Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (27 page)

Angeli’s comparison of Bettini to a confused bee is mocking enough, but he is not yet done with the Jesuit. He quotes the passage in which Bettini calls the method of indivisibles “counterfeit philosophising” (“similitudinem philosophantium”), and exclaims, “far, far be it from me to wish to make my geometrical theorems useless.” Seeing an opening, Angeli pounces: “Note here, reader, how this author, on falling in with indivisibles, cries out as if he were met with demons: Far, far be it from me, etc.” Bettini is here a hysterical exorcist trying to fend off demonic indivisibles with furious incantations. But as to substance, Angeli concludes, “he adds nothing new but spite.”

The hyperbolic Bettini was perhaps easy quarry, but neither was the more formidable Tacquet spared Angeli’s sharp pen. In the preface to his
De infinitis parabolis
of 1659, Angeli describes how a few days after the publication of his previous book, in which he faced down the Jesuit Bettini, he wandered into the Venetian bookstore Minerva. There he came across
Cylindrica et annularia
, the work of another, “most deserving mathematician of the same Society.” Flipping through the pages, he by chance came across a passage in which the author “carps on indivisibles,” claiming that they are neither legitimate nor geometrical. Angeli claims that he had never previously heard of the book or known of its critique of indivisibles, but that is highly unlikely. He was extremely well versed in the mathematical output of his contemporaries, and later in the preface, he cites the Frenchmen Jean Beaugrand and Ismael Boulliau, the Englishman Richard White, and the Dutchman Frans van Schooten, as well as his fellow Italians. It stretches credulity that he was not familiar with the work of Tacquet, the leading Jesuit mathematician of the day, or Tacquet’s views on indivisibles, until he stumbled upon them in a Venetian bookstore. Angeli’s plea of ignorance is rather a rhetorical pose, aimed at presenting him as an impartial scholar reacting to outrageous claims made by Bettini and Tacquet. The long and bitter history that had pitted him against the Jesuits for decades is left unmentioned.

Angeli goes on to claim that there is nothing particularly troubling in Tacquet’s criticism of indivisibles. His arguments are old, he writes, and were already raised by Guldin and satisfactorily answered by Cavalieri years ago. But Tacquet did provide Angeli with an opportunity to proclaim how influential the method of indivisibles had become by the late 1650s. “Who does this reasoning convince?” Tacquet asks rhetorically, pointing to what he considered the inherent implausibility of the method of indivisibles. “Whom does it convince?” Angeli repeats incredulously. Everyone, he responds, except the Jesuits.

Angeli here is trying to turn the tables on the Jesuits: rather than the indivisiblists being a lonely and diminishing band under attack from powerful enemies, it is the Jesuits who are lone holdouts against a method that is being universally accepted. Indeed, at first reading, the list cited by Angeli is impressive, and appears to support his case. But a closer inspection tells a very different story: Yes, Beaugrand, Boulliau, White, and van Schooten did indeed adopt Cavalieri’s method, but they resided in faraway lands, north of the Alps. Of the three Italians whom Angeli cites—Torricelli, Rocca, and Raffaello Magiotti—only Torricelli had in fact published on indivisibles, whereas Rocca and Magiotti had remained mum; and in any case, by 1659 all three were dead. Despite his protestations to the contrary, Angeli was, in his own land, alone.

Satisfied with his rhetorical salvos, Angeli then confronts Tacquet’s dark warning that unless it were destroyed first, the notion that the continuum is composed of indivisibles would destroy geometry. Cavalieri insisted that the question of the composition of the continuum was irrelevant to the method of indivisibles, and Angeli here follows his teacher—but only up to a point. Like Cavalieri, he, too, argues that Tacquet was wrong, and that “even if the continuum is not composed of indivisibles, the method of indivisibles nevertheless remains unshaken.” But he adds a twist: “if in order to approve the method of indivisibles, the composition of the continuum from indivisibles is necessarily required, then certainly this doctrine is only strengthened in our eyes.” In other words, unlike his cautious teacher, Angeli is perfectly willing to accept that the continuum is indeed composed of indivisibles. The power and effectiveness of the method of indivisibles is proof enough of its correctness, and if it leads to the conclusion that the continuum is composed of indivisibles, then that, too, must be correct. The fact that this doctrine leads to contradictions and paradoxes bothers him not at all.

