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How did Galileo discover the law of free fall?

Nagarjuna G.
Homi Bhabha Centre for Science Education
Tata Institute of Fundamental Research, Mumbai, INDIA


The essay is an epistemological reconstruction of the conceptual change that took place in the mind of Galileo. This is based on two of Galileo’s works: the first one is De Motu (On motion), which was published posthumously.1 The second one is Dialogues Concerning the Two New Sciences published in 1636. De Motu is very important for our dicussion because it throws a lot of light on how the conceptual change from Aristotelian view to the modern view took place. In this text Galileo informs us about the internal dialogue that went on in his mind before arriving at his discoveries. The episode also illustrates the basic steps of scientific imagination and how the conceptual change took place.

The conceptual change that we shall see here consists of the transformation from the Aristotelian conceptual scheme to the basic modern conceptual scheme. Unlike most modern scientists, who never tell us how they arrived at their ideas, Galileo informs us how the discovery happened in his writings in the form of a dialogue. Whether he wrote simple prose or in the explicit from of a dialogue there always existed an inherent dialogue in his writings. This dialogical style of presentation is a very good case for introducing history of science in the class room.

The style adopted by Galileo has some inherent strengths particularly for communicating history of science. For example, the characters in his Dialogues, Salviati (who presents Galilean view point) and Simplicio (who argues for Aristotle), strictly speaking, do not belong to the same historical period. But, they talk and argue for their respective theories. Galileo studied and tried to learn Aristotle, Archimedes and Euclid. He discovered the basic cognitive conflicts that arose when he tried to master and assimilate the differing view points of the masters. As a restult of this intellectual ‘chemical reaction’ emerges the basics of modern science. Galileo tells us how this happened by re-enacting them in the dialogues through the innovative characters. It may not be possible to bring this ‘intellectual chemistry’ to the class room for each and every episode from the history of science, but a few of them deserve the place. I wish to present here the reconstructed episode of how Galileo discovered the law of free fall, as an example. Another reason why I selected this episode is that it can be discussed in a high school and the argument can be understood without invoking higher mathematics.

Epistemic Nature of the Discovery

Before going through the details of how the discovery was made, let us pay some attention to the epistemic nature of the discovery. In an orientation course for secondary school teachers, we asked the question: “How did Galileo discover the law of free fall?” Not many replied. But a few of them who happened to have read about Galileo from the popular science books told us that Galileo performed the experiment by dropping stones, feathers etc., from the famous leaning tower of Pisa. Galileo never told us anywhere in his writings that he had ever performed this experiment, but he did indeed tell us how he arrived at the law of free fall. The discovery appears to be made out of systematic thought experiments rather than real world experiments. From what he told us it appears that this story about the leaning tower of Pisa is incorrect.

Also if we look at the epistemic nature of the law, it becomes clear that it would not have been possible to infer the law from such an experiment. Let us look at the discovery made by Galileo:

[I]n a medium totally devoid of resistance all bodies would fall with the same speed”. [The Two New Sciences, p.72.]

Which means that all bodies irrespective of their size and density will fall at the same rate only in void. (Note that the term used by Galileo was ‘speed’ at this point and not ‘rate of speed’.) In order to even make sense of the statement at that time, it is necessary to convince his Aristotelian colleagues that void is possible.

Is this an empirical statement? Can such a statement be arrived at by inductive reason alone? Or is this a restult of theoretical construction arising out of imagination or reason. By no stretch of ‘induction’, this kind of generalisation can be obtained, because at that time it was not technically feasible to conduct an experiment of this kind. Also the concept of ‘void’ is a theoretical construction. The general statement that heavier bodies fall quickly than lighter ones is inductively possible. So the statement that all bodies irrespective of their ‘heaviness’ would fall at the same rate is clearly counter-inductive and also counter-intuitive within the Aristotelian world-view. We need to have a high degree of theoretical construction to make sense of such a discovery.

The epistemic strength of the discovery cannot be appreciated by students unless we manage to convey to them that the discovery is impossible without theoretical imagination. It is therefore important to inform the students with examples that most important scientific generalizations are counter-intuitive, beginning from the law of inertia to Einstein’s notion of space and time. A discussion of this kind provides an ideal context for introducing some fundamentals of inductive and deductive reason. I would only say at this time that Galileo’s works, particularly De Motu offers ample chance to introduce many problems of philosophy of science. Since the text is not very technical it is an ideal choice for bringing history and philosophy of science into class rooms at the secondary and higher secondary stage.

The Two Conflicting World Views

Now to some details of the discovery. What was the problem that Galileo was addressing? Galileo was addressing the problem: What causes motion? He was not the first person to raise this problem in the history. While many Greek metaphysicians raised this question it was Aristotle who began raising this question at the level of specific phenomena. He attempted explanations to isolated phenomena such as the motion of an arrow, the movement of a cart on the road, of falling of bodies, of projectiles etc., based on a general theory. What was his theory of motion?

