Galileo

by Mario Livio

  In this monumental poetic work (running more than fourteen thousand lines), Dante tells the story of a poet’s imaginative journey through the afterlife, drawing inspiration from a wide range of philosophies. After an epic tour through the Inferno and Purgatory to Paradise, the poet finally reaches that “love that moves the Sun and other stars.”

  The invitation to present the lectures demonstrated the Academy’s respect not only for Galileo’s mathematical skills but also for his literary scholarship. Galileo was undoubtedly delighted to receive this request for two main reasons. First, mapping Dante’s disorienting description of hell in The Divine Comedy gave Galileo his first opportunity to attempt to build a bridge between a literary magnum opus and scientific reasoning. In later years, an important part of what was to become Galileo’s continuous philosophy and ultimate legacy was the demonstration that science is an integral part of culture, and that it can enhance, rather than diminish, even the poetic experience. As a means to this goal, he went against the long-standing tradition of writing science in Latin and wrote instead in Italian. Working in the other direction, in his extensive scientific writing, Galileo drew from his literary resources to convey ideas and associations in a colorful, stimulating fashion.

  Second, Galileo shrewdly recognized the importance of these lectures for his personal career. He was fundamentally asked to act as an arbiter between two contradictory commentaries and views on the location, structure, and dimensions of the Inferno, offered by two interpreters of Dante’s work. One was the beloved Florentine architect and mathematician Antonio Manetti, biographer of the famous architect Filippo Brunelleschi. The other was the intellectual Alessandro Vellutello of Lucca. Vellutello argued that Manetti’s amphitheater-like edifice could not be stable, and he offered an alternative model in which hell occupied a much smaller volume around the Earth’s center. Much more than a purely highbrow dispute was at stake. Florence had suffered a humiliating military disaster at Lucca in 1430. After an unsuccessful besieging of that city, Brunelleschi, acting that time as an army engineer, came up with the idea of diverting the river Serchio, so as to surround Lucca with a lake and force it to surrender. The plan backfired cataclysmically when a dike failed, and the river flooded the camp of the Florentine army instead. This painful historical memory was surely on the minds of the members of the Florentine Academy when they asked Galileo to demonstrate that Manetti “had been slandered by Vellutello.” Moreover, Vellutello’s commentary represented a disowning of Manetti’s authority—and, by association, the Florentine Academy’s—in the interpretation of Dante. In other words, Galileo was entrusted with saving the academy’s prestige, and he realized that by handing Manetti a victory over Vellutello, he could be regarded as a champion of Florentine pride.

  Galileo started his first lecture with a direct reference to astronomical observations (probably having in mind the fact that most of the positions he was seeking at the time were in mathematics and astronomy) but emphasizing that deciphering the architecture of hell would require theoretical considerations. He then swiftly moved on to describing Manetti’s interpretation, using the same analytical skills that would become his trademark in all of his scientific investigations. The dark scenery of Dante’s hell occupied a cone-shaped portion of the Earth, with Jerusalem at the center of the cone’s dome-shaped base and the cone’s vertex being fixed at the Earth’s center (Figure 2.1 shows Botticelli’s depiction). Contrary to Vellutello’s claim that Manetti’s structure occupied a full one-sixth of the Earth’s volume, Galileo used the geometry of solids he had learned from reading the works of Archimedes to demonstrate that, in fact, it filled less than seven-hundredths of the bulk—in his words: “less than one of the 14 parts of the whole aggregate.” He then methodically proceeded to tear apart Vellutello’s model by showing that not only would parts of his proposed architecture have collapsed under their own weight, but also that the design did not even agree with Dante’s chilling description of the descent to hell. In contrast, Galileo argued that in Manetti’s construction, “its thickness is sufficient… to sustain it.” Galileo finished his Inferno lectures by thanking the academy, to which he felt “most obligated,” wisely adding that he thought he had demonstrated “how much subtler is the invention of Manetti.”

  Figure 2.1. Sandro Botticelli’s Chart of Hell, based on Dante’s Inferno.