Angeli, the flamboyant Jesuat, took on the Jesuits like no one had dared since the days of Galileo himself. He called them names, ridiculed their exorcism practices, and pretended never to have heard of the most illustrious mathematician among them. But nothing demonstrated the clash between Jesuit and Jesuat like their contradictory approaches to the question of the composition of the continuum. For the Jesuits, the notion that the continuum is composed of indivisibles led to paradoxes, and on that account alone it must be banned from mathematics. A method based on it, even if effective and fruitful, was unacceptable because it undermined the very reason for which mathematics was studied: for its pure logical structure. Angeli’s view was the exact opposite: because the method of indivisibles was effective, he reasoned, its underlying assumptions must be true, and if they involved paradoxes, then we would just have to live with them. One approach emphasized the purity of mathematics, the other emphasized practical results; one approach insisted on absolute perfect order, the other was willing to coexist with ambiguities and uncertainties. And never the twain shall meet.

THE FALL OF THE JESUATS

Thanks to the protection of his order and the Jesuit-unfriendly Venetian Senate, it appeared that Angeli would get away with his open defiance of the Society of Jesus. He kept up his work, and over the next eight years he published an additional six mathematical books, in all of which he used and advocated the method of indivisibles. His greatest triumph came in 1662, when he was appointed to the chair of mathematics at the University of Padua, a position once held by Galileo. The Jesuits, so powerful elsewhere in Italy, could only fume as the upstart Jesuat was raised to one of the most prestigious mathematical posts in all Europe. They never responded to his taunts, or denounced him openly, but quietly, patiently, bided their time.

The Jesuits were in a tight spot. As long as Angeli continued with his insolence, there was always a danger the forbidden doctrine might be revived in Italy and their decades-long campaign would come to naught. But what could they do? Angeli was safe in Venice, and if they ever thought they could persuade the authorities there to silence him, then surely his appointment to the Padua professorship showed them this was unlikely. So the Jesuits changed tactics: in Venice they might be of little account, but in Rome they were still ascendant. And so, in order to silence the last Italian voice advocating the infinitely small, they turned to the papal Curia.

The evidence for what happened next is circumstantial, the documents relating to the events buried to this day deep in the Vatican archives. But what we do know is this: On December 6, 1668, Pope Clement IX issued a brief suppressing three Italian religious orders: one was a community of Canons Regular, residing on the island of San Giorgio in Alga, in the Venetian lagoon; the second were the Hieronymites of Fiesole, a popular order that, at its height, had forty houses across Italy; the third were the Jesuats of St. Jerome. As the brief put it, “no advantage or utility to the Christian people was to be anticipated from their survival.”

The Canons Regular was a tiny community, and confined to a single Venetian island, and it is plausible that the bureaucrats at the Vatican indeed came to the conclusion that it served no real purpose. The Hieronymites of Fiesole was a much larger order, but in their case the term
suppression
is misleading. For while it is true that they ceased to exist independently, the order was not in fact dissolved, but rather merged with its sister order of the Hieronymites of Pisa; the houses themselves continued to exist much as before. But for the Jesuats of St. Jerome the suppression was a death sentence: from one day to the next, the order simply ceased to exist, its houses dissolved and its brothers dispersed. It was a stunningly violent and unexpected end to an old and venerable order. Founded by the Blessed John Colombini in 1361 to tend for the poor and the sick, it had survived for exactly three centuries and seven years.

The official reason given, and the one quoted in all public sources today, is that “abuses had crept into the order.” But this explanation is no more helpful than the claim that the order’s survival served no purpose. Some scholars note that the Jesuats were often referred to as the “Aquavitae Brothers,” a designation that may suggest lax morals and loose living. But this was far from the case: the nickname was given to them because of their dedication to treating victims of the plague, which they did by administering an alcoholic elixir produced in their houses. There is no evidence that the Church hierarchy objected to the Jesuats’ medical practices or tried to put an end to them.