We can get a glimpse of his theory by looking at the statemnet: “A change contrary to the arrangement of nature causes motion”. In order to understand this statement we need to first look at “the arrangement of nature?” Let us recall that according to the ancient wisdom of Greeks the world is made of the four elements: earth, water, air and fire. The elements according to Aristotelian world-view have fixed positions. Earth lies at the bottom of all things being heavier than the rest, and fire the lightest of all is closer to the sky. Water lies above the earth and air below the fire as shown in the figure.

Figure 1: Aristotelian world-view

Given this arrangement of nature, any variation in this order would cause motion. For example, when we use our muscle force to throw a stone (earth) up in to the air we disturb the stone from its natural order. The stone will come back to its ‘earthly’ place (on its own), because that is where it belongs. Smoke (fire) generated on the ground rises up, again because that is where it belongs.

On the other hand the Galilean world-view is radically different. His world-view is borrowed from the Greek Atomists’ legacy. The elements in this view do not have any fixed positions, though they get organized according to their relative densities of their atoms. So the difference from one element to the other is only with regard to how densely the atoms are arranged in them. As shown in the picture, there are no fixed boundaries from one element to the other, and there is only a degree of difference from one level to the other. Motion in this world depends on the relative density of the elements. There is nothing like a fixed position to any of the elements. For example, if there were no water and air, fire would, so to speak, fall closer to the earth, and if there were no earth and air, water and fire would still form the same picture.

Figure 2: Galilean world-view

To make the distinctions clearer let us give them names. Aristotelean view may be called ‘categorical’ view and the Galilean view ‘relativistic’. Also note that in the former view we only need to name things, because each substance differs from the other essentially. In the latter, however, there is no essential difference, but only relative, because all substances are made of atoms. (Some atomists, though, believed that there are different kinds of atoms, and so they allowed essentialism to creep in at the level of atoms.) Let us also note that in the latter, Galilean, world-view it is possible to make measurements, because properties are relative and comparable.

Before we jump to next section, I would like to make a few comments. There is a normal tendency to characterize Aristotle as an inductivist. But if we look at the world-view, it is clearly not a set of inductive generalizations. The model is sufficiently abstract and requires higher levels of theoretical imagination. There are however several statements that Aristotle invokes while dealing with specific phenomena, and it is in these contexts he tends to be more an inductivist. I think this clarification is important because when cognitive psychologists claim that children tend to be Aristotelian, their claims actually apply only to the specific empirical statements, and certainly not the abstract cosmological picture of Aristotle.

We need to keep in mind the opposing world-views while understanding this episode. There are several other aspects to the conceptual schemes of the two opposing world-views, and we shall cover them as we proceed. Now let us turn to the details of the discovery.

Relativistic Analysis

In the first chapter of De Motu Galileo clarifies (defines) the terms ‘heavier’ and ‘lighter’ in a characteristically anti-Aristotelian way. The vantage point from which this is done becomes clear in the first few chapters of De Motu, though greater understanding will be achieved as we go further into the details of other chapters. These terms are going to become the contraries of the principle (cause) of natural motion, and are therefore immediately relevant.

[W]e sometime say that a large piece of wood is heavier than a small piece of lead, though lead, as such, is heavier than wood. And we say that a large piece of lead is heavier than a small piece of lead, though lead is not heavier than lead.

These words could lead to confusion if it is not clear as to what is being said and from what point of view. Galileo, therefore, thinks it is “necessary to settle this,” and “avoid pitfalls of this kind”. Can there be a way of saying that something is heavier or lighter absolutely? Galileo’s answer is ‘no’. Heaviness and lightness can however be defined relatively. He defines three notions, and not just two, viz., “equally heavy”, “heavier” and “lighter” in the following way.

Equally heavy: “[T]wo substances which, when they are equal in size [i.e., in volume], are also equal in weight.”[p. 10].2
Heavier: “[O]ne substance should be called heavier than a second substance, if a piece of the first, equal in volume to a piece of the second, found to weigh more than the second.”[p. 14.]
Lighter: “[O]ne substance is to be considered lighter than a second substance, if a portion of the first, equal in volume to a portion of the second, is found to weigh less than the second.”[p. 14.]

The similarity of these statements with that of Archimedes can be readily seen. The first one in both cases is regarding the state of equilibrium, and the other two are about the two possible variations of disequilibrium. The three statements actually constitute one single principle, by means of which all effects are to be explained with regard to differences in heaviness or lightness. In the above definitions, heaviness has been defined in terms of weight. These definitions therefore also make a conceptual distinction between weight and heaviness, which later becomes the distinction between weight and mass.

