The theoretical origins of the pendulum in the work of Galileo

Abstract

Pendulums are scientific instruments of great value to contemporary science. What is little known about them is that these instruments have a long history. There are records of the use of pendulums dating back at least to Antiquity. However, it was only with Galileo, together with Sanctorius and the invention of the pulsilogium, that pendulums began to be regarded as scientific objects. Galileo is the main figure responsible for establishing one of the fundamental properties of the instrument: isochronism. In this article, we intend to describe the process of theorizing the pendulum, with a primary focus on Galileo’s work. We have seen that other scholars had already been trying to understand the dynamics of the pendulum; however, it is with Galileo that the problem of the pendulum begins to be studied in depth. The Italian physicist is recognized for presenting a rigorous theory of the pendulum and for seeking mathematical laws to describe it. We will present Galileo’s correspondence with Guidobaldo del Monte, as well as excerpts from Galilean texts in which the physicist sets out the elements of the theory of pendulums.

Keywords:
History of the pendulum; The theorem of strings; The law of isochronism; Correspondence between Galileo and Del Monte; The theoretical development of the pendulum

1. Introduction

Pendulums are scientific instruments of great importance, especially in physics. They are essential for the study of periodic phenomena. The simple pendulum serves as a theoretical model for the study of what is called Simple Harmonic Motion (SHM). From a practical point of view, pendulums are used to measure time, as well as the acceleration of gravity [¹]. Both theoretical and practical experiments are commonly carried out in physics laboratories using pendulums to demonstrate the principles of energy conservation. Sometimes, they are also used in electrical studies, such as in the case of the electrostatic pendulum [²], among other applications. Despite its relevance to science, what is rarely discussed in physics text, however, is the historical evolution of pendulums. How were they introduced to physics? When did pendulums become objects of scientific investigation? It is common to attribute to Galileo the primacy in the study of this instrument, as well as its introduction as an object of physical inquiry [³].

When discussing this object, there is a myth associated with the name of Galileo Galilei (1564–1642), who, while attending mass at the Cathedral of Pisa, perhaps out of boredom, began to observe the ceiling and became interested in the movement of the chandelier (Figure 1[⁴]). What caught the attention of the student from Pisa was that, due to the action of the wind, the object moved with a regular and periodic motion: ‘the young medical student began to observe a chandelier high above his head, suspended by a long and thin chain, gently swaying back and forth in the spring breeze’ [³, ⁵, ⁶]. Galileo became very curious about the regularity of the chandelier’s motion and began to time its swings with his pulse. As he himself says: ‘How long does it take for the swings to repeat?’ he asked, timing them with his pulse [⁶]. After that, when he returned home, Galileo built a prototype of a simple pendulum and began studying its properties. This is the legend that has circulated since the time of Vincenzo Viviani (1622–1703), Galileo’s first biographer, who popularized this story in his biography [⁶]. However, we cannot confirm or refute the validity of this story.

Figure 1
Pisa Cathedral chandelier, one of those that Galileo may have observed in the 16th century¹[⁴].

Pisa Cathedral has undergone several renovations from the time Galileo attended it to the present day. It is sometimes claimed that the chandelier associated with Galileo’s name has been preserved; however, there are some doubts as to whether this chandelier even existed during Galileo’s time, as stated by [⁷]. Furthermore, any association of the chandelier with Galileo is more a matter of speculation than anything else, given that it is unlikely, if Viviani’s story is true, that Galileo ever reported to anyone which specific chandelier he was observing during that Mass in 1582, the date usually assigned to the supposed occurrence of this event. Or perhaps we should extend this event to the period between 1581 and 1583, when Galileo was a student in Pisa, since Viviani himself does not provide any dates for this episode in Galileo’s life.

At the dawn of modern science, the new scientific instrument, the pendulum, became an object of great scientific curiosity. The pendulum was fundamental to the studies of Galileo, Huygens, Newton, Hooke, and many other scientists during this period of the Scientific Revolution [⁸]. With the emergence of modern mechanics, the pendulum became established as an important tool for the study of periodic motion, which was becoming increasingly important in science. Galileo was the most prominent figure in the attempt to describe the motion of this object. He was also one of the first physicists to present a mathematical model for the simple pendulum, which helped him in the study of motion, force, and gravity [⁹, ¹⁰].

In the history of physics, the pendulum plays a uniquely important role. From the early years of the seventeenth century, when Galileo announced the formulation of the laws governing pendular motion, to the early years of this century, when it was replaced by devices of superior precision, the pendulum served as an object of study or as a tool for investigating questions in astronomy, gravitation, and mechanics [[¹⁰], p. 441].

Furthermore, Galileo was one of the first to use the experimental method to extract information regarding the object’s properties, such as the period of oscillation, the amplitude as a function of the angle, and the dependence of the pendulum’s length on the period, among other questions he investigated [⁸, ¹⁰, ¹¹].

Galileo’s first significant scientific discovery was the property of simple pendulums, ideally consisting of a weight suspended by a light string: as long as they do not oscillate too widely, their period (the time it takes for each swing) is independent of the oscillation amplitude (the length of the swing arc). Contrary to the anecdote [about the chandelier in the Pisa Cathedral], Galileo most likely arrived at his discovery [about pendulums] through his interest in music², which led him to experiment with pendulums of various lengths to study their rhythms. [³, p. 51].

There are also reports and some evidence in his writings suggesting that Galileo studied music and that his experiments with various types of pendulums date back to this same period [³], although this is not certain. There is another line of investigation that may offer a more plausible explanation as to when and how Galileo first became interested in pendulums; it is precisely this perspective that we will investigate in this article. The invention of the pulsilogium can be considered the initial trigger that sparked Galileo’s interest in pendulums. Although there is no historical evidence to support this line of thought, it can be assumed to be the most plausible, given that there is little evidence of Galileo’s work with pendulums before 1602. In this article, we intend to present a historical study on the origin of this instrument, which has been so important since the beginnings of early modern physics.

The central question of this study can be summarized as follows: to what extent can Galileo be considered the true precursor in the scientific study of the pendulum? Through a historiographical and bibliographical analysis, using various references, we will seek to identify the historical sources that confirm or contest Galileo’s role as one of the first scholars to theorize the properties of pendulums, as well as the relevance of contributions from other researchers of the time. Furthermore, we will discuss Galileo’s work in formalizing the mathematical properties of the pendulum, particularly his contribution to the understanding of its periodic motion and its relationship with gravity [⁸, ¹²]. This study aims not only to review the existing literature but also to bring to light original hypotheses and analyses authored by the author himself; the intention is to decipher certain hidden aspects of this history through a new interpretation of the known facts.

It is fundamental for this historical discussion to understand how Galileo presents his results. However, considering that his reasoning is grounded in the geometry of proportions, the standard language of physics at the time, yet far removed from contemporary algebraic notation and infinitesimal calculus, it was decided to allocate Galileo’s technical demonstrations and original diagrams to the Appendix. This final section constitutes optional reading, serving as a complementary resource to the discussion contained in the main body of the article, which prioritizes the historical and conceptual reconstruction of the object under study. Additionally, the Appendix offers a modern demonstration of the theorem of chords and the law of isochronism, serving as an alternative to simplify the understanding of the Galilean proofs.

2. The Beginning of Galileo’s Study on Pendulums

It is said that Galileo recognized the potential usefulness of pendulums in medicine when the physician Santorio Sanctorius (1561–1636), who was also a professor of medicine at the University of Padua, invented the pulsilogium, a simple pendulum of standard length designed to measure the pulse of patients in hospitals. According to [³, p. 64], the first known application of the pendulum was in medicine, used to determine patients’ heart rates using a pulsilogium. The pulsilogium was not Galileo’s invention, but he likely became acquainted with this physician’s invention while he was a professor in Padua. Therefore, the popular story that Galileo’s interest in pendulums was sparked by observing a chandelier in the Cathedral of Pisa is probably not entirely accurate. The event may have occurred, but it was not the only reason that led the Italian physicist to actively dedicate himself to research with pendulums. The pulsilogium, first mentioned in 1602 by a colleague of Sanctorius in Padua, served as a source of inspiration for Galileo and triggered a series of pendulum experiments throughout seventeenth-century Europe [¹³].

