# Law of Free Fall in Renaissance Science

**DOI:**https://doi.org/10.1007/978-3-319-02848-4_939-1

## Abstract

The formulation of the laws of free fall and projectile motion is usually regarded as Galileo’s most important scientific achievement. The new science of motion was however met with skepticism not only by Aristotelians but also by mechanical philosophers. After the publication of the *Dialogo sopra i due massimi sistemi del mondo* (1632) and the *Discorsi e dimostrazioni matematiche intorno a due nuove scienze* (1638), interesting discussions took place in Europe concerning the validity of the law of fall. The issues that were mostly debated were (a) the possibility of deriving that law from a causal explanation of gravity, (b) Galileo’s views concerning the composition of continuous magnitudes, and (c) the alleged lack of empirical support in favor of the law.

## Introduction

The first published formulation of Galileo’s law of free fall is found in the *Dialogo sopra i due massimi sistemi del mondo* (1632). In the second day of the work, Galileo claims that, in a naturally accelerated motion, the speed grows in proportion with the time elapsed and that the falling body acquires a new degree of speed in each of the infinite instants that compose a finite interval of time. Starting from these principles, Galileo formulates a law of fall which is valid in the void for all bodies, independently of their weight and material. The law states that the spaces traversed by a falling body are to each other as the squares of the times elapsed (s_{1}:s_{2} = t_{1}^{2}:t_{2}^{2}) or, which amounts to the same, that the distances covered by a falling body in equal and successive intervals of time grow according to the series of the odd numbers starting from one (1, 3, 5, …). Galileo claims that there is a “most purely mathematical proof of this statement” (Galileo 1967, 222), but he does not produce it. It is only in his last work, the *Discorsi e dimostrazioni matematiche intorno a due nuove scienze* (1638), that he offers a mathematical derivation of the law of free fall and of the semi-parabolic trajectory of projectiles.

Galileo’s new science of motion is rooted in a thorough criticism of some of the fundamental tenets of scholastic natural philosophy, such as the theory of natural places and the ensuing distinction between heavy and light bodies, the view that a force is needed to keep bodies in motion, the dichotomy between natural and violent motion, the idea that motion in the void is impossible, and the assumption that the speed of fall is proportional to the weight of the falling body.

Galileo’s derivation of the law of fall and of the parabolic trajectory of projectiles can certainly be regarded as a crucial episode in the process of mathematization of nature that took place in the seventeenth century. It is however important to point out that Galileo’s science of motion was met with skepticism not only by Aristotelian natural philosophers but also by some representatives of the mechanical philosophy. Following the publication of the *Dialogo* and the *Discorsi*, interesting discussions took place concerning the validity of the law of fall. The issues that were mostly debated were the possibility of deriving that law from a causal explanation of gravity, Galileo’s views concerning the composition of continuous magnitudes, and the alleged lack of empirical support in favor of the law.

In this entry we shall first provide an account of Galileo’s discovery of the law of fall and then describe its impact and legacy.

## Galileo’s Law of Free Fall: Heritage, Original Aspects, and Rupture with the Tradition

Galileo’s law of fall is often translated in the equation: s **=** ½ gt^{2}. This formula is, however, anachronistic, as Galileo relied on the medieval theory of proportions, which only permits the expression of ratios between homogeneous magnitudes (Giusti 1990, xxix; Damerow et al. 2004, 254–256). This explains the resemblance between the diagrams accompanying Galileo’s demonstrations and those used in the fourteenth century by Nicholas Oresme to describe the uniform variation of qualities (Palmerino 2010b).

In his *Tractatus de configurationibus qualitatum et motuum*, Oresme offered a diagrammatic representation of the theory of the intension and remission of intensive qualities (such as color, sound, heat, and velocity) which had been developed by the *Calculatores* at Merton College, Oxford (Maier 1949; Sylla 1986; Damerow et al. 2004, 17–21).

