Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (12 page)

Through it all, even in their advanced work, the Jesuits never deviated from their commitment to Euclidean geometry. It was the core of their teaching and the foundation of their mathematical practice. This was not a stylistic choice, but a deeply held ideological commitment: the whole point of studying and teaching mathematics was that it demonstrated how universal truth imposed itself upon the world—rationally, hierarchically, and inescapably. Ideally, the Jesuits believed, the truths of religion would be imposed on the world just like geometrical theorems, leaving no room for avoidance or denial by Protestants or other heretics and leading to the inevitable triumph of the Church. For the Jesuits, mathematics must be studied according to the principles and procedures of Euclid, or it should not be studied at all. A mathematics that ran counter to these practices not only was useless to their purposes, but it would challenge their unconquerable faith that truth, handed down through the hierarchy of the universal Catholic Church, would inevitably prevail.

THE SLOW-WITTED BEAST

Christopher Clavius died in Rome on February 12, 1612, at the height of his power and prestige. The struggles of his early years were far behind him, and “our Clavius,” as he was referred to in Jesuit documents, was one of the Society’s cherished treasures. He was the undisputed founder and leader of the brilliant mathematical school, which not only brought honor to the Jesuits but also increased their political clout when they tried to establish themselves as the intellectual authority of the Catholic Church. Even their great rivals the Dominicans could boast of no comparable accomplishment. As recently as 1610, Clavius was called upon to confirm or deny Galileo’s astounding telescopic observations, including his contention that there were mountains on the moon and that Jupiter was orbited by four moons. Clavius’s intervention was decisive: he supported Galileo, thereby ensuring that the discoveries were accepted almost universally as fact.

The Jesuits’ reverence for the aging Clavius comes through in the words of Jesuit astronomer Giambattista Riccioli, who commented in 1651 that “some would rather be blamed by Clavius than praised by others.” His many admirers outside the Society included the Danish astronomer Tycho Brahe, the Italian mathematicians Federico Commandino and Guidobaldo del Monte, and as eminent a figure as the Archbishop of Cologne, who wrote in 1597 that Clavius is regarded as “the father of mathematics” and is “venerated by the Spanish, the French, the Italians, and most Germans.” But Clavius did not lack for detractors either. Some, as would be expected, were Protestants, such as the German astronomer and mathematician Michael Maestlin, best known as Kepler’s mentor, who was harshly critical of the calendric reform. So was the French humanist Joseph Justus Scaliger, who despised all Jesuits and referred to Clavius as “a German beast with a big belly, slow witted and patient.”

Others, however, were Catholic. Cardinal Jacques Davy Duperron also found the livestock analogy useful, referring to Clavius as the “fat horse of Germany,” and the French mathematician François Viète, who got into a fierce fight with Clavius over the merits of the new calendar, denounced him as “a false mathematician and a false theologian.”

Such vitriolic denunciations, considered beyond the pale of academic discourse today, were not unusual in the sixteenth and seventeenth centuries. But even so, the references to Clavius as a slow-witted “beast” or “horse” were not idle insults for a man known for his girth. They bespoke a deeper criticism of Clavius, one that could not easily be brushed aside. Jacques-Auguste de Thou expresses it very clearly in his
History
of 1622, when he cites Viète’s view of the Jesuit: Clavius, de Thou writes, was a master expositor who possessed a talent for explaining the discoveries of others, but made no original contributions to the disciplines over which he presided. He was, in this view, nothing more than a beast of burden, capable of immense expenditures of energy on behalf of his cause but incapable of original insight.

And this, it must be said, is not an entirely unjust assessment. Clavius was, unquestionably, a great promoter of the mathematical sciences, raising their profile both inside and outside the Society. He was an effective organizer, who powered through political and organizational obstacles to establish his mathematical institute at the Collegio Romano. He was a master teacher, beloved and revered by generations of students, quite a few of whom became leading mathematicians in their own right. He was one of the leading pedagogues of the age, whose detailed mathematical curriculum profoundly shaped the teaching of mathematics in Europe for years to come. And perhaps most influentially, he was an author of textbooks, issuing repeated editions of his books on geometry, algebra, and astronomy.