We do not know exactly when Galileo began his studies of the pendulum; however, it is certain that Galileo exchanged a set of letters with Guidobaldo del Monte (1545–1607) around 1602. We will discuss this in greater detail in the next section. This correspondence very likely indicates that either Galileo had been studying the pendulum well before 1602 and wrote nothing about it, or that he began his investigations of the pendulum in 1602. The understanding of pendular motion required of Galileo a refinement of the ancient tradition of his predecessors concerning the relationship between the pendulum and the balance. Galileo analyzed motion on inclined planes and on circular arcs through analogies with the balance, treating the problem of pendular motion as a question of mechanical equilibrium and geometric compensation. This became more evident, especially after 1602.

Already at that time, around 1602, Galileo seemed to have in hand the law of isochronism, which shows that he was very likely already studying the pendulum: “the marvelous property of the pendulum is that it makes all its vibrations, large or small, in rigorously equal times” [¹²]. This property can also be visualized from another perspective; Galileo observes that the time of oscillation does not depend on the amplitude of the pendulum. Below is another way of stating the law of isochronism,

For if we take two equal weights and suspend them by strings of equal length, and displace them from the vertical line [position of equilibrium] by unequal distances, we shall see that they will traverse their paths, one longer, the other shorter, in the same time; and this occurs because the motion along the larger arc is faster than along the smaller arc [¹⁴, p. 98].

Galileo realized that the motion of a pendulum is essentially a free fall interrupted by the string that supports it. If lead (very dense) and cork (very light) swing at the same rhythm, this means that the action of gravity acts in the same way on any kind of matter, regardless of its weight, as he says: “if we take two weights, one of lead and the other of cork … we shall see that the lead moves in perfect synchrony with the cork, even after hundreds of oscillations” [¹⁴]. Galileo believed (at that time) that the arc of a circle was the path of descent (under the action of gravity) that is fastest between two points that are not on the same vertical, except for the straight line. Later, it would be shown that this path is, in fact, the brachistochrone³[¹⁵, ¹⁶]. He assumes that the motion of a pendulum preserves impetus⁴[¹⁷] better than an object rolling on any inclined plane, as he says,

A heavy body, when descending along the circumference of a circle, moves with greater ease and speed than along any other line, except the perpendicular straight line; for, along such an arc, the heavy body approaches the center of the Earth more rapidly than along any other path, except the straight path. And this occurs because, on an arc, the descent is less impeded than on any inclined plane [¹⁴, p. 98].

This conclusion was a consequence of the problem with which Galileo dealt when comparing the time of descent along an arc of a circle with that along an arbitrary polygon. One may question how these comparisons were made by Galileo; regarding this, [¹⁸, p. 105] states, ‘the law of chords was probably first conjectured as a step towards reaching a physical explanation of isochronism and only later, if ever, was it experimentally verified’. Therefore, for Galileo, pendular motion was the most perfect among all those in which the body is under the action of gravity; for, whereas on an inclined plane the acceleration is retarded, on the arc of a circle the acceleration varies smoothly. This was what Galileo regarded as the most efficient way for nature to move a heavy body. Hence the importance of the study of the pendulum within the context of the Galilean theory of motion. Galileo, in his Discourses and Mathematical Demonstrations Concerning Two New Sciences (Theorem XXI, Proposition XXXVI), states that,

If a chord is drawn from the lowest point of a vertical circle so as to subtend an arc no greater than a quadrant, and if from the endpoints of this chord two additional chords are drawn to any point on the said arc, the descent along these two chords takes less time than along the first [¹⁹, p. 238].

Immediately after presenting a demonstration of this result, Galileo reaffirms his conclusion in a scholium, in which he says: “From the foregoing, it is possible to infer that the path of fastest descent [lationem omnium velocissimam] from one point to another is not the shortest path, that is, a straight line, but the arc of a circle” [¹⁹, p. 239]. We know that this Galilean result would later be contested, for the circle is not the curve of fastest descent; rather, it is the cycloid. This was first proven by the Bernoulli brothers in 1696, on the basis of earlier studies by Christiaan Huygens (1629–1695) on the tautochronous⁵ curve [²⁰].

For Galileo, the principle remains valid if, instead of a single straight line, the object were to descend along a sequence of chords (as if it were a broken path, approaching a curve). His conclusion was that the straight line is the shortest path in distance, but the straight line is not the shortest path in time. To this end, consider the quadrant BAEC, (Figure 2[¹⁹]), with side BC vertical, and let AC be the circular arc corresponding to the quadrant. Divide it into any number of equal parts, and, for two consecutive points, draw the segments AD, DE, EF, FG, and GC, with C being the lowest point of the quadrant. From C, draw the chords AC, CD, CE, CF, and CG. Galileo asserts that the descent along ADC is faster than along the chord AC alone.

Figure 2
Galileo’s diagram illustrating the fall of bodies along the chords of a quadrant of the circle BAEC, which he uses to compare the times of descent [¹⁹, p. 239].

Now a body starting from rest at D will traverse the path DEC in less time than along the chord DC alone; in the same way, after the descent along ADE, the time required to traverse EFC is less than that required for EC alone and, so on. Similarly, if a path of three chords, ADEC, is chosen, it will take less time than a path of two chords, ADC, both starting from rest at A. And, if a path of four chords, ADEFC, is chosen, the descent along this path will be faster than along a path of three chords, ADEC, the body having started from rest at point A and, so on successively. In this way, Galileo confirms that “the more the inscribed polygon approaches a circle, the less is the time required for the descent from A to C” [¹⁹]. Thus, by increasing the number of chords ad infinitum, the time of descent will tend toward the smallest possible value, when compared with any other polygonal path, as the polygon tends toward the circle itself.

Moreover, for Galileo the pendulum serves as a proof to support an argument against Aristotle’s theory, for the Greek philosopher held that velocity was proportional to weight and inversely proportional to the resistance of the medium. By observing that different weights (or the same weight at different amplitudes) take the same time to traverse a certain distance, Galileo concludes that motion does not depend solely on the weight of the body. If weight were the only factor responsible for the speed of a body’s fall, a lead pendulum should complete the arc much earlier than one made of cork. Since they move together in the same time, this reinforced Galileo’s argument that gravity accelerates all bodies equally, regardless of their weight. Possibly, the isochrony of the pendulums of different weights may have helped Galileo become even more convinced regarding the law of free fall.

The same Vincenzo Viviani was the one who popularized that famous Galilean experiment of the Tower of Pisa, in which Galileo drops weights from the Tower of Pisa and demonstrates to the public that heavier weights fell at the same time as lighter ones; according to the story, the experiment took place around 1589 to 1592, during the period when he was a professor at the University of Pisa [⁵]. Much like the story of the chandelier, we cannot confirm or refute Viviani’s account; however, we know that by around 1602, when he was researching pendulums, Galileo was already in possession of some discovery concerning the time of fall of bodies and, in a letter to his colleague Paolo Sarpi, Galileo claimed to have discovered that the distance traveled by a falling body is proportional to the square of the time [²¹]. He most likely already knew, at that time, the property of the independence of weight in the fall of bodies. It is also possible that Galileo made use of something from his studies of pendulums to arrive at his conclusions regarding the free fall of bodies.

3. Correspondence Between Galileo and Del Monte

The complexity of the pendulum’s motion represented a formidable challenge for early modern natural philosophers; it is true that the problem was not solved by Sanctorius’s efforts and, certainly, during the same period, Galileo and Guidobaldo Del Monte (Figure 3[²²]) were working on the problem. However, Galileo was not as successful in his own investigations into pendulum motion, as we are told by [⁷]. Nevertheless, we consider it valuable to analyze and compare both Galileo’s published writings and his correspondence. In his book, Discourses on Two New Sciences, Galileo presented the main results he managed to verify and certainly did not include those in which he was unsuccessful, since, according to [⁷], one cannot rely exclusively on Galileo’s published writings, where he attempted to conceal the problems he was unable to solve. Regarding Del Monte, his work on the investigation of the pendulum was of a more experimental than theoretical nature, prompted primarily by his correspondence with Galileo.

