by Mario Livio
The first part of this statement is Galileo’s discovered law: the distance that a free-falling body travels is proportional to the square of the time of travel. That is, a body that free-falls for two seconds (from rest) travels a distance that is four times (2 squared) longer than one that free-falls for one second. In three seconds, a free-falling body covers a distance that is nine times (3 squared) that of a body that falls for one second, and so on. The second statement in Galileo’s letter is, in fact, a direct consequence of the first. Imagine that we’ll call the distance traveled during the first second of fall “1 Galileo”; then the distance covered during the following one second will be the difference between 4 Galileos (the distance covered in two seconds) and 1 Galileo (the distance covered in the first second), which is 3 Galileos. Similarly, the distance through which the body will fall in the third second will be 9 Galileos minus 4 Galileos, which leaves 5 Galileos. Consequently, the distances passed during successive periods of one second will form the odd number sequence of 1, 3, 5, 7… Galileos.
The last statement in the quote from Galileo’s letter to Sarpi was actually incorrect. In 1604 Galileo still thought that the speed of a free-falling body increases in proportion to the distance from the point from which the free fall had started. Only much later did Galileo realize that in free fall, the speed increases in direct proportion to the time of fall and not to the distance. That is, the speed of an object that has been free-falling for five seconds is five times the speed of one that has been falling for only one second. In his later Two New Sciences, he therefore correctly asserted: “Uniformly accelerated motion I call that to which, commencing from rest, equal velocities are added in equal times.”
The importance of these discoveries for the history of science cannot be overemphasized. Whereas in the physics of Aristotle there were elements (such as earth and water) whose “natural motions” were supposed to be downward, Aristotle’s theory contained also elements (such as fire) whose “natural motions” were upward, and air, whose natural motion depended on its location or surroundings. To Galileo, the only natural motion on Earth was downward (that is, toward the Earth’s center), and it applied to all bodies. Entities that had been observed to float upward (such as air bubbles in water) did so only because of the lift force exerted on them by a medium of a higher density, as explained by the laws of hydrostatics originally formulated by Archimedes. We can recognize in these ideas some of the ingredients of Newton’s theory of gravitation. What Galileo did not have an answer to—nor did he even try to answer—was why bodies fall at all. This was left for Newton. Galileo concentrated instead on discovering the “law,” or what he regarded as the essence of free fall, rather than on causal explanations for free fall.
There was another aspect in which Galileo’s ideas differed fundamentally from those of Aristotle. The Greek philosopher’s theory of motion had never been put to any serious experimental tests, partly because of his (and Plato’s) conviction that the correct way to discover truths about nature was to think about them rather than to perform experiments. To Aristotle, the only possible way to understand phenomena was to know their purpose. Galileo, on the other hand, employed a clever combination of experimentation and reasoning. He realized relatively early that progress is often achieved through correct decisions as to which questions should be asked, and also via studying artificial circumstances (as in the case of balls rolling down inclined planes) instead of examining only natural motions. This truly marked the birth of modern experimental physics.
Two elements in particular stand out as revolutionary in Galileo’s new theory of motion: First, the universality of the law, which applies to all bodies in accelerated motion. Second, the extension of the formulation of mathematical laws from ones describing only static configurations that don’t involve motion, as in Archimedes’s law of the lever, to motion and dynamical situations.
A CONVERSION
Another facet of Galileo’s years in Padua was the most significant for his future. Whereas many of his fruitful investigations were indeed in mechanics, the most important revision in his outlook on science was in astronomy. As noted earlier, in a publication entitled Treatise on the Sphere, or Cosmography (probably written in the late 1580s), Galileo still described and appeared to follow in detail the Ptolemaic geocentric system, without even mentioning the Copernican heliocentric model. This book probably reflected the requirements imposed by the university’s curriculum, and it was used primarily for tutoring students. Two letters written in 1597, however, in which Galileo expresses for the first time his increasing conviction in Copernicanism, provide evidence for a radical change in his views.