“The fat horse of Germany.” Christopher Clavius around 1606. Engraving by E. de Boulonois, after a painting by Francisco Villamena. From I. Bullart,
Académie des Sciences
(Amsterdam: Daniel Elzevier, 1682).
(Photograph courtesy of the Huntington Library)

But was he a creative mathematician? His textbooks offer little evidence of this. His
Euclid
is essentially a latter-day exposition of the ancient text, though it has been pointed out that it contains some new results in the theory of combinations. His edition of Sacrobosco’s
Tractatus de Sphaera
does make use of some observations and theories that postdate the medieval original, but at a time when the traditional geocentric worldview was being challenged by Nicolaus Copernicus, Tycho Brahe, and Johannes Kepler, Clavius’s text is a strict defender of the old orthodoxy. And although he knew of Viète’s groundbreaking work, which is the foundation of modern algebra, Clavius’s
Algebra
contains no trace of it, summarizing instead the ideas of earlier Italian and German algebraists whose work pales by comparison. All of which is to say that de Thou’s depiction of Clavius as an unoriginal mathematician who never strayed far from the well-beaten path of his predecessors is supported by the evidence. And while this depiction was undoubtedly meant to insult the old Jesuit, it appears particularly severe in our own age, when mathematicians are judged almost exclusively by their creativity and originality.

To judge Clavius by this standard, however, would be an injustice. Clavius never wanted to make any original contributions to mathematics, and would have been quite happy if no one else did, either. “The theorems of Euclid and the rest of the mathematicians,” he explained in the “Prolegomena,” “still today as for many years past, retain … their true purity, their real certitude, and their strong and firm demonstrations.” Mathematics is worth studying, for Clavius, not because it offers a field for open-ended investigations and new discoveries, as it does for a modern mathematician, but precisely because it never changes: its results are as true today as they were in the faraway past, and as they will be in the distant future. Mathematics, more than any other field, offers stability, order, and unchanging eternal truths. New discoveries are not only irrelevant for this goal, but potentially disruptive, and should by no means be encouraged. Judged from this standpoint, Clavius may indeed have been one of the great mathematicians of his age, but he was a very different kind of mathematician from those we know today.

In sticking close to the old and established truths of mathematics, Clavius was being true to the intellectual traditions of his order and the decrees of its leaders. No one, General Acquaviva had warned, should even suspect the Jesuits of innovating, and while this entrenched conservatism applied to all fields of knowledge, it was particularly critical in mathematics. The entire case that Clavius had made for raising the status of mathematics in Jesuit schools rested on the fact that mathematics, more so than any other science, was fixed, orderly, and eternally true. Other fields might be pursued for other reasons: theology, because it was the study of the word of God; philosophy, because it was the study of the world and was essential for understanding theology. But why study mathematics? Only because it provided a model of perfect rational order and certainty, and an example of how universal truths governed the world. If mathematics was to become a field of far-reaching innovation, in which new truths were proposed and then subjected to challenge and debate, then it would be worse than useless. It would be dangerous, as it would compromise the very foundations of the truth it was meant to buttress.

Clavius’s imprint remained strong within the Jesuit mathematical tradition. For centuries, the Society’s mathematicians chose to stick to tried-and-true methods, following Euclid as much as possible and avoiding treacherous new fields. But in the very years that Clavius was establishing the powerful Jesuit school of mathematics, a very different mathematical practice was gaining ground that would put all his cherished principles to the test. Where the Jesuits insisted on clear and simple postulates, the new mathematicians relied on a vague intuition of the inner structure of matter; whereas the Jesuits celebrated absolute certainty, the new mathematicians proposed a method rife with paradoxes, and seemed to revel in them; and whereas the Jesuits sought to avoid controversy at all cost, the new method was mired in intractable controversies seemingly from its very inception. It was everything that the Jesuits thought mathematics must never be, and yet it flourished, gaining new ground and new adherents. It was known as the method of indivisibles.

