International Catholic University

Galileo: Science and Religion

Galileo the Physicist

Peter Hodgson

1 Introduction

Galileo had an insatiable curiosity about the world, and he believed that we can learn about it by careful observation and by accurate measurement. The results of measurements can be used to find relationships between the quantities measured, and these in turn can be used to develop and test theories about the behaviour of phenomena. He applied this method to a wide range of phenomena.

This combination of physical insight and mathematical description is the essential feature of modern science. It is not enough to have a physical description of a phenomenon, for this may be plausible but wrong. Neither is it adequate to have a set of mathematical rules that describe the measurable features accurately but give no physical insight, for there may be several sets that do this equally well. We only have a real understanding when both are combined. Even then, it may appear later that it is not a complete understanding, and that our knowledge has to be developed to cover a still wider range of phenomena.

It is not always easy to reconstruct exactly what Galileo did and his motives for doing so. It is extremely difficult, if not impossible, for us to think ourselves back into the mindset of the past, to know the mental background, to know what was taken for granted and what was implicitly denied, to understand what sort of arguments were accepted as valid, and the criteria used to separate truth from falsity. If we are to distinguish between genuine discovery and mere copying, we need to know what had already been discovered and how widely these previous discoveries were known. It is nearly always possible to find many precursors who had some partial or even quite accurate knowledge of what is claimed as a new discovery. Quite often the person recognised as the discoverer has taken some idea already known, rephrased it more clearly, demonstrated it by well-chosen experiments, and then has publicised it in a particularly arresting and cogent way.

In the case of Galileo, we have his extensive writings, but when we read them, especially in translation, we are faced with the problem of knowing just what the words mean. There is an ever-present danger that we use our present knowledge of physics to read into his experiments motives and ideas that were not his. The terms he used to describe his work are often translated by modern terms that have precise meanings that are unlikely to correspond to those in Galileo's mind.

The whole process of scientific discovery is mysterious even to the scientists themselves. When we are wrestling to understand some strange phenomenon we think about it for weeks and months, trying out one idea after another, until finally the light dawns. All these intermediate stages are soon forgotten, so it is impossible even for the scientists themselves to re-create the process. Certainly it is not present in the dry, logical account that is written up for publication.

These difficulties are particularly acute in the case of Galileo, for he inherited the mainly qualitative science of the Middle Ages, and was largely responsible for transforming it into the quantitative science we know today. The concepts used to describe motion, for example, were not clearly understood, and only gradually achieved the clarity we know today. The ideas defining the concepts of impetus and inertia developed over the years from the work of Buridan to that of Galileo, and the degree of continuity and discontinuity is still disputed.

In some cases it is doubtful whether Galileo actually performed the experiments he describes; they are more in the nature of thought experiments designed to clarify his ideas and to convince others of their truth. He undoubtedly made some experiments in order to understand the phenomena he was studying, and when he believed that he had attained this understanding he deduced the consequences for other situations which he had not studied.

Many of his experiments were technically difficult for him, since he did not have available the simple measuring instruments that we now take for granted. It was comparatively easy to measure distances, for example in his studies of falling bodies, but it was very difficult to measure short times at all accurately. The inevitable uncertainties in his results then made it more difficult to be sure that any relationship that he found is the only one possible. There is also the ever-present difficulty of removing or allowing for the effects of extraneous influences.

In the course of his work Galileo designed and made many ingenious new instruments, and frequently had them manufactured in his workshop for sale. An example is an early form of thermometer, and various quadrants and magnetic compasses. He also applied his knowledge of levers to simple machines and in his book on mechanics he described the windlass, the capstan, the screw and the Archimedean screw.

2 The Pendulum

Galileo's earliest biographer Viviani claimed that in 1582, when he was still a medical student in Pisa, he observed the motion of the swinging lamp in the cathedral. Using his pulse to measure the time of swing, he found that it is independent of the amplitude of the swing, providing that this is small. This is a rather surprising result, as it implies that it takes the same time for the pendulum to reach the nadir of its swing however far it is drawn aside before release. According to Viviani, this suggested to him that a pendulum could be used to measure the pulse rate. There is, however, no other evidence for this story, as Galileo first mentioned the isochronous nature of the pendulum in a letter in 1602. He also showed that the period of swing is independent of the material of the pendulum and that the period is proportional to the square root of the length of the string. Galileo also compared the swing of the pendulum with the motion of a ball that runs down one inclined plane and up another one opposite to it. In another investigation, he found that the times of descent are equal for all chords from the highest or to the lowest points of a vertical circle.

