superior to any others, and he made sure that their merits were widely recognised and that his career benefitted.

2 The Discovery of the Moons of Jupiter

He turned the telescopes to the heavens, and was rewarded by a series of outstanding discoveries. He looked at the planets, and noticed that there were one or two stars on either side of Jupiter, almost in a line. On subsequent nights he found that the stars had moved relative to Jupiter, and that their number changed. He realised that he was seeing four of the moons that orbit Jupiter just as the moon orbits the earth. By observing them for several weeks he was able to determine their periods of rotation. He called them the 'Medicean stars' in honour of Cosimo de Medici, and published an account of his discovery in a pamphlet called Sidereus Nuncius, or Starry Messenger. At first his discovery was ridiculed, but most people were soon convinced when they looked through one of his telescopes. He presented telescopes to several powerful princes, and they naturally asked their own astronomers to examine them and assess their merits. By this clever move, the astronomers were forced to examine his claims, whether they believed them or not, and indeed they soon endorsed them.

Galileo was particularly excited by this discovery because it provided an example of several moons orbiting a planet, very similar to Copernicus's suggestion that the planets orbit the sun. It did not, of course, prove Copernicus's theory, but showed that it was not necessary for everything to rotate about a single centre, and also answered those who said that if the earth moves it will lose its moon.

This spectacular discovery made Galileo famous throughout Europe, and he followed it by a whole series of new observations. He found hundreds more stars in the familiar constellations, and showed that the Milky Way is made up of thousands of individual stars. He turned his telescope to the moon and observed the circular craters that we now know are due to the impact of meteorites. By observing the behaviour of the shadows of their edges as the moon waxed and waned he was able to show that they had a central depression surrounded by a high rim, and estimated that they were about four miles high, a reasonably accurate value.

This discovery was important because Aristotle had said that the heavenly bodies were perfectly spherical, with no rough surfaces. The Aristotelians tried to explain Galileo's observations by saying that the moon is completely surrounded by a smooth transparent shell that covers all the craters. Galileo sarcastically replied that he would believe this if they would allow him to cover the moon with high and transparent mountains.

These new results supported previous observations of changes in the skies. Thus in 1604 there appeared a new star that excited great public interest. Galileo gave three lectures on the phenomenon, admitting that he was not at all sure that it was really a star; for all he knew it might be due to the condensation of vapours in faraway space. Studies of its parallax showed that it was much further from the earth than the moon and so provided another example of imperfections in the celestial realm.

These new discoveries were highly uncongenial to the Aristotelians, who redoubled their efforts to discredit his work, maintaining that what he saw was due to imperfections in his telescopes. Galileo defended himself vigorously, first developing the opposing views and supporting them by real arguments, and then demolishing the whole structure with undisguised relish.

His discoveries were soon accepted by other astronomers, particularly by the Jesuits of the Collegio Romano, who became very supportive of his work. Galileo visited Rome in 1611 to lecture on his discoveries and was feted by them.

In the same year Galileo studied the planet Saturn, and found that sometimes it had the appearance of three stars in the form of a larger central body with two equal satellites on opposite sides. At other times these satellites disappeared. His telescope was not sufficiently powerful to show that this was due to Saturn's rings seen from various angles, as was shown by Huygens in 1657.

In 1611 Galileo observed the sunspots, that provide another example of imperfections in the celestial realm. He correctly surmised that they are clouds of vapour on the sun's surface, and by observing their motion was able to deduce that the sun rotates with a period of about a month. He did not discover the sunspots; indeed they had been known to the Chinese for centuries, and other European scientists had also noticed them. Among these was the Jesuit astronomer Scheiner who sent his observations to Mark Welser, who published them anonymously under the pseudonym Appelles. In order to maintain the incorruptibility of the heavens Scheiner suggested that sunspots are little planets orbiting the sun. Welser sent Scheiner's letters to Galileo asking for his opinion, and Galileo replied at some length. He had little difficulty in demolishing Scheiner's theory, but did so in a very courteous way. Nevertheless Scheiner was very offended, probably because he mistakenly thought that Galileo had him in mind when he criticised one of his opponents without mentioning him by name; this was to cause Galileo much trouble later on.

