Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (20 page)

The first decree by the Revisors General on the composition of the continuum dates from 1606, when the office was only five years old. Responding to a proposition sent in from the Society’s schools in Belgium that “the continuum is composed of a finite number of indivisibles,” the Revisors, quickly and without comment, ruled that the proposition was an “error in philosophy.” Only two years later another missive from Belgium brought the very same doctrine before the Revisors. This time they were somewhat more expansive, though just as firm: “everyone agrees that this must not be taught, since it is improbable and also certainly false and erroneous in philosophy, and against Aristotle.” Only a decade before, Suárez had merely expressed some concern about whether the notion that the continuum was composed of indivisibles was philosophically viable, and offered some alternatives. The Revisors, in contrast, banned it outright as “false and erroneous.”

What had changed? The Revisors themselves offer no clue, and the summaries they left provide no details on the sources of propositions brought before them, except for their country of origin. But we do know that those early years of the seventeenth century saw a significant uptick in interest in the infinitely small among mathematicians. In 1604, Luca Valerio of the Sapienza University in Rome published a book on calculating the centers of gravity of geometrical figures in which he employed rudimentary infinitesimal methods. Valerio was well known to the Jesuits, having studied under Clavius for many years, and even receiving doctoral degrees in philosophy and theology from the Collegio Romano. His work could not have gone unnoticed by the Jesuit fathers, who likely felt they needed to better define their position on this new approach. We also know that in 1604, Galileo, then at the University of Padua, was experimenting with indivisibles in formulating his law of falling bodies. Galileo thought very highly of Valerio: years later he nominated him for membership in the Lincean Academy, and in his
Discourses
of 1638 he refers to Valerio as “the Archimedes of our age.” Whether the two drew upon each other or developed their ideas independently, their work marked a significant change in the status of infinitesimals: instead of an ancient doctrine definitively discussed by Aristotle and his later commentators, infinitesimals now seemed to be entering the arena of contemporary mathematics.

For the Jesuits, this was a critical change. Clavius had only recently won his battle to establish mathematics as a core discipline in the Jesuit curriculum, and the order’s mathematicians were beginning to be recognized as leaders in the field. When, in the early seventeenth century, infinitesimals began seeping into mathematical practice, the Jesuits felt compelled to take a stand on the new methods. Are they compatible with the Euclidean approach so central to the Society’s mathematical practice? The Revisors’ answer was a resounding no. Yet despite the stern pronouncements, the problem never seemed to go away. Mathematically trained Jesuits across Europe were closely following developments at the forefront of mathematical research, and were well aware of the growing interest in infinitesimals. Conscious of the sensitivity of the subject, they kept appealing to the Revisors with different versions of the doctrine, each one deviating slightly from those that had already been banned. Consequently, when the Revisors turned their attention once more to the infinitely small, the catalyst once again was developments in the field of mathematics.

Johannes Kepler (1571–1630) is remembered today as the man who first plotted the correct elliptical paths of the planets through the heavens. Nor did Kepler go unappreciated in his own day. In the early seventeenth century he was the only scientist in the world whose fame rivaled Galileo’s, and though a Protestant, he held the most coveted mathematical position in the world: court astronomer to the Holy Roman Emperor in Prague. In 1609 Kepler published his masterpiece
Astronomia nova
(
The New Astronomy
), in which he demonstrates that the planets move in ellipses and not perfect circles, and codifies his observations in two laws of planetary motion. (Kepler’s third law was published later, in his
Harmonices mundi
of 1619.) To calculate the precise motion of the planets at varying speeds along their orbit, Kepler made rough use of infinitesimals, assuming that the arc of their elliptical path was composed of an infinite number of points. Six years later Kepler further developed his mathematical theory in a work dedicated to calculating the exact volume of wine casks, where he calculated a whole range of areas and volumes of geometrical figures using infinitesimal methods. To calculate the area of a circle, for example, he assumed that it was a polygon with an infinite number of sides; a sphere was composed of an infinite number of cones, each with its tip at the center and its base on the surface of the sphere, and so on. Titled
Nova stereometria doliorum vinariorum
(“A New Stereometry of Wine Caskets”), it was a mathematical tour-de-force that hinted at the power of the approach that Cavalieri would later systematize and name. Once again the Jesuits felt compelled to respond, and once again the job fell to the Revisors General in Rome. In 1613 they denounced the proposition that the continuum was composed of either physical “minims” or mathematical indivisibles. In 1615 they reiterated their condemnation, rejecting, first, the opinion that “the continuum is composed of indivisibles” and, several months later, the opinion that “the continuum is composed of a finite number of indivisibles.” This doctrine, they opined, “is also not permitted in our schools … if the indivisibles are infinite in number.”