*Discorsi.*As noted by Richard Arthur, Galileo “appropriates the idea initiated by the Merton School that a motion has a certain intensity at any given instant” and that all the “degrees of velocity” add up to the overall velocity of the motion of fall (Arthur 2016, 89). In the

*Discorsi*, Galileo defines naturally accelerated motion as one “which, abandoning rest, adds on to itself equal momenta of swiftness in equal times” (Galilei 1989, 162). This definition is followed by a postulate and by a number of propositions that are “proven demonstratively” (Galilei 1989, 165). Proposition I, Theorem I is precisely the Mean Speed Theorem, which Galileo demonstrates with the help of Fig. 2 above. Here the vertical line AB represents the time of fall, the horizontal lines EB and FB the final speeds of the accelerated and uniform motion respectively, and the lines parallel to EB stand for the degrees of speed. The areas of the triangle and of the rectangle are identified by Galileo with the total speed, which is proportional to the space traversed, represented by the separate line CD (Damerow et al. 2004, 238–242).

We know that Galileo had discovered the time-squared law as early as 1604. In a letter to Paolo Sarpi written in October of that year, he curiously derived that law from the false assumption that the speed of fall grows in proportion to the space traversed, rather than to the time elapsed. Scholars have provided different interpretations of Galileo’s mistake and of the way in which he managed to recognize and correct it (Drake 1974; Galluzzi 1979, 272; Mahoney 1985, 206–207; Giusti 1990, xxv–xxvi; Blay and Festa 1998, 75–76; Damerow et al. 2004, 165–175; Palmerino 2010b, 419–425; Arthur 2016, 90–91).

It is interesting to point out that a similar mistake was made by René Descartes, who in 1618 explored the issue of free fall together with Isaac Beeckman. In a letter to Mersenne, dated 13 November 1629, Descartes maintained that falling bodies are impelled downward by gravity, which adds to them a new impetus in each successive moment of time (Mersenne 1945–1988, 2: 316–317; Descartes 1964–1974, 1: 71–72).

*Dialogue*, in 1634, he did not perceive the difference between the law he had formulated in 1629 and Galileo’s law. In a letter to Mersenne dated 14 August 1634, Descartes declared that he had recognized in the

*Dialogue*“some of his own thoughts,” such as that “if a ball employs three moments to descent from A to B, it will employ only one moment to fall from B to C” (Mersenne 1945–1988, 4: 298; Descartes 1964–1974, 1: 304–305). Descartes did not realize that while in his own figure the vertical line AC represents the distance fallen, and the spaces AB and AC are supposed to be traversed in times that are to each other in a proportion of 3 to 4, in Galileo’s triangle of speeds the vertical line is taken to represent time, and the spaces traversed are supposed to be in the same proportion as the squares of the times elapsed (Damerow et al. 2004, 37; Jullien and Charrak 2002, 147–148).

But what is especially interesting from our point of view is the fact that by 1634, Descartes had lost interest in the formulation of a law of free fall. As we will see in the following section, he believed, like other contemporaries, that a causal analysis of free fall should be given priority above the mathematical description of the phenomenon.

## Impact and Legacy of Galileo’s Law of Fall

In 1633, Marin Mersenne published a short pamphlet entitled *Traité des mouvemens, et de la cheute des corps pesans* in which he presented some “veritable experiences” (*expériences véritables*) in support of Galileo’s law of free fall (Mersenne 1633, 2). Having measured the time employed by a laden ball falling from different heights, Mersenne had verified that the spaces grow as the squares of the times (Dear 1984, 136–137; Dear 1995, 129–132; Palmerino 2010a).

The last pages of the *Traité* are devoted to exploring the nature of gravity. Mersenne takes into account three possible causes of free fall, notably “positive and real heaviness,” the pressure of the air on the falling object, and terrestrial attraction, all of which seem however incompatible with Galileo’s law of free fall. Mersenne concludes that “it is sufficient to explain the phenomena of nature, since the human spirit is not capable of possessing its causes and principles” (Mersenne 1633, 21–24). This stance was in agreement with that of Galileo, who in the *Dialogue* had confessed his ignorance concerning the nature of gravity (Galileo 1967, 234).

Mersenne’s attitude toward Galileo’s law of free fall was to shift over the years, from “pragmatic adoption” (Dear 1984, 215) to skepticism. In the *Novarum observationum … tomus III*, published in 1647, Mersenne still stressed that bodies dropped from modest altitudes appeared to follow the odd number law but pointed out that the validity of that law could not be demonstrated as long as the true cause of gravity remained unknown (Mersenne 1647, 133).