To disprove Scheiner's theory Galileo observed that the sunspots are approximately circular when they are near the centre of the sun's disk and progressively become more elliptical as they approach the edge. This is just what would be expected if they are situated on the surface on the sun; if they were spherical planets they would keep the same circular shape as they moved around the sun. Galileo also showed that he could explain the velocities of the spots, and found that they are inconsistent with Scheiner's theory.

Galileo also saw that the trajectories of the sunspots change through the year: they appear to move in a straight line only at six-monthly intervals and at other times they appear to move in concave or convex arcs. This is what would be expected if the sun's axis is tilted with respect to the plane of the earth's orbit around the sun. While it is true that these motions could be described in the reference frame of a stationary earth, this would require the attribution of four independent motions to the sun (McMullin, 1967, p.40), whereas the motions are described quite naturally on the heliocentric theory. Thus although this does not amount to a strict proof, Galileo was justified in using the motion of the sunspots as an argument in favour of the heliocentric theory.

3 The Heliocentric Theory

Copernicus' book De Revolutionibus, putting forward the heliocentric theory, remained Aristotelian in all except its central idea, and is written so that, apart from some introductory sections, it can only be understood by professional astronomers. Initially, Copernicus was concerned, like Ptolemy, to find the best way to calculate the motions of the planets, and used the heliocentric hypothesis as a calculating device. As the work proceeded, he found that it accounted naturally for many observations that could be fitted by the geocentric model only by making specific assumptions in each case. Eventually he came to believe that the Copernican theory is true. As Galileo remarked in a letter to Mgr Dini in 1615, "From many years of observation and study, he was abundantly in possession of all the details observed in the stars, for it is impossible to come to know the structure of the universe without having learned them all very diligently and having them very readily available in mind; and so, by repeated studies and very long labours, he accomplished what later earned him the admirations of all those who study him diligently enough to understand his discussions. Thus, to claim that Copernicus did not consider the earth's motion to be true could be accepted perhaps only by those who have not read him, in my opinion; for all six parts of his book are full of the doctrines of the earth's motion, and of explanations and confirmations of it". (Finocchiaro 1989, p.60).

Copernicus was highly regarded by the professional astronomers, and they realised the many advantages of the new system. Many of them began to use his methods, even if they continued to reject his heliocentrism.

Subsequently, Tycho Brahe proposed a new cosmology, in which all the planets revolve around the sun, which in turn revolves around the earth. Providing the spheres of the fixed stars is sufficiently far away, this is mathematically equivalent to the Copernican heliocentric system. It was adopted by many astronomers as a way of using the ideas of Copernicus while avoiding the apparent absurdities of a moving earth.

The Copernican system, particularly when refined by Kepler's ellipses, provided a more accurate way of analysing planetary motions, but at the expense of the physical explanation provided by the Aristotelian spheres. A whole new set of questions arose and demanded answers: why do the planets move in elliptical orbits, what keeps them going around the sun and so on. Kepler suggested that the sun emits rays that push the planets around, and that the ellipticities of the orbits are due to magnetic effects. This idea did not fit the data and was soon discarded, but it contained the germ of the idea of forces that was eventually to provide the solution. Galileo ignored this possibility, considering Copernican heliocentrism as the only alternative to Aristotelian geocentrism.

The heliocentric theory provided natural qualitative explanations of several phenomena, such as the retrograde motions of the planets, the phases of Venus and the angular closeness to the sun of the inner planets Mercury and Venus. In the Ptolemaic system these observations were included by the special choice of the parameters of the ellipses. Although some astronomers including Copernicus became convinced of the correctness of the heliocentric theory, they had no conclusive arguments. It was easy to defuse the opposition likely to be encountered by the heliocentric theory by maintaining that it was just a convenient mathematical scheme with no pretensions to reality. The Lutheran theologian Osiander, who saw the manuscript of Copernicus through the press, inserted an anonymous preface to this effect, without Copernicus knowing.