  Unfortunately, in perhaps wishing too much to please his audience, Galileo fell into his own trap. He didn’t realize that Manetti’s architectural structure was also prone to catastrophic collapse (not that any of his listeners noticed). Galileo may have discovered his blunder shortly after delivering the Inferno lectures, since he stopped referring to them for many years, and his biographer, Viviani, never even mentioned the lectures, despite having lived in Galileo’s house during the master’s last years.

  Only in his last book, on the Two New Sciences, did Galileo return to the interesting problem of the strength and stability of constructions when they are scaled up in size. The key insight he had gained by then was that whereas the volume (and therefore the weight) increases a thousandfold when the size is blown up to be ten times larger, the resistance to cracking (which happens along two-dimensional surfaces) increases only a hundredfold, and therefore falls behind the increase in the weight. Galileo wrote in Two New Sciences: “The larger machine, made of the same material and the same proportions as the smaller one, in all other conditions will react with the right symmetry to the smaller one, except for its strength and its resistance against violent invasions; the bigger the ship, the weaker it will be.” Then, most likely alluding to his Inferno mishap, he noted that “some time ago” he also had made a mistake when estimating the strength of scaled-up objects. Perhaps the most remarkable point about Galileo’s flawed Inferno incident was the fact that even many years after having delivered a scientific talk about a poetic work, Galileo felt compelled to revisit his conclusions, revise his old ideas based on newly acquired acuity, and publish the new, correct results in an entirely different context for the problem.

  Galileo was indeed a Renaissance man, but one may wonder whether in our age of narrowly focused specialization and career-driven attitudes such people still exist, and whether individuals who are curious about a wide range of topics or polymaths with broad interests are even needed. Drawing on about a hundred interviews with extraordinarily creative men and women across many disciplines, University of Chicago psychologist Mihaly Csikszentmihalyi suggested that the answer to both questions is in the affirmative. His conclusion: “If being a prodigy is not a requirement for later creativity, a more than usually keen curiosity about one’s surroundings appears to be. Practically every individual who has made a novel contribution to a domain remembers feeling awe about the mysteries of life and has rich anecdotes to tell about efforts to solve them.” Indeed, creativity often means the ability to borrow ideas from one field and transpose them into another. Charles Darwin, for example, took one of the pillars of his theory of evolution—gradualism—from his geologist friends. This was the notion that just as the surface of the Earth is shaped very slowly by the actions of water, the Sun, the wind, and geological activity, so too do evolutionary changes occur over hundreds of thousands of generations.

  Recognizing that a shout-out for “Renaissance persons” can inspire creativity in the modern world does not mean giving up on specialization. With sources of information literally at our fingertips, even those ten thousand hours or so (that are supposedly required to become an expert in a given topic according to author Malcolm Gladwell, though this has been disputed by authors of the original study) can be shortened through more efficient learning practices and techniques. This time saving, combined with the fact that humans are living longer than ever before, means that there is nothing today (in principle, at least) to prevent people from being both experts and Renaissance persons.

  Returning to Galileo’s life, the reputation he had gained through his Inferno lectures and a strong recommendati
on he had eventually received from Clavius proved to be extremely fruitful. In the summer of 1589, Filippo Fantoni left the chair of mathematics at the University of Pisa, and Galileo, the former dropout from that university, was appointed to that post.

  CHAPTER 3 A Leaning Tower and Inclined Planes

  Galileo’s first appointment as a professor and chair of mathematics at Pisa lasted only from 1589 till 1592, yet one particular story associated with that period has generated an iconic image of Galileo. It is a picture of him dressed in his imposing academic gown, dropping balls of different weights from the top of the leaning tower of Pisa.

  The original tale comes from Viviani, who in 1657 put together what he described as his recollections from a conversation he had with Galileo in the latter’s final years:

  A great many conclusions of Aristotle himself on the subject of motion were shown by him [Galileo] to be false which up to that time had been held as most clear and indubitable, as (among others) that speeds of unequal weights of the same material moving through the same medium did not at all preserve the ratio of their heaviness assigned to them by Aristotle, but rather, these all moved with equal speeds, he [Galileo] showing this by repeated experiments made from the height of the leaning tower of Pisa in the presence of other professors and all the students.

