Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Alexander, Amir

  The Swedish victory at Breitenfeld stunned Europe, instantly transforming a war that had already lasted for thirteen years and was to continue for seventeen more. Up to that point the Habsburg Imperial armies had outclassed and defeated all their rivals. They had crushed the Bohemian nobles in the Battle of White Mountain in 1620 and had defeated the Danes, who had intervened in support of their Protestant coreligionists. The union of Protestant princes, under the leadership of Johann Georg of Saxony, proved no match for the Imperial generals, Count Tilly and Albrecht von Wallenstein. But in 1630, Gustavus Adolphus of Sweden ended his war with Poland and landed his battle-tested army in northern Germany. The Holy Roman Emperor, Ferdinand II, hoped to keep the Swedes weak and isolated, but his expectations were dashed in early 1631, when Gustavus came to terms with Cardinal Richelieu of France. Cardinal though he was, Richelieu was more intent on frustrating Habsburg designs for European domination than on promoting the interests of his mother Church, so he promised to finance Gustavus’s campaign on a grand scale. The results of this unorthodox alliance became clear on the field of Breitenfeld, where Gustavus’s veterans—well armed, well trained, and united behind the inspired leadership of the king—crushed the empire’s most powerful army.

  For more than a year after the battle, the Swedes swept through Germany like a force of nature. They defeated the Habsburg armies again in April 1632, at the River Lech, killing Count Tilly and marching south into Bavaria. Gustavus’s army was now deep in the heart of German Catholicism, and was busy ransacking its cities and desecrating its churches. The Jesuit colleges, proud symbols of a resurgent Catholicism, were favorite targets of the Swedes. They were plundered mercilessly, their books and treasures dispersed, the learned fathers chased away. Meanwhile, Johann Georg of Saxony, taking advantage of the Swedes’ domination, invaded Bohemia, occupying and looting the former imperial capital of Prague and disbanding the city’s famed Jesuit college. That November, the imperial army, now commanded by the crafty von Wallenstein, took a stand once more, near the town of Lützen, but the Swedish veterans prevailed once again. Only Gustavus’s death in that battle finally slowed down the Swedish juggernaut, bringing relief to Catholics. When the news spread across Catholic Europe, church bells rang from Vienna to Rome, and the sons and daughters of the Church gathered in special celebratory Masses to thank the Lord for their deliverance from a ruthless enemy.

  The sudden crisis in Catholic fortunes in Germany hit the papal Curia like thunder on a clear day. At a stroke, the Pope’s policy of playing France against the Habsburgs in order to preserve his own freedom of action became untenable. It was one thing to make overtures to Richelieu when the Catholic empire reigned supreme, and with the Protestants seemingly on the run. It was quite another to do so when Richelieu had allied himself with the heretics and the fate of Catholic Germany hung in the balance. Urban still wavered, but he could not do so for long: when he appeared unwilling to ally himself with the Habsburgs and throw everything into a life-and-death struggle against the marauding Swedes, there were those in Rome ready to remind him of his duty.

  Chief among them was Cardinal Gaspar Borgia (1580–1645), the Spanish ambassador to the Holy See and the leader of the opposition in the Curia to the Pope’s pro-French policy. Just as important, he was the grandson of Francis Borgia, the Duke of Gandia, who had been a devoted follower of Ignatius of Loyola and ultimately the third superior general of the Society of Jesus. The bonds between the Borgia clan and the Jesuits remained strong, and Gaspar had been a natural ally of the Society in Rome’s culture wars. The cardinal and the Jesuits had stood shoulder to shoulder in the 1620s in a struggle against what they saw as Urban’s tolerance of dangerous and heretical opinions, first and foremost those of Galileo and his friends. Politically, the cardinal and the Jesuits were staunch advocates of a Habsburg alliance, which would unite the empire, Spain, and the Papacy in a holy war against the Protestant schismatics.