The first letter, dated May 30, 1597, was addressed to Jacopo Mazzoni, a philosopher and former colleague of Galileo’s at Pisa. Mazzoni had just published a book entitled On Comparing Aristotle and Plato, in which he argued that he had found proof that the Earth did not revolve around the Sun, thus invalidating the Copernican scenario. The argument was based on Aristotle’s assertion that the top of Mount Caucasus, at the intersection of Europe and Asia, was illuminated by the Sun for a full one-third of the night. From this assumption, Mazzoni concluded incorrectly that since in the Copernican model an observer at the mountain’s top (when the mountain was on the side of the Earth not facing the Sun) would be farther from the center of the world (the Sun), than in the Ptolemaic model (where the center of the world was assumed to be the Earth’s center), the horizon of a Copernican observer should span much more than 180 degrees, contrary to experience. In his letter to Mazzoni, Galileo used precise trigonometry to show that the motion of the Earth around the Sun would not result in any detectable change in the visible portion of the celestial vault. Then, after refuting this presumed challenge to Copernicus, Galileo added a critical statement, saying that he “held [the Copernican model] to be much more probable than the opinion of Aristotle and Ptolemy.”
Galileo’s second letter was even clearer in expressing his views on Copernicanism. It came on the heels of a publication by Johannes Kepler. The great German astronomer is best remembered today for three laws of planetary motion that bear his name, which served as an impetus for Newton’s theory of universal gravitation. Kepler was an accomplished mathematician, a speculative metaphysicist, and a prolific author. As a child, he was inspired by the spectacle offered by the comet of 1577. After studying mathematics and theology at the University of Tübingen, he was introduced to the theory of Copernicus by mathematician Michael Mästlin. Kepler seems to have been immediately convinced by the Copernican system, partly perhaps because the idea of a central Sun surrounded by the fixed stars with a space between the Sun and the stars appealed to his profound religious beliefs. He thought that the universe reflects its creator, with the unity of the Sun, the stars, and the intervening space symbolizing the Holy Trinity.
In 1596 Kepler published a book known as Mysterium Cosmographicum (Cosmic Mystery), in which he proposed that the structure of the solar system is based on the five regular solids known as the Platonic solids—the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron—being embedded one within the other. Since the five solids and the sphere of the fixed stars created precisely six spaces, Kepler thought that his model explained why there were six planets. (Only six were known at that time.) While the model itself was quite crazy, Kepler did assert in his book the Copernican view that all the planets orbited the Sun. His mistake was not in the details of the model but, rather, in his assumption that the number of planets and their orbits were some fundamental quantities that had to be explained from first principles. Today we know that the orbits of planets are just accidental results of the conditions that happened to prevail in the protosolar nebula.
The two copies of Kepler’s book intended for astronomers in Italy somehow landed on Galileo’s desk. On August 4, 1597, after having read only the preface, Galileo sent Kepler a letter in which he stated that he believed the Copernican model to be correct. He went eve
n further, saying that he had been a Copernican “for several years,” and adding that he found in the Copernican model a way to explain a number of natural phenomena that couldn’t be explained by the geocentric scenario. But, he admitted, he had “dared not publish” any of those theories, as he had been deterred by the fact that Copernicus “appeared to be ridiculed and hissed off the stage.”
In Kepler’s reply of October 13, 1597, he urged Galileo to hasten and publish those explanations supporting the Copernican model, if not in Italy, then in Germany. Such a publication, however, didn’t materialize. Since Galileo was never particularly shy or hesitant when it came to publishing what he regarded as solid results, this lack of action on his part suggests that at the time, before he made any observations with the telescope, Galileo might have had nothing more than hunches stimulated perhaps by his discoveries in mechanics. He was probably also thinking already about an explanation for sea tides, which he was later to develop into one of his main arguments for the motion of the Earth. As with the case of Mazzoni, those clues could have included Galileo’s intuitive feeling that a few of the objections raised with regard to the motion of the Earth could be refuted. It is also possible, however, that Galileo’s passiveness was political—a consequence of the fact that at that stage of his career, with Europe immersed in the Counter-Reformation era, he was somewhat reluctant to appear in Catholic Italy as an ally of Kepler, a known Lutheran.