The most significant point to note is the method of selectively controlling one of the parameters to understand the proportional relations between them. Here the volume of the substances under comparison are controlled in the sense that they are kept constant in all the cases. What is the methodological necessity of this? Since a change in volume effects a corresponding change in heaviness, unless volume is kept constant no relative [objective] comparison is possible. In common comparisons, since we do not bring in a third factor into consideration the resulting judgements can be confusing.

So far we see two distinct influences on Galileo, one, that of the Atomists’ understanding of density, and the other of Archimedes’ principle of equilibrium.

Galileo argues that the more categorical terms “heavier” and “lighter” are less preferred to relativized terms “heavier” and “less heavy” on the ground that nothing is devoid of weight. Later in Chapter 12, he says that even fire, which is less heavy than all “will move downward if air is removed under it, that is, if a void or some other medium lighter than fire is left under it.” The other pair of categorical terms “downward” and “upward” have also been relativized to “nearer the center” and “farther from the center”. Thus in Chapter 2, of version II he says:

Up to now we have spoken of “the heavy and the less heavy” not of “the heavy and the light”; and of “nearer the center and farther from the center,” not of “downward and upward.” ... Yet, if at times, out of a desire to use ordinary language (for quibbling about words has no relevance to our purpose), we speak of “the heavy and the light,” and of “downward and upward,” these expressions should be understood as meaning “more and less heavy” and “nearer the center and farther from the center.

Having considered the natural order of the arrangement of bodies, Galileo considers the question of what causes natural motion in Chapter 3. Both upward and downward motions (i.e., going farther from the center and coming nearer to the center) are caused, according to Galileo, by considerations of relative heaviness or lightness (lesser heaviness). Any change contrary to the arrangement of nature causes motion. A main consideration that goes into the argument is that the motion of bodies should not be studied independently of the medium in which it takes place. Should this mean that there is no motion if there is no medium? Surely not for Galileo, but for Aristotle ‘yes’. We shall shortly see that his position is that though the heaviness of bodies is relative to the media, the actual heaviness can be measured only in vacuum. The information that a body is heavier is not sufficient to determine the character of its motion. The further information that at which place, in which medium, and whether the medium is heavier or less heavy than the body, is also necessary.

Later he goes on to state three principles of hydrostatics: [1] that bodies of equal heaviness as the medium move neither upward nor downward - a state of equilibrium; [2] that bodies lighter than the medium do not sink in it, and cannot be submerged totally, but move upwards; [3] that bodies heavier than the medium get totally submerged and move downwards. These principles are the same in structure as that of the propositions 3, 4, and 7 stated by Archimedes in On Floating Bodies3, except that Archimedes does not concern himself with the movement of bodies upwards or downwards, for his concerns are purely with statics and not kinematics or dynamics. In proposition 7, however, Archimedes speaks of the descending of the heavier body.

Galileo being interested in solving the problem of natural motion by using the model developed by Archimedes, makes it a special point to use the terms denoting movements of bodies in media. As noted earlier Archimedes systematically avoids any kinematic concerns unlike Aristotle.

In this respect Galileo’s return to kinematic questions keeping in mind the model of statics, which has achieved sufficient abstraction such that the mathematization of physical phenomena would be possible. Though he barrowed the theoretical model we shall regard it as a development because we perceive it as the finding of new applications (discoveries) of the model already invented or constructed by Archimedes. The Alexandrian school had not merely achieved remarkable strides in mathematics, they had also used the experimental method. Whoever would build models must ultimately come ‘down’ to controlled experiments to realize a ‘world’ constructed in thought.

Before going further on the subject, Galileo explains the analogy between the case of a balance and the case of bodies moving naturally, by reducing the latter to the former. Though he has been a student of and occupied a chair in mathematics, he has a desire to make matters clear by conveying the message through physical analogies. His acute concern for communication has very few parallels in intellectual history. His objective is “a richer comprehension of the matters under discussion, and a more precise understanding on the part of [his] readers” and he therefore restrains himself from using mathematical elucidation.

Coming to the analogy, he says in Chapter 6, whatever happens in a balance also happens in the case of bodies moving naturally.


Figure 3: Structure of Balance

Let ab represent a balance, as shown in the figure, and c its center bisecting ab; and let e and o be weights suspended from points a and b.

Now in the case of weight e there are three possibilities: it may either be at rest, or move upward, or move downward. Thus if weight e is heavier than weight o, then e will move downward. But if e is less heavy, it will, of course, move upward, and not because it does not have weight, but because the weight of o is greater. From this it is clear that, in the case of the balance, motion upward as well as motion downward takes place because of weight, but in a different way.

This is indeed a breakthrough, in the sense that it differs drastically from the Aristotelian order of things. For Aristotle, motion upward takes place because of two reasons: one by force, and otherwise naturally (i.e., without force) if the bodies are lighter or have no weight like fire. Bodies go downwards because of weight, or if the body is by nature light then it will go down by force. For Galileo, having relativized the notion of heaviness and lightness, both take place due to the same cause, namely, heaviness. Thus the analogy of the balance systematically helps Galileo to break conceptually with Aristotle. However, as he qualifies in the end, there is a difference in the way in which weight causes motion in both cases. The difference lies in a careful distinction between internal and external weight.

