by Mario Livio
Figure 12
The drama surrounding the palimpsest is only fitting for a document that gives us an unprecedented glimpse of the great geometer’s method.
The Method
When you read any book of Greek geometry, you cannot help but be impressed with the economy of style and the precision with which the theorems were stated and proved more than two millennia ago. What those books don’t normally do, however, is give you clear hints as to how those theorems were conceived in the first place. Archimedes’ exceptional document The Method partially fills in this intriguing gap—it reveals how Archimedes himself became convinced of the truth of certain theorems before he knew how to prove them. Here is part of what he wrote to the mathematician Eratosthenes of Cyrene (ca. 276–194 BC) in the introduction:
Figure 13
I will send you the proofs of these theorems in this book. Since, as I said, I know that you are diligent, an excellent teacher of philosophy, and greatly interested in any mathematical investigations that may come your way, I thought it might be appropriate to write down and set forth for you in this same book a certain special method, by means of which you will be enabled to recognize certain mathematical questions with the aid of mechanics [emphasis added]. I am convinced that this is no less useful for finding the proofs of these same theorems. For some things, which first became clear to me by the mechanical method, were afterwards proved geometrically, because their investigation by the said method does not furnish an actual demonstration. For it is easier to supply the proof when we have previously acquired, by the method, some knowledge of the questions than it is to find it without any previous knowledge.
Archimedes touches here on one of the most important points in scientific and mathematical research—it is often more difficult to discover what the important questions or theorems are than it is to find solutions to known questions or proofs to known theorems. So how did Archimedes discover some new theorems? Using his masterful understanding of mechanics, equilibrium, and the principles of the lever, he weighed in his mind solids or figures whose volumes or areas he was attempting to find against ones he already knew. After determining in this way the answer to the unknown area or volume, he found it much easier to prove geometrically the correctness of that answer. Consequently The Method starts with a number of statements on centers of gravity and only then proceeds to the geometrical propositions and their proofs.
Archimedes’ method is extraordinary in two respects. First, he has essentially introduced the concept of a thought experiment into rigorous research. The nineteenth century physicist Hans Christian Ørsted first dubbed this tool—an imaginary experiment conducted in lieu of a real one—Gedankenexperiment (in German: “an experiment conducted in the thought”). In physics, where this concept has been extremely fruitful, thought experiments are used either to provide insights prior to performing actual experiments or in cases where the real experiments cannot be carried out. Second, and more important, Archimedes freed mathematics from the somewhat artificial chains that Euclid and Plato had put on it. To these two individuals, there was one way, and one way only, to do mathematics. You had to start from the axioms and proceed by an inexorable sequence of logical steps, using well-prescribed tools. The free-spirited Archimedes, on the other hand, simply utilized every type of ammunition he could think of to formulate new problems and to solve them. He did not hesitate to explore and exploit the connections between the abstract mathematical objects (the Platonic forms) and physical reality (actual solids or flat objects) to advance his mathematics.
A final illustration that further solidifies Archimedes’ status as a magician is his anticipation of integral and differential calculus—a branch of mathematics formally developed by Newton (and independently by the German mathematician Leibniz) only at the end of the seventeenth century.
The basic idea behind the process of integration is quite simple (once it is pointed out!). Suppose that you need to determine the area of the segment of an ellipse. You could divide the area into many rectangles of equal width and sum up the areas of those rectangles (figure 14). Clearly, the more rectangles you use, the closer the sum will get to the actual area of the segment. In other words, the area of the segment is really equal to the limit that the sum of rectangles approaches as the number of rectangles increases to infinity. Finding this limit is called integration. Archimedes used some version of the method I have just described to find the volumes and surface areas of the sphere, the cone, and of ellipsoids and paraboloids (the solids you get when you revolve ellipses or parabolas about their axes).
In differential calculus, one of the main goals is to find the slope of a straight line that is tangent to a curve at a given point, that is, the line that touches the curve only at that point. Archimedes solved this problem for the special case of a spiral, thereby peeping into the future work of Newton and Leibniz. Today, the areas of differential and integral calculus and their daughter branches form the basis on which most mathematical models are built, be it in physics, engineering, economics, or population dynamics.
Archimedes changed the world of mathematics and its perceived relation to the cosmos in a profound way. By displaying an astounding combination of theoretical and practical interests, he provided the first empirical, rather than mythical, evidence for an apparent mathematical design of nature. The perception of mathematics being the language of the universe, and therefore the concept of God as a mathematician, was born in Archimedes’ work. Still, there was something that Archimedes did not do—he never discussed the limitations of his mathematical models when applied to actual physical circumstances. His theoretical discussions of levers, for instance, assumed that they were infinitely rigid and that rods had no weight. Consequently, he opened the door, to some extent, to the “saving the appearances” interpretation of mathematical models. This was the notion that mathematical models may only represent what is observed by humans, rather than describing the actual, true, physical reality. The Greek mathematician Geminus (ca. 10 BC–AD 60) was the first to discuss in some detail the difference between mathematical modeling and physical explanations in relation to the motion of celestial bodies. He distinguished between astronomers (or mathematicians), who, according to him, had only to suggest models that would reproduce the observed motions in the heavens, and physicists, who had to find explanations for the real motions. This particular distinction was going to come to a dramatic head at the time of Galileo, and I will return to it later in this chapter.