Figure 3
Guidobaldo del Monte (1545–1607), Marquis del Monte, was an Italian mathematician, philosopher and astronomer of the 16th century. Guidobaldo was a critic of Galileo’s principle of the isochronicity of the pendulum, a major discovery which Guidobaldo thought was impossible. He was an important scholar of Renaissance mechanics, famous for his book in which he studied aspects of the theory of equilibrium and other aspects of mechanics [²²].

Let us examine some excerpts from the correspondence between Galileo and Del Monte so that we may draw some conclusions. Unfortunately, part of this correspondence has been lost; we do not have access to the first two letters exchanged between them; only the third has survived, dated November 29, 1602. Apparently, Galileo had already mentioned to Guidobaldo, in the first letter, his conjecture concerning pendular isochronism, and Guidobaldo most likely rejected this idea in his reply (the second letter). Now, in this third letter, Galileo once again attempts to convince his renowned colleague of the property in question and proposes some experiments. It is likely that the beginning of this correspondence dates from 1602, as stated by Drake [²³, p. 68–69]: ‘It is doubtful that [the correspondence] began much before 1602, since it is evident that Galileo had not previously mentioned [his results] to Guidobaldo regarding the subject they were discussing’.

Most Illustrious Sir and Revered Master

I beg you to excuse my insistence in persuading you that the proposition of the movements made in equal times in the quadrant of the circle is true; for, having always seemed admirable to me, I now fear it may be considered impossible by you, $\dots$ [¹⁴, p. 97].

Galileo begins the letter addressed to del Monte by stating that it would be difficult to convince his reverend master of the validity of the proposition concerning the isochronism of the pendulum. He was well aware that Guidobaldo was embedded in the Renaissance tradition, based on the analogy of the pendulum with the movement of a weight on a scale, and that his way of thinking did not allow him to accept any reasoning that could not be verified in practice. Galileo was not the only one to use experimentation to support his results; Guidobaldo would also use this same resource in an attempt to refute the arguments of his young pupil. In that letter, Galileo proposes to Guidobaldo an experiment with pendulums.

He instructs us to consider two cords (strings), AB and EF, of the same length, and attaches them with two nails at points A and E, and at their lower ends he fastens two identical lead balls. Once the pendulums are constructed, Galileo displaces both pendulums from equilibrium, away from the vertical: in the case of cord AB, he moves it so as to form arc BD, and in the case of cord EF, he moves it so as to form arc FG (see Figure 4[¹⁴]). After this, he asks that the weights be released simultaneously. The pendulum on cord AB begins to describe the arc BCD, while the pendulum on cord EF begins to describe the arc FIG, in which, BCD FIG Even so, the mass at B does not take more time to traverse the entire arc BCD than the other mass, F, takes to traverse the arc FIG.

Figure 4
Two small masses B and F attached to two strings AB and EF and when set in motion travel along arcs BCD and FIG, this is the model of a pendulum contained in a letter from Galileo to Guidobaldo Del Monte from November 1602 [¹⁴, p. 98].

To show that this is true, Galileo resorts to an experiment carried out by him, perhaps in the year 1602, we do not know for certain. The greatest problem in this type of experiment is the measurement of time. Galileo makes the pendulums oscillate 500 times, and, according to him, the regularity of their motions is observed. Again, he lets the pendulums oscillate 1,000 times, and the regularity persists. In the time interval that Galileo counted, for example, 100 oscillations of the pendulum on cord AB along arc BCD, a second observer would have counted, for the second pendulum EF, the same 100 oscillations along arc FIG [¹⁴]. This is, therefore, for Galileo, a proof that the oscillation time of the two pendulums, even with different amplitudes, is the same.

[$\dots$] a most evident sign that each of the largest [oscillations] BCD takes as long as any of the smallest [oscillations] FIG Now, if the whole BCD be traversed in as much time as FIG, then its halves, also, which are the falls along the unequal arcs of the same quadrant, will be traversed in equal times. But even without worrying about counting, you will see that mobile F will not make its smallest reciprocations more frequent than mobile B will make its largest ones, but they will always go together [¹⁴, p. 97–98].

This excerpt from the letter dated November 29, 1602, sent by Galileo to Del Monte, shows that Galileo was trying to persuade his renowned academic colleague of the proposition concerning motions performed in equal times along the quadrant of a circle, which relates to the motion of the pendulum. At that time, Galileo apparently already had a demonstration of the so-called law of strings, that is, the proposition that the natural descent of a moving body along all inclined planes inscribed as chords in a circle and passing through its lowest or highest point takes the same time. In other words, the fundamental property Galileo seeks to demonstrate in this discussion is that, even though the arc BCD of pendulum AB is much longer than the arc FIG of pendulum EF, the time it takes for pendulum AB to complete one oscillation along its arc is the same as the time it takes for pendulum EF to complete one oscillation along its arc. This is called the isochronism of the pendulum (see Figure 4).

Therefore, in this letter, Galileo presents an attempt to demonstrate the property of isochronism of pendulums through the use of inclined planes [⁴, ⁶, ¹⁰, ²⁴]. The idea he expresses in this correspondence shows that the Italian physicist had probably already been thinking about the pendulum for some time, although we may suppose that Sanctorius’ pulsilogium exerted some influence on him. Could it have been Guidobaldo who first suggested to Galileo the idea of using the analogy of a concave surface to compare with the motion of the pendulum, which Galileo reportedly embraced, as stated by, “Galileo’s idea begins with the observation that the movement of a pendulum can be identified with the swing of a body along a concave spherical surface, since both movements effectively occur under the same geometric constraints” [⁷].

Galileo saw a possibility in describing the movement of the pendulum by approximating it through two conjugate inclined planes placed one in front of the other (see Figure 5). If a spherical body, for example, is released from one of the upper ends of one of the inclined planes, it will follow a periodic motion that is only not perpetual due to the action of friction and air resistance. The aforementioned law of strings, which the physicist used in his attempt to describe pendulum motion, is a way of approximating displacement along an arc by displacement along a chord.

Figure 5
Conjugated inclined plane with a sphere A to approximate the pendulum’s motion. Source: author.

Guidobaldo apparently did not accept the conclusions drawn by Galileo from the experiment he proposed. In fact, Guidobaldo’s letter to Galileo has not been preserved, and we do not know his response to Galileo’s November 1602 letter. However, through his preserved notes and notebooks, it is known that Guidobaldo carried out an experiment to test the validity of Galileo’s assertion, as he says, Guidobaldo’s letter has been lost, but he could not believe that one body could travel many miles while another moved only a few centimeters [²³].

Guidobaldo argued that if an experiment were conducted with pendulums having sufficiently long strings (let us suppose 100 miles, as Galileo himself suggested), and one were released from a height such that it would travel an arc of only a few centimeters, while a second pendulum identical to the first were released from a height such that it would travel an arc of several miles in length, then, based on this thought experiment, Guidobaldo supposed that Galileo’s property of isochronism could not hold in such a situation. Galileo opposed his colleague’s opinion in the 1602 letter, and referring to this experiment, he says,

As for the fact that it does not seem reasonable that, in a quadrant 100 miles long, two pendulums would traverse it – one completely and the other only by the width of a palm – in the same period of time, I admit that it is admissible. [$\dots$] But, returning to our subject, I believe I have demonstrated this conclusion, which is no less certain than the others. [¹⁴, p. 98].

Moreover, Guidobaldo designed some more refined experiments to test in practice what Galileo asserted. He probably observed that the proposition did not hold for very large angles, and, as a result, he began to doubt the property that his Florentine⁶ colleague had presented to him. The Concave Basin experiment was one of the best-known experiments carried out by Guidobaldo. For this experiment, he took a concave surface (such as the bottom of a basin or a channel in the shape of a circular arc) and released a sphere from point A at the top of the basin to see whether the sphere would reach the same height at point B on the opposite side, and how long this would take (see Figure 6). Guidobaldo wanted to verify whether the descending motion was perfectly compensated by the ascending motion, and whether the total time followed any rule that could confirm Galileo’s hypothesis. However, he encountered practical difficulties: the ball would stop before reaching the other edge, or the times seemed inconsistent.