3

Mathematical Disorder

THE SCIENTIST AND THE CARDINAL

In December of the year 1621, Galileo Galilei, mathematician and philosopher at the court of Grand Duke Ferdinando II of Tuscany, received a letter from an admirer, the twenty-three-year-old Milanese monk Bonaventura Cavalieri. Galileo had met Cavalieri in Florence some months before, was impressed with the young monk’s mathematical acumen, and invited him to continue their exchanges by post. Cavalieri did just that; his letter, filled with admiration and praise for the Florentine sage, reported on his latest mathematical work and asked for Galileo’s opinion on the radical new direction he was taking.

Galileo at the time was at the height of his power and fame, and was used to ambitious young men seeking his advice and patronage. It had been twelve years since, while still a mathematics professor at the University of Padua, he had built a telescope and pointed it at the skies. What he saw there changed humans’ view of the universe forever: innumerable stars invisible to the naked eye, mountains and valleys on the supposedly spherical disk of the moon, dark spots on the allegedly perfect surface of the sun. Most remarkable were the four tiny specks he observed circling Jupiter, which he inferred were moons orbiting the planet just as our moon orbits Earth. Galileo had quickly compiled his findings in a booklet he titled
Sidereus nuncius
(
The Starry Messenger
) and sent it to the leading scholars and astronomers of the day. The impact was immediate, and seemingly overnight the obscure professor became known across Europe as the man who had opened up the heavens. On visiting Rome in 1611, Galileo regaled the Pope with tales of his discoveries, and was invited to a friendly private audience with the Jesuit cardinal Bellarmine. At the Collegio Romano, Father Clavius, ever suspicious of innovation, initially demurred, quipping that in order to see such things one must first have put them inside the telescope. But he, too, came around when Jesuit astronomers confirmed the Florentine’s discoveries and gave them their blessings. The venerable mathematician, now in the last year of his life, was in attendance when the Jesuits celebrated Galileo with a day of lavish ceremonies at the Collegio.

Like many a professor then and now, Galileo was not enamored of his university teaching duties. Sensing an opportunity to be rid of this burden once and for all, he dedicated the
Sidereus nuncius
to the ruler of his native Florence, the grand duke Cosimo II de’Medici of Tuscany, hinting broadly that he would like nothing better than to join that prince’s household. To sweeten things further, he named the newly discovered satellites of Jupiter, in honor of the grand duke and his family, “the Medicean Stars,” thus inscribing the Medici name in the heavens for all time. The gambit worked: by 1611, Galileo had left Padua for the Medici court in Florence, where he was appointed chief mathematician and philosopher to the grand duke. Happily for Galileo, his new position came with no teaching obligations, even though he was officially named the head mathematician of the University of Pisa. It also came with a salary several times greater than he had enjoyed as a lowly professor.

Now famous, Galileo did not rest on his laurels. A flamboyant man with a quick wit and a sharp pen, he enjoyed the rough-and-tumble of scientific disputes. Not long after moving to Florence, he wrote his
Discourse on Floating Bodies
, which was, in effect, a direct attack on the principles of Aristotelian physics. By 1613 he had published his
Letters on Sunspots
, which tells the tale of his debate with the mysteriously named “Apelles” over the discovery and nature of this solar phenomenon. In it, Galileo argues that he was the first to observe sunspots (a claim that may have been sincere but was in any case mistaken). He also argues, correctly, that the spots are on the surface of the sun or very close to it, and that they demonstrate that the sun is rotating on its axis. Finally, moving beyond the immediate matter at hand, he claims the sunspots provide crucial support to the Copernican system, which placed the sun, and not the Earth, at the center of the cosmos. As Galileo knew well, these claims were sure to rile traditional Aristotelian scholars, who believed that the heavens were perfect and that the spots must be an atmospheric effect near the Earth. Things got even testier when “Apelles” was revealed as the Jesuit scholar Christoph Scheiner of Ingolstadt, who was deeply offended by Galileo’s ridicule. It was the first sign of friction between Galileo and the Jesuits, and it came only two years after he was publicly honored at the Collegio Romano. But it was far from the last, as the tensions between Galileo and the Society of Jesus would only grow in the years that followed. When, nearly twenty years later, the Florentine scientist would be put on trial by the Inquisition, accused and ultimately convicted of heresy, it was the Jesuits who led the charge.