Issue number 25 : 20/05/2004
A scientific publication by SGF and NEODyS

The history of NEOs' science

Part 3: The solution of the Problem of Motion

The sensate esperienze and the certe dimostrazioni of Galileo Galilei

Galileo Galilei (1564-1642) is unanimously considered the most important physical scientist of the XVIIth century, and indeed the founder of modern science. His role in the controversy on the Copernican hypothesis and in its final victory was decisive, but his major contribution was not of an astronomical nature. Galileo revolutionised the way of understanding nature and its phenomena, as well as the methods of investigation and analysis. His proud, desperate and ultimately losing battle against dogmatism and authority enabled the birth of a new way of thinking, both in the practice of science (the experimental method), and in the philosophical and epistemological analysis of the foundations of knowledge.
This intellectual revolution was well explained by Galileo himself at the beginning of his book Saggiatore, published in 1623 after its author had been sentenced by the Catholic Church for the first time (1616), and had been forbidden to teach the Copernican ideas. Arguing against Lottario Sarsi (a pseudonym of the Jesuit priest Orazio Grassi), Galileo writes:

"It seems to me...that Sarsi has a firm belief that in order to philosophise it is necessary to found upon the authority of some celebrated author, ...and perhaps [Sarsi] considers philosophy as a book and a phantom of a man, like the Iliad or the Orlando Furioso...Mr Sarsi, things are not that way. Philosophy is written in this immense book that is continuously open in front of us (I mean the universe) but it is impossible to understand it without knowing its language, and the characters, in which it is written. It is written in mathematical language, and the characters are triangles, circles, and other geometrical figures, without which means it is impossible for man to understand a single word; without these [means] it is like wandering in a dark maze."
So, nature and its phenomena may be understood, if we only know how to investigate them; and the way of understanding is to use mathematics extensively, with the conviction that natural phenomena may be analysed in mathematical terms. However, it is not only mathematics that is to play a crucial role in this endeavour; it is necessary to integrate tightly the sensate esperienze (meaningful experiments) and the certe dimostrazioni (rigorous demonstrations). The former will make the scholar able to ask questions of nature, and the latter will provide the answers. It is a double proof of trust in reason, somehow anticipating the spirit of enlightenment of the following century. It is also a profession of faith in the capacity of man, any man. Writing to Paolo Gualdi, Galileo declares:

"I wrote it in vulgar tongue [the treatise Istoria intorno alle macchie solari] because I require that everybody might understand it...and I want them [the readers] to realise that nature, having provided them with eyes in order to admire its works...has also provided the brain to understand them."

Science, therefore, is no longer reserved only to knowledgeable persons, but is accessible to anybody moved by the wish to read the book of nature. But, what might be a suitable method to apply?

The experimental method

The conjugation of sensate esperienze and of certe dimostrazioni is the basis of the experimental method. Galileo, besides being a subtle observer and a scrupulous thinker, was also what we would now term a good experimental physicist. He frequently resorted to experiments, crafting his instruments himself in a workshop at home. As a matter of fact, another innovative character of the work of Galileo is the unification of science and technique. The craftsmen of that time were often very skilled, as was Galileo. In essence, Galileo thought that the machines, instruments and observational techniques of the technicians deserve to play an important role in the scientific research. Research is guided by reason, but reason must conform to the facts, and not dictate the rules to which nature must conform.
I said above that the major contribution of Galileo to the birth of science was not of an astronomical nature. I will discuss later his final contribution that was in fact fundamental for the final success of the Copernican theory, but first I will examine briefly his studies on mechanics and dynamics, studies that put down the foundations for the work of Newton.
Sometimes it is said that Galileo, despite his great work, did not discover anything new or exceptional outside mechanics: it was not he to invent the telescope (he was not, apparently, an expert in optics), and his termoscopio was only an outline of the real thermometer invented later on by his disciples. Even in the field of dynamics one could maintain that many of the ideas often attributed to Galileo were already available in some form, if not already explicitly stated by others, including the isochronism of the pendulum, for example. However, it must be noted that the excellence of Galileo lies not only in the quantity and quality of his discoveries, but mainly in the innovation that he introduced in the methods of investigation.
Concerning the discoveries, let us examine only his contributions to the comprehension of dynamics. While still young, Galileo became interested in the problems connected to gravity, immediately facing Aristotle's lessons. Aristotle maintained that the bodies moved down (for example a stone) or up (for example smoke) depending on their gravitas or levitas (heaviness or lightness), intrinsic qualities of all bodies. Their motion was at a constant velocity, directly proportional to the gravitas and inversely proportional to the resistance of the medium (air, water), and taking place always in the direction dictated by their nature (heavy or light). This is termed the "theory of natural motions and places". Moreover, Aristotle and the peripatetic school maintained that an object subject to a push ("theory of violent motions") moved with a velocity directly proportional to the applied push and came to rest when the push ended. Both these interpretations of the phenomena are basically in agreement with experience, at least if the measurements are not accurate.
Therefore, according to the theories by Aristotle, a stone falls down because it is "heavy", and not because there is something that makes it move: it is an active, not a passive, motion. The stone can indeed only fall down, towards its natural place. In opposition to this Galileo, based upon the tradition dating back to Archimedes, maintained that in reality the speed of fall is not proportional to the ratio between weight and density of the medium, but to the difference between the respective densities. In other words, an object that falls down in the air may well not fall at all, but rather be pushed up, if placed in water. This break with tradition, besides excluding the theory of the "natural places", has an important consequence: the possible - even necessary - existence of the vacuum. This concept, like those of zero and infinity, was totally foreign to Aristotle, who judged it absurd. On the contrary, according to Galileo the vacuum is not only conceptually necessary, but it is also the only condition in which the real motion of an object reveals itself, free from the resistance of the medium.
Although agreeing at first with Aristotle, Galileo suddenly realised that the peripatetic description disagreed with the experiments: a body free to fall does not have a constant falling velocity, at least at the beginning of its trajectory, but is subject to an acceleration. In fact, the presence of the medium eventually causes the falling velocity to become constant, as is more evident in water than in air. By observing and studying the behaviour of pendulums, Galileo understood that the acceleration was a fundamental ingredient of motion, not an accident, and that the resistance of the medium represented simply an incidental perturbation. Only a motion in the vacuum may be considered as a "pure state".