Figure 6
Guidobaldo’s concave basin experiment. Source: author.

Galileo expected that his former patron would be impressed by the discovery of isochronism. However, Guidobaldo del Monte, a practical and experienced mechanic, was not persuaded by the geometric elegance. He replied that he had carried out the test, probably using a circular edge or a basin, and found that the descent time for a large arc was visibly greater than for a small arc. Thus, Guidobaldo concluded that Galileo’s theory failed in practice [²³, p. 67–71].

The experiment you tell me you have done with the bowl may be quite uncertain, perhaps because its surface is not very smooth, perhaps because it is not perfectly circular, and perhaps because it is impossible to observe the exact moment the motion begins in a single pass. However, if you still wish to use this concave surface, let ball B start from a great distance, for example, from point B; this ball will reach point D and will initially make wide movements, but in the end, small ones, and these latter will be no more frequent than the first ones [¹⁴, p. 98].

The Florentine physicist may have noticed in these experiments that the weight of the suspended body does not affect the period; only the length of the string alters this quantity. For this reason, such an empirical observation may have suggested to him a connection between the phenomenon of the pendulum and that of free fall. Indeed, if it were necessary to associate the pendulum with the inclined plane: ‘With regard to the pendulum experiments conducted by Galileo in 1602, it is evident in his letter to Guidobaldo that he had observed that only the length, and not the weight of the sphere, affected the period of oscillation. This likely suggested a connection with free fall, since in De motu he had argued that velocity was independent of weight’ [²³, p. 72–73]. In this letter, Galileo also presents other propositions concerning the motion of falling bodies along inclined planes assumed as chords in a circle.

One of the properties that Galileo claims to be able to prove is the following: consider a circle BDA (Figure 7 [¹⁴]) with vertical diameter AB; from A, draw the chords AF, AE, AD, and AC. If equal bodies fall starting from B along the vertical line BA, from C along the line CA, from D along DA, from E along EA, and from F along the chord FA, and if these bodies start simultaneously from points B, C, D, E, and F, they will all arrive together at point A. In this way, the inclination of the chords CA, DA, EA, and FA does not affect the falling time of the bodies. This is basically the first statement of Galileo’s law of strings. However, although the physicist claims to be able to prove this result, in these 1602 letters there is no demonstration; it is only in his 1638 book that Galileo presents this law as a demonstrated theorem.

Figure 7
A circle used by Galileo in his Letter to Guidobaldo Del Monte, in which he takes several chords AF, AI, AE, AD, $\dots$, and an analogy is made between the chords and inclined planes [¹⁴, p. 98].

Regarding isochronism, the statement he proposed is basically the following: if a given chord does not exceed a quadrant, with SA, SI, and IA being the chords in question, which can be taken at any point of the quadrant, then a body starting from rest at point S will traverse the path SIA more quickly than the chord IA, starting from point I. However, Galileo admits that he does not have a proof of this result, as he says: ‘up to this point, I have demonstrated [the results] without exceeding the limits of mechanics; but I cannot demonstrate that the arcs SIA and IA are traversed simultaneously, which is what I am seeking’ [¹⁴].

Galileo appears to admit that his experiment is of a more theoretical nature and that he was not overly concerned with its validity in the real world. In practice, he argued that we must abstract from the imperfections of natural bodies in order to uncover the mathematical law behind them; if a scholar were to be fixated on every small irregularity of matter, he would never arrive at the universal laws of physics. Galileo acknowledges that matter is “contingent” and sometimes does not follow geometry perfectly; nevertheless, it is the most appropriate way to determine the laws of nature.

As to your question, I agree with your opinion, namely, that when we begin to concern ourselves with matter, because of its contingency, the propositions demonstrated by geometers in the abstract begin to be altered. As to these propositions, thus disturbed, since no certain knowledge of them can be attributed, the mathematician is absolved from speculation. I have been very long-winded and tedious; please forgive me and consider me as your most devoted servant. With my utmost reverence to you [¹⁴, p. 99].

From what we have seen, the law of strings remains valid even when we have an arc approximated, for example, by two chords, where the two chords share a common point at which they intersect. Galileo had not yet reached a conclusion about the type of curve of fastest descent; he did not comment on this hypothetical property of the circle in these letters to Guidobaldo. Thus, from what has already been observed, if the falling time is independent of the angle, considering any two chords, the time will be the same. This result was an attempt by Galileo to approximate the motion of a body along an arc of a circle using a polygonal curve. However, his efforts did not lead to any definitive conclusion regarding the isochronism of the pendulum,

[$\dots$] since the approximation of an arc by a single string [or two] was obviously too crude to draw any consequences from it about the time of movement along an arc, Galileo laboriously proceeded to consider increasingly refined polygonal approximations. His excessive efforts to prove the isochronism of pendulums swinging along this line, which are documented by a series of folio pages of his notes on motion, have not, however, produced a conclusive result [⁷, p. 234].

Perhaps inspired by the ancient method of exhaustion, especially by the Archimedean approach of approximating the area of a circle by lower and upper estimates using inscribed and circumscribed polygons, which, by extrapolation through the use of the method of exhaustion, made it possible to determine a good estimate for the value of $\pi$, unfortunately for Galileo his method did not produce the desired results. It is certain that his attempt by means of this method was doomed to failure: “Galileo approximated a quarter of a circular arc by a sequence of up to 16 chords and laboriously calculated the corresponding times of motion along those chords” [⁷]. For years, he continued trying to prove the law of the isochronism of pendulums by means of this method of polygonal approximation.

The (optional) material presented in the appendices aims to present the proof of the so-called law of chords, something he would only reveal in his final book of 1638, since Galileo omitted this in his letters to Guidobaldo. A pertinent question would be how Galileo intended to prove the isochronism of the pendulum using his theorem of chords. We already know that Galileo’s law of chords states that the time of descent along any chord of a circular arc of arbitrary radius is the same for all chords. In the case of the pendulum, Galileo argued that, since a circular arc can be seen as an infinite succession of very small chords, the time a body would take to traverse any arc should also be the same, regardless of whether the arc is large or small, and this, according to him, would prove the law of isochronism.

Manuscript evidence further indicates that Galileo, around the time of the composition of this letter to Guidobaldo, was also aware of the law of falling, i.e. the quadratic dependence of the time of falling on the distance fallen, as well as the so-called law of the pendulum, i.e. the quadratic dependence of the period of a pendulum on the length of its suspension. The realization of the challenging structural similarity between the isochronism of the pendulum’s motion and the law of strings, on the one hand, and between the law of the pendulum and the law of falling, on the other, must have strongly suggested the existence of a close relationship between the motion of the pendulum and accelerated motion along inclined planes [⁷, p. 231].

Perhaps prompted by the criticisms of his reverend master, Galileo even attempted to carry out some real experiments to address the theoretical problem of the pendulum. According to [⁷], there is found, in a folio of Galileo’s experimental notes, a page on which he writes about an experiment conducted using real pendulums, in which he timed the period of a pendulum and compared it with the fall time of a ball on an inclined plane. In the theoretical construction of his theory, Galileo assumed a direct relationship between pendular motion and accelerated motion along inclined planes; if a plane were taken that had the same length as a chord of a quadrant of the pendulum’s arc, he thought, the pendulum would traverse the quadrant in the same time that a body would traverse the plane. Although having made precise and accurate measurements, the experiments did not allow Galileo to reach the same conclusions as his theoretical results.

Galileo even attempted to address the theoretical problem of this relationship in an experiment. A folio page of Galileo’s notes on the motion documents that Galileo had timed the period of a real pendulum, as well as the rolling of a ball on an inclined plane. The supposed relationship between the motion of the pendulum and accelerated motion along inclined planes suggested to him that the period of the pendulum should be related to the time of fall over a vertical distance equal to the length of that pendulum. However, even though his measurements were quite accurate from a modern point of view, the result of his considerations of the experiment did not present him with a decisive clue to his real problem [⁷].

