Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Alexander, Amir

  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.

  St. Vincent’s experience typifies the Jesuit attitude toward indivisibles in the years that followed the prohibitions of 1615 and Valerio’s fall from grace. Indivisibles were prohibited, to be sure, but policing their use was not a high priority for the Society. When confronted with new mathematical methods that made use of the infinitely small, the Jesuits took action, reminding their members that these approaches were not permitted. Beyond that, however, they did little to curb the spread of the new approach. The fact that a distinguished Jesuit such as St. Vincent had developed a method inspired by indivisibles, and that he sent it to his Roman superiors to be approved, attests to the fact that he expected to be allotted some wiggle room. His application to publish his work was denied, but beyond that, St. Vincent was not punished for his transgression. He continued to hold distinguished positions in Jesuit colleges for the rest of his long life and, taking advantage of an opportune moment, even managed in the end to publish his work. In later years, when the Jesuits pursued the infinitely small with an air of grim finality, they would not be so forgiving.

  THE ECLIPSE OF THE JESUITS

  St. Vincent, as it turns out, was fortunate in his timing, which fell in the midst of a lull in the Jesuit campaign to curb infinitesimals. After the prohibition of 1615, the Revisors did not return to the issue for another seventeen years. The reason had much to do with the changing fortunes of the Society itself. Following their victory over the Galileans in 1616, the Jesuits reigned supreme in Rome. Pope Paul V had sided with them publicly in the fight with Galileo, humbling their critics and establishing them as the arbiters of truth. The advocates of Copernicanism had been effectively silenced, as had, it seemed, any talk of the infinitely small. Valerio had been humiliated, and Galileo was in no position to challenge the authority of the Collegio Romano. In 1619, Pope Paul V showed his favor to the Jesuits by beatifying Ignatius’s acolyte, the intrepid missionary Francis Xavier, just as he had done for Ignatius himself ten years before. In 1622, Paul’s successor, Pope Gregory XV, completed the process, making St. Ignatius and St. Francis Xavier the first Jesuit saints of the Catholic Church. In celebration, the Jesuits began plans for construction of a magnificent new Church of St. Ignatius on the grounds of the Collegio Romano.

  But in the autocracy that was the papal Curia, the one source of authority and power that mattered was the personal favor of the Pope. This was rarely a problem for the Jesuits, who championed papal supremacy in the Church, and whose elite swore a personal oath of obedience to the Pope. Not surprisingly, most popes thought it in their best interest to bestow their favor on the Society. Nevertheless, there were exceptions: Pope Paul IV (1555–59), for example, a founder of the rival order of the Theatines, was hostile to the Jesuits, and during his reign the Society suffered. Now, seven decades later, this history seemed to be repeating itself. In July of 1623, only two years after his election, Pope Gregory XV died, throwing all political calculations in Rome into chaos. It took a month of intense maneuverings for the College of Cardinals to decide on a successor, but when the dust finally settled, it was clear to all that a new day had dawned in Rome: the man elected was the Florentine cardinal Maffeo Barberini, who chose the name of Urban VIII. For the Jesuits, the choice could hardly have been worse.

  There were many reasons the Jesuits would view the election of Barbe
rini with apprehension. For one thing, he was from Florence, a city that prided itself on its tradition of independence and where, consequently, the Jesuits held relatively little sway. It was, after all, the protection of Florence’s ruler, Grand Duke Cosimo II, that had saved Galileo from far more severe consequences in his tangle with the Jesuits in 1616. Barberini had also served for many years as papal nuncio to France, and was known to be close to the French court. Indeed, it was French influence, in the person of Cardinal Maurizio of Savoy, that had secured Barberini’s election as Pope. The Jesuits, in contrast, tangled regularly with the French monarchy and its advocates at the Sorbonne over the issue of papal supremacy, and on more than one occasion were banned from teaching in France for refusing to take an oath of obedience to the king. At the papal Curia, the Jesuits were staunch supporters of France’s rivals, the Habsburgs—Holy Roman emperors and kings of Spain—whom they viewed as the best hope for restoring the unity of Christendom. But perhaps most troubling to the Jesuits was the fact that Barberini was a personal friend of Galileo, and had openly professed his admiration for his fellow Florentine, his discoveries, and his opinions. In the Roman culture wars of the previous decade, Barberini had come down firmly on the side of Galileo and the Lincean Academy, the enemies of the Jesuit fathers at the Collegio Romano.