Figure 14
Somewhat surprisingly perhaps, Archimedes himself considered as one of his most cherished accomplishments the discovery that the volume of a sphere inscribed in a cylinder (figure 15) is always 2/3 of the volume of the cylinder. He was so pleased with this result that he requested it be engraved on his tombstone. Some 137 years after Archimedes’ death, the famous Roman orator Marcus Tullius Cicero (ca. 106–43 BC) discovered the great mathematician’s grave. Here is Cicero’s rather moving description of the event:
Figure 15
When I was a quaestor in Sicily I managed to track down his [Archimedes’] grave. The Syracusans knew nothing about it, and indeed denied that any such thing existed. But there it was, completely surrounded and hidden by bushes of brambles and thorns. I remembered having heard of some simple lines of verse which had been inscribed on his tomb, referring to a sphere and a cylinder modeled in stone on top of the grave. And so I took a good look around all the numerous tombs that stand beside the Agrigentine Gate. Finally I noted a little column just visible above the scrub: it was surmounted by a sphere and a cylinder. I immediately said to the Syracusans, some of whose leading citizens were with me at the time, that I believed this was the very object I had been looking for. Men were sent in with sickles to clear the site, and when the path to the monument had been opened we walked right up to it. And the verses were still visible, though approximately the second half of each line had been worn away. So one of the most famous cities in the Greek world,
and in former days a great center of learning as well, would have remained in total ignorance of the tomb of the most brilliant citizen it had ever produced, had a man from Arpinum not come and pointed it out!
Cicero did not exaggerate in describing Archimedes’ greatness. In fact, I have deliberately put the bar for the title of “magician” so high that progressing from the giant Archimedes, we have to leap forward no fewer than about eighteen centuries before encountering a man of similar stature. Unlike Archimedes, who said he could move the Earth, this magician insisted that the Earth was already moving.
Archimedes’ Best Student
Galileo Galilei (figure 16) was born in Pisa on February 15, 1564. His father, Vincenzo, was a musician, and his mother, Giulia Ammannati, was a witty, if rather ill-disposed woman who couldn’t tolerate stupidity. In 1581, Galileo followed his father’s advice and enrolled in the faculty of arts of the University of Pisa to study medicine. His interest in medicine, however, withered almost as soon as he got in, in favor of mathematics. Consequently, during the summer vacation of 1583, Galileo persuaded the mathematician of the Tuscan Court, Ostilio Ricci (1540–1603), to meet with his father and to convince the latter that Galileo was destined to become a mathematician. The question was indeed settled soon thereafter—the enthusiastic youth became absolutely bewitched by the works of Archimedes: “Those who read his works,” he wrote, “realize only too clearly how inferior are all other minds compared with Archimedes’, and what small hope is left of ever discovering things similar to the ones he discovered.” At the time, little did Galileo know that he himself possessed one of those few minds that were not inferior to that of the Greek master. Inspired by the legendary story of Archimedes and the king’s wreath, Galileo published in 1586 a small book entitled The Little Balance, about a hydrostatic balance he had invented. He later made further reference to Archimedes in a literary lecture at the Florence Academy, in which he discussed a rather unusual topic—the location and size of hell in Dante’s epic poem Inferno.
Figure 16
In 1589 Galileo was appointed to the chair of mathematics at the University of Pisa, partly because of the strong recommendation of Christopher Clavius (1538–1612), a respected mathematician and astronomer from Rome, whom Galileo visited in 1587. The young mathematician’s star was now definitely on the rise. Galileo spent the next three years setting forth his first thoughts on the theory of motion. These essays, which were clearly stimulated by Archimedes’ work, contain a fascinating mixture of interesting ideas and false assertions. For instance, together with the pioneering realization that one can test theories about falling bodies using an inclined plane to slow down the motion, Galileo incorrectly states that when bodies are dropped from towers, “wood moves more swiftly than lead at the beginning of its motion.” Galileo’s inclinations and general thought process during this phase of his life have been somewhat misrepresented by his first biographer, Vincenzio Viviani (1622–1703). Viviani created the popular image of a meticulous, hard-nosed experimentalist who gained new insights exclusively from detailed observations of natural phenomena. In fact, until 1592 when he moved to Padua, Galileo’s orientation and methodology were primarily mathematical. He relied mostly on thought experiments and on an Archimedean description of the world in terms of geometrical figures that obeyed mathematical laws. His chief complaint against Aristotle at that time was that the latter “was ignorant not only of the profound and more abstruse discoveries of geometry, but even of the most elementary principles of this science.” Galileo also thought that Aristotle relied too heavily on sensory experiences, “because they offer at first sight some appearance of truth.” Instead, Galileo proposed “to employ reasoning at all times rather than examples (for we seek the causes of effects, and these are not revealed by experience).”