4. Mentions of the Pendulum in Galileo’s Later Works

There are other manuscripts and unpublished texts by Galileo where we find experiments conducted by him seeking to determine the period of pendulums; this was another problem faced by the Italian physicist. He only suspected that the period of the pendulums was governed by the law of quadratic dependence between the period of a pendulum and the length of its suspension. Galileo knew, from the many experiments he conducted over many years, that this law always held true; however, he lacked sufficient means to prove that the law of the pendulum’s period was such that $T^2\propto \frac{L}{g}$. We find some mentions of the pendulum period problem in Galileo’s most important writings. In the Dialogue Concerning the Two Chief World Systems, of 1632, we find several passages in the book where Galileo addresses the problem, discusses some of his experiments with pendulums, and presents some demonstrations of certain laws relating to the pendulum.

SALV. Tell me: of two different pendulums, does not the one attached to the longer string make its vibrations less frequently? SAG. Yes, if they were moved along equal distances from the vertical. SALV. This removal from the vertical, being more or less, does not matter, since the same pendulum always makes its reciprocations, whether they be by very long or short arcs, under the same times, whether the pendulum is removed from the vertical much or very little; or, if they are not exactly equal, they are insignificantly different, as experience can show you. Yet, even if they were quite different, that would not go against our opinion [²⁵, p. 256–257].

In this book, Galileo describes another pendular mechanism, this time involving a real pendulum: the composite pendulum experiment. This experiment is basically as follows: given a perpendicular AB, a string is fixed at point A, and a weight C is attached to the lower end, forming the pendulum AC; consider another weight, E, and assume it is fixed along the same line AC, above C, forming the pendulum AE. If the string AC is moved from its equilibrium position, the system begins to move. Considering (Figure 8[²⁵]), since pendulums of different lengths have different periods (or frequencies), weight E tends to oscillate faster than weight C. This causes E, by oscillating more rapidly, to interrupt the trajectory of C, which, if it were alone, would oscillate freely. As Galileo states: ‘the weight E, as it hangs at a shorter distance and is also less removed from the perpendicular, will wish to return sooner and make its vibrations more frequent than weight C, so that it will prevent the latter from running towards point D as much as if it were free’ [²⁵].

Figure 8
Diagram of a pendulum in the book Dialogue between the Two Chief World Systems where Galileo discusses pendulums [²⁵, p. 256].

Galileo asserts that if we observe not only E and C, but every point along the string (imagining infinite small segments), each segment behaves as a pendulum of a different length. The closer to A, the shorter this ‘local pendulum’ is and the faster its natural oscillation; the closer to C, the longer the ‘local pendulum’ is and the slower its natural oscillation. This entanglement of ‘pendulums within the pendulum’ creates a situation in which the upper parts of the string pull or delay the lower parts; each segment attempts to return to equilibrium at its own rhythm.

In the case of a flexible string, this generates internal tensions that cause it to curve, preventing it from remaining straight since each part is at different angles and speeds; furthermore, it dissipates the energy of weight C throughout the oscillations, causing it to stop sooner than a simple pendulum would. Each link of a chain or fiber of a string has its own weight: the links near the top tend to swing faster, while the links at the bottom tend to swing slower. This discrepancy in speeds along the string creates an internal tension that ’brakes’ the system over time. Such a phenomenon becomes more evident if, instead of a string, a chain is used as the support for weight C: ‘if, instead of a string, we take a chain, we shall see this effect more evidently, especially when we remove weight C from the perpendicular’ [²⁵].

The result is an even more curved and ’uncoordinated’ motion, clearly demonstrating how weight C cannot maintain large amplitudes of oscillation for long. The idea is that in an ideal pendulum (rigid, frictionless, and without air resistance), the oscillation would continue for a long time; however, in a real pendulum made of string or chain, dissipation, internal tensions⁷, and friction between links arise even without air resistance. These internal forces cause the large-amplitude motion to slow down and gradually lose energy. In performing this exercise, Galileo is dealing with the complexities that arise with the real pendulum, since when we study the ideal pendulum, none of this needs to be taken into account.

There is another passage in Galileo’s Dialogue where he enunciates somewhat more precisely his law of the amplitude independence of pendulums,

$\dots$truly remarkable$\dots$ that the same pendulum makes its oscillations with the same frequency, or with very little difference – almost imperceptibly – whether they are made through large or very small arcs along a given circumference. I mean that if we move the pendulum away from the perpendicular by only one, two, or three degrees, or, on the other hand, seventy or eighty degrees, or even a whole quadrant, it will make its oscillations when released with the same frequency in both cases [²⁵, p. 450].

Galileo continues his discussion on pendulums in his 1638 book, Discourses and Mathematical Demonstrations Relating to Two New Sciences, in which he revisits his earlier idea of relating the motion of a pendulum along a circular arc to the motion of a falling body on an inclined plane. Since allowing a falling body to descend along a plane involves many impediments (air resistance and friction) that prevent drawing proper conclusions, the Florentine physicist decided to replace the use of the inclined plane with the pendulum for the study of motion under the action of gravity; this latter system offered much less resistance from the medium, as the only contact is with the air and with the string at the point of fixation.

As he states: ’moreover, in order to take advantage of the slowest possible movements, in which the resistance of the medium alters the effect dependent on simple gravity much less’ [¹⁹]. Galileo understood that an inclined plane or a pendulum are nothing more than ways to delay the fall of bodies under the action of gravity; thus, in general terms, both the pendulum and the plane would be convenient means to study the vertical fall of heavy bodies, ’since the behavior of heavy bodies of different weights would be observable both on an inclined plane and in a vertical descent’ [¹⁹].

It is in this book that Galileo definitively enunciates his law of free fall and uses his experiments with pendulums to support his arguments; he had previously supported these same ideas in his letters to Del Monte. Therefore, to refute the idea that the speed of fall is proportional to weight, the physicist proposes an experiment with two pendulums of equal length, one of lead and the other of cork, purposefully choosing materials with different masses. If Aristotle’s theory were correct, the lead ball should complete its oscillations much sooner than the cork, as the lead, being heavier, would travel the arc at a higher speed.

However, by observing that both pendulums traveled the path in the same time and maintained the same rhythm for hundreds of swings, there is clear evidence of the independence of the oscillation period in relation to the mass of the body. This contradicts Aristotelian theory and indicates that, for any amplitudes, the pendulum’s motion depends primarily on the length of the string and the acceleration of gravity. As he says: ’then, desiring to eliminate the impediments that might be due to the contact between the moving bodies and the plane, I decided to use two balls, one of lead and the other of cork, the former being more than a hundred times heavier than the second. I attached both to fine strings of equal length, about 4-5 fathoms, and fixed them above’ [¹⁹, p. 128].

Next, Galileo demonstrates that he is truly convinced of the property of the isochronism of pendulums and also of the law of the period of a simple pendulum. It is in the Discourses that Galileo reaffirms his law of isochronism, for he guarantees that for any heavy body supported by a string of equal length, it will travel any arc in the same time; he also states that it is true that a body in vertical motion descending along the chords of any arc will, necessarily, travel them in equal times. This is the general form of that first statement of the law of chords found in the letters of 1602; now, he is no longer restricted to a quadrant, it can be an arc of 180^∘, 120^∘, 60^∘, 30^∘, 1/2^∘, etc.

SAVE. We will see if we can extract solutions to our difficulties from these pendulums. As to the first doubt, which is, whether truly and more exactly the same pendulum makes all its vibrations, the greatest, the intermediate and the smallest, at precisely equal times, I will refer you to what I heard from our Academician. He clearly proves that a moving body descending along the chords of any arc would necessarily traverse all of them in equal times, not only that corresponding to 180 degrees (i.e., the diameter), but also those corresponding to arcs of 100, 60, 10, 2, ½, degrees, or even a fraction of a degree, on the understanding that all the chords converge to the lowest point touching the horizontal plane [¹⁹, p. 139–140].