  Marginalized since the election of Urban VIII in 1623, Borgia and his allies believed that the crisis of Catholic Germany had given them the opening they needed. On March 8, 1632, in the Hall of the Consistory in the Vatican, in the presence of the Pope and the cardinals, Borgia launched his attack. Breaking all rules of protocol and decorum, he startled Urban by reading an open letter harshly criticizing the Pope’s policies. He denounced the unholy French-Swedish alliance and demanded that Christ’s vicar on earth make his apostolic voice heard like the trumpet of redemption, uniting all Catholics in a titanic struggle against the heretics. Scandalized by the insult to the Pope, Cardinal Antonio Barberini (who was also Urban’s brother) lunged at Borgia, but was repelled by the coterie of pro-Spanish and pro-Habsburg cardinals surrounding the Spanish envoy. Borgia finished reading the letter.

  For a Pope to be lectured to by a mere cardinal, and be accused of letting the enemies of the faith have their way, was an insufferable humiliation. In the months that followed, Urban VIII tried to preserve his dignity and authority by punishing some of the prelates who had turned against him in his own house, and sending out indignant letters of protest to Madrid. But the game was up, and the Pope knew it. With the changing fortunes of the war, and the changing balance of political power in Rome, Urban VIII had run out of options. He backed away from his informal alliance with Richelieu and sided openly with the Habsburgs in their battle to save Bavaria and Bohemia from the Swedes. The Pope’s dramatic course reversal was felt just as strongly in Rome, where—according to the Florentine ambassador Francesco Niccolini—surveillance of orthodoxy and vigilance against heretics and innovators were the instruments of power of the newly ascendant Spanish party. Urban now distanced himself from the freethinking ways of the Linceans, effectively removing his protection of Galileo. Cardinal Borgia was now the most powerful man in the Eternal City, second only to the Pope—if that. And the Jesuits, who in the nine years since Urban’s accession had been exiled from the power centers of the Curia, now came in from the cold.

  In 1632 the Jesuits were like a political party returning to power after a long exile, determined to put their stamp on the cultural and political life of Rome and all Catholic lands. Their first target was the man who had humiliated them for years with his sharp pen and venomous taunts, and happily for them, he provided them with a perfect opportunity to strike. In May of 1632, Galileo published his Dialogue on the Two Chief World Systems, the final incarnation of his book on the causes of the tides, which he had discussed with Urban years before. The Dialogue may not have violated the letter of Bellarmine’s edict of 1616 against advocating Copernicanism, since at book’s end it defers to the authority of the Church on the question of the Earth’s motion. But it did, as anyone reading the book could see, violate the spirit of the edict, by eloquently presenting all the arguments in favor of Copernicanism while undercutting and ridiculing the counterarguments. Galileo, however, was apparently unconcerned, confident in the protection of the Pope and the Pope’s Lincean cardinal nephew.

  But Galileo’s timing could not have been worse. In place of the Rome he had known, in which his friends the Linceans moved in the highest circles while the Jesuit fathers fumed helplessly on the sidelines, was a new Rome in which the positions were reversed: the Jesuits were ascendant, and Galileo’s friends were running for cover. The Dialogue had been cleared for publication by official Vatican censors, but Galileo’s enemies argued successfully that the license had been obtained under false pretenses. Galileo was charged with holding Copernicanism to be true and advocating it, thus violating Church doctrine as well as the personal injunction handed him by Cardinal Bellarmine sixteen years before. He was interrogated three times by the Inquisition, in 1632 and 1633, the last time under threat of torture. He was found to be “vehemently suspect of heresy” and sentenced to imprisonment at the pleasure of the Inquisition. After he publicly recanted his opinions, his sentence was commuted to permanent house arrest, and he spent the remainder of his life confined to his villa in Arcetri.

  Rivers o
f ink have been spilt debating the causes of Galileo’s persecution, with suggestions ranging from an irrepressible conflict between science and religion to the prickliness of a thin-skinned Urban VIII, who allegedly thought that Galileo had ridiculed him in the Dialogue and was out for revenge. It is a debate that has been ongoing for nearly four centuries and will undoubtedly continue for many more, and any of the causes discussed might have played some role in the events. Nevertheless, it cannot be a coincidence that the tragic reversal of Galileo’s fortunes coincided precisely with the political crisis in Rome and the return to favor of his Jesuit enemies. His downfall was brought about in large measure by the enmity of the resurgent Society of Jesus.

  At the very same time that the Jesuits were gearing up to indict and try Galileo for his Copernicanism, they relaunched another fight, not as publicly visible but just as crucial to their plans for reshaping the religious and political landscape: the war against the infinitely small. Relaunched in 1632, the campaign against this dangerous idea would be pursued with fierce and dogged determination for the next several decades. It would not end until infinitesimals were effectively extinguished in Italy, and significantly curtailed in other Catholic countries.

