Galileo at the height of his fame. Portrait by Ottavio Leoni (1578–1630). (RMN-Grand Palais / Art Resource, NY)
Galileo was treading on dangerous ground. He was not only challenging the authority of Aristotle, held dear by the Church theologians, but also going against the clear meaning of Scripture, which in several places implied that the sun revolves around the Earth. A more squeamish soul might have steered clear of such a potentially explosive issue, but Galileo was anything but. Instead of waiting for the attacks of his adversaries, he decided to take the battle onto their home ground by publishing his own theological treatise. The “Letter to the Grand Duchess Christina” was addressed to Christina of Lorraine, mother of the ruling grand duke of Tuscany, who had expressed her concerns to Galileo that his system was inconsistent with the revealed word of God. Circulated in 1615, but published only many years later, the letter contains Galileo’s response, which became known as the doctrine of “the two books.” The book of nature, he reasoned, and the book of Scripture can never be in conflict. One contains all we see around us in the world, the other contains divine revelation, but both are ultimately derived from the same source: God Himself. Therefore, if there appears to be conflict between the two, the only possible explanation is that we do not properly understand the one or the other.
As long as we do not have scientific “proof” of a particular thesis, Galileo conceded, we should always accept the authority of Scripture, understood in its most simple and direct meaning. But if we do possess a scientific proof, then the roles are reversed, and Scripture must be reinterpreted to accord with the book of nature. Otherwise, Galileo warned, we will be required to believe something that is manifestly false, bringing ridicule and discredit upon the Church. This, Galileo insisted, was precisely why the Church should accept Copernicanism. He could prove, he insisted, that Earth and the planets do indeed revolve around the sun, and the Church would only discredit itself by contradicting manifest truth. The traditional understanding of Scripture must be replaced with interpretations that are consistent with scientific truth, Galileo argued, and included his own readings of critical biblical passages to show that they were perfectly consistent with Copernicanism.
Beautifully written and eminently persuasive, the “Letter to the Grand Duchess Christina” is a cogent defense not just of Copernicanism, but of the compatibility of faith and free scientific research. But seventeenth-century religious authorities were not inclined to look kindly on an uninvited intrusion onto their turf. Granted, Galileo was a gifted astronomer, but he had no business pronouncing on theology, a field in which he was strictly an amateur. The task of reminding the interloper of his place fell to Clavius’s old friend, the venerable Jesuit theologian cardinal Robert Bellarmine. In April of 1615, the cardinal issued an opinion on a work by one of Galileo’s ardent followers, the Carmelite monk Paolo Foscarini. Although nominally addressed to Foscarini, the opinion was clearly intended to put Galileo on notice. If there were scientific proof of Copernicanism, Bellarmine conceded in his letter, then passages in Scripture should be reconsidered, since “we should rather have to say that we do not understand them than to say something is false which had been proven.” But since no such proof “has been shown to me,” he continued, one must stick to the manifest meaning of Scripture and the “common agreement of the holy fathers.” All of these agreed that the sun revolves around the Earth.
Bellarmine undoubtedly had a point. Galileo could and did bring up many strong arguments in support of the Copernican system, but despite his brave proclamations, he could not actually prove it. His supposed “proof,” based on the ebb and flow of the tides, was weak and, as some contemporaries pointed out, deeply flawed. In the absence of proof, Bellarmine’s insistence that Scripture be taken at face value seems eminently reasonable. He furthermore did not prohibit Galileo from studying the Copernican system as a hypothesis that fit well with observations. He only insisted that Galileo not hold that Copernicanism was, in fact, true, and that it described the actual motions of the sun and planets.
Less than a year after Bellarmine wrote the letter, his opinion became the official Church position, putting strict limits on Galileo’s ability to advocate for Copernicanism. But in 1616 the Church was not ready to give up on its erstwhile hero, who was not only the most celebrated scientist in Europe but also a good Catholic. In a sign of the high esteem in which he was held in Rome, Galileo was granted an interview with Pope Paul V, who assured him of his goodwill, and with Bellarmine, who explained the terms of the ban and confirmed them in writing. It was Galileo’s alleged violation of this injunction that brought him before the Inquisition sixteen years later.