Once the Revisors had issued their decision, a well-oiled machinery of enforcement sprang into action. The numerous Jesuit provinces across the globe were informed of the censors’ verdict, and they then passed it on to lower and then lower jurisdictions. At the end of this chain of transmission were the individual colleges and their teachers, who were instructed on the new rules on what was permissible and what was not. Once a decision by the Revisors in Rome descended the Jesuit hierarchy and reached an individual professor, he was now responsible for carrying it out to the fullness of his ability and of his own free will, regardless of his previous views on the subject. It was a system based on hierarchy, training, and trust—or, as an unfriendly observer might suggest, on indoctrination. Either way, there was no doubt that it was remarkably effective: the Revisors’ pronouncements became law in the many hundreds of Jesuit colleges worldwide.

THE FALL OF LUCA VALERIO

The Revisors’ decree of 1615 against infinite indivisibles may have been directed against the admirers of Kepler. But whatever the intent, it was the Jesuits’ former associate Luca Valerio who fell victim to the Society’s new and harsher stance. Three years had passed since Galileo proposed Valerio for membership in the prestigious Lincean Academy, which served as the institutional center of the Galileans in Rome. The academy was an exclusive club made up of a select group of leading scientists and their aristocratic patrons, but Valerio seemed like a perfect fit: not only was he a mathematician renowned for bold ideas and a professor at the ancient Sapienza University, but he was also an aristocrat and a personal friend of the late pope Clement VIII (1592–1605), who had been his student. He brought with him sparkling social prestige, as well as personal creativity and institutional respectability, and the Linceans promptly elected him on June 7, 1612. From the moment of his election, Valerio became a leader among the Linceans, given overall editorial responsibility for all the academy’s publications.

Valerio, who had studied for years under Clavius, remained on good terms with his former mentors and colleagues at the Collegio Romano, and that, too, made him valuable to the Linceans. At a time of increasing tensions between the Galileans and the Jesuits of the Collegio Romano, Valerio served as a means of communication and possible compromise between the two camps. Indeed, there was nothing Valerio wanted more than to heal the rift that had opened between his two groups of friends. It was not to be. Showing no concern for Jesuit sensitivities, Galileo published his
Discourse on Floating Bodies
—which attacked the principles of Aristotelian physics—debated the nature of sunspots, and circulated his views on the proper interpretation of Scripture. For the Jesuits of the Collegio Romano, this intrusion into theology was the last straw. They determined to strike back against the man they had once honored with a full day of ceremonies but whom they now viewed as a bitter enemy.

The Jesuits had learned from their past mistakes. Time and again they had been outmatched by Galileo’s brilliant polemics, coming off as rigid and didactic pedants standing in the way of scientific progress. So, instead of engaging in public debate, they turned to the arena in which their power was unchallengeable: the hierarchy and authority of the Church. In 1615 cardinal Bellarmine issued his opinion against Copernicanism, which soon became official Church doctrine. He followed this up with a personal warning to Galileo to desist forever from holding or advocating the forbidden doctrine. It was an impressive demonstration of the Jesuits’ ability to harness the Church apparatus to their cause, and a stinging defeat to the Galileans. As regards infinitesimals, nothing so public as the decree against Copernicanism took place. But it is probably not a coincidence that the Revisors’ ruling against indivisibles in April 1615 coincided precisely with Bellarmine’s issuing his opinion against Galileo.

Valerio felt besieged. The two great intellectual schools that he had hoped to reconcile were now openly at war. The middle ground on which he stood was quickly melting away, and he was being pulled in opposite directions. The Revisors’ decree of April 1615 on the composition of the continuum was a reminder that, as a mathematician identified with infinitesimal methods, he could not long remain above the fray. When the Revisors repeated their decree in November, this time adding that it applied even “if the indivisibles are infinite in number,” he may well have concluded that he himself was their target. We do not know what he was told in private by either the Jesuits or the Linceans, but the pressure must have been unbearable. Finally, in early 1616, with the tide turning decisively against the Galileans, Valerio made his decision: he tendered his resignation to the Lincean Academy, siding openly with the Jesuits.