The professional astronomers were well aware of the large number of minor improvements that had been made in the Ptolemaic system over the previous centuries without significantly improving the fit to the unsatisfactory ancient data and which proved quite unable to fit the greatly improved data of Tycho Brahe, increasingly turned to the Copernican theory as the basis of their calculations. Many of then still rejected the heliocentric theory as a real account of celestial motions and used the Copernican theory simply as a method of calculation. The astronomers gradually improved the Copernican theory and found that it is much more tightly constrained that the Ptolemaic theory, so that it is not possible to adjust the parameters of the planetary orbits independently of each other. And so, gradually, impelled by their very practical concerns, the belief of the professional astronomers changed. By around 1616, the time of the Church's first action against Galileo, the case for Copernicanism was respectable but still weak, but by the time of his recantation in 1633 the tide had turned and geocentrism was almost a lost cause. According to Kuhn, 'by the middle of the seventeenth century it is difficult to find an important astronomer who is not Copernican; by the end of the century it is impossible'. It took much longer for heliocentrism to be generally accepted; Milton, for example, in his great work treated the Ptolemaic and Copernican systems on an equal footing.

It is important to recall that not one of the arguments for Copernicanism was conclusive. Galileo's favourite argument from the tides is fallacious. Bellarmine said that if the heliocentric theory was proved correct (and by proof he meant, following Aristotle, certain knowledge through causes) then it would be necessary to study carefully how it could be reconciled with Scripture. However the proofs that first convinced the astronomers were only accessible to them; it is the cumulative effects of a large numbers of indications, individually not coercive, the unity of indirect reference akin to the illative sense of Newman. This may also be described as the interpretation of signs, which can only be done by the prepared mind. When eventually the definitive proof of the heliocentric theory came two hundred years later with the measurement of stellar parallax by Bessel in 1838, the battle was long over, and it is doubtful if there was any great stir among either scientists or theologians.

If it had been purely a matter for astronomers, the Copernican view would probably have gradually prevailed, without drama. However the prestige of Aristotelian cosmology, and especially its integration with Christian theology, made this impossible.

Galileo's discoveries brought the whole heliocentric debate out of the domain of the professional astronomers and into the area of public discourse. Using his telescopes, people could see the evidence for the celestial phenomena that were contrary to the Aristotelian view such as the mountains on the moon, the sunspots, the moons of Jupiter and the phases of Venus. None of this proved the heliocentric theory, but by weakening the Aristotelian cosmology it made it more worthy of consideration. Telescopes soon became very popular, and Galileo had to make many more to satisfy popular demand. At the same time he announced his discoveries in well-written booklets in the vernacular language. In contrast to the impenetrable tome of Copernicus, these were immediately accessible to non-professionals. Sure of his position, Galileo poured scorn on his opponents, thus further inflaming the opposition. Heliocentrism became popular among those who opposed Aristotle for other reasons, even if they had little understanding of the astronomical arguments.

Galileo maintained that he was more faithful to Aristotle than the Aristotelians. Aristotle believed in the importance of observation and reason, and Galileo believed that if he had the opportunity to look through a telescope, he would soon be convinced by what he saw, and would revise his cosmology accordingly. Galileo had nothing but contempt for those who looked for the truth about the physical world only by searching through musty old texts instead of opening their eyes to the world around them.

4 The Motion of the Earth

The central theme of Galileo's scientific endeavours was his continuing efforts to prove that the earth moves in two distinct ways: its daily rotation on its axis and its annual motion around the sun. These themes link together his astronomical discoveries and his work on terrestrial dynamics.

To achieve this aim he had to overcome three distinct sets of obstacles. Firstly the commonsense conviction based on direct experience that the earth is immoveable, secondly the opposition of the Aristotelian philosophers, whose whole view of the universe was centred on a stationary earth, and thirdly the belief of many theologians that a moving earth was contrary to Scripture.