In the years that followed, Galileo had to all appearances put this distressing affair behind him. He was still a famous man, admired by scientists and laymen alike, and secure in his position in the Medici court. His skirmishes with Church authorities made him suspect in some circles, the Jesuits above all. But they also made him a hero of more liberal segments of Italian society, who resented the Church’s insistence that it was the arbiter of all truth, and resented even more the domineering ways of the Jesuits. The bastion of this “liberal party” was the Accademia dei Lincei (Academy of the Lynx-Eyed) in Rome, of which Galileo was the most illustrious member. Founded in 1603 by the aristocratic Federico Cesi, the academy was a gathering place for some of the most brilliant intellectuals in Rome, both ecclesiastics and laymen. During the troubled years of 1615–16, the Linceans stood shoulder to shoulder with Galileo, and their support no doubt helped him get off as lightly as he did. They would prove just as important in the following years, as Galileo began once again to express forbidden views of the heavens.
PARADOXES AND INFINITESIMALS
In 1621, Galileo, still cautious, was likely pleased to receive an inquiry about a seemingly safe mathematical topic. Suppose, Cavalieri suggested, that we have a plane figure and we draw a straight line inside it, and suppose, furthermore, that we then draw all the possible lines inside the figure that are parallel to the first. “In that case,” he writes, “I call the lines so drawn ‘all the lines’ of that plane figure. Similarly, given a three-dimensional solid, all the possible planes inside a solid that are parallel to a given plane are ‘all the planes’ of that solid.” Is it permissible, he inquires of Galileo, to equate the plane figure with “all the lines” of the figure and the solid with “all the planes” of the solid? Furthermore, if there are two figures, is it permissible to compare “all the lines” of one with “all the lines” of the other, or “all the planes” of one with “all the planes” of the other?
Cavalieri’s question seems simple, but it goes straight to the paradoxical heart of the infinitely small. On an intuitive level, the plane does indeed seem to be composed of parallel lines, and a solid appears to be composed of parallel planes. But as Cavalieri notes in his letter, we can draw an infinite number of parallel lines through any figure, and an infinite number of planes through any solid, which means that the number of “all the lines” or “all the planes” is always infinite. Now, if each of the lines has a positive width, however small, then an infinite number of them will add up to an infinitely large figure—not to the one we started out with. But if the lines have no width (or zero width), then any accumulation of them, no matter how large, will still have zero width and zero magnitude, and we are left with no figure at all. The same applies to “all the planes” of a three-dimensional solid: if they have a thickness, however small, they will inevitably combine to a solid of infinite size; but if they have no thickness, then any accumulation of them will always add up to zero.
It is the old question of the composition of the continuum that had confounded philosophers and mathematicians since the days of Pythagoras and Zeno. And to this familiar if troublesome one, Cavalieri now added another query: is it allowable to compare “all the lines” of one figure with “all the lines” of another? This, he notes in his letter, involves comparing one infinity with another,
a move that was strictly forbidden by the traditional rules of mathematics. This is because, according to the “axiom of Archimedes,” two magnitudes have a ratio if and only if one can multiply the smaller magnitude so many times that it will be bigger than the larger magnitude. This, however, is not the case with infinities, since, however many times one might multiply infinity, one will always arrive at the same unchanging result: infinity.
Unfortunately we do not have Galileo’s response to his young colleague, since only one side of the correspondence has survived. Letters from Cavalieri in the following months suggest that Galileo, at the very least, encouraged Cavalieri to continue his investigations. This is likely what Cavalieri had expected, since Galileo was already reputed to hold unorthodox views on the composition of the continuum. As early as 1604, while working out the law of falling bodies, he had experimented with the notion that the surface area of a triangle, representing the distance traveled by a body, was composed of an infinite number of parallel lines, each representing the body’s speed at a given instant. Some years later, in 1610, Galileo was still occupied with the paradoxes of the continuum, and announced his intention to devote an entire book to the matter. The book never materialized, probably because of the dramatic events that recast his life in those years, but three decades later he offered a fairly detailed exposition of his views in his last great work, Discourses and Mathematical Demonstrations Relating to Two New Sciences, considered by many today to contain his most important scientific contributions. The Discourses, as the work came to be known, was written in Galileo’s villa in Arcetri, outside Florence, during the long years of house arrest that followed his condemnation by the Inquisition in 1633. Although published in Holland in 1638, it is based on studies that Galileo conducted many decades before, when he was a professor at the universities of Pisa and Padua.