  No sooner had Urban VIII settled in his office than he began behaving in ways that confirmed the Jesuits’ worst fears. He appointed Monsignor Giovanni Ciampoli as his personal secretary, and the young duke Virginio Cesarini as master of the papal secret chamber. Both men were Linceans who had plotted with Galileo on how to “bring down the pride of the Jesuits.” Following Cardinal Bellarmine’s death in 1621 the Jesuits were left without a representative in the College of Cardinals, and the new Pope seemed happy to leave things that way. When in 1627 the Jesuits petitioned him to have Bellarmine elevated to sainthood, Urban was in no hurry to respond. Instead, he set up a new barrier in the process, ruling that fifty years must pass between a candidate’s death and his canonization. But most troubling to the Jesuits was that Urban VIII’s admiration for Galileo and his eagerness to speak about it had not diminished with his accession. In 1623, when Galileo published The Assayer, his latest and most powerful salvo in his war of words with the Collegio Romano, the Pope welcomed it warmly. He publicly accepted a specially bound copy of the book, personally dedicated by Galileo, from Prince Federico Cesi, founder of the Linceans, and asked Ciampoli to read it to him at table. As Cesarini, who was present at these occasions, assured Galileo, the new Pope was amused and full of admiration. Taking advantage of the favorable constellation, the Linceans quickly inducted the Pope’s nephew Francesco Barberini into their academy, a move whose wisdom was soon confirmed when the Pope bestowed the cardinal’s purple on young Francesco. Now the cardinal nephew, perhaps the second most powerful man in Rome, was himself a Lincean, while the Jesuits went for decades without a cardinal of their own.

  Nor did the new Pope’s foreign policy endear him to the Jesuits. Indeed, Urban VIII was something of a throwback to the Renaissance popes, concerned more with securing his independence as an Italian prince than with enhancing his standing as the spiritual leader of all Christians. In the midst of the Thirty Years’ War (1618–48), the bloodiest of all religious struggles, the Pope was naturally expected to side firmly with the Habsburgs, whose imperial domains were being ravaged and who bore the brunt of the struggle against Protestantism. Instead, Urban tried to free himself from the suffocating Habsburg embrace by allying himself with France and its enigmatic chief minister, Cardinal Richelieu. Rather than support the war on the heretics, Urban built up his own military power and bullied neighboring Italian potentates—all good Catholics. He added the Duchy of Urbino to his domains, becoming the last Pope to expand the Papal States, and launched the “wars of Castro” against the Farnese dukes of Parma. In 1627, when there was no male heir to the ancient Gonzaga line of Mantua, he actually supported the succession of a Protestant, Charles Gonzaga, Duke of Nevers, over the Habsburg claimant.

  Even in their season of discontent, the Jesuits, ever pious, ever active, tried to direct Church policy away from what they considered a disastrous course. Their plan hinged on a perennial bone of contention between Rome and Paris: the question of papal supremacy. In 1625, Antonio Santarelli, professor at the Collegio Romano, authored a spirited defense of papal power in a book with the imposing title Treatise on Heresy, Schism, Apostasy, the Abuse of the Sacrament of Penance, and the Power of the Roman Pontiff to Punish These Crimes. His main thesis was that popes reigned supreme over secular monarchs, and even had the power to remove them if they acted in ways harmful to the faith. This doctrine was hardly new, and seemed quite self-evident to the Jesuits, who believed in a strict hierarchy of Church, state, and society, with the Pope at its apex. Back in 1610, Cardinal Bellarmine himself had published his Treatise on the Power of the Supreme Pontiff in Temporal Affairs, making much the same claims on behalf of papal power. But as the cardinal found out, what was a simple truism in Rome was sedition in Paris, where the Bourbons were busy building an absolute monarchy in which royal authority would reign uncontested. Bellarmine’s book was publicly condemned by the Parlement of Paris, the Jesuits were banned from teaching in France for several years, and a brief diplomatic crisis flared between the Bourbon court and the Vatican. By publishing Santarelli’s book in 1625 in Paris, the Jesuits were likely hoping to instigate a similar crisis, which would force the Pope, willingly or not, back into the arms of the Habsburgs.