For motion upward will occur for e on account of weight of o, but motion downward on account of its own weight.

This remarkable distinction, we think, must have become crucial for the development of both the conceptions of inertia as an ‘internal’ cause, and force as an ‘external’ cause. This distinction would not have been possible without the analogy of balance because the analogy constrains us to think of only one other inverse factor that is affecting the motion, and not any of the numerous other logically possible factors. Since the body that is moving upwards and the weight that is causing are joined together by the lever, and because it is joined, the most immediate cause must be just the other weight that is external, but joined, to the body. If the joint is cut, not only the heavy body, but also the lighter body would move downwards, due to their own weights. Therefore downward natural motion must be because of its own weight.

Galileo did not arrive at the conclusion all at once. It took time because he started with natural motion, and now he has to call it the so called natural motion. He must have achieved this break while analyzing the consequences carefully from the analogy. The analysis in the initial version goes on as follows.

Continuing his considerations of balance he enunciates the general proposition that the heavier cannot be raised by the less heavy. This follows from the principles of balance. If water is less heavy than wood then wood cannot float on water. And, most importantly, wood goes up above the state of equilibrium because, water as a weight on the other side of the balance, by analogy, is lifting or raising the piece of wood.

It is therefore clear that the motion of bodies moving naturally can be suitably reduced to the motion of weights in a balance. That is, the body moving naturally plays the role of one weight in the balance, and a volume of the medium equal to the volume of the moving body represents the other weight in the balance.

Thus the other weight is not of the entire water, but only that portion of it which is equivalent to the volume of the moving body. The motion therefore is caused by force, especially the upward motion.

Soon Galileo reaches the unambiguous conclusion that no upward motion is natural, i.e., it must be forced. Since all bodies have weight they all have an internal cause, which is nothing but the weight (relative) for downward motion.[p. 177.] In the memorandum Galileo adds:

Downward motion is far more natural than upward. For upward motion depends entirely on the heaviness of the medium, which confers on the moving body an accidental lightness; but downward motion is caused by the intrinsic heaviness of the moving body. In the absence of a medium everything will move downward. Upward motion is caused by the extruding action of a heavy medium. Just as, in the case of a balance, the lighter weight is forcibly moved upward by the heavier, so the moving body is forcibly pushed upward by the heavier medium.[Notes 4 on p. 22.]

That all upward motion is forced is a significant move away from, if not against, Aristotle. In order that one arrives at this statement we need to reject that some bodies levitate and some gravitate by nature. In Aristotle’s scheme of things, if there existed only one element, say fire, in the universe, it would have occupied that layer which is in proximity to the lunar sphere, while in Galileo’s scheme of things it would reach the center of the universe, for fire is, in this hypothetical case, the heaviest. That every body (every element) has weight and that weight is relative are some of the initial steps Galileo takes away from Aristotle’s thesis. Taking clues from the kind of motion that takes place in the case of balance he goes another step forward, yet another step away from Aristotle, by proposing that all upward motion is forced. In order to reject Aristotle’s thesis that wood in water rises up naturally, and propose that in such a case the body (in this case wood) is being lifted up by another weight external to the body, the analogy with the balance is crucial. Here lies the genius of Galileo. This is no ordinary achievement, despite the simple logic.

The crucial contribution does not consist in saying that bodies with weight would naturally move downward, for this was also the thesis of Aristotle. It also does not consist in saying that lighter bodies go up because of force. Neither Galileo nor Aristotle would say this, because for Galileo there is nothing like a light body, but only less heavy, and for Aristotle not all upward motion is by force, it is by force only for heavier bodies. The contribution consists in proposing that all bodies, irrespective of their heaviness, if they move upwards the motion is forced, and the force is external to the body. Our concern here is not to see whether what Galileo says is true or not, but to understand how the conceptions are transforming.

Galileo continues his journey, being convinced that his path is right.

And since the comparison of bodies in natural motion and weights on a balance is a very appropriate one, we shall demonstrate this parallelism throughout the whole ensuing discussion of natural motion. Surely this will contribute not a little to the understanding of the matter.[p. 23.]

Eliminating the Plurality of Causes

Galileo has so far postulated the cause of ‘natural’ motion, which is heaviness or relative density for both upward and downward motion. If the cause of motion is heaviness then what would be the cause of change in motion? Since a change in cause should produce a corresponding change in effect, a difference in heaviness should produce a difference or change in motion.