The Discourses is written as a conversation between three friends, Salviati, Sagredo, and Simplicio, who would have been familiar to the book’s readers. Only a few years before, the same trio had starred in the Dialogue on the Two Chief World Systems, Galileo’s immensely popular book on the Copernican system and the work that had led to his trial and condemnation by the Inquisition. Although much of the sparkle and wit that characterized Dialogue is absent from the Discourses, the three friends retain their former roles: Salviati as Galileo’s spokesman, Simplicio as the voice of Galileo’s outdated Aristotelian critics, and Sagredo as the wise arbiter, who regularly sides with Salviati. In the first of the four days of the dialogue, the three friends discuss the question of cohesion: what is it that holds materials together, and prevents them from breaking up under outside pressures? Salviati begins by discussing ropes, showing that their strength is due to the fact that they are composed of a large number of threads packed and twisted together. He then extends his discussion to wood, whose inner strength, he argues, is also due to the fact that it is composed of tightly packed fibers. But what, he asks, of other materials, such as marble or metals? What force is it that holds them together with such remarkable strength?
The answer, according to Salviati, is “horror vacui”—nature’s abhorrence of a vacuum. We know from experience, he argues, that horror vacui is an extremely powerful force: two perfectly smooth surfaces of marble or metal can hardly be separated, since pulling them apart would produce a momentary vacuum. This powerful force, he continues, is active not only between bodies, but also inside each body, holding it together. Just as a rope is composed of separate threads, and wood of separate fibers, a block of marble or a sheet of metal is also composed of innumerable atoms arranged side by side. There is, however, this difference: whereas a rope is composed of a large but finite number of threads, and a piece of wood of an even larger but still finite number of fibers, a block of marble or a sheet of metal is composed of an infinite number of infinitely small atoms, or “indivisibles.” Separating them is an infinite number of infinitely small empty spaces. The vacuum in these infinite spaces is the glue that holds the object together and is responsible for its internal strength.
This was Salviati’s (Galileo’s) theory of matter, and as he himself admitted, it was a difficult one. “What a sea we are slipping into without knowing it!” Salviati exclaims at one point. “With vacua, and infinities, and indivisibles … shall we ever be able, even by means of a thousand discussions, to reach dry land?” Indeed, can a finite amount of material be composed of an infinite number of atoms and an infinite number of empty spaces? To prove his point that it could, he turned to mathematics.
Figure 3.1. The paradox of Aristotle’s wheel. From Galileo, Discourses, Day 1 (Le Opere di Galileo Galilei, vol. 8, Edizione Nazionale [Florence: G. Barbera, 1898], p. 68)
Salviati investigated the question of the continuum by way of a medieval paradox known as Aristotle’s wheel, although it had nothing to do with that ancient philosopher, and its revelation was anything but Aristotelian. Imagine, Salviati suggests to his friends, a hexagon ABCDEF and a smaller hexagon HIJKLM within it and concentric with it, both around the center G. And suppose, furthermore, that we extend the side AB of the large hexagon to a straight line AS, and the parallel side of the smaller hexagon into the parallel line HT. Next we rotate the large hexagon around the point B, so that the side BC comes to rest on the segment BQ of the line AS. When this happens the smaller hexagon will also rotate until the side IK comes to rest on the segment OP of the line HT. There is a difference, Salviati points out, between the line created by the rotating large hexagon and the line created by the smaller hexagon: the larger hexagon is creating a continuous line, because the segment BQ is placed right next to the segment AB. The smaller hexagon’s line, however, has gaps, because between the segments HI and OP there is a space IO where the hexagon in its rotation never touches the line. If we complete a full rotation of the large hexagon along the line AS, it will create a continuous segment whose length is equal to the hexagon’s perimeter. At the same time, the smaller hexagon will travel a distance approximately equal in length along the line HT, but the line it creates will not be continuous: it will be composed of the six sides of the hexagon, with six equal gaps between them.