  The plot misfired badly. The French were indeed incensed with Santarelli’s treatise, but they focused their fury strictly on the Jesuits rather than on the Pope. The book was condemned to the flames by the Parlement of Paris, and denounced by the faculty of the Sorbonne and other French universities. On March 16, 1626, the leaders of the French Jesuits were called before the Parlement and asked to sign a public disavowal of Santarelli’s “evil doctrine.” Faced with the destruction of their entire French mission if they refused, the Jesuits humbly signed. If this was not embarrassment enough, a greater one awaited them in Rome: on May 16 the Pope summoned Mutio Vitelleschi, general of the Society of Jesus, to appear before him and, in front of the cardinals and prelates of the Curia, castigated him for undermining his French policy. “You are not content to blacken me in France, you also wish to tear me apart in Italy,” the Pope thundered. It was an extraordinary humiliation for the leader of the Jesuits, and a public repudiation of the order as a whole. Only four years after the canonization of St. Ignatius, the once-invincible Jesuits were relegated to the farthest margins of the Roman Curia.

  While the Jesuits suffered, their enemies prospered. Galileo, who was in Rome to oversee the publication of The Assayer, met several times with the Pope in the spring of 1624 for friendly discussions on natural philosophy. He returned to Florence in June bearing a letter that declared him the Pope’s “beloved son,” and with warm encouragement for his next book—which he then called “Treatise on the Flux and Reflux of the Sea,” but which would ultimately become the Dialogue on the Two Chief World Systems. Galileo even believed that he had been given implicit permission to reopen the question of the motion of the Earth, a misjudgment that would cost him his liberty nine years later. In the fall of 1628 the freethinking attitudes of Galileo and his friends even reached deep into the very center of Jesuit power: in a splendid ceremony in the great hall of the Collegio Romano, with numerous cardinals in attendance, the Marchese (Marquis) Pietro Sforza Pallavicino (1607–67), godson of the Cardinal of Savoy, defended his doctoral dissertation in theology. The young aristocrat was a rising star in Roman intellectual circles, with a bright future that would ultimately lead him to the college of cardinals. Even at the age of twenty-one he was already a member of the sparkling literary “Academy of the Desirous,” and a friend to Galileo. On this occasion, he proved himself as one: his dissertation was a defense of the orthodoxy of the doctrine of atomism, which Galileo advocated in The Assayer a
nd which was the target of Jesuit charges of unorthodoxy and even heresy. Contrary to the claims of Galileo’s nemesis Father Orazio Grassi of the Collegio Romano, atomism, according to Pallavicino, was unobjectionable and perfectly consistent with the official doctrine of the Eucharist. Only a few months later, the Jesuit-trained Pallavicino became a full-fledged member of the Lincean Academy.

  With their fortunes at a historically low ebb, and their authority and prestige under attack, the Jesuits effectively suspended their campaign against the infinitely small. After all, at a time when a defense of despised atomism could be publicly recited in their own most hallowed halls, how could the Jesuits credibly condemn the related doctrine of mathematical indivisibles? So they lay low for seventeen years, issuing not a single condemnation while the mathematics of the infinitely small gained ground. It was during those years that Cavalieri developed his method of indivisibles and ultimately secured the prestigious chair of mathematics in Bologna. It was also during those years that Torricelli was first introduced to the new mathematics by his mentor, Benedetto Castelli, and embarked on the career that would make him the most influential practitioner of the new mathematics. All the while, the Jesuits observed, took notes, and patiently bided their time.

  THE CRISIS OF URBAN VIII

  On September 17, 1631, the Protestant armies of Sweden and Saxony met the Catholic forces of the Holy Roman Empire in battle near the village of Breitenfeld in Saxony. As the Imperials charged, the inexperienced Saxons panicked and fled the field, exposing the flank of their Swedish allies to a potentially devastating strike. But the Swedes held their ground: they covered their flank and then launched their own attack. Under the cool direction of their king, Gustavus Adolphus, they routed the emperor’s men, inflicting thousands of casualties on Count Tilly’s previously invincible army. At a stroke, the road to the heart of Catholic Germany lay open before the Protestant Swedes.