  THE CENSORS, PART II

  And so it was that on a Roman summer day, August 10, 1632, the Revisors General met at the Collegio Romano to pass judgment on the doctrine of the infinitely small. As was always the case, the proposition before the Revisors had allegedly been “sent in” for them to rule on, but unusually, the record of the meeting does not specify which province asked for the judgment. It is more than likely that, in this case, the initiative for the ruling came not from some anxious provincial but from the Jesuit hierarchy in Rome. Indeed, for the Jesuit leaders, the matter was urgent, as the insidious doctrine appeared to be seeping into the heart of the order itself.

  Only a few months earlier, Father Rodrigo de Arriaga of Prague published his Cursus philosophicus, a textbook on the essential philosophical doctrines to be taught in the Society’s colleges. Arriaga was not just any Jesuit father. Sent to Prague after the Imperial forces dislodged the Protestants in 1620, he had played a leading role in securing Jesuit control of Bohemian schools and universities. He soon became dean of the faculty of arts at Prague University, and was now rector of the Jesuit college there. So great was his intellectual authority and his fame as a teacher that he was the subject of a popular quip: “Pragam videre, Arriagam audire”—“To see Prague, to hear Arriaga.”

  Arriaga’s Cursus philosophicus was an immediate hit among Jesuit scholars and educators, as well as many others outside the order. On the surface it seemed like a traditional, even old-fashioned text, modeled on medieval commentaries on Aristotle. Indeed, Arriaga was famous for reviving the old Scholastic disputation format, in which questions are posed on a canonical text and then discussed and answered. But to the shock of Arriaga’s superiors, this seemingly sedate book expounded some very radical opinions. An entire section was devoted to a discussion of the composition of the continuum, and the conclusions reached were not at all what the Jesuit authorities would find acceptable. After carefully weighing all the arguments for and against the doctrine of indivisibles, and discussing them at length, Arriaga came to a conclusion: it is more probable than not that the continuum is indeed composed of separate indivisibles.

  We do not know why Arriaga ventured into these troubled waters. Quite possibly he was influenced by his friend Gregory St. Vincent. The two taught together in Prague, and when, after the Battle of Breitenfeld, they were forced to flee the city before the advancing Saxons, it was Arriaga who saved St. Vincent’s manuscripts from being dispersed and lost. Perhaps Arriaga was impressed by St. Vincent’s use of infinitesimals and wanted to provide a philosophical justification for his friend’s controversial mathematics. Perhaps Arriaga also thought that his stature in the order meant he was immune to censorship, and he may well have been right, had the Jesuits remained in the political wilderness they had occupied for most of the previous decade. But in 1632 the Jesuits in Rome were back in charge, and determined to end the toleration of heretics and to enforce their strict theological dogma. In this new Rome, Arriaga’s challenge to his superiors could not go unanswered.

  The document recording the Revisors’ meeting that day, signed by Father Bidermann and his four colleagues, is preserved in the archives of the Society of Jesus in Rome. As befits the gravity of the issue at hand, the text is unusually formal. It takes up a complete page in the Revisors’ records, instead of the usual line or two, and it begins with an imposing legalistic header: “Judgment on the Composition of the Continuum by Indivisibles.” With a barely disguised reference to Arriaga, it continues:

  This has been handed to us to examine, proposed by a certain Professor of Philosophy, on the composition of the continuum. The permanent continuum can be constituted of only physical indivisibles or atomic corpuscles having mathematical parts identified with them. Therefore the said corpuscles can be actually distinguished from each other.

  The medieval technical terminology used by the Revisors is rather opaque, but the meaning of the proposed doctrine is nevertheless clear: any continuous magnitude, physical or mathematical, is composed of irreducible, indivisible parts that can be identified one by one. What was before the Revisors was the doctrine of the infinitely small, the basis of the infinitesimal mathematics of Galileo, Cavalieri, and Torricelli.

  The Revisors’ judgment was unsparing:

  We judge this proposition to be not only contrary to the common doctrine of Aristotle, but that it is also improbable, and in our Society it has always been condemned and prohibited. It cannot in the future be seen by our professors as permitted.