Can there be kinds of change in motion, such as slowness and speed? Accordingly should we need to postulate two separate causes, one for slowness of motion, and one for swiftness of motion? For Aristotle, slowness has one cause, namely, density of the medium and swiftness has another, namely, rarity of the medium. Galileo, on the other hand, argued for one cause for both slowness and speed, just as he argued for one cause for both upward and downward motions. This unification is a necessary move for what he would be finally driving at, which is one cause for motion as well as for change in motion. This ultimate unification is one of the primary contributions of Galileo.

What Aristotle says is that a body would be faster in air than in water, because the former is more incorporeal (less dense) than the latter. (Note that Aristotle did describe in relative statements. But he restricts relative description to the two media, water and air. His statements with respect to earth and fire are absolute. He also says that density of the medium impedes the movement of a body.

We see the same weight or body moving faster than another for two reasons, either because there is a difference in what it moves through, as between water, air, and earth, or because, other things being equal, the moving body differs from the other owing to excess of weight or of lightness.

Now the medium causes a difference because it impedes the moving thing, most of all if it is moving in the opposite direction, but in a secondary degree even if it is at rest; and especially a medium that is not easily divided, i.e. a medium that is somewhat dense.[Physics 215a25-31.]

From this it is clear that Aristotle believed in a twofold cause to the motion of the body, one external to the body in the form of resistance of the medium, and other internal in terms of the weight of the body. One of them (weight) to be accounted for the speed and the other (density of the medium) for the slowness of the moving body. Galileo differs from him in a very subtle but significant way.

Galileo says that both downward motion in the rarer media and upward motion in denser media would be swifter, and upward motion in the rarer media and downward motion in denser media would be slower. These descriptions, one can easily see, are transformations obtained by appropriate changes of the opposite terms.

From the above arguments it follows that density of the medium does not always decrease motion, because upward motion of rarer bodies in denser media is swifter. Similarly rareness of the medium causes swifter motion only in the downward direction and not in the upward direction. Therefore the view of Aristotle that slowness of natural motion is due to the density of the medium is incorrect because certain things such as an inflated bladder, which when left in deep water (or any other denser medium), moves up swiftly. In a place where downward motion takes place with difficulty, an upward motion necessarily takes place with ease.[p.24]

Therefore, dismissing Aristotle’s opinion, so that we may adduce the true cause of slowness and speed of motion, we must point out that speed cannot be separated from motion. For whoever asserts motion necessarily asserts speed; and slowness is nothing but lesser speed. Speed therefore proceeds from the same [cause] from which motion proceeds. And since motion proceeds from heaviness and lightness, speed or slowness must necessarily proceed from the same source.[pp. 24-25]

This is the method of unifying the causes that Galileo consistently, and (there is evidence to show that he) consciously, adopts in solving problems of physics.

Compare the pattern of reasoning that leads him to infer that lightness is nothing but less heavy and heaviness is a character of all bodies, with what he says here in the case of motion and change of motion and the causes of motion and change of motion. Substituting ‘lightness’ with ‘slowness’ and ‘less heavy’ with ‘lesser speed’, ‘heaviness’ with ‘speed’ and ‘bodies’ with ‘motion’, the statement underlined above reads: Slowness is nothing but lesser speed and speed is a character of all motion. This is a typically Galilean method of solving the problem.

Though it falls short of finality, the remarkably Galilean turn, necessary for the emergence of modern physics, takes place here. This has been made possible by a specific pattern of thinking in terms of contraries. This pattern has an added advantage over, and is not accessible to, the categorical way of thinking. The main point is that the contraries cannot become two separate qualitatively or quantitatively distinct categories, but belong to one scale and are therfore commensurable. This point is entirely missing in Aristotle, despite the fact that he shows awareness and inclination towards principles characterized by contraries.

Aristotle realizes, as Galileo does, that media as well as weight affect the motion of bodies. Galileo would not deny that corporeal (dense) nature of the media would resist motion, but he would not accept it for all kinds of motion, because of the reasons mentioned above. It may appear as though Galileo is introducing a new taxonomy unknown to Aristotle. Was Aristotle wrong in taxonomizing kinds of motion? It appears not. Galileo does not disagree with Aristotle on this point. He, however, disagrees with him on the corresponding taxonomizing of the kinds of causes of motion. That is causes need not follow the same taxonomic pattern of effects, and this for the very important reason that one cause is sufficient to explain (by generating) all varieties and effects of (natural) motion. Suppose for every kind of effect there should exist a corresponding cause, then there should be as many causes as there are effects. The simplicity and systematicity of causal explanation, however, consists in reducing a large number of effects to a single unifying causal principle.

Extending the Argument

Already Galileo has laid a secure foundation for the construction of an alternative interpretative framework to Aristotle’s, for the investigation of the problems of motion. However many specific and deeply held beliefs of Aristotelian science need to be demolished before erecting the alternative structure.