The commonsense belief in an immoveable earth can be supported by rational arguments. If the earth is moving with the speed necessary to carry it round the sun, then surely the high winds would demolish all buildings and blow everything away. It is well known, it was said, that an object dropped from the high mast of a ship falls nearer to the stern because the ship moves while it falls, and therefore the effects due to any motion of the earth should be even more marked. Galileo responded by showing that this statement is false; the object lands at the foot of the mast, and this is because the object shares the forward motion of the ship all the time. He then pointed out that if one is in a closed cabin in a steadily-moving ship we can play ball and jump around just as we could if the ship were stationary. Indeed there is no way of finding out if the ship is moving or not. We can experience the up and down motion of the waves, and any changes in the forward speed, but these are all accelerations. This absence of any effects due to a uniform velocity is known as the principle of Galilean relativity. Thus the absence of any such effects is no argument against the translational motion of the earth.

It may be remarked that these arguments are strictly true only for rectilinear motion. On the earth, however, the situation is different because the objects on the earth's surface have circular trajectories, although the difference is very small for short distances. Thus the top of the mast is moving slightly more rapidly than its foot, due to the earth's rotation, and this implies that an object dropped from the top of the mast will hit the deck a small distance to the east of the base. The effect is very small but not negligible, and was indeed first detected by G.A. Guglielmini in Bologna in 1789 and confirmed by J.F. Benzenberg in 1802 and 1804, and by F. Reich in 1831. The most accurate study was made by E.H. Hall at Harvard in 1902, and he found a deviation of 1.50+0.05 mm to the east for a drop of 23 m, compared with a calculated value of 1.8 mm. Further confirmation of the earth's rotation came from the detection of stellar parallax by Calandrelli in 1806 (W.A. Wallace, 1995).

Another argument against the rotation of the earth is that objects would fly off into space if the earth were rotating, as it is well-known that objects can fly off a rotating wheel. Whether they actually do so depends on the rotational velocity; in modern terminology they fly off if the centrifugal force mv2/R due to the rotation is greater than the force holding them on the rotator, the gravitational force mg in the case of the earth. So the answer to this objection is that the earth does not rotate fast enough for this to happen. Since the period of rotation T = 2pR/v, the centrifugal force is 4pmh/T2 where R is the radius of the earth, T the period of its rotation and g the acceleration due to gravity. The ratio of these is gT2/4pR, approximately 287. Thus the earth would have to rotate about seventeen times faster before bodies flew off.

Many of the arguments Galileo used against the Aristotelians have already been mentioned. He was able to show that many Aristotelian beliefs, especially those concerning the immutability of the heavens, are demonstrably false. The attempts of the Aristotelians to evade his arguments, by saying for example that what Galileo saw was due to defects in his telescopes, were quite unconvincing. None of this had any direct bearing on the motion of the earth, but it did expose defects in the Aristotelian system and thus weaken the belief in the centrality and immovability of the earth.

In the context of his times, the argument that a moving earth contradicted Scripture was the most serious. It could be met in two ways, depending on one's beliefs concerning the way Scripture should be interpreted. The simplest way, supported by a considerable theological tradition, is to say that Scripture does not contain any statements about the nature of the world, and if it appears to do so then it is a matter of using colloquial speech. Thus we still say that the sun rises in the morning, without committing ourselves to geocentrism. More succinctly, in the words of Cardinal Baronius, Scripture teaches us how to go to heaven, not how the heavens go.

Galileo might have been wise to keep to this argument, but in the aftermath of the Council of Trent a more rigorous interpretation was imposed, namely that the literal sense of Scripture must be strictly adhered to unless there was definite proof to the contrary. Thus heliocentrism was excluded unless it could be shown to be true. If this were done, then the theologians would be obliged to re-interpret Scripture in accord with the new cosmology. Galileo was willing to accept this stricter view because he believed that he could indeed demonstrate the correctness of heliocentrism. In this belief he was over-optimistic, and this led to his downfall.