An occurrence in the autumn of 1604 gave Galileo an opportunity to present publicly, if not quite a Copernican view, then at least a clear anti-Aristotelian position. On October 9 astronomers in a few Italian cities were startled to discover a new star—a nova—that rapidly became brighter than all the stars in the sky. Meteorologist Jan Brunowski observed it on October 10 and informed Kepler, who embarked on continuous, fruitful observations that lasted for almost a year. (That is why the object is known today as Kepler’s supernova.) Baldessar Capra, who a few years later would have the dispute with Galileo over the compass/calculator, noticed the new star together with his tutor Simon Mayr and a friend Camillo Sasso, on October 10. The Italian friar and astronomer Ilario Altobelli informed Galileo, who first observed it in late October and then gave three lectures about the nova to huge audiences sometime between November and January. Galileo’s main point was simple: since no displacement or shift had been observed in the position of the new star against the background of the distant stars—a phenomenon known as parallax—the star had to be farther than the Moon. However, that region, according to Aristotle, was supposed to be inviolable and immune to change. Therefore, the new star (which, by the way, we know today represented the explosive death of an old star, a phenomenon known as a supernova) contributed to shattering Aristotle’s conception of an immutable stellar sphere.
That imaginary sphere had started cracking already in 1572, when the Danish astronomer Tycho Brahe discovered another “new” star—also an exploding, dying star known now as Tycho’s Supernova. Somewhat unfortunately, perhaps, Galileo added another element to his “explanation” of the nova, which was completely wrong. He suggested that the new star represented a reflection of sunlight by “a large amount of vapor” ejected from Earth and projected past the Moon’s orbit. If true, this would have dealt an even more fatal blow to the Aristotelian distinction between the degradable terrestrial matter and the eternally incorruptible celestial stuff, but in reality, this fanciful, supplemental idea was totally unnecessary, and Galileo himself had doubts about it.
Not everybody agreed that the new star all but destroyed the Aristotelian cosmos. It often takes more than one or two observations, regardless of how convincing they might be, to persuade people to abandon beliefs cherished for centuries. A few didn’t even believe the nova to be located in Aristotle’s presumed pristine celestial quintessence, mistrusting the parallax measurements. Others, such as the authoritative Jesuit mathematician and astronomer Christopher Clavius, confirmed the null parallax determination—that is, no shift having been observed—but refused to accept its implications as compelling. Still others, such as the Florentine philosopher Lodovico delle Colombe, with whom Galileo was to have grave disputes in later years, came up with alternative explanations for the nova’s appearance. Wanting to preserve the incorruptibility of the heavens, delle Colombe suggested that the nova was not really a new star or an intrinsic change in the brightness of a star, but, rather, only a newly observable star. That is, a star becoming visible because of a swelling in the heavenly material acting like a lens. Galileo bothered to answer only a few of the critics, deeming the others unworthy of a response. In one case, his answer was cast in sarcastic dialogue form, which he composed with friends and published under a pseudonym.
Overall, the superb results in mechanics, the contemplation of new theoretical vistas in astronomy, and the artistic and free-spirited allure of Venice made life in Padua very enjoyable for Galileo. However, his financial troubles, which forced him to undertake a burdensome load of teaching, apparently weighed heavily on his mind. The difficulties and stress eventually caused him to start seeking better-paying opportunities with individual patrons as opposed to universities. Later, he candidly explained his motivation for moving from Padua in two letters written in 1609 and 1610:
Greater leisure than I have here, I do not believe I could have elsewhere so long as I am forced to derive the sustenance of my household from public and private teachings.… To obtain any salary from a Republic, however splendid and generous, without rendering public service, is not possible, since to draw benefits from the public it is necessary to satisfy the public. In a word, I cannot hope for such benefits from anybody but an absolute ruler.… Hence… I desire that the primary intention of His Highness shall be to give me ease and leisure to bring my works to a conclusion without my being occupied in teaching.