In the eighth and ninth chapters Galileo advances a large number of arguments, most of them in the form of what we today call thought (gedanken) experiments, against Aristotle’s thesis that there is a direct proportionality between largeness (greater weight) of a body and its speed in natural motion.

Aristotle’s view as presented in De Caelo[273b30-274a2] is as follows:

A given weight moves a given distance in a given time; a weight which is as great and more moves the same distance in a less time, the times being in inverse proportion to the weights. For instance, if one weight is twice another, it will take half as long over a given movement.

Similar statements asserting that larger and/or heavier bodies move quicker have been made in De Caelo 290a1-2; 277b4-5; 309b11-15; 394a13-15; and in Physics 216a13-16. The law as stated is also believed to be true of the weightless element fire. He says in De Caelo:[277b4-5]

The greater the mass of fire or earth the quicker always is its movement towards its own place.

It is clear from these statements that ratios of the speeds of their motion downwards for earth, and upwards for fire, is proportional to the sizes of the bodies. Since Aristotle had no notion of mass, we interpret the term ‘mass’ as ‘massive’ or ‘larger’ or ‘voluminous’.

Galileo’s view, contrary to Aristotle’s, is that bodies of the same kind i.e., made of the same substance, whatever be their size move at the same rate in the same medium. To help understand this rather surprising conclusion Galileo asks us to conduct a series of thought experiments. It should be pointed out that while this statement of Galileo is sufficiently surprising, this remains a statement that plays the role of only a ‘rung’ of the ladder he climbed, for ultimately he arrived at a far more surprising statement, viz., that all bodies irrespective of their kind, and irrespective of their size fall at the same rate in vacuum (or void). We shall see how he, step after step, arrives at this conclusion. It took him less than four decades to reach from one rung to the next.

His arguments in the form of thought experiments have the following pattern. First, he considers the case of a body moving downwards and upwards in a changing medium, and second, the case of a combination of two bodies moving downwards and upwards, and finally, on the basis of the above two arguments he proves that Aristotle’s thesis leads to contradiction.

First: Consider a medium like water on which one large and another small piece of wood are afloat. Imagine that the medium is gradually made successively lighter, so that finally the medium becomes lighter than the wood and both pieces slowly begin to sink. Now following the principles of hydrostatics “who could ever say that the large piece would sink first or more swiftly than the small piece?”

As already argued both the pieces being made out of the same material (wood) they would have same heaviness (specific gravity), which is same for wood whatever be its size. since heaviness is the only determining factor of natural motion, as already argued, there would be no difference in their motion.

For, though the large piece of wood is heavier than the small one, we must nevertheless consider the large piece in connection with the large amount of water that tends to be raised by it, and the small piece of wood in connection with the correspondingly small amount of water. And since the volumes of water to be raised by the large piece of wood is equal to that of the wood itself, and similarly with the small piece, those two quantities of water, which are raised by the respective pieces of wood, have same ratio to each other in their weights as do their volumes ... i.e., the same ratio as that of the volumes of the large and the small piece of wood. Therefore the ratio of the weight of the large piece of wood to the weight of the water that it tends to raise is equal to the ratio of weight of the small piece of wood to the weight of the water that it tends to raise.[pp. 27-28.]

Consider now the case of a large piece of wax floating on water and suppose by some means, such as mixing some sand with the wax, it be made successively heavier than water so that it would begin to sink slowly. If we take say one-hundredth part of that wax, considering the principles of hydrostatics, who would ever believe that the piece would not sink at all or would sink hundred times more slowly than the whole piece of wax?

In the former experiment the medium was considered for change and in the latter the floating body. This aspect of experimental science changing the parameters symmetrically, but successively, reveals an important truth about a law of nature. In this case it is illustrated that it is the difference of specific weight that matters, and not whether the difference is with the body or with the medium. The source of difference does not matter, what matters is the difference. Working out the argument by varying the conditions symmetrically now on the ‘left side’ and now on the ‘right side’ of the balance, and obtaining an invariant result remains a remarkable feature of Galileo’s thinking pattern, also true of the structure of scientific thinking. If the situation can be reduced to the balance, then how and why should it matter which side is considered for variation. Having shown that it is the difference in specific weight that matters and not the source of difference, he goes to the next step of the argument.

Second: Consider there are two bodies of which one moves more swiftly than the other, then the

combination of the two will move more slowly than that part which by itself moved more swiftly, but the combination will move more swiftly than that part which by itself moved more slowly.[p. 28.]

For example, take the combination of a piece of wax and an inflated bladder both moving upward from deep water.

[W]ho can doubt that the slowness of the wax will be diminished by the speed of bladder, and, on the other hand, that the speed of the bladder will be retarded by the slowness of the wax, and that some motion will result intermediate between the slowness of the wax and the speed of the bladder?[p. 29.]