It is in the following passage that Galileo confirms his law of isochronism, he says that experience shows that all arcs are traversed in the same time, and these times are shorter than the times of fall along the corresponding chords. Furthermore, the physicist confirms that the fastest path, that is, which demands less time, is the arc of the chord, in other words, he is saying that the circular pendulum is the object that moves fastest (in the shortest time) on the path between two points, and the most incredible, according to Galileo, is that the arc of the pendulum’s chord is not the short path in distance, since the shortest path, in distance, between two points is a straight line, even so, the arc is the fastest path in time.

As regards bodies descending along the vertical arcs of the strings, which are not greater than a quadrant, that is, 90 degrees, experience shows that all the arcs are traversed in the same times, but shorter than the times of the strings; the effect of which is wonderful, inasmuch as, superficially, it seems that the opposite would have to happen, for, given that the starting and ending points of the movement are the same, and given that the straight line is the shortest line between two points, it would seem reasonable that the movements along the straight strings would be the shortest; but that doesn’t happen. In fact, the shortest time, and therefore the fastest movement, is that along the arc of the string [¹⁹, p. 139–140].

Another fundamental law of pendulums that Galileo reaffirmed was the law of the period [²⁶]; this was already known to him since at least those letters of 1602: the period is proportional to the sub-double ratio [square root] of the length. Certainly, the constant of proportionality (the acceleration of gravity g) was not known to the Italian physicist, but it would have been of great use had he known it. In fact, since this law was obtained by Galileo, most likely through the use of experiment, he would not have had many clues to lead him to conclude that the constant of proportionality was g; in any case, by the standards of rigor of the time, it was completely acceptable and sufficient to say that $T\propto \sqrt{L}$,

As to the proportion of the times of the mobiles hanging on cords of different lengths, they are in the sub-double proportion [i.e., the square root] of the proportion of the lengths of the cords. In other words, the lengths of the strings are in doubled ratio to the times, that is, they are like the squares of the times. Thus, for example, for the vibration time of one pendulum to be twice the vibration time of another pendulum, the length of the string of the first must be four times the length of the string of the latter. Thus, during one vibration of a pendulum, another will make three, if the string of the first is nine times that of the last. It follows that the lengths of the strings have the same ratio to each other as the squares of the number of vibrations made in the same time [¹⁹, p. 139–140].

Del Monte was a natural philosopher greatly esteemed by Galileo (Figure 9[²⁷]), not only for having helped him on several occasions, but also for his erudition. Indeed, Del Monte taught Galileo some mathematical knowledge. However, to a certain extent, Del Monte could be considered more of an Aristotelian than a modern philosopher [²⁴]. Galileo’s view that natural phenomena can be quantified through ideal models and then applied to explain the real world was disapproved of by his older contemporary. Del Monte strongly disapproved of Galileo’s attempts to prove the law of isochronism, a property he never accepted as valid, and consequently considered Galileo’s efforts fruitless. Galileo’s work was once questioned by a Venetian artilleryman, who questioned the physicist’s predictions about the parabolic motion of projectiles, something he could not empirically verify. Galileo, perhaps anticipating this kind of criticism, said in his dialogues:

Figure 9
Portrait of Galileo Galilei (1564–1642), Italian physicist famous for many scientific discoveries at the beginning of Modern Science [²⁷].

I admit that these conclusions, proven in the abstract, will be different when applied to the concrete and will be fallacious, since neither the horizontal motion will be uniform, nor the natural acceleration will be in the assumed ratio, nor the trajectory of the projectile will be a parabola [¹⁹, p. 251].

Del Monte did not support Galileo’s position, because if a conclusion is valid in the theoretical and abstract domain and, in the same way, if something related to an ideal phenomenon (approximately real) is proven, why, upon leaving the realm of abstraction, is the validity of what was proven lost? This was one of the difficulties faced by the new science in its confrontation with the thought of Aristotelian philosophers. Giovanni Renieri, the Venetian gunner who tried to apply Galileo’s theory to his practice, complained to Torricelli, in 1647, that his cannons did not behave according to Galileo’s predictions, and Torricelli replied that ’his master spoke the language of geometry and was not bound to any empirical results’ [²⁸, p. 43].

The correspondence between Galileo and del Monte, in particular, is little known; generally, it is studied by few historians of science and is not widely known even by this group of specialists. The excerpts highlighted in this text help to understand part of Galileo’s journey toward the understanding of the physical laws of the pendulum.

5. Final Considerations

This study examined the historical evolution of pendulums, from their earliest appearances in the works of Oresme, Leonardo da Vinci, and others, a context that was analyzed in the first part [¹⁷]. The second part was devoted, basically, to the theorization of the pendulum carried out in Galileo’s scientific work; that is, the entirety of this work consisted of two articles. It is true that the theoretical and scientific use of pendulums only began in the seventeenth century. In 1602, Sanctorius was the first to design a practical pendulum device, the pulsilogium, an instrument whose objective was the scientific use of the properties of pendulums. When we state here that Sanctorius was the inventor of the pulsilogium, this does not imply that he was the inventor of the pendulum, since, as discussed previously, pendulums are very ancient devices [¹⁰].

As mentioned previously, there was some controversy regarding who was the first scholar to use pendulums as instruments for scientific purposes. The invention of the pulsilogium is contested by three contemporary scholars: Sanctorius, Sarpi, and Galileo. Historians still have doubts about who may have been the first to create this device. However, in this work, we adopt the hypothesis that Sanctorius was probably the first inventor of the pulsilogium; nevertheless, it is not possible to know who influenced whom, since both were professors at the University of Padua. It is equally possible that: (i) Galileo shared his theoretical ideas with Sanctorius, who then invented the device; or (ii) Sanctorius independently invented his instrument, and shortly thereafter the invention was communicated to Galileo, who then began to study it theoretically. Regarding this dispute, we will make no judgment, unlike what [²³, p. 69] did, when he stated that “the pulsilogium described in 1603 by Santorio Sanctorius, then a physician in Venice and later professor of medicine in Padua, was probably inspired by discussions about such experiments with his friend Galileo”.

Precisely, it is impossible to know exactly when Sanctorius invented the instrument, and what led Galileo to become interested in the study of the pendulum, specifically in 1602, the probable year of the pulsilogium’s invention. This is a question that continues to intrigue historians: when did Galileo begin his studies of pendulums? According to his biographer Vincenzo Viviani [⁶], Galileo had already begun studying the pendulum at least twenty years earlier, in the 1580s. Evidence on this point is scarce. Nevertheless, regardless of when Galileo began his investigations, he was the first to promote the systematic study and theorization of this instrument, the pendulum.

Before Sanctorius invented his pulsilogium, some theoretical work had already been done on the dynamics of pendulum motion, something that, in the spirit of Aristotelian physics, would have been inconceivable [¹⁰]. The emergence of a new science of motion was necessary, something that would not appear until Galileo. Sanctorius, Del Monte, and Benedetti, all contemporaries of Galileo, collaborated on the problem of weight on a balance, which, as we have seen, had some parallels with pendulums. However, it was only with Galileo that theorizing about pendulums became concrete. The Italian physicist focused on the object as an instrument of physics that required an adequate physical and mathematical description; this is how Galileo began his experimental work with pendulums [²⁶].

The law of isochronism of pendulums is attributed to Galileo and was probably never formulated before him, neither by Ibn Yunus nor by anyone else. Although Galileo was not entirely successful in proving his law of isochronism, he contributed significantly to establishing the pendulum as an object of interest in physics. After many years of work, Galileo managed to assemble a set of statements or hypotheses that, based on his experimental and theoretical studies, were quite probable. In his final book, Dialogue Concerning the Two New Sciences, these hypotheses began to be considered physical laws for pendulums. In this work, Galileo enunciated the main properties of the pendulum and sought to provide a justification for certain theoretical elements that accounted for aspects of pendulum motion, such as the string law.