Mersenne’s change of mind can only be understood against the background of the debate concerning Galileo’s science of motion that took place in Europe after the publication of the *Dialogo* and the *Discorsi*. An important actor in this debate was Pierre Gassendi, who between 1640 and 1646 wrote six Latin letters, the *Epistolae de motu impresso a motore translato* (1642) and the *Epistolae de proportione qua gravia decidentia accelerantur* (1646), in which he adduced new experimental evidence in favor of Galileo’s science of motion and attempted to show how the *vis attrahens* of the earth could produce an acceleration according to the law of the odd numbers (Galluzzi 2000; Palmerino 2004).

Contrary to Gassendi, René Descartes believed that Galileo’s law of fall was incompatible with a mechanistic explanation of gravity. According to the theory presented in the posthumously published *Le Monde* and in the *Principia philosophiae* (1644), bodies were pushed downward by the particles of subtle matter contained in the terrestrial vortex. Descartes confessed his incapacity to translate this complex mechanism into an exact mathematical law, but expressed the conviction that the clashes between subtle matter and falling bodies produced an acceleration in jumps, rather than a uniform and continuous acceleration (Palmerino 1999; Jullien and Charrak 2002).

Between 1644 and 1646, three authors tried to formulate an alternative to Galileo’s law of fall.

In 1645 the Jesuit Pierre Le Cazre reacted to Gassendi’s *Epistolae de motu* with a *Physica demonstratio*, in which he claimed to have verified experimentally that the speed of fall grew in proportion to space. On the basis of an incoherent piece of mathematical reasoning, Cazre concluded that the distances covered by a falling body in equal and successive intervals of time grew according to the series of ever doubling numbers (1, 2, 4, 8, …) (Palmerino 2003).

A more coherent and interesting account of free fall was proposed by another Jesuit, Honoré Fabri. In the *Tractatus physicus de motu locali*, published in 1646 by his pupil Pierre Mousnier, Fabri identified the cause of free fall in the impetus produced by gravity. Fabri explained that while gravity remained constant, a new degree of impetus, and hence a new degree of speed, was acquired in each of the finite instants composing time. According to Fabri’s “discretist” account of free fall, each new degree of speed is enough to traverse a new minimum of space, which means that the spaces traversed in equal and successive instants of time grow according to the series of natural numbers (1, 2, 3, 4) (Palmerino 2003; Elazar 2011; Arthur 2016).

Giovan Battista Baliani reached a similar conclusion in the second edition of his *De motu*, which was also published in 1646. While in the first edition (1638) he had subscribed to the Galilean law of free fall, he now argued, like Fabri, that gravity adds to falling bodies a new impetus in each successive finite instant of time and that the acceleration of fall takes place according to the series of the natural numbers (Baliani 1998).

But what about the experiments that seemed to confirm the validity of Galileo’s law? To Mersenne, who asked this question, Fabri answered that in order to obtain empirical confirmation of the law of natural numbers one should have taken as one’s unit of measure a minimum of time. Given that “in actual experience one can but take a part which contains several instants of time (…) one should not be surprised if the proportion found by experience corresponds more or less to that of Galileo” (Mersenne 1945–1988, 12: 289).

Fabri’s answer shed light on a peculiar characteristic of Galileo’s law of free fall, notably the fact that it was valid irrespective of the unit chosen to measure time. Interestingly, precisely this property was regarded by some as a proof of the superiority of Galileo’s law over its rivals. In their letters to Mersenne, Theodore Deschamps, Evangelista Torricelli, and the young Christiaan Huygens stressed the fact that the odd number law was the only one to possess the property of scalar invariance (Palmerino 1999). But this argument was not enough to convince Mersenne. What was at stake in the debate was the continuity of acceleration, and the odd numbers law was invariant precisely because it described a continuous acceleration.

Many of the difficulties which Galileo had encountered in the derivation of the law of free fall were superseded by the invention of the infinitesimal calculus. In the *Principia* (1687) Newton successfully applied the new method to the study of accelerated motion and managed to “avoid long and tiresomely rigorous demonstrations in the style of the ancients, as well as the traps posed by indivisibles” (Blay 1998, 59). This, however, did not put an end to the debate concerning the cause of free fall.As is well known, Leibniz as well as Descartes’ followers rejected Newton’s theory of gravity, which involved action at a distance, for being incompatible with the principles of mechanical philosophy.

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