In fact, by all indications the Jesuats were a flourishing order. Pope Gregory XIII (1572–85), who patronized the Jesuits, also supported the Jesuats, and entered their founder, the Blessed John Colombini, in the official Church calendar, fixing July 31 as his feast day. The order expanded rapidly in the sixteenth and seventeenth centuries, and established dozens of houses across Italy. They must have been popular with the upper classes in Italian cities, since both the Cavalieris of Milan and the Angelis of Venice saw fit to send their gifted sons to be educated in the order. The fact that two members of the order occupied academic chairs in Bologna and Padua, among the most prestigious universities in Europe, added intellectual luster to the Jesuats that few orders could match. Although it is not easy to learn what life was like inside the Jesuat houses, nothing that we know suggests moral decrepitude. Cavalieri’s letter to Galileo in 1620 about life in the Jesuat establishment in Milan, in which he complained about being besieged by old men who expected him to study theology, does not give one a sense that this was in any way a party house. Nor does one get that feeling about the house in Bologna, where Cavalieri resided for the last eighteen years of his life, ill with gout, and where he engaged in mathematical discussions with the young Angeli. The inescapable impression is that these were establishments with a serious focus on academic learning and religious ministry, and the rapid advancement of both Cavalieri and Angeli to positions of authority in the order indicate that intellectual achievement was highly prized. Before 1668 the Vatican generally saw no reason to intervene in Jesuat affairs, except in 1606, when, for the first time, it allowed clergymen to enter the order—a change that suggests a rise rather than a diminishment in the order’s standing. There is nothing in all this that would explain why this old and venerable brotherhood was singled out for sudden annihilation.

But the Jesuats did stand out in one way: they counted among their members the most prominent Italian mathematicians promoting the doctrine of infinitesimals. First Cavalieri and then Angeli, each in turn, was the leading advocate for indivisibles in his generation, and both received the full backing of their order. Not only were they promoted rapidly through the ranks, but many of their books were personally approved by the general of the Jesuats. Inevitably, when Angeli and Cavalieri entered into a bitter conflict with the Jesuits over the infinitely small, the fight became not just their own, but that of their order as well. Whether by design or happenstance, the Jesuats of St. Jerome became the chief obstacle for the Jesuits in their drive to eradicate the infinitely small.

It is possible that if the Jesuits had found a way to silence Angeli while leaving his brothers in peace, they would have done so. But it is just as likely that they were eager to make an example of the smaller order, a warning to all those within the Church who would dare challenge the Society of Jesus. In the end, the result was the same. Unable to persuade the Venetian authorities to discipline the insolent professor, they turned instead to the papal Curia in Rome, where their influence was decisive. They could not punish Angeli directly, so they let their fury rain on the order that sheltered him and his late teacher. When faced with the wrath of the mighty Society of Jesus, the Jesuats never stood a chance. The order that had survived three hundred years of political and religious upheaval, whose brothers administered the waters of life to victims of the plague, and two of whose members rose to the heights of mathematical distinction, simply evaporated at the stroke of a papal pen.

Surreally, the man at the eye of the storm remained unmoved—at least geographically. Although the brotherhood that had been his home since his youth had suddenly dissolved around him, Angeli was still professor of mathematics at the University of Padua, and still protected by the Venetian senate. He remained in Padua for the next twenty-nine years, until his death in 1697. But even though he still professed himself an admirer of Galileo, and although he had previously published no fewer than nine books promoting and using the method of indivisibles, Angeli did not publish a single word on the topic ever again. The Jesuits had won.

TWO DREAMS OF MODERNITY

By the 1670s the war over the infinitely small had ended. With Angeli at long last silenced, and all the Jesuits’ rivals driven underground or melting away, Italy was a land cleansed of infinitesimals, and the Jesuits reigned supreme. It was, for the Society, a great triumph, and it came at the end of a difficult campaign that had claimed many victims along the way. Some of them were famous in their day, such as Luca Valerio and Stefano degli Angeli, but many more will remain forever anonymous. The cold hand of the Society of Jesus drew a curtain over this lost generation of Italian mathematicians and left them in the dark.