  In other words, contrary to the view held by all Aristotelians that the heavier the ball, the faster it would fall, Viviani claimed that by dropping balls from the Leaning Tower (sometime between 1589 and 1592), Galileo had demonstrated that two balls of the same material but of different weights hit the ground simultaneously.

  As if this story wasn’t dramatic enough, later biographers and historians just kept adding more details that weren’t included in Viviani’s original account or in any other contemporary sources. For example, British astronomer and popularizer of science Richard Arman Gregory wrote in 1917 that members of the University of Pisa assembled at the foot of the Leaning Tower “one morning in the year 1591,” even though Viviani never mentioned the precise year or the time of day. Gregory also added that one ball was “weighing a hundred times more than the other”—again a detail not given by Viviani. Author Francis Jameson Rowbotham, who wrote about the lives of great scientists, great musicians, great authors, and great artists, added in his 1918 vivid description that Galileo “invited the whole University to witness the experiment.”

  Others were equally inventive. Physicist and historian of science William Cecil Dampier Whetham tells us in 1929 that Galileo dropped “a ten-pound weight and a one-pound weight together,” repeating the same values of weights that had been mentioned in an earlier biography by Galileo scholar John Joseph Fahie. All of these science historians and others took the tower story to mark a turning point in the history of science: a change from reliance on authority to experimental physics. The event had become so famous that in a fresco painted in 1816 by Tuscan painter Luigi Catani, Galileo is shown performing the experiment even in the presence of the grand duke. But did this demonstration really take place?

  Most present-day historians of science think that it probably did not. The skepticism stems partly from Viviani’s known tendency for ahistorical embellishments, partly from his occasional errors in recording the chronology of events, and perhaps mostly from the fact that Galileo himself never mentions this very specific experiment in his extensive writings, nor does it appear in any other contemporary documents. In particular, philosopher Jacopo Mazzoni, who was a professor at Pisa and a friend of Galileo, published a book in 1597, in which, while he generally supported Galileo’s ideas on motion, he never mentioned an experiment by Galileo at the tower of Pisa. Similarly, Giorgio Coresio, a lecturer at Pisa who described in 1612 experiments that involved dropping objects from the top of the Tower of Pisa didn’t attribute any of those to Galileo. We should note that Coresio made the strange claim that the experiments “confirmed the statement of Aristotle… that the larger body of the same material moves more swiftly than the smaller, and in proportion as the weight increases so does the velocity.” This statement becomes especially puzzling when we realize that already in 1544, historian Benedetto Varchi mentioned experiments that had shown Aristotle’s prediction to be wrong.

  Galileo was seventy-five when Viviani came to live in his house; Viviani was eighteen, so embellishments could have come from both sides. I would argue, however, that from the perspective of appreciating Galileo’s science, it is really not very important whether he performed this particular demonstration or not. The fact remains that during his years at Pisa, Galileo embarked on serious experimentation with free-falling bodies. This stands up irrespective of whether he dropped balls from the Leaning Tower or not. In Pisa, he also started composing a treatise analyzing various aspects of motion. This monograph, De Motu Antiquiora (The Older Writings on Motion), was published only in 1687, after Galileo’s death, but its contents traces the development of his early ideas, and definitely puts Galileo (already during his early Pisan years) at the forefront of both experimental and theoretical investigations of motion in general, and of free-falling bodies in particular. In De Motu (On Motion) Galileo stated that he had confirmed by repeated experiments (without mentioning the Leaning Tower) that when two objects are dropped from a high place, the lighter one moves faster at first, but then the heavier object overtakes it and reaches the bottom first. This peculiar result has been shown by later experiments to be probably due to a nonsimultaneous release of the two objects. Basically, experiments have demonstrated that when each ball is held in one hand, the hand holding the heavier object becomes more tired, and it has to clasp the object with more force, resulting in a delayed release. Incidentally, the Flemish physicist Simon Stevin of Bruges dropped two balls of lead, one ten times the weight of the other, “from a point about 30 feet high,” some years before Galileo’s supposed Leaning Tower demonstration, and he published his results (“they landed so evenly that there seemed to be only one thump”) in 1586.