Now, what is true of hexagons, according to Salviati, is true of any polygon, even one with 100,000 sides. Rolling it along will create a straight line equal in length to its circumference, whereas a smaller but similar polygon inside it will trace a line of equal length, but composed of 100,000 segments interspersed with 100,000 empty spaces. But what will happen if we replace those finite polygons with a polygon with an infinite number of sides—in other words, a circle? As the lower part of Aristotle’s wheel shows, rolling the circle one full revolution will trace a line BF equal to the circle’s circumference, and the inner circle, meanwhile, will trace a line of equal length along CE while completing its own revolution. In this the circles are no different from the polygons. But here’s the problem: the length of the line CE is equal to that of BF, a line created by a circle with a greater circumference. How can the smaller circle create a line longer than its own circumference? The answer, according to Salviati, is that the seemingly continuous line CE is, just like the line created by the rotating polygons, interspersed with empty spaces that contribute to its length. The line created by the smaller polygon of 100,000 sides is composed of 100,000 segments separated by 100,000 gaps; it follows that the line traced by the smaller circle is composed of an infinite number of segments separated by an infinite number of empty spaces.
By pushing Aristotle’s wheel to its illogical limit, Galileo arrived at a radical and paradoxical conclusion: a continuous line is composed of an infinite number of indivisible points separated by an infinite number of minuscule empty spaces. This supported both his theory of the structure of matter and his view that material objects are held together by the vacuum that pervades them. It provided a new way of thinking about the material world and also pointed to a new vision of mathematics, one in which, according to Salviati, any “continuous quantity is built up of absol
utely indivisible atoms.” The inner structure of the mathematical continuum is indistinguishable from the threads of rope, the fibers inside blocks of wood, or the atoms that make up a smooth surface: it is composed of tightly compressed indivisibles with empty spaces between them. For Galileo, the mathematical continuum was modeled on physical reality.
Galileo’s approach was troubling to contemporary mathematicians, as it went directly against the well-established paradoxes that had guided the treatment of the continuum since antiquity. He did, nonetheless, have at least one prominent supporter in fellow Lincean Luca Valerio (1553–1618), who had been inducted into the Accademia dei Lincei at Galileo’s urging. Valerio was professor of rhetoric and philosophy at the Sapienza University in Rome, and widely acknowledged as one of the leading mathematicians in Italy. In his De centro gravitatis of 1603 and Quadratura parabola of 1606 he had experimented extensively with indivisibles, which enabled him to determine the centers of gravity for plane figures and solids.
But Valerio learned his mathematics among the Jesuits of the Collegio Romano, under the tutelage of Clavius himself, and with this prominent group, Galileo’s mathematical atomism found no favor. For the Jesuits, indivisibles represented the exact inverse of the proper and correct approach to mathematics. The Jesuits, it will be recalled, valued mathematics for the strict rational order it imposed upon a seemingly unruly universe. Mathematics, and in particular Euclidean geometry, represented the triumph of mind over matter and reason over the untamed material world, and reflected the Jesuit ideal not only in mathematics but also in religious and even political matters. By anchoring his mathematical speculations in an intuition of the structure of matter rather than in self-evident Euclidean postulates, Galileo turned this order on its head. The composition of the mathematical continuum, according to Galileo, could be derived from the composition of ropes and the inner structure of a piece of wood, and it could be interrogated by imagining a wheel rolling over a straight surface. In place of the Jesuit approach, Galileo proposed that geometrical objects such as planes and solids were little different from the material objects we see around us. Instead of mathematical reason imposing order on the physical world, we have pure mathematical objects created in the image of physical ones, incorporating all their incoherence. Clavius, needless to say, would not have been pleased.
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Page 10