Part of the difficulty is to know what constitutes a proof, and this was not specified by the theologians, and indeed it is difficult to see how they could have done so. Except in some very clear and obvious cases, this is a matter of some difficulty, and it depends on one's whole philosophy of science. Furthermore, the acceptability of a proof depends on the knowledge of the reader; someone ignorant of science would be unable to judge whether a proof is valid or not.

Galileo could have avoided all his troubles by saying that heliocentrism was just a calculational device that bore no relation to reality, and indeed he was strongly urged to do this. But scientists, and Galileo was no exception, know that they are investigating an objectively-existing world, and such a subterfuge is unacceptable.

Galileo was thus committed to proving the motion of the earth. Concerning its daily rotation, it is very unlikely that all the stars could move so regularly around the earth with the necessary very high velocities precisely adjusted to keep them in the same relative positions. It is dynamically so much simpler to attribute the observed appearance to the rotation of the earth. Concerning the planets, Ptolemy found that they required two circular motions to describe their positions, and one of these always had a period of exactly one year. Is this just a coincidence? Is it not much more plausible to attribute this to the yearly motion of the earth around the sun? Scientists accept such arguments as being very strong; it is not necessary to prove that there is no other conceivable explanation. And yet such considerations do not amount to a conclusive proof.

It is an ironic comment that the rotation of the earth was subsequently demonstrated by Foucault's pendulum, an experiment that in principle could have been carried out by Galileo. It is interesting to speculate how this would have been received by the Aristotelians, as its interpretation requires the concept of inertial motion. The translational motion of the earth was proved by James Bradley's discovery of stellar aberration in 1729.

Another argument depends on the apparent brightness of the planets. According to the heliocentric model, the distance of the planet Mars from the earth should vary by a factor of seven or eight, resulting in a brightness variation of about sixty, compared with the observed ratio of four or five. This was, however, a visual estimate, and Galileo was able to show that more accurate telescopic observations gave a brightness ratio in full agreement with the Copernican model.

The proof that Galileo considered the strongest was his explanation of the tides. It had long been known that the tides follow a daily cycle with high and low tides about every twelve hours, a monthly cycle due to the time-lag of about fifty minutes each day, the half-monthly cycle with high tides at new and full moon and the half-yearly cycle with higher tides at the equinoxes than at the solstices. It was also necessary to account for the very different magnitudes of the tides at different places, from the absence of tides in lakes, the small tides in the Mediterranean to the much larger tides in the oceans.

Marcantonio de Dominis had already attributed the daily cycle to the attraction of the oceans by the moon, but this was usually rejected because it would give only one tide a day when the moon is over the ocean. This argument is however mistaken because, just as the moon attracts the ocean nearest to it more than the earth, so it attracts the earth more than the ocean at the other side of the earth, giving two high tides each day. Galileo firmly rejected the possibility that the moon is responsible for the tides, considering it to be the invocation of 'occult qualities and similar idle imaginings' (Shea, 1977, p.181). Just the same objection was later to be directed against Newton when he proposed his theory of gravitation. Galileo also rejected the lunar explanation of the tides as another instance of 'explaining by naming' (as when, for instance, we explain free fall by 'gravity'). This is a valid objection until we have a quantitative theory.

Instead, Galileo sought a physical explanation of the tides and attributed the tides to the double motion (rotation and revolution) of the earth, modulated by the extent and orientation of the sea. He observes that just as the water at the edge of a basin rises and falls as the basin is moved back and forth, so do the oceans rise and fall as the earth moves. The two motions of the earth, rotation and revolution, are indeed uniform, but when combined they sometime reinforce and sometimes oppose each other, so the result is a non-uniform motion. This explanation is ingenious, but it is incorrect in principle and fails to account for the known association between the tides and the moon. Galileo had apparently forgotten his earlier argument that since the air and the oceans share the motion of the earth they can have no effect on the tides. Furthermore, even if his argument were correct, it would imply that high tides should occur at noon and low tides at midnight, which is not the case.