  De Motu marked the beginning of Galileo’s serious criticism of Aristotle, and it formed the basis for his subsequent experiments with balls rolling down inclined planes. It also demonstrated that science sometimes progresses incrementally rather than as a result of revolutions. While Galileo’s ideas about free-falling bodies departed significantly from those of earlier natural philosophers, at their initial stages they still did not quite square with the results of his experiments. The concepts inherited from Aristotle suggested that bodies fall at a constant speed, which is determined by the weight of the body and the resistance of the medium. To many, the fact that Aristotle had said so was sufficient to accept this as the truth. In De Motu, Galileo held that falling bodies accelerate (speed up), but only initially, and then they settle down to a constant proper speed, which is determined by the relative densities of the body and the medium. That is, he suggested that a ball made of lead moves faster [in Galileo’s words, “far out in front”] than one made of wood, but that two balls of lead fall with the same speed, no matter how much they weigh. This was a step in the right direction, but not quite correct. For instance, Galileo realized that this description did not agree with the fact that free fall appeared to be continuously accelerating, but he thought that the speeding up itself might be gradually decreasing, eventually approaching a constant speed.

  Only in his later book Discourses and Mathematical Demonstrations Concerning Two New Sciences (Discorsi) published in 1638, did Galileo arrive at a correct theory of free fall, according to which, in a vacuum, all bodies, irrespective of their weights or densities, uniformly accelerate in precisely the same way. Galileo put this explanation in the mouth of Salviati, Galileo’s alter ego in the fictional dialogue in Discorsi: “Aristotle says, ‘A hundred-pound iron ball falling from the height of a hundred braccia hits the ground before one of just one pound has descended a single braccio.’ I say that they arrive at the same time.” This crucial realization of Galileo’s—the result of a vigorous experimental effort—was an essenti
al prerequisite to Newton’s theory of gravitation.

  In modern times, in 1971 Apollo 15 astronaut David Scott dropped from the same height a hammer weighing 2.91 pounds and a feather weighing 0.066 pounds on the Moon (where there is virtually no air resistance), and the two objects struck the lunar surface simultaneously, just as Galileo had concluded centuries earlier.

  Another problem with De Motu was that Galileo’s early measurements, particularly of time, were still not sufficiently precise to allow for any definitive conclusions. He nevertheless had the foresight to make the following remark:

  When a person has discovered the truth about something and has established it with great effort, then, on viewing his discoveries more carefully, he often realizes that what he has taken such pains to find might have been perceived with the greatest ease. For truth has the property that it is not so deeply concealed as many have thought; indeed, its traces shine brightly in various places, and there are many paths by which it is approached.

  In later years, questions such as “What is truth?” and “How is truth shown?” (especially in scientific theories) were to become essential in Galileo’s life. These same questions have become perhaps even more critical today, when even indisputable facts are sometimes labeled “fake news.” It is certainly true that, at their inception, the sciences were not immune to false beliefs, since they were sometimes connected to fictitious fields such as alchemy and astrology. This was partly the reason why Galileo decided later to rely on mathematics, which appeared to provide a more secure foundation. With the development of practices that allowed experiments to be reproduced (Galileo being one of the pioneers), scientific assertions have become increasingly more reliable. Basically, for a scientific theory to become accepted, even tentatively, it needs not only to agree with all the known experimental and observational facts, but also the theory must be able to make predictions, which can then be verified by subsequent observations or experiments. Not to accept the conclusions of studies that have passed all of these rigorous tests, with the associated uncertainties being clearly stated (as in climate change models, for instance), is tantamount to playing with fire—as is literally demonstrated by the weather extremes currently occurring around the globe and causing massive fires.