3 The Dynamics of Free Fall and of Projectiles

Galileo's earliest work on mechanics was in his De Motu of 1592, which is devoted to a discussion of the fall of bodies in media of different densities. In this work he was much influenced by the ideas of the Jesuits at the Collegio Romano, whose lecture notes he used extensively (Wallace, 1984). They held, with Aristotle, that the aim of science is the understanding of natural phenomena in terms of evident principles, and Galileo continued to accept this throughout his life. However he strongly opposed the arid textual Aristotelians found in the universities and it is against them that his polemics are directed. De Motu is largely a detailed analysis of the writings of Aristotle on the subject, and after about forty pages of discussion he exclaims: "Heavens! At this point I am weary and ashamed of having to use so many words to refute such childish arguments and such inept attempts at subtleties as those which Aristotle crams into the whole of Book 4 of De Caelo, as he argues against the older philosophers. For his arguments have no force, no learning, no elegance or attractiveness, and anyone who has understood what was said above will recognise their fallacies." (Drabkin, p.58). Later on he remarks that "Aristotle was ignorant not only of the profound and more abstruse discoveries of geometry, but even of the most elementary principles of this science"(p.70). A few pages later, discussing how projectiles are moved, he remarks: "Aristotle, as in practically everything that he wrote about locomotion, wrote the opposite of the truth" (p.76).

At that time, however, Galileo apparently thought that each of the cases he discussed was characterised by a constant velocity rather than a constant acceleration. He also accepted the current but false belief that if a light and a heavy body are dropped together, the light body will initially move more rapidly than the heavier, and so devoted several pages to ingenious arguments to explain why this happens. If indeed there is experimental evidence for this effect, it probably occurs because a heavy body has to be held more tightly than a light one, and so tend to be released a little later.

Galileo developed his views on motion throughout his life and his mature conclusions are described in his Discoursi of 1638 that in many respects is a precursor of Newtonian mechanics. The transition from medieval to Newtonian mechanics is thus largely due to him.

Since ancient times there was much discussion concerning the rate of fall of bodies towards the earth. Study of this natural motion was an essential preliminary to the discussion of the forced or unnatural motion of projectiles. Aristotle suggested that the speed of free fall V is proportional to the weight W of the body and inversely proportional to the resistance R of the medium. It is however not justified to conclude that V = kW/R, where k is a constant, since he also believed that the motion is accelerated: the velocity increases as the body approaches its natural place. Some medieval commentators suggested that the velocity is proportional to the distance d fallen, which would give V = kdW/R, but this connection was not made. The philosophers in Oxford and Paris in the fourteenth century succeeded in formulating the odd-number rule for the distances traversed by a uniformly accelerated falling body in equal times and this is equivalent to saying that the distance traversed is proportional to the square of the time, an achievement usually attributed to Galileo. The medievals however had no means of measuring acceleration and there was no justification for assuming that freely-falling bodies are uniformly accelerated, so this was no more than speculation.

These discussions went on for centuries without much progress because few if any actual experiments were made, the complications due to the resistance of the medium were not properly understood, and there was no clearly defined concept of acceleration. Thus what is now a trivial exercise for thirteen year olds was still a problem exercising the minds of the leading philosophers of nature.

It was not difficult to show that the above expressions for the velocity have unacceptable implications. They imply, for instance, that in a vacuum, when the resistance is zero, the velocity would be infinite (which, incidentally, provided Aristotle with his argument for the impossibility of a vacuum), and that two bodies of equal weight, falling side by side, would double their velocity if joined together. Furthermore, if the medium is denser than the body, the latter will rise and not fall.

We can now see that the resistance in a particular medium is not a well-defined quantity, as it depends on the size, shape and surface roughness of the body, and also on its velocity and internal motion. Even today this remains a very complicated problem. Since the resistance increases with the velocity it eventually equals the gravitational force so that after falling a certain distance the velocity becomes constant. This is obvious if we consider a metal ball and a feather falling through treacle, but also applies to free fall in air: the probability of survival of a cat that has the misfortune to fall out of the window becomes a constant for falls from more that about seven floors. The attainment of a terminal velocity was recognised by Galileo in his De Motu, although he thought that it is achieved more rapidly than it is.