However, Galileo was convinced that his model was correct and he tried to explains the obvious objections by reference to the configurations of the seas, but this was not convincing. He considered the alternative explanations to be just fantasies, and he brushed them aside.

Thus, although Galileo was able to advance many very cogent arguments for the motion of the earth, the one he emphasised most strongly is fallacious, and none of them amounted to a strict proof. He thus failed to meet the over-rigorous conditions that he had accepted in order to show that heliocentrism is not inconsistent with Scripture.

The final theoretical proof of heliocentrism for physicists was provided by Newton's laws of motion and his theory of gravitation. From these, Kepler's three laws of planetary motion can be deduced and all the motions of the planets calculated to high accuracy. Thereafter there was no doubt that the heliocentric theory is correct.

In recent years it has been said that since Einstein has shown that all motion is relative the whole question of whether the earth goes round the sun or the sun around the earth is now meaningless. It is not necessary to embark on an analysis of the concept of inertial motion to show that this argument is fallacious; it is sufficient to ask what the solar system would look like to someone on a spaceship at some distance away from it.

5 The Comets

Three comets appeared in 1618, and naturally excited great interest. Galileo was urged by his friends to write about them, but he was bedridden at the time with arthritis and was unable to make any observations. Eventually he yielded to their request, particularly because it was being said that the motions of the comets provide a strong argument against the Copernican system. Among the many publications on the comets, he singled out for detailed criticism one by Fr Grassi, a professor at the Collegio Romano. Grassi published an answer to Galileo's remarks, and Galileo responded with a lengthy analysis eventually published as Il Saggiatore (The Assayer), a combination of brilliant rhetoric and erroneous science.

Galileo began with a summary of ancient theories of comets, showing that they are all unsatisfactory, except the view of Pythagoras that they are refractions of our vision of the sun, like the rainbow or the aurora borealis: "In my opinion this effect has no other origin than that a part of the vapour-laden air surrounding the earth is for some reason unusually rarefied, and being extraordinarily sublimated rises above the cone of the earth's shadow so that its upper part is struck by the sun, and made to reflect its splendour, thus causing the aurora borealis". He thought that comets move in straight lines, maintaining that this accounts for all the basic observations. There were however as Grassi showed several difficulties that he was unable to answer satisfactorily.

Galileo got himself into these difficulties because he thought that the heliocentric theory would be threatened if comets are real bodies moving along orbits. He was further inhibited by his belief in the simplicity of nature, and would only accept linear and circular orbits, neither of which is appropriate for comets. Rather than admit other orbits he preferred to deny that comets are real and reduced them to optical phenomena. He rejected the possibility of a very large orbit, apparently forgetting that this is required by the heliocentric theory. Indeed, in this discussion of comets Galileo behaved just like a conservative Aristotelian.

6 The Scientific Achievements of Galileo

The main achievement of Galileo was to inaugurate a new way of thinking about the world. He rejected the traditional method of seeking the answers to physical problems by studying the works of masters such as Aristotle and replaced it by quantitative measurement and analysis. Instead of philosophical discussions about the nature of motion he measured as accurately as possible how long it took for bodies to fall a certain distance, and then tried to find a mathematical relation between them. He was not the first to emphasise the importance of experiment; others like Robert Grosseteste in the thirteenth century had laid the foundations of experimental science. But he was the first to stress the importance of establishing mathematical relationships between the results of measurements. He was thus a pioneer in the mathematicisation of nature, which ultimately led to the science of theoretical physics.

Galileo maintained that "philosophy is written in that great book which ever lies before our eyes, I mean the universe, but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is humanly impossible to comprehend a single word of it, and without which one wanders in vain though a dark labyrinth".