Similarly the combination of wood and bladder in air will fall more slowly than the wood alone, but more swiftly than the bladder alone. Similarly when two equal bodies moving equally come close and join together they would not double their speed, contrary to Aristotle, for the same reason. It follows from this that the same kind of body, whatever be its weight should move at the same speed. Galileo having also shown what happens when two bodies combine, goes on to prove that Aristotle contradicts himself. The proof is as follows:

Figure 5: Galileo’s and Aristotle’s Calculations on Fall of Bodies

Suppose a body a whose weight is 20, and two different media, bc and de. Let the volume of a, b and d be equal, and their weights 20, 12, and 6 respectively. The ratio of the speed of body a in the medium bc and de will be equal to the excess of weight of a over the weight of the medium bc to the excess of weight of d to the weight of the medium de, which is 8 : 14. If the speed of a in bc is 8, its speed in de will be 14. Aristotle would have calculated the ratio as 8 : 16, because bc is doubly denser than de. Since Galileo calculates the arithmetical difference, the difference in speed is lesser than Aristotle’s calculation. It therefore follows that the speed does not increase at the same rate even if there is a similar rate of decrease in the density of the medium. Realizing this apparently minor point of difference is more than vital for the development of the modern science of motion, where medium does not play an essential role in the motion of bodies, because the impeding effect of the medium is less than what Aristotle expected. To see how Galileo extrapolates these simple calculations to bring back the void into physics, let us calculate the ratios of speeds by decreasing the values of the weight of de gradually, keeping the rest of the things constant.

If the medium de has the weight 4, for the unit volume, the speed of a in de according to Galileo would be 16, and the difference between the speeds of a in bc and de would be 8, which means only twice as fast as in bc. Aristotle’s calculation for the same situation shows that it would be thrice (12/4) (See the table in the figure). When we decrease the weight of de further to 3, the difference in speed will be 9 for Galileo, but for Aristotle four times. When the weight of de is further decreased to 2, Aristotle would calculate a difference of six times that a’s speed in bc than de, and at a further decrease it becomes 12 times. And finally one more step and Aristotle would be in great trouble, because when the weight of de becomes 0, Aristotle gets at what is impossible to comprehend, 20/0. For a similar situation Galileo gets a convenient 8 : 20.

On one hand Aristotle uses his calculations to abandon motion in void, which was presumed to be of zero weight. On the other hand Galileo goes to a radical conclusion that the pure form of motion takes place only in the void. It is thus very clear how eventually for Galileo the medium became an impeding and accidental factor, while for Aristotle it was an essential factor of motion. What was held to be necessary became accidental, and what was held to be impossible became not only possible, but became the purest possible.

The above proof also brings home the point that it is one thing to know that two quantities are inversely proportional and quite another thing to know the quality of proportionality, i.e., whether arithmetic or geometric. Aristotle’s charge that it is impossible for one number to have the same relation to another number as a number has to zero, has been proved by Galileo as untenable. In conclusion to this proof, Galileo says:

Therefore, the body will move in a void in the same way as in a plenum. for in a plenum the speed of motion of a body depends on the difference between its weight of the medium through which it moves. And likewise in a void [the speed of] its motion will depend on the difference between its own weight and that of the medium. But since the latter is zero, the difference between the weight of the body and the weight of the void will be the whole weight of the body. And therefore the speed of its motion [in the void] will depend on its own total weight.[p. 45.]

It becomes clear from this passage that Galileo considers the void as a medium with zero weight. As noted earlier, things weigh proper in the void because in any media other than the void they will always be lighter.

In De Motu Galileo thought that different bodies would fall in the void at different rates, though irrespective of their size or weight.

For example, in the case of a body whose weight is 8, the excess over the weight of the void (which is 0) is 8; hence its speed will be 8. But if the weight of a body is 4, the excess over the [weight of the] void will, in the same way, be 4; and hence its speed will be 4. finally, using the same method of proof in the case of the void as we used in the case of the plenum, we can show that bodies of the same material but of different size move with the same speed in a void.[pp 48-49.]

This conclusion is not correct which Galileo realizes much later. In Discoursi (First Day) he reaches the correct conclusion that all bodies, irrespective of weight, density and size fall in the void at the same rate. To arrive at this conclusion Galileo has to correct another of his premises. In De Motu he thought that in natural fall bodies would fall at a constant speed, while later he corrects this to say that they undergo uniform acceleration. Both these corrections are very vital for the further development of the science of motion.

The Final Discovery

For the final conclusion Galileo had to wait for nearly four decades. The discovery was announced in The Dialogues Concerning the Two New Sciences. Simplicio, who argues for Aristotle in the The Dialogues, expresses disbelief that “a bird-shot falls as swiftly as a cannon ball”,[Ibid p. 64.] and Sagredo, the third interlocutor, requests Salviati to explain how “a ball of cork moves with the same speed as one of lead”.[The Two New Sciences p. 68.] Salviati then describes the method of approaching the result.