To make any progress it is therefore necessary to consider the situation where the resistance of the medium can be neglected. Galileo noticed that bodies of different materials and shapes fall with very different velocities in dense materials, but at about the same velocity in air. This is supported by his experiments with a pendulum. He then conjectured that in a vacuum all bodies would fall with the same speed. The motion is then independent of the medium and of the nature, size and shape of the body. It is impracticable to make measurements in a vacuum but this is approximately true for the fall of smooth heavy spherical bodies falling through short distances in air. Ideally, we could make measurements in air of decreasing density and then extrapolate to zero density. There is always, as in most experiments in physics, a final leap from the best we can achieve to the ideal situation. Thus most of physics refers to an ideal Platonic world and not to the real world, a distinction that is seldom recognised, sometimes with disastrous results. This is directly contrary to Aristotelian physics, which is concerned with what normally happens, which implies that it is not possible to learn anything about natural motion by considering a situation, however ideal, that dos not actually exist.

Throughout his work Galileo made use of the principle that the laws of nature are simple. Thus the laws of motion must be expressed by simple formulae. In some contexts this led him astray, as when he rejected Kepler's conclusion that the planetary orbits are ellipses and not circles.

Galileo's views on motion went through several stages. At first, as described in De Motu, he believed that natural motion has a natural uniform speed proportional to the difference between the density of the moving object and that of the medium. As the effective density is diminished by the medium, so is the natural uniform speed. Non-natural motions are due to an impressed force, and this is responsible for the initial acceleration. These views constituted a coherent philosophy of motion, but Galileo could not find a single example of this uniform motion and so concluded that acceleration is a feature of all motion. He considered the possibility that the velocity is directly proportional to the distance, but soon rejected this possibility. Then, from the mean speed theorem, which implies that the acquired velocity is proportional to the time taken, he deduced that the distance covered is proportional to the square of the time taken. We can obtain this result more easily using Newton's notation: x' = g, x" = gt, x = gt2/2.

There has been some controversy about whether Galileo actually performed any experiments related to free fall. It is suggested that since he already knew that the distance travelled is proportional to the square of the time the actual experiment was undertaken to confirm that result. It is certainly true that many of Galileo's earlier statements about motion were the result of thought experiments. He believed that ideas suggested by simple observations are often misleading, and that they must be tested by mathematical reasoning. Later on he realised the need for carefully planned experiments.

Galileo soon found that it was very difficult, if not impossible, for him to measure with sufficient accuracy the time taken for bodies to fall. He therefore hit on the ingenious idea of timing them as they rolled down planes inclined at different angles. The times to be measured are much longer, and so could be measured more accurately. He could make measurements for a series of increasing angles, and then extrapolate to find the rate for free fall. The experiment is indeed quite practicable, as shown by Settle (1961).

Even then it was not easy to measure the times. Eventually he did this by weighing the amount of water that spurted out of a pipe from a small hole near the bottom of a large jar of water. He stopped the hole with his finger, removed it when the ball started and stopped the flow when the ball had traversed the prescribed distance. After many hundreds of trials he found that the distance traversed is proportional to the square of the time taken for all angles, although of course the constant of proportionality varied with the angle. Although Galileo did not recognise this, it was not possible to extrapolate the constant of proportionality to obtain that for free fall, what we now know as the acceleration due to gravity. because when the balls rolled down the plane some of their potential energy was converted into rotational energy, and not to kinetic energy, to use the modern terminology. A simple calculation shows that this does not invalidate the time squared law, but it reduces the value of the acceleration by a factor 5/7.

Galileo also made further studies of motion that do not require time measurements. He let balls roll down an inclined plane, and at the end of the plane they were deflected horizontally and then allowed to fall freely until they hit a horizontal plane. The time squared law implies that the path of free fall is a semi-parabola, so that by seeing how the length and angle of the inclined plane was related to the point of contact on the horizontal plane Galileo could verify the correctness of the law.

One of the most familiar stories about Galileo, also due to Viviani, is that he dropped two different weights from the top of the leaning tower of Pisa and that, to the dismay of the watching Aristotelians, they hit the ground at the same time, thus disproving Aristotle's law. If ever he did the experiment, however, and if he succeeded in releasing them at exactly the same moment, which is not as easy as it sounds, careful observation would have shown that, due to air resistance, the heavier body would have hit the ground slightly before the lighter body. This is still quite different from the proportionality given by Aristotle.