It is difficult for us to realise the magnitude of this achievement. Aristotle had a rather low view of mathematics, and many others, from Swineshead to Hegel, doubted its usefulness in studying physical problems. Indeed, it is not at all obvious that mathematics can be applied to the physical world: when we look around us there is so much that is apparently chaotic and unpredictable. One can also argue that mathematics does no more than provide a formal description of phenomena, whereas physics described the real processes taking place, as illustrated for example by Ptolemy's epicycles and Aristotle's physics. Why then should we expect the abstractions of mathematics to apply to material things? Even now, scientists such as Einstein and Wigner have expressed their astonishment at the 'unreasonable effectiveness of mathematics'. There have indeed been many unsuccessful attempts to apply mathematics to subjects like medicine and psychology. It was a stroke of genius for Galileo to select the one phenomenon that shows the effectiveness of mathematics most clearly, namely the free fall of bodies, and even then he had to abstract from irrelevant considerations such as the shape, colour and material of the body and the retarding effect of the medium and also overcome the difficulties of time measurement. Mathematics is now essential for physics; indeed Rontgen remarked that "the physicist in preparing for his work needs three things, mathematics, mathematics and mathematics".

It is not correct to describe Galileo's scientific method either as Platonist or as hypothetico-deductive (Wallace 1981, p.129 et seq). He believed that nature is simple and that we can attain true knowledge of it. He did not accept the view that a scientific theory just saves the appearances, enabling us to calculate results that are more or less in accord with observations and measurements but tell us nothing about the real nature of the world. He was thus a realist, and he built on and combined the methods of Archimedes and Aristotle to forge a new method, and this is one of his chief claims to be regarded as the founder of modern science.

The hypothetico-deductive method can be expressed in the form 'if p, then q; but q, therefore p'. This is fallacious because there may be other p's that also imply q. So how can we attain certainty in scientific reasoning? Galileo starts from a form of reasoning known as ex suppositione, used by Archimedes and Aristotle, and further elaborated by Aquinas. This may be expressed in the form: "if p, then 'if p then q', then q". This requires considerable explanation, and can be illustrated by the example of a lunar eclipse used by the medieval commentators. We can show geometrically that a lunar eclipse occurs when the earth is between the sun and the moon, so that it intercepts the light from the sun that would otherwise fall on the moon. This is genuine, certain, theoretical knowledge. From this we can infer that if the required conditions for an eclipse actually occur, then there will be an eclipse, unless some other effect intervenes to prevent it. It is important to notice that the first part of the reasoning is exact, theoretical and geometrical and refers to an ideal world, whereas the subsequent inference refers to the real world where our knowledge is approximate and subject to various uncertainties, due to a multitude of other effects that could conceivably interfere with what usually happens. Nevertheless, we have certainly attained genuine knowledge of the phenomenon, and have taken into account the possibility of interfering effects. As described by Archimedes, the demonstration uses formal (mathematical) causality and the relationships refer to quantifiable aspects of the subject, and any defects arise because mathematical entities are not exactly realised in the real world. For Aristotle the demonstration uses efficient causality and the defects are due to the matter involved or to the agent.

Galileo combined these modes of reasoning, which takes the following form in the case of a projectile shot horizontally. First he shows mathematically that if such a projectile is shot in a vacuum, then its motion will be a combination of a uniform horizontal motion (s=vt) and an accelerated downward motion (s=gt /2), giving a semi-parabolic trajectory. Then, if a real projectile is shot in this way, and accepting the existence of air resistance and other disturbing influences, but knowing that they are small, we deduce that the resulting trajectory will be quite closely a semi-parabola, and that the time of fall will be quite closely that given by our calculation. One again, we have genuine, true knowledge, though it is not exact. In addition to his calculations, Galileo went beyond Archimedes by making careful experiments, and verified the correctness of his results, thus showing that it is possible to apply mathematics to natural phenomena in this way, and that the disturbing influences are indeed small. The argument is no longer qualitative, as it was for Aristotle, but quantitative and indeed exact, if we understand (as did Galileo) that 'exact' means 'exact within the limitations of measurement' and in the limit when the disturbing effects are reduced to zero. In this stress on exactness Galileo differed from Plato, although Plato also stressed the importance of mathematical forms, for him these applied exactly only to an ideal world, and only imperfectly to the real world. Galileo thus developed and used the method that is implicitly used by working scientists today.