These hypotheses can be enumerated according to [²⁹]: (1) Weight-independence property: the period is independent of weight; (2) Amplitude-independence property: the period is independent of amplitude; (3) Length property: the period varies directly with length; (4) Property of isochronism: for any pendulum [of the same length], all oscillations take the same period of time. Del Monte was, in fact, somewhat correct in rejecting the supposed isochronous property of pendulums proposed by Galileo. As enunciated by him, the isochronism of the pendulum held for any angle, and we know today that such a statement is incorrect, just as the circle is not the fastest path between two points. Some time after Galileo’s death in 1642, his proposition would be duly corrected by Huygens, who rightly stated that the isochronism of a simple pendulum is only valid for small angles.

This work falls within the internalist tradition of the history of science, as we have been primarily concerned with describing and characterizing the internal aspects of the development of pendulum mechanics. Of course, we cannot neglect the external aspects, relating to the historical context of the time. In this period, from the late sixteenth to the early seventeenth centuries, measuring time remained a major challenge for scientists and for people in general. The problem of longitude had long been a significant issue for navigation [³⁰]. Galileo was aware of this problem, and in science itself, the precise measurement of time was becoming increasingly necessary. He often experienced the difficulty of measuring time for his experiments with pendulums and inclined planes [¹¹]. When Galileo discovered the isochronous property of pendulums, he realized that it could be used to measure time accurately.

For small angles, the oscillations of any pendulum take the same period of time. This is the fundamental property behind all pendulum clocks. However, Galileo believed that the pendulum possessed the isochronous property for all amplitudes, meaning that regardless of whether the angle was large or small, the property would always hold. Galileo was the first to conceptualize the construction of a pendulum clock in 1637 [³⁰]. However, he was unable to complete his invention; in fact, the mechanism he developed was impractical. After Galileo’s death, it was the Dutchman Christiaan Huygens (1629–1695) who, in 1656, invented the first functional pendulum clock and refined the idea based on Galileo’s studies of the isochronism of the pendulum. Overall, this paper meets the outlined objectives, being a historical study on the theoretical origins of the pendulum.

Acknowledgments

The author would like to thank the editor and the reviewers of the RBEF for their careful review of the initial manuscript. Their dedication and valuable comments significantly contributed to the improvement of this work.

Appendix

Some considerations on the geometric constructions of Galileo’s pendulum motion

The string law is a central result in Galilean theory, used to justify the isochrony of pendulums. In this appendix, we will examine it in more detail. In the 1638 Discourses and Mathematical Demonstrations Concerning Two New Sciences (Discorsi e dimostrazioni matematiche, intorno à due nuove scienze), we find a statement of the string law by Galileo, in which he describes the motion of bodies under the influence of gravity, specifically, motion along curved paths such as circles and the descent times along these paths.

Theorem VI. If any inclined planes are drawn from the highest or lowest point of a vertical circle that meet the circumference, the descent times along these strings will each be equal.

Consider Figure 7, a geometric representation of Galileo’s work in his 1602 letter to Del Monte, it contains a circle and several strings AF, AI, AE, AD, etc., but the precise statement of Galileo’s theorem appears in the Discursi. Galileo postulated that, on trajectories that form an arc of a circle, the descent times do not depend on the shape of the trajectory. What matters is that, along any trajectory connecting the highest to the lowest point of the circle, the total time to travel the distance will always be the same, as long as there is no air resistance.

This theorem presents a mechanical connection between the motion of chords in a circle and the motion of a pendulum. The pendulum describes a circular arc; that is, a body of mass m traverses a trajectory along a circular arc. With this theorem, Galileo attempted to approximate the pendulum’s motion using an inclined plane, a system he used to study and simplify the motion of free fall. In one of the passages of his letters to Guidobaldo Del Monte, he mentions being convinced of the veracity of the theorem of chords and also, based on this theorem, of its extrapolation to the isochronism of pendulums. However, in this letter from 1602, Galileo confesses to not having a demonstration for this result: ‘Up to this point [that is, up to the theorems of chords] I have demonstrated without transgressing the terms of mechanics; but I cannot demonstrate how the arcs SIA and IA were traversed in equal times, and that is what I seek’.

There are other related results presented by Galileo before and after his Theorem VI, let’s look at these, and then look at Galileo’s demonstration of his string law (Galileu, 1638).

Theorem V. The descent times along planes of different lengths, inclinations, and heights maintain a ratio that is equal to the product of the ratio of their lengths by the square root of the inverse ratio of their heights.

To prove this theorem, Galileo uses the construction in Figure 10[¹⁹] below. Draw planes AB and AC, with different inclinations, lengths, and heights. The theorem states that the ratio between the descent time along AC and the descent time along AB is equal to the product of the ratio of AC to AB by the square root of the inverse ratio of their heights. Let AD be a perpendicular to which the horizontal lines BG and CD are drawn; let AL also be a mean proportional to the heights AG and AD; from point L, draw a horizontal line that meets AC at F; consequently, AF will be a mean proportional between AC and AE. Now, since the time of descent along AC is to that along AE, as the length AF is to AE; as AE is to AB, it remains clear that the time along AC is to the time along AB as AF is to AB.

Figure 10
Galileo’s geometric construction for the proof of Theorem V [¹⁹, p. 244].

Let us now continue with Galileo’s proof of the chord theorem.

Theorem VII. If, from the highest or lowest point of a vertical circle, any inclined planes are drawn that meet the circumference, the times of descent along these chords are equal.

On the horizontal line GH, construct a vertical circle, as indicated by the construction of Figure 11, [¹⁹]. From its lowest point – the point of tangency with the horizontal – draw the diameter FA, and from the highest point, A, draw inclined planes to B and C, any points on the circumference; then the times of descent along them are equal. Draw BD and CE perpendicular to the diameter; let AI be a proportional average between the heights of the planes, AE and AD; and since the rectangles FA.AE and FA.AD are respectively equal to the squares of AC and AB, while the rectangle FA.AE is to the rectangle FA.AD as AE is to AD, it follows that the square of AC is to the square of AB as the length AE is to the length AD. But since the length AE is to AD as the square of AI is to the square of AD, it follows that the squares on lines AC and AB are to each other as the squares on lines AI and AD, and therefore the other length AC is to the length AB as AI is to AD. But it has been shown previously that the ratio of the time of descent along AC to that along AB is equal to the product of the two ratios, AC to AB and AD to AI; but this latter ratio is the same as that of AB to AC. Therefore, the ratio of the time of descent along AC to that along AB is the product of the two ratios, AC to AB and AB to AC. The ratio of these times is, therefore, unity.

Figure 11
A circle with diameter AF and at point F, we find a tangent GFH. This construction was used by Galileo to prove his string theorem [¹⁹, p. 245].

Using the principles of mechanics [ex mechanicis], one can obtain the same result, namely that a falling body will require equal times to travel the distances CA and DA, indicated in the following Figure 12[¹⁹]. Discard BA equal to DA and drop the perpendiculars BE and DF; it follows from the principles of mechanics that the component of the momentum [momentum ponderis] acting along inclined plane ABC is to the total momentum [i.e., the momentum of the body in free fall] as BE is to BA; similarly, the momentum along plane AD is to it’s total momentum [i.e., the momentum of the body in free fall] as DF is to DA or to BA. Therefore, the momentum of this same weight along plane DA is to that in plane ABC as length DF is to length BE; therefore, this same weight will travel in equal times through spaces, according to the second proposition of the first book. It travels through spaces along planes CA and DA, which are to each other as lengths BE and DF. However, it can be shown that CA is to DA as BE is to DF. Hence, the body in free fall will travel the two trajectories CA and DA in equal times.

Figure 12
Geometric representation used by Galileo to demonstrate the string law. From the lowest point of the circle, A, several chords (AC, AD, etc.) are drawn, corresponding to inclined planes of different inclinations. From points on the circumference, perpendiculars are projected to the base, forming auxiliary rectangles (such as HB, GI, EF), used in the demonstration [¹⁹, p. 246].