Galileo also considered the fall of bodies in a medium. As an example he chose two balls of lead and ebony. Supposing lead to be ten thousand times and ebony to be a thousand times as heavy as air he concluded that if they are allowed to fall from a tower two hundred cubits high, the lead ball will outstrip the ebony ball by less than four inches. Since a cubit is about 20 inches, he is saying that the lead ball outstrips the ebony by less than one thousandth of the height of the tower. This follows approximately from the assumption that the velocity of the lead ball is reduced by a factor (1 - 1/10,000) and that of the ebony ball by a factor (1 - 1/1000). No justification for this conjecture is given, and it is clear that the result is hypothetical and not the result of any experiment.

Physicists, and Galileo was no exception, are sometimes prone to imagine that they have such a firm grasp of a particular phenomenon that they can confidently say what is going to happen without making any experiments. Frequently their confidence is justified, especially when they are making qualitative predictions, but sometimes they are wrong. Many instructive examples could be given from the history of science. Quantitative speculations, like that mentioned above, are much more shaky.

When he came to consider the motion of projectiles, Galileo was faced with the problem of combining the unnatural motion due to the action of the thrower with the natural motion due to the tendency of all bodies to move towards their natural place. Aristotle believed that they cannot be combined so that, for example, the shot from a gun is first impelled by its unnatural motion along a linear path in the direction of the barrel and then, when that motion is exhausted, begins to fall vertically downwards following its natural motion towards the earth. Galileo began by considering the simple case when a body is thrown horizontally; the natural motion is its vertical fall according to the time squared law and the unnatural motion is the horizontal motion that, as will be shown below, has a uniform velocity. He enunciated the important principle that these two motions can be combined vectorially, so that the resulting motion is the sum of the two independent motions, which is a semi-parabola.

Galileo considered motion on a horizontal plane as a limiting case of his inclined plane experiments. He found that if he had two inclined planes arranged so that the ball rolls down one and then up the other, then whatever the angle of the second plane the balls always rolls up to very nearly the same height as it had on the first plane when it was released. Now reduce the angle of inclination of the second plane until it is infinitesimally close to the horizontal; in the limiting case it will roll on forever with constant velocity. In practice it will eventually come to rest due to air resistance and friction, but as always Galileo abstracted from such non-essential disturbances. By a horizontal plane he evidently meant a plane that is parallel to the earth's surface, which is essentially flat for practical purposes when small distances are involved.

4 Hydrostatics and Floating Bodies

The ancient Greeks discussed the reasons why some bodies float and others sink, and how this depends on their shapes and densities. The achievement of Archimedes in devising a method to determine the presence of base metal in the king's golden crown is well-known. It is surprising to us that such simple problems, now easily understood by children, were the subject of impassioned and confused debate for centuries by highly intelligent men. This reminds us that it was far more difficult than we think to establish the conceptual framework within which such problems can be clearly discussed, and that this was largely due to the work of Galileo.

Galileo became involved in such problems during a series of philosophical discussions held in the villa of Filippo Salviati near Florence. An Aristotelian philosopher maintained that the action of cold is to condense, but Galileo said that since ice is lighter than water the action of cold is to rarefy. The philosopher replied that ice only floats because of its shape, whereas Galileo maintained that it floats whatever its shape. At the next meeting one of the philosophers, Ludovico delle Colombe, showed that thin plates of ebony float on water, whereas other shapes sink, thus showing that whether bodies float or sink depends on their shape. This discussion, initially friendly, soon escalated into a bitter feud, and Galileo was told by the Grand Duke to avoid controversy and to confine himself to written comments. So he wrote his Discourse on Floating Bodies, which was published in 1612.

Following Archimedes, he said that the upward force on a body immersed in a fluid is equal to the weight of the displaced fluid, i.e. dvg, where d is the density of the fluid, v the volume displaced and g the acceleration due to gravity. Similarly the weight of the body is DVg. Thus the net upward force is

(dv - DV)g. If the body is completely submerged, then V = v, and this becomes

(d - D)Vg. Thus whether a body sinks or rises depends on the difference between the densities of the body and the fluid.

According to this, ebony should always sink, as it is slightly denser than water. Why then do thin plates of ebony float? Galileo did notice that such plates depress the surface of the water, thus effectively increasing the volume of fluid displaced sufficiently to keep the ebony floating. However the full explanation requires the concept of surface tension, which was not at that time known.

