The Galilean Cutoff: the Law of Falling Bodies and Modern Science

History of the Galilean Cutoff

The domination of Aristotle's physics

Historians of science agree that the Galilean Cutoff marks the beginning of modern physics. From ancient Greece, about 4 centuries before Christ, until the end of the Middle Ages (15th century), the vision of the world was stable: that of the great Greek philosopher and scientist Aristotle (384-322 BC). His work in the fields of metaphysics and physics was so considerable that it influenced many civilizations.

Aristotle's thought spread through the writings of many thinkers and various schools. Aristotle's works describe an "intuitive" world divided into two parts: the sublunar world (below the Moon, including Earth) and the supralunar world (beyond the Moon, the rest of the Universe). The sublunar world is changing, imperfect, and corruptible; it is the realm of fire, air, water, and earth. The supralunar world is the realm of ether, immutable, perfect, and incorruptible.

The world is therefore geocentric, and the Earth is immobile in space; everyone accommodated this because it corresponded to the observations of the time. This vision of the world was propagated by all Aristotelian commentators (in Aristotle's time), the Neoplatonists (from about 300 to 540), the Arabs in the middle of the Middle Ages (from about 980 to 1200), the Jews (from about 1130 to 1200), the Byzantines (from about 1040 to 1140), and the Medieval Commentators (from about 1180 to 1280). Of course, some mathematicians, physicists, astronomers, philosophers, and science popularizers of the late Middle Ages were skeptical about this geocentric world, in particular Nicole Oresme (1320-1382) and Jean Buridan (1292-1363).

The Copernican rupture

The Aristotelian world had reached its limits, but no one questioned it because it perfectly suited the "Holy Scriptures" of the Middle Ages. With Nicolaus Copernicus (1473-1543), the world was about to shift. The Polish canon, physician, and astronomer developed a theory that placed the Sun at the center of the Universe (heliocentrism), around which the planets revolve. The Earth was no longer central and immobile but a planet like the others.

A profound paradigm shift

The law of falling bodies

His 1543 book entitled "De Revolutionibus Orbium Coelestium" was "widely" circulated. This profound paradigm shift, both philosophical and scientific, imposed by Copernicus is called the "Copernican Revolution," but it was Galileo (1564-1642) who truly propelled science into the modern world.

When we think of Galileo, we imagine him pointing his telescope at the uneven surfaces of the Moon, the stars of the Milky Way, Jupiter's moons, the phases of Venus, or Saturn. But it is not the use of the telescope in astronomy that marks the beginning of modern physics.

The Galilean Cutoff is marked by the formulation of the Law of Falling Bodies in the early 17th century, around 1604. It states: The speed acquired is proportional to the duration of the fall and is independent of the mass and nature of the body. This is the first time a physical law was expressed with the parameter "time," according to the French physicist Etienne Klein (1958-); this is the Galilean Cutoff. Time becomes a mathematical variable and will play a decisive role in modern physics.

Galileo's law of falling bodies is revolutionary because it goes against our senses. Aristotle's intuitive theory seems more correct to us because it explains that heavy bodies fall faster than light ones. But it is false.

Dialogue Concerning the Two Chief World Systems

A work in Italian to convince

To better spread his discoveries, Galileo wrote in 1632 "DIALOGO," a dialogue on the two chief world systems: that of Ptolemy (90-168) and that of Copernicus (1473-1543). He wrote this work in Italian to be better understood, rather than in Latin, which was the language of publications at the time. He used three characters for this. For this book, Pope Urban VIII (1548-1644) condemned Galileo on June 22, 1633.

Excerpts from Galileo's book featuring three characters over four days: Simplicio, defender of Aristotle's theory; Sagredo, an honest, open, and cultured man; and Salviati, who represents the ideas Galileo wishes to spread.

The dialogue on falling bodies

Simplicio: Aristotle demonstrated that, in the same medium, objects of different masses fall at different speeds and that these speeds are proportional to the masses of the objects. […] You surely do not intend to prove to us that a cork ball falls at the same speed as a lead ball?

Salviati: I strongly doubt that Aristotle based this on an experiment. […]

Simplicio: His own words show, however, that he observed the phenomenon, since he says "We see that the heavier…". This "we see" alludes to an experiment.

Sagredo: But I, who have tried it, signor Simplicio, assure you that a cannonball of one hundred or two hundred pounds, or more, will not have gained the lead of a palm, upon reaching the ground, over a musket ball of half a pound, even if the height of the fall is one hundred cubits!

Simplicio: I find it hard to believe that a tear of lead can fall as fast as a cannonball.

Salviati: […] I would not want, signor Simplicio, that, like so many others, you focus on some thing I have said that deviates from the truth by the thickness of a hair, to avoid seeing the error as big as a hawser that Aristotle committed. Aristotle writes: "An iron ball of one hundred pounds falling from a height of one hundred cubits reaches the ground before a one-pound ball has descended a single cubit." I say, on the contrary, that they arrive at the same time. You have only to make the experiment, and you will find that when the large ball touches the ground, the other is only two fingers away. And you would now, behind these two fingers, hide the ninety-nine cubits of Aristotle, and, highlighting my minimal error, pass over his enormous error in silence.

Simplicio: Be that as it may, I cannot believe that in a vacuum, if motion were possible there, a tuft of wool would fall as fast as a piece of lead.

Salviati: Gently, signor Simplicio […], listen instead to this reasoning that will enlighten you. We are investigating what would happen to objects of very different masses in a medium of zero resistance. […] Only a space absolutely empty of air would allow us to perceive an answer. Since such a space does not exist, we will observe what happens in low-resistance media, by comparison with more resistant media; and if we find that different objects have less and less different speeds as the media are easier to traverse, […] then we may admit with great probability, it seems to me, that in a vacuum the speeds would all be equal. […] The experiment consisting of taking two objects of very different masses and dropping them from a certain height to observe whether their speeds are equal involves some difficulties. Indeed, if the height is great, the medium will hinder the light object much more, and over a long distance the light object will then lag behind. […] However, if one takes two objects of the same shape and made of the same material, and reduces the mass of one while also reducing its surface, no reduction in speed occurs.[…] I therefore conclude that if the resistance of the medium were completely eliminated, all objects would fall at equal speed.

Galileo's inclined plane experiment

A device to slow the fall

To study the fall of bodies with the means of his time, Galileo devised an ingenious experiment: he slowed the fall by rolling balls on a smooth inclined plane. In this way, the effect of gravity is reduced, and perfectly polished and spherical bronze balls roll slowly, driven by their own weight. This device allowed him to measure durations easily observable with the naked eye, which would have been impossible during free fall.

The formulation of the laws of fall

After many repeated trials, Galileo was able to formulate the laws of falling bodies. He established in particular that the speed acquired is proportional to the duration of the fall, and that the distance traveled is proportional to the square of the time. These results, obtained through a rigorous experimental method, mark a rupture with Aristotle's qualitative physics and lay the foundations of uniformly accelerated motion.