Galileo’s father died in 1591, prompting the young man, who had now to support his family, to take an appointment in Padua, where his salary was tripled. The next eighteen years proved to be the happiest in Galileo’s life. In Padua he also began his long-term relationship with Marina Gamba, whom he never married, but who bore him three children—Virginia, Livia, and Vincenzio.
On August 4, 1597, Galileo wrote a personal letter to the great German astronomer Johannes Kepler in which he admitted that he had been a Copernican “for a long time,” adding that he found in the Copernican heliocentric model a way to explain a number of natural events that could not be explained by the geocentric doctrine. He lamented the fact, however, that Copernicus “appeared to be ridiculed and hissed off the stage.” This letter marked the widening of the momentous rift between Galileo and the Aristotelian cosmology. Modern astrophysics was starting to take shape.
The Celestial Messenger
On the evening of October 9, 1604, astronomers in Verona, Rome, and Padua were startled to discover a new star that rapidly became brighter than all the stars in the sky. The meteorologist Jan Brunowski, an imperial official in Prague, also saw it on October 10, and in acute agitation he immediately informed Kepler. Clouds prevented Kepler from observing the star until October 17, but once he started, he continued to record his observations for a period of about a year, and he eventually published a book about the “new star” in 1606. Today we know that the 1604 celestial spectacle did not mark the birth of a new star, but rather the explosive death of an old one. This event, now called Kepler’s supernova, caused quite a sensation in Padua. Galileo managed to see the new star with his own eyes late in October 1604, and the following December and January he gave three public lectures on the subject to large audiences. Appealing to knowledge over superstition, Galileo showed that the absence of any apparent shift (parallax) in the new star’s position (against the background of the fixed stars) demonstrated that the new star had to be located beyond the lunar region. The significance of this observation was enormous. In the Aristotelian world, all changes in the heavens were restricted to this side of the Moon, while the far more distant sphere of the fixed stars was assumed to be inviolable and immune to change.
The shattering of the immutable sphere had started already in 1572, when the Danish astronomer Tycho Brahe (1546–1601) observed another stellar explosion now known as Tycho’s supernova. The 1604 event put yet another nail in the coffin of Aristotle’s cosmology. But the true breakthrough in the understanding of the cosmos didn’t descend from either the realm of theoretical speculation or from naked-eye observations. It was rather the outcome of simple experimentation with convex (bulging outward) and concave (curving inward) glass lenses—hold the right two of those some thirteen inches apart and distant objects suddenly appear closer. By 1608, such spyglasses started to appear all over Europe, and one Dutch and two Flemish spectacle makers even applied for patents on them. Rumors of the miraculous instrument reached the Venetian theologian Paolo Sarpi, who informed Galileo around May of 1609. Anxious to confirm the information, Sarpi also wrote to a friend in Paris, Jacques Badovere, to inquire whether the rumors were true. According to his own testimony, Galileo was “seized with a desire for the beautiful thing.” He later described these events in his book The Sidereal Messenger, which appeared in March 1610:
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.
Galileo demonstrates here the same type of creatively practical thinking that characterized Archimedes—once he knew that a telescope could be built, it didn’t take him long to figure out how to build one himself. Moreover, between August 1609 a
nd March 1610, Galileo used his inventiveness to improve his telescope from a device that brought objects eight times closer to an instrument with a power of twenty. This was a considerable technical feat in itself, but Galileo’s greatness was about to be revealed not in his practical know-how, but in the use to which he put his vision-enhancing tube (which he called a perspicillum). Instead of spying on distant ships from Venice’s harbor, or examining the rooftops of Padua, Galileo pointed his telescope to the heavens. What followed was something unprecedented in scientific history. As the historian of science Noel Swerdlow puts it: “In about two months, December and January [1609 and 1610, respectively], he made more discoveries that changed the world than anyone has ever made before or since.” In fact, the year 2009 has been named the International Year of Astronomy to mark the four hundredth anniversary of Galileo’s first observations. What did Galileo actually do to become such a larger-than-life scientific hero? Here are only a few of his surprising achievements with the telescope.
Turning his telescope to the Moon and examining in particular the terminator—the line dividing the dark and illuminated parts—Galileo found that this celestial body had a rough surface, with mountains, craters, and vast plains. He watched how bright points of light appeared in the side veiled in darkness, and how these pinpoints widened and spread just like the light of the rising sun catching on mountaintops. He even used the geometry of the illumination to determine the height of one mountain, which turned out to be more than four miles. But this was not all. Galileo saw that the dark part of the Moon (in its crescent phase) is also faintly illuminated, and he concluded that this was due to reflected sunlight from the Earth. Just as the Earth is lit by the full Moon, Galileo asserted, the lunar surface bathes in reflected light from Earth.