These considerations mark an important advance on the ideas of Aristotle, who said that there were two kinds of motion, one due to a natural inclination to rise, like fire, and the other an inclination to fall, like heavy bodies. Galileo showed that whether a body rises or falls depends only on whether it is less dense or more dense than the surrounding medium, so that all bodies obey the same law. This progressive unification of apparently different phenomenon is characteristic of the advance of science.

5 The Atomic Constitution of Matter

Galileo's discussion of floating bodies led him to speculate that motion through water is rather like pushing oneself through a crowd or thrusting a stick into a heap of sand. He thus thought of liquids as composed of multitudes of tiny particles, too small to be visible.

He also thought that fire provides evidence for the atomic constitution of matter. Democritus maintained that broad plates are able to float by heat-particles rising in the water, whereas narrow plates sink because there are too few such particles impinging on them. Aristotle rejected this argument, saying that if it were true heavy objects would float more easily in air than in water. Galileo considered this argument to be incorrect because bodies weigh more in air than in water,and also there is no reason to suppose that fire-atoms move more rapidly in air. He suggested that a thin, broad plate of a material sightly denser than water be placed on the bottom of a vessel filled with water. On heating, the fire atoms, if they can support it on the surface, should be able to raise the plate. However this does not happen, and so he concludes that the fire atoms are not able to provide the full explanation for the floating of such bodies (Shea, p.28).

Galileo also considered evaporation and boiling, and said that he could see millions of small spherical globules of fire rising through water when it is heated, and passing through the surface into the air. He rejected the view that the globules are water changed by the fire into vapour because the level of the water never falls, however long it is boiled.

This is an instructive example of a failure of Galileo's method. He believed that he could, on the basis of a few experiments, understand the principles governing the behaviour of a particular phenomenon. After that, simply by deductive reasoning, he could say with confidence what would happen in a large variety of circumstances that had not been experimentally investigated. Often this method worked well, but if the understanding is in any way faulty, it inevitably leads to false conclusions. It is very easy to see only what we want to see, and of course we want to see behaviour in accord with our own theories.

His speculations about heat were more successful when he said that it is not the fire-corpuscles that give the sensation of heat but their motion, and thus explains how a stone or a stick can be heated by rubbing. This changes them into 'very subtle flying particles', or perhaps releases fire-corpuscles, and these produce the sensation of heat.

Galileo also considered the strength of materials. He wanted to understand why it is that when machines are constructed "the larger machine, made of the same materials and in the same proportions as the smaller, will correspond to it with perfect symmetry in all respects except that of strength and resistance to breakage; the larger it is, the weaker it will be". This phenomenon is also notable in the animal world: an elephant is more massively constructed than a gnat. At first sight this seems to reveal a discrepancy between matter and geometry, between physical and mathematical divisibility. To understand this requires a consideration of the strength of materials, and this led Galileo to think about the ideas of continuity, the vacuum and the atomic structure of matter.

It has been suggested by Redondi that Galileo's atomistic explanation of sensory perception has heretical implications for the dogma of Eucharistic transubstantiation, and that this is the real reason for his trial and condemnation. This suggestion is based on a single document that in fact has nothing to do with the trial, and so there is no basis for Redondi's claim (Carugo and Crombie, 1988).

6 The Velocity of Light

In his Dialogue Concerning Two New Sciences Galileo describes an experiment to determine the velocity of light. Two people, each with a lantern, stand several miles apart. The first uncovers the lantern and then immediately the light is seen by the second he uncovers his lantern. The first then measures the time that elapses from the moment he uncovers his lantern to when he sees the light of the second lantern. This time, divided by twice the distance between the two people, give the velocity of light. However it was found that the time was immeasurably small, and so the experiment failed. We now know that the velocity of light is so great that such experiments are bound to fail.

An interesting sequel is that the first reliable measurement of the velocity of light was made by Romer by observing the eclipses of the satellites of Jupiter. These were found to occur rather later than expected when Jupiter was far from the earth, compared with the times when Jupiter was near. This is due to the time taken by the light to travel from Jupiter to the earth, and from this the velocity of light was determined.

Galileo also tried to develop a method of determining longitude at sea by observations of the satellites of Jupiter. Although this is possible in principle, it was found to be impracticable because at that time it was not possible to measure the time at sea with sufficient accuracy. This was achieved later by Harrison. The method has, however, proved useful in surveying on land.