Galileo indeed moved to Florence in September 1610 at the invitation of the Grand Duke Cosimo II de’ Medici of Tuscany, but not before he manufactured the instrument with which he was about to produce his most breathtaking discoveries. His confidants in Venice regarded the trading of intellectual freedom (which Galileo had in abundance in Padua) for financial stability and release from teaching weariness as a grave mistake. History has shown that even the long hand of the Inquisition had rarely reached the Republic of Venice in any consequential way, while a move to Florence made Galileo vulnerable to control by the Church. Knowing what we know today about Galileo’s fate, we therefore have to conclude that Galileo’s Venetian friends were absolutely right. Intellectual freedom is indeed invaluable. This is especially important today, when truth and facts appear to be under siege.
CHAPTER 4 A Copernican
If until 1609 Galileo’s experiments concentrated on objects falling downward toward the Earth’s center, starting that year he turned his attention upward to the heavens. Here is how that celestial adventure unfolded. In late 1608 Galileo’s Venetian friend Paolo Sarpi heard a rumor about a spyglass—an optical gadget invented in the Netherlands—that could make distant objects appear closer and larger. Realizing that such an instrument could have interesting applications, Sarpi alerted Galileo in 1609. Around the same time, he also wrote to a friend in Paris to inquire whether the rumor was true.
In his publication The Sidereal Messenger, Galileo described the circumstances:
About 10 months ago, a report reached my ears that a certain Fleming had constructed a spyglass by means of which visible objects, though very distant from the eye of the observer, were distinctly seen as if nearby. Of this truly remarkable effect, several experiences were related, to which some persons gave credence while others denied them. A few days later, the report was confirmed to me in a letter from a noble Frenchman at Paris, Jacques Badovere, which caused me to apply myself wholeheartedly to investigate means by which I might arrive at the invention of a similar instrument. This I did soon afterwards, my basis being the doctrine of refraction.
The last sentence in this description could be a bit
misleading, since it gives the impression that Galileo was guided by the theoretical principles of optics, a topic in which his knowledge was, in fact, rather scanty. In reality, Galileo’s approach was much more experimental. He discovered through trial and error that by placing spectacle lenses in a tube, a plano-concave one at one end and a plano-convex one at the other, he could easily achieve a magnification of about three or four. Since Venice was an aspiring maritime power, Galileo immediately realized the bargaining potential that such a device (in his words: “of inestimable value in all business and every undertaking at sea or on land”) could give him in salary negotiations with Venetian senators. He therefore swiftly embarked on learning how to polish higher quality lenses, and experimented with lenses of different sizes and distances apart. Amazingly, within less than three weeks, he was in Venice, equipped with an eight-powered telescope and, through Sarpi’s connections, about to demonstrate this “perspicillum,” as he called it, to Venetian decision-makers.
The ability to spot distant ships long before they could be seen with the naked eye sufficiently impressed the senators, who initially agreed to increase Galileo’s salary from 520 to 1,000 scudi per year. To his disappointment, however, once the senate realized that the telescope was not an exclusive Galileo invention (even though he never claimed it to be), but, rather, a device already known elsewhere on the Continent, the salary increase was limited to one year, after which it was to be frozen. Furious about this turn of events, as well as about the fact that the senators did not seem to appreciate that his telescope was far superior to those that were circulating in Europe at the time, Galileo sent a telescope to the Grand Duke of Tuscany, Cosimo II de’ Medici, in the hope of securing a court appointment in Florence. This might seem like a long shot on his part, but Galileo did have reasons for optimism. He had been Cosimo’s tutor in mathematics during some summers between 1605 and 1608, and it was Cosimo’s father, Ferdinando I de’ Medici, who appointed Galileo to the professorship of mathematics at the University of Pisa in 1589.