The Linceans were stunned. Membership in the academy was a great honor, and never before had anyone turned his back on it. That such a thing could happen at all was a measure of just how precarious the Galileans’ position had become in the face of the Jesuit onslaught. Undeterred, the Linceans responded decisively: they promptly refused Valerio’s resignation, on the grounds that it conflicted with the oath taken by every member of the academy. Valerio therefore remained technically a Lincean, but only in name: in a meeting on March 24, 1616, his fellows censored him for betraying his oath of loyalty and offending both Galileo and “Lyncealitas,” the Lincean principle of mutual solidarity. They then barred Valerio from any future meetings of the academy and, for good measure, deprived him of his voting rights.

Valerio had misread the signs. The Galileans may have been put on the defensive, but they were still powerful enough to strike back at their former colleague. His life and career, so brilliantly successful for so long, ended up a Greek tragedy. Recognized early for his mathematical prowess, he had scaled the heights of scholarly fame in Italy, admired and honored by both conservatives and innovators. But when he could no longer bridge the growing divide between the two, he made a choice, and it turned out to be the wrong one. Isolated, humiliated, and a pariah to his former friends, Valerio retired, and died less than two years after his expulsion by the Linceans, an early victim of the Jesuit war against the infinitely small.

GREGORY ST. VINCENT, SJ

Valerio had studied and trained at the Collegio Romano for many years, but he was not a Jesuit himself. Sometimes, however, Jesuit officials had to deal not with outsiders but with their own members, Jesuit intellectuals who pushed back against the strictures placed upon them by their superiors, trying their best to pursue their work freely while still adhering to the letter, if not the spirit, of the law. In such cases the Jesuits usually took a milder approach, preferring to remind a wayward member of the common Jesuit bond and the ideal of voluntary obedience. By relying on their hierarchical order and the deeply ingrained value of obedience, the Jesuits managed to exert much greater control over their members than they would likely have achieved through disciplinary action, force, or intimidation.

Even so, challenging the order’s decrees had its price, as the mathematician Gregory St. Vincent of Bruges found to his dismay. A Fleming, like his younger contemporary Tacquet, St. Vincent (1584–1667) was one of the most creative mathematical minds ever to come out of the Society of Jesus. In 1625, while teaching at the Jesuit College in Louvain, St. Vincent developed a method for calculating areas and volumes of geometrical figures, which he called “ductus plani in planum.” His greatest triumph, he believed, was solving an ancient problem that had stumped the greatest geometers of every age: constructing a square equal in area to a given circle, or more simply, “squaring the circle.” St. Vincent decided to publish his results, and good Jesuit that he was, he sent his manuscript to Rome to obtain permission. Since St. Vincent was a distinguished mathematician, and his text technical and challenging, his request made it up through the ranks all the way to Mutio Vitelleschi, the general superior of the Society of Jesus.

Vitelleschi hesitated: most mathematicians of the era believed (rightly, as it turned out) that squaring the circle was impossible, or at least impossible by classical Euclidean methods. Those who claimed to have accomplished the feat were usually dismissed as quacks, and there was a significant danger that a Jesuit scholar who claimed he had succeeded in squaring the circle would tarnish the Society’s reputation. More troubling was the fact that St. Vincent’s “ductus plani in planum” method looked suspiciously like it was based on the forbidden doctrine of infinitesimals. Not wishing to decide a technical matter himself, Vitelleschi passed the matter to Father Grienberger, Clavius’s student and successor at the Collegio Romano, and the highest mathematical authority in the order. Grienberger read the treatise closely, but he, too, was unconvinced, and ruled against publication. Undeterred, St. Vincent requested and received permission to travel to Rome, where for two years he tried to convince Grienberger that his method was valid and did not violate the strictures against infinitesimals. He failed. In a 1627 letter, Grienberger informed Vitelleschi that while he did not doubt the correctness of St. Vincent’s results, he nevertheless had serious concerns regarding his method. The Fleming returned to Louvain empty-handed, and published nothing for the next twenty years. Only in 1647, taking advantage of the recent death of General Vitelleschi, did St. Vincent finally bring his work to print. This time, circumventing the authorities in Rome, he settled for a curt permission by the Jesuit provincial of Flanders, allowing the work to be printed.