Having once established the falsity of the proposition that one and the same body moving through differently resisting media acquires speeds which are inversely proportional to the resistances of these media, and having also disproved the statement that in the same medium bodies of different weight acquire velocities proportional to their weights ... I then began to combine these two facts and to consider what would happen if bodies of different weight were placed in media of different resistances; and I found that the differences in speed were greater in those media which were more resistant, that is, less yielding. This difference was such that two bodies which differed scarcely at all in their speed through air would, in water, fall the one with a speed ten times as great as that of the other. Further, there are bodies which will fall rapidly in air, whereas if placed in water not only will not sink but will remain at rest or will even rise to the top: for it is possible to find some kinds of wood, such as knots and roots, which remain at rest in water but fall rapidly in air.[Ibid, p. 68.]


[I]n a medium of quicksilver, gold not merely sinks to the bottom more rapidly than lead but it is the only substance that will descend at all; all other metals and stones rise to the surface and float. On the other hand the variation of speed in air between balls of gold, lead, copper, porphyry, and other heavy materials is so light that in a fall of 100 cubits a ball of gold would surely not outstrip one of copper by as much as four fingers. Having observed this I came to the conclusion that in a medium totally devoid of resistance all bodies would fall with the same speed.[Ibid, p. 72.]

If the differences in speed are more in denser media, then the differences in speed will have to be lesser in rarer media like air. And if it is void, a medium with zero density, no difference should be observed. This is the reason that led him to the discovery that in a void, all bodies irrespective of their weight, size or shape, would fall at the same rate.

Since Galileo has no equipment with which to obtain a total vacuum to demonstrate his findings the truth of his claim, he therefore gives plausibility arguments.

Since no medium except one entirely free from air and other bodies, be it ever so tenuous and yielding, can furnish our senses with the evidence we are looking for, and since such a medium is not available, we shall observe what happens in the rarest and least resistant media as compared with what happens in denser and more resistant media. Because if we find as a fact that the variation of speed among bodies of different specific gravities is less and less according as the medium becomes more and more yielding, and if finally in a medium of extreme tenuity, though not a perfect vacuum, we find that, in spite of great diversity of specific gravity [peso], the difference in speed is very small and almost inappreciable, then we are justified in believing it highly probable that in a vacuum all bodies would fall with the same speed.[Ibid, p. 72.]

Let us note the logical modality of the claim made in the last sentence. Galileo was fully aware that the empirical world as presented to us through our perception is a world of possibilities. He therefore does not announce the discovery in the logical modality of necessity. Also the kind of observations in support of his claim suggest that Galileo did not construct pure idealized systems without taking inputs from experience. His theoretical constructions were constrained by the empirical inputs. This demonstrates his strength which lies in balancing empirical observations with mathematical reasoning.


This episode, as we saw, gives us ample chance to look at the way Galileo unfolds the discovery. His use of statics and hydrostatics extensively in another physical context for studying motion, illustrates in a simple way how a theoretical model can be applied in a different context. Today while studying motion we do not impute either statics or hydrostatics. But they were relevant in the context of discovery, particularly in the construction of idealized systems (physical systems). The explanation of motion by reducing the number of causes informs us how important parsimony is in science. Above all the episode is very useful in understanding the introduction of relativity and its role in quantifying the physical phenomena. This is only a small episode that can be extracted out of Galileo’s texts. It is possible to reconstruct in a similar way how Galileo discovers the horizontal component of motion using the analogies of pendulum and inclined planes. As we saw in this episode Galileo himself employs only the vertical component while analyzing motion in De Motu, and the horizontal component was discovered towards the end of his life, published in the Two New Sciences.

This work is based on my case study on Galileo in my Ph.D. thesis, which also contains the details of how Galileo discovers the horizontal component. The case study presented in the thesis was to illustrate the role of inverse reason in the growth and development of scientific knowledge, particularly in the context of discovery. I believe that the details from the context of discovery are very relevant in the context of learning. My arguments in this regard are presented in the other paper in this volume.


[1]   Nagarjuna G. The Role of Inversion in the Genesis, Development and the Structure of Scientific Knowledge. Ph.D. Thesis, Indian Institute of Technology, Kanpur, India., 1994.

[2]   Galilei Galileo. De Motu. The University of Wisconsin Press, Madison, 159? Posthumously published, written by Galileo around 1590s, Translated by Drabkin, I.E. 1960.

[3]   Galilei Galileo. Dialogues Concerning Two New Sciences. Dover Publications, Inc., New York, 1636. Translated by Henry Crew and Alfenso De Salvio, 1914.

[4]   T.L. Heath. The Works of Archimedes. Dover Publications, Inc., New York, 1897. Translated.

[5]   W.D.  Ross (ed.). The Works of Aristotle. Oxford University Press, London, 1908-52.

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