7 Galileo's Scientific Method

When he was a young professor in Padua, Galileo was strongly influenced by the writings of the Jesuits teaching at the Collegio Romano, particularly Menu, Vella, Rugierius and Vitelleschi, and he based his lectures on their work. These Jesuits accepted Aristotle's definition of science and treated logic and physical questions in a realist way, following Aquinas. This formed the solid basis of Galileo's subsequent work (Wallace, 198. 1984, 1986)

Galileo realised more clearly than anyone before him that the primary task of the physicist is to understand the world as it is, to penetrate behind the apparent complexity of phenomena to the often surprisingly simple reality beneath. Thus when he considered freely falling bodies he wanted to establish the laws obeyed by all bodies of whatever shape or material. He therefore considered fall in a vacuum but, as this cannot be realised in practice, he chose the best approximation, namely the fall in air of smooth hard balls. The physicist is almost never able to make an experiment in an ideal situation, so it is necessary to consider all the unwanted influences that could affect the final result, and to allow for them. This often requires a subsidiary experiment to study and quantify these influences. This evaluation of perturbing effects is a vital component of the art of scientific investigation.

Galileo also distinguished between primary and secondary qualities. He pointed out that all bodies have a shape and a size, that it is in a particular place at a given time, that it is moving or stationary and so on. These are primary or essential qualities and cannot be separated from the body. On the other hand there are other secondary qualities such as colour, taste and smell that, although grounded in the properties of the body, are in themselves sensations that exist only as they are perceived by the observer. Scientific research is thus essentially concerned with studying the primary qualities of bodies.

In his research, Galileo combined the insights of Aristotle and Plato, and went beyond them. Like Aristotle, he insisted on the primary importance of experience, of the knowledge that comes to us through the senses. This knowledge, however, cannot be taken at its face value, but must be tested by combining it with other experiences and uniting them all by a general principle or theory. This theory cannot be deduced from the experiences; it is a creation of the human mind. The theory should not only agree with the original experiences, but usually also predicts a range of other experiences that enable it to be tested. In order to specify these new experiences we need not only Aristotelian logic but also mathematics. The theory and the mathematics refer to an ideal world and are thus Platonic in nature. Theories are tested by making experiments, and the conditions are chosen so as to be as close as possible to the ideal situation. If the results disagree with the theory, then the theory must be modified so as to be consistent with the new experiences and then tested again. It is not necessary to make a very large number of experiments; due to the uniformity and rationality of nature a few well-chosen experiments suffice.

There are many practical difficulties in carrying out this programme. What experiences do we start with? Usually this is indicated by an existing theory. If this has a mathematical character it is necessary not only to observe but also to measure. The construction of the theory depends on the insight of the scientist and cannot be specified by a set of rules; it may therefore be wrong in a fundamental way or, more frequently, it may be inadequate in one respect or another. Its consequences are likely to be very extensive, and it is not easy to choose the ones that make the sharpest test and yet are relatively easy to carry out. If there is a disagreement, is it due to a defect in the experiment or does it show a real defect in the theory? If the latter, then how should the theory be modified, and so on.

As the theories become more sophisticated and agree with a wide range of experience they may be said to give genuine, though still limited, knowledge about the world. As confidence grows, it become less necessary to make experimental tests, and this is certainly true of the laws of motion. However it always remains possible that new experiences show inadequacies in the theory that require it to be modified. There are many examples of this in the history of science.

Galileo was thus neither a pure Aristotelian nor a pure Platonist (McTighe, 1967). He could with justice claim that he was a better Aristotelian than many of the professed Aristotelians that criticised him. He, like Aristotle, observed nature, and did not seek the answer to questions only in books. If Aristotle had been able to look through a telescope he would have certainly modified his views. Likewise Galileo was a good Platonist by his stress on the importance of mathematics, which Aristotle undervalued. However Plato considered that the material world is an imperfect copy of the ideal world, whereas Galileo believed in the possibility of an exact mathematical description.

Galileo never formulated a fully-articulated theory of the scientific method; indeed this is still the subject of controversy today. He was a pioneer with a vision of the future and had to develop his tools as he tackled new problems. He was primarily interested in solving problems, not in explaining the methods he used to solve them, which he made up as he went along. Yet in so doing, he was inevitably throwing doubt on the traditional Aristotelian natural philosophy, and this could have the most far-reaching and serious consequences. Structures of thought are linked together far more tightly than is generally supposed, so that it is not possible to modify one section without affecting the others. This was clearly seen by many of Galileo's opponents and ensured their opposition, even if they were unable to mount any effective criticism of his actual scientific work.

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