Galileo unified what had previously been separated, and separated what had previously been entangled. Through his astronomical discoveries and his application of mathematics to celestial phenomena he overcame Aristotle's sharp separation and unified terrestrial and celestial phenomena. Everything is now subject to the same laws, and these are expressible in mathematical form. This unification, Galileo recognised, is not easily attained. Initially, each realm of phenomena has to be studied with its own concepts and laws, and eventually they may be united with other phenomena, as electric and magnetic phenomena were unified by Maxwell. This work is still in progress: gravitation and quantum mechanics still await unification. Many of what are sometimes regarded as Galileo's greatest discoveries were anticipated in one way or another by scholastic philosophers and other Renaissance mathematicians, and his achievement was to unify them in a way that led to the development of theoretical physics.

In addition to his work of unification, Galileo separated science from theology, in the sense that theology, and in particular the Bible, should not be used as a source of scientific knowledge. Rather, what we find out by observation and experiment can sometimes assist the interpretation of the Bible. Galileo is sometimes credited with the elimination of metaphysics from science, but no scientific activity can ever be free from metaphysical assumptions. In particular, Galileo had a strong belief in the order and simplicity of nature and in its real existence, and these and other beliefs presupposed by science are Christian beliefs that played a determining role in the origin of modern science in the High Middle Ages. Likewise, Galileo separated physics from the principles of Aristotelian philosophy.

Although Galileo was largely responsible for the overthrow of Aristotelian physics, he retained his belief in Aristotle's natural philosophy, particularly the need to attain a physical understanding of nature and to use the laws of logic. He was a strong believer in the simplicity of nature and so continued to believe that the orbits of the planets are circular. As a result, although he greatly admired the work of Kepler, he never showed much interest in Kepler's three laws of planetary motion, which indeed provide strong support for the heliocentric theory. This belief in universal circular motion led him into great difficulties when he considered the explanation of comets. Neither did he spend time on the optical theories underlying the operation of the telescope, but constructed them by a process of trial and error.

Galileo's second great achievement was the construction of greatly improved telescopes, and the astronomical discoveries he made with their aid. These made him famous throughout Europe and changed his life. He soon became convinced that the heliocentric theory of Copernicus was correct and used his astronomical discoveries to develop arguments in its favour. Individually, none of these arguments was conclusive, and at least one was incorrect, but together they were sufficient to convince the scientifically-trained mind.

If he had been content, like most scientists, to publish his results in weighty Latin tomes, he would not have become embroiled in controversy. Instead, he vigorously publicised his work, and wrote in the vernacular in a way that could be understood by the general reader. Partly this was necessary in order to obtain employment, but in addition he felt that he had a duty to publicise his new view of the world. When his jealous enemies, unable to defeat him on scientific grounds, invoked the aid of theology, he countered with a treatise on the interpretation of Scripture. The theologians, angered by his incursion into their domain, and concerned by the threat to their whole Aristotelian world-view, used their political power in an attempt to suppress his views. They were, for the most part, well motivated, and saw themselves as defenders of a hallowed synthesis of natural and supernatural knowledge. They had, however, no understanding of Galileo's scientific achievements, or of the strength of his arguments, which made it absolutely essential for them to re-think their whole world view.

The attempt to suppress Galileo's scientific work of course failed, and his writings became widely known throughout Europe. His work on dynamics, in particular, culminated in the achievements of Newton, who unified Galileo's laws of motion and those of Kepler with his theory of gravitation. More generally, Galileo inaugurated a new style of scientific thinking that was to bear much fruit in the following centuries and identifies him as one of the founders of modern science.

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