Furthermore, the fact that CA is to DA as BE is to DF can be demonstrated as follows: join C and D; through D, draw line DGL parallel to AF and intersecting line AC at I; through B, draw line BH, also parallel to AF. Then the angle ADI will be equal to the angle DCA, since they subtend equal arcs LA and DA, and since the angle DAC is common to the sides of the triangles, CAD and DAI, and on which the common angle is in proportion to each other; in consequence, CA is to DA as DA is to IA, that is, as BA is to IA, or as HA is to GA, that is, as BE is to DF.

We will now present a modern demonstration of Galileo’s law of chords and the law of isochronism [²⁹], transposing the original geometric proportions into the current kinematic model. Consider a circle of radius R and several chords that originate from the lowest point of the circle (Figure 7). Each chord represents a different inclined plane. Although the length of each chord (that is, the distance traveled by the body) varies with the angle, the ratio between this length and the component of gravitational acceleration acting along the chord remains constant. In other words, although a more steeply inclined chord is longer (closer to the diameter), the body accelerates more rapidly; whereas a less steeply inclined chord is more curved in length, yet the body accelerates more slowly. Consider the vertical circle ABC in Figure 13[³¹], let BA be the vertical diameter and AC any chord of the circle. Suppose that two identical bodies, starting from rest, descend simultaneously along the chords BA and AC, starting from B and C, respectively; by Galileo’s theorem of chords, both will arrive together at A.

Figure 13
A circle with center O, radius r, diameter BA and an inclined plane AC, this scheme was designed to demonstrate the property under study. Adapted from [³¹].

Note that by joining chord BA and chord AC with line BC, we form triangle ABC. We will prove that angle $\widehat{C}$ is a right angle. To this end, consider Thales’ theorem concerning a triangle inscribed in a semicircle, which states: Given a triangle inscribed in a semicircle, Figure 14[³¹], if one of its sides coincides with the diameter AB of the semicircle, then the triangle is right-angled.

Figure 14
Demonstrative diagram of Thales’ theorem. Adapted from [³¹].

Draw the line segment CM, where M is the midpoint of segment AB (M is the center of the semicircle). By drawing CM, we divide triangle ABC into two triangles, BCM and ACM. Moreover, note that $BM=AM=CM=r$; therefore, triangles BMC and ACM are isosceles, where $\beta$ and $\alpha$ are, respectively, their base angles. In this way, the sum of the interior angles of triangle ABC is,

$$\begin{array}{c}\left(\alpha +\beta \right)+\alpha +\beta ={180}^{\circ }\\ \alpha +\beta ={90}^{\circ }\end{array}$$

From this we conclude that $\widehat{C}={90}^{\circ }$, which proves Thales’ theorem. Returning to Figure 13, note that triangle ABC is right-angled. If this is the case, knowing that $\theta$ is the inclination of chord AC, then there exists a relationship between the lengths $H$ and $l$.

$$l=H\cdot \cos \left({90}^{\circ }-\theta \right)=H\cdot \sin \theta$$

Since g is the acceleration due to gravity, looking at the right triangle we can write that $cos\left({90}^{\circ }-\theta \right)=a/g$, that is, $a=gsin\theta$, where we say that a is the projection of the acceleration of the fall along the cord AC; therefore, the equations that describe the falling motion of the bodies are given by,

$$\begin{array}{c}H=\frac{1}{\text{2}}g{t}_H^2,l=\frac{1}{\text{2}}a{t}_l^2\\ l=\frac{1}{\text{2}}\left(g\sin \theta \right)t_l^2\to H\cdot \sin \theta \\ =\frac{1}{\text{2}}g\cdot \sin \theta t_l^2\to H=\frac{1}{\text{2}}g{t}_l^2\end{array}$$

Equating the two relations we obtained for H, we finally have,

$$\frac{1}{\text{2}}g{t}_H^2=\frac{1}{\text{2}}g{t}_l^2\to t_H=t_l$$

Let us now consider the property concerning the period t, which Galileo says is proportional to the square root of the length of the pendulum’s cord,

(1) $$S=\frac{1}{\text{2}}a{t}^2$$

As we know, the acceleration of a body with mass m descending a frictionless inclined plane is $a=g.sin\theta$, since resolving the force weight is $P=mgsin\theta$, which is the resultant force. Substituting the acceleration $a=g.sin\theta$ into the previous expression, we have,

(2) $$S=\frac{1}{\text{2}}g\cdot \sin \theta t^2$$

Consider the vertical circle with center O and vertical diameter $BA$ (with B at the top and A at the bottom), Figure 13. Take any chord AC extending from the lowest point A, which makes an angle $\theta$ with the horizontal, and denote its length by $S=AC$. At this point, Galileo uses a result, which states that for any chord $AC$ extending from the lowest point $A$ of the circle with radius r, the relation holds,

(3) $$S=AC=2r\sin \theta$$

Therefore, the ratio of S to the slope $sin\theta$ is constant, since,

$$\frac{S}{\sin \theta }=2r$$

From (2), we can isolate the time to obtain,

(4) $$t^2=\frac{2S}{g\cdot \sin \theta }$$

From the result considered previously, we know that,

$$t^2=\frac{2\cdot \left(2r\cdot \sin \theta \right)}{g\cdot \sin \theta }=\frac{4r}{g}$$

Therefore, $t^2$ (and, consequently, t) is independent of $\theta$; it is a constant that depends only on R and g. Therefore, all times are equal,

(5) $$t=2\cdot \sqrt{\frac{r}{g}}$$

Finally, I add some notions of the concepts of Galilean mechanics; according to [¹⁸, p. 103], we have the definitions of weight (gravitas), defined by Galileo as the ‘tendency to move naturally downward’; and of moment (momentum), also a tendency to move downward caused not only by weight, but also exacerbated by speed and the geometry of Archimedean simple machines (something akin to mechanical advantage).

Data Availability

The entire dataset that supports the results of this study was published in the article itself.

References

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• 1
  This chandelier is known as the ‘lampada di Galileo’ in honor of the Italian physicist. However, there are serious doubts about this story raised by Viviani, as noted by Antonio Favaro that the specific lamp today referred to as the “lampada di Galileo” did not exist in the cathedral of Pisa at the time when Galileo was a young student attending the cathedral [⁷].
• 2
  In this regard, there is no historical evidence to support the claim that Galilean pendulum studies began due to Galileo’s interest in music. Therefore, such an assertion rests solely with the aforementioned author.
• 3
  The term brachistochrone comes from the Greek brakhistos (shortest) and khronos (time); that is, the word can be literally translated as the curve of fastest descent or the path of shortest time. The Brachistochrone problem can be stated as follows, which is the curve connecting two points (A and B) at different heights, but not on the same vertical line, along which a body, acting only under the force of gravity, slides in the shortest possible time? [²⁰].
• 4
  In the article preceding this one, titled ‘Comments on the origin of pendulums: The controversy surrounding the pulsilogium’, in reference [¹⁷], there is an explanation of what was understood by impetus at the time.
• 5
  The term tautochrone originates from the Greek tauto (ταὐτό), meaning ‘the same’, and khronos (χρόνος), meaning ‘time’; therefore, the word tautochrone can be literally translated as the ‘curve of the same time’. For this reason, this curve can also be called an isochrone curve. A tautochrone curve is one in which the time taken by an object sliding without friction to reach the lowest point is the same, regardless of the height from which it begins to descend [¹⁹].
• 6
  Galileo sometimes presented himself as a Florentine (born in Florence), but in fact, he was a Pisan (born in Pisa).
• 7
  It should be noted that these terms, energy, dissipation, internal tension, friction, used in the explanation of this phenomenon, do not align with the nomenclature and parlance of the Renaissance physics in which Galileo was immersed. The use of these terms was intended to facilitate the explanation of the phenomenon.

Edited by

• Editor-in-Chief:
  Marcello Ferreira https://orcid.org/0000-0003-4945-3169

Publication Dates

• Publication in this collection
  27 July 2026
• Date of issue
  2026

History

• Received
  04 Feb 2026
• Reviewed
  12 May 2026
• Accepted
  12 June 2026