Figure 1. The experience of the inclined plane Let the ball P roll down the slope starting from different points. By measuring the time needed to reach the terminal point A, one gets the law that binds distances to times through the acceleration, which is proportional to the component ft of the force of gravity acting along the slope.
In order to study more easily these phenomena, Galileo made ample use of inclined planes (Figure 1). His best known experiment consisted in letting a ball roll down the slope of an inclined plane starting from different points along the slope and measuring the times needed to reach the terminal point. By doing so, he discovered two important things: the length of the paths are proportional to the squares of the needed times, and the descent times along planes of equal height, but different lengths (i.e., with different inclinations) are proportional to the length of the planes. By applying these observations to the vertical motion (i.e., to the case of free fall) one gets the time law of motion of the bodies, that we know as:

where x is the distance covered, t the time taken, and g the acceleration of gravity.
It is important to remark that the studies made by Galileo on the motion of heavy bodies, especially in free fall, are not presented in the way with we are familiar. On the contrary, they are often filled with cloudy and often incorrect reasoning, mainly because Galileo had recognised in his experiments a concept that had never been considered before: the instantaneous velocity. We will see later that the problem of the velocity of an object in a temporal instant without duration was solved only by Newton. Galileo did not make any step towards infinitesimal quantities, and his theory of motion provided the basis for Newtonian dynamics, but did not complete the program of mathematisation of the laws of nature that he was pursuing.
Nonetheless, Galileo finally reached two goals of extreme relevance: the principle of the relativity of motions, and the principle of inertia. For the first of those we can again call upon Galileo to speak:

"Close yourself with some friends in the largest room below deck of a big ship, and manage to have there available flies, butterflies and similar small flying animals; put there also a big vase filled with water, with small fishes; have also tall jars dripping in smaller ones: and when the ship is at rest, observe diligently how those flying creatures go in any direction with the same velocity; you will see the fishes wandering with indifference towards any border of the vase; the drops will all enter the jars below... Once these observations will be done, let the ship move with any possible velocity; provided that the motion is uniform and not floating here and there, you will not see any change in the mentioned things, nor will you be able by yourself to decide whether the ship is moving or not..."

This is the very famous passage that, in modern terms, affirms the equivalence of the physical laws in all reference systems animated by rectilinear, uniform motions. A basic principle in physics, of which the Einstein's special relativity is an extension to all natural phenomena, including electrodynamics and optics. Galileo's relativity principle is valid in all reference frames. This means that no frame is more "correct" than others. This statement, therefore, makes a clean sweep of all privileges, not only for the position of the Earth, but also of man on Earth. It is in some way a premonition of the difficulties that Charles Darwin will have in removing the privileged position of man in the biological world.
The second goal reached by Galileo, the principle of inertia, is implicit in the aforementioned relativity principle. In fact, Galileo was never fully aware of the importance of this principle, nor did he give a precise formulation to it; however, the readiness and confidence with which he handled the consequences of the principle show clearly that he understood its relevance. The principle of inertia would eventually be stated explicitly by Newton, and would become his first law of dynamics: that a body not subject to any force moves with rectilinear, uniform motion (or remains at rest).
The study of dynamics allowed Galileo to solve many physical problems. He showed that motion is a passive state and is modified only by the presence of impressed forces. He also showed that it is possible to describe mathematically the motion of an ensemble of objects. The problem erroneously formulated by Aristotle, which had resisted to all attacks for two thousand years, was now solved in principle. Galileo laid down the foundations of the solution, and Newton invented the mathematical tools needed to treat it and then formulated the underlying laws.