  And that was that. The doctrine of the infinitely small was now banned by the Society of Jesus for all eternity. Neither philosophers such as Arriaga nor mathematicians such as St. Vincent could henceforth be allowed to promote their subversive theories from within the Society. The order’s position was now set, and any member Jesuit who challenged this dogma would face censure from the highest ranks of the order.

  Once the Revisors pronounced their verdict, notices of their decision were immediately sent out to every Jesuit institution around the world. The process was routine, and usually the responsibility of the scribes and secretaries who managed the ordinary correspondence between Rome and the provinces. But when it came to the infinitely small, this was not deemed sufficient: the command to cease and desist from teaching, holding, or even entertaining the doctrine came directly from the superior general of the Jesuit order. Only a few years before, General Mutio Vitelleschi had been publicly humiliated by the Pope following the Santarelli affair. Now, with the Society poised to reshape the Catholic intellectual landscape, he was determined that, on matters of consequence, the order speak with a single, all-powerful voice.

  Vitelleschi moved quickly. Only six months after the Revisors issued their edict, he found himself writing to Father Ignace Cappon at the Jesuit College in Dôle, in eastern France, to complain that his repeated instructions were not being followed. “As regards the opinion on quantity made up of indivisibles,” he wrote impatiently, “I have already written to the provinces many times that it is in no way approved by me and up to now I have allowed nobody to propose it or defend it.” Indeed, he had made every effort to suppress the doctrine: “If it has ever been explained or defended, it was done without my knowledge. Rather, I demonstrated clearly to Cardinal Giovanni de Lugo himself that I did not wish our members to treat or disseminate that opinion.” The fight to wipe out any vestiges of the offending doctrine from the Society was one the general would undertake himself.

  The lines were now firmly drawn. On the one side were the Jesuits, determined to wipe out the doctrine of the infinitely small. On the other side was a small band of mathematicians who still saw Galileo as their undisputed leader, despite his public humiliation. And the struggle raged. In 1635, Cavalieri published his Geometria indivisibilibus, givi
ng the most systematic exposition of the method of indivisibles. Three years later, Galileo published his Discourses in Holland, which included his treatment of the paradox of Aristotle’s wheel and his discussion of infinitesimals. Galileo’s towering fame ensured that his views on the continuum would be taken very seriously by scholars across Europe, and his praise of Cavalieri established the Jesuat monk as the leading authority on the subject of indivisibles. In response, the Jesuits struck back with repeated condemnations of the infinitely small. From this time onward a steady stream of denunciations of the offending doctrine began issuing from the pens of the Revisors General in Rome.

  On February 3, 1640, for example, the Revisors were asked to comment on the proposition that “The successive continuum … is composed of separate indivisibles,” and ruled that “the doctrine is prohibited in the Society.” Less than a year later, in January 1641, they were once again confronted with ideas “that were invented or innovated by certain moderns,” including the proposition “that the continuum is made of indivisibles,” and a variation that claimed “that the continuum is made of indivisibles that expand and contract.” Both propositions, the Revisors ruled, were “contrary to the common and solid opinion.” On May 12, 1643, they came down hard on an author who allegedly preferred the opinions of Zeno to those of Aristotle: “We do not approve or concede of these,” they wrote, “as it is against the Society’s constitution and rules, as well as the decrees of the General Congregation.”

  A pattern set in. In Rome the Jesuits were strong enough to quash any talk of the forbidden doctrine; but farther away, some mathematicians could still defend and promote it. Torricelli, for example, was in Rome in the 1630s, toiling in private on the new mathematics but publishing not a word. But it took only two years after his installment as Galileo’s successor at the Medici court in Florence before he published his Opera geometrica, which contains the fruits of his silent labor in Rome. Even from the safety of his court position, however, Torricelli tried to avoid open conflict with his powerful critics. Unlike Galileo, he did not directly engage with them in his book, argue over the merits of his method, or ridicule their motives or reasoning. He let his powerful results speak for themselves, and that they undoubtedly did: the Opera geometrica was admired and emulated by mathematicians from Germany to England. In Italy the long shadow cast by the Society of Jesus was sufficient to ensure that no mathematician there expressed similar enthusiasm for Torricelli’s work. But the Jesuits nonetheless took note of his success, and prepared to strike back.