Galileo and the Copernicanism

Galileo was not prosecuted due to his researches in physics, but because he was a Copernican. In fact, the two aspects cannot be fully separated. As remarked by Sebastiano Timpanaro, "the Discorsi delle nuove scienze are not less Copernican than the Dialogo dei massimi sistemi. The theologists did not condemn the former book because they did not understand it."
The astronomical activity of Galileo is well known. I will underline only the most important discoveries, without entering the controversial history of the trials to which Galileo was subjected. The main discoveries of Galileo were:

In fact, none of these discoveries was pivotal in turning the scale in favour of Copernicus; even the phases of Venus, obviously unexplainable by the Ptolemaic system, were perfectly compatible with the scheme by Tycho Brahe. However, the consequences of these observations were of enormous importance: let us examine them briefly.
The discovery of the "Medicean planets", now called Galilean satellites (Io, Europa, Ganymede, and Callisto), proved that the Earth was not at the centre of all motions. It also showed that there are objects in the sky that are invisible to the naked eye and, even more important, showed that human abilities could be enhanced with the use of appropriate instrumentation.
The explanation of the nature of the Milky Way as an immense array of stars added more value to the previous discovery. The universe showed itself as it is: of enormous dimensions both in terms of size, and also in the number of objects it contains. This discovery was very badly received, because it recalled to everyone's mind the claim of Giordano Bruno that the universe was infinite and populated with other worlds and, perhaps, intelligent beings. This viewpoint cost Bruno his life.
The nature of the lunar surface, spread with mountains and valleys (which we now know to be impact craters), showed that the profile of celestial objects did not differ hugely from that of the Earth. The reign of incorruptibility had been destroyed forever. This fact was immediately confirmed by the existence of dark spots on the Sun.
Finally, the phases of Venus. If this planet were really in orbit around the Earth, at a greater distance than the Sun, it could never be located between the Earth and the Sun; therefore, the face of Venus could never appear as a small crescent. This represented a direct proof that the Ptolemaic scheme was wrong but, as aforesaid, did not jeopardise the Tychonic system, which was geometrically (we would say now topologically) identical to the one of Copernicus.

The Galilean revolution

The fundamental importance of the work of Galileo is its contribution to the scientific rationality, and the first contribution was to show the intimate connection between the activity of scientists and technicians. The Renaissance was no longer a time for positions of principle, for endless discussions on the ultimate causes of the phenomena, for speculations about the essence of nature. The technicians of the Renaissance asked to know how the natural phenomena behaved and how to confront them and, possibly, take advantage from them. It was therefore necessary for science to ally to technique and, in doing so, to deeply transform itself from an abstract philosophical discipline to a very real investigation of the phenomena and their natural causes.
This is the meaning of the revolution provoked by Galileo: the nova scientia (new science) will not do without theory (Galileo himself pushed for a complete formulation in mathematics of all natural philosophy), but will reject those theories that will not agree with the sensate esperienze. Theories will now be accepted only if they will be able to provide verifiable, or falsifiable, laws.
Science was therefore made free from any metaphysical hypothesis. Thereafter, it did not need philosophical guarantees on its investigation methods, it was warranted only by the practical verification of the correctness of its results. It is a profound detachment that originated the separation between philosophy and the natural sciences, and then between science and the humanities. In fact, a real detachment has never occurred: science has become the starting point for new philosophical systems that have brought in all fields the same rationality that has been proven so useful in the study of nature.
A second important consequence of the work of Galileo is a new interpretation of the concept of truth. The truth is no longer provided by the authority of those who declare it, nor by the dogmatic certainty of principles. Truth must be accepted as provisional, must be subject to controls and verifications, to changes and, more often, to upheavals. It becomes a temporary truth, but able to trigger very rapid changes that sometimes will provoke crises involving the same principles from where it started. It is, therefore, a dialectic truth, feeding itself and adapting to circumstances. This new kind of truth was well understood by Galileo:

"...I am sure that if Aristotle came back to this world, he would accept me among his fellows...more than many others. And if Aristotle would see the new things discovered on the sky, while he stated that it is immutable and unchangeable, because no changes had been observed until then, he would undoubtedly now say the contrary...".

This statement is echoed by the character of Galileo as written by Bertold Brecht:

"I do not care to show that I am right, but to ascertain whether I am right...Yes, we will put everything in doubt...And what we will find today, we will erase it tomorrow from the blackboard...".

It is wrong and misleading, for the development of a healthy civic consciousness, to teach that science is infallible. On the contrary, systematic doubt is its nutriment, and any triumph of science can only be temporary.
There is a last noteworthy consequence in the work of Galileo. If what is accepted today as the truth will not be accepted tomorrow, if we can "erase it from the blackboard", then the building of science cannot be done by a single scientist. There will no longer be thinkers of whom one could say ipse dixit (he said it), but groups of scientists collectively doing research. Science, therefore, becomes additive: every generation builds on the preceding ones, changing what needs to be changed and overthrowing what needs to be overthrown. In the words of Laplace:

"Sciences grow infinitely thanks to the work of successive generations: the most perfect piece of work generates new discoveries, thus putting the foundations for the works that will eclipse it."