Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Alexander, Amir

  Legem impone subactis—impose your rule upon the subjects!—was the motto of the Accademia Parthenia at the Collegio Romano, open to those at the college who were exceptionally devoted to the Jesuit ideals and way of life. Accompanying the motto was its equally transparent coat of arms, known as an “impresse.” At the top, seated on a throne, is the female figure of Theology. Flanking her, on a lower plane, are her servants Philosophy and Mathematics, reclining and awaiting her command. And so it was in the Society’s schools, where theology reigned as the “queen of the sciences” and imposed her rule upon subordinate subjects. It is a system of knowledge that seems alien to us today, even stifling, designed as it was to establish absolute truths and quash dissent. But the Jesuits believed that the purpose of education was not to encourage the free exchange of ideas, but to inculcate certain truths. And in that, they were undeniably successful.

  AN UNAPPRECIATED MAN

  So things stood in the first decades of the Society of Jesus, when mathematics, if addressed at all, was taught only to the extent that it was useful for other, higher disciplines. And so things would likely have remained were it not for the work of one man who made it his life’s mission to bring mathematics to the center of the Jesuit curriculum. It was thanks to his efforts that, by the dawn of the seventeenth century, the Jesuits had become not only skilled teachers of mathematics, but also leading scholars in the field, numbering among their own some of the most prominent mathematicians in all Europe. His name was Christopher Clavius.

  Little is known of Clavius’s early years—even his true birth name remains in doubt—but we do know that he was born on March 25, 1538, in the city of Bamberg, in the south German province of Franconia. As the seat of a Catholic prince-bishop, but surrounded by the Protestant territories of Nuremberg, Hesse, and Saxony, Bamberg was on the front lines of the struggle for the soul of the Holy Roman Empire. It was cities such as Bamberg that were targeted by the Jesuit Peter Canisius as he barnstormed across the empire, reviving the sagging spirits of the faithful and exhorting them to take a stand against the encroaching Protestant tide. It is easy to imagine the young Clavius attending one of Canisius’s giant Masses at Bamberg Cathedral and being moved by his fiery preaching, but we don’t know this for a fact. What we do know is that, in 1555, as his home city was fending off the forces of the Protestant margrave Albert Alcibiades, Clavius was in Rome. On April 12 he was received as a novice into the Society of Jesus by Ignatius of Loyola himself.

  Clavius was just seventeen when he joined the Society, but he was thirty-seven by the time he professed his final solemn vows. Even considering the lengthy and rigorous Jesuit training regimen, twenty years is an unusually long time for a bright young man to rise from novice to fully formed Jesuit, especially for one who was recognized for his promise early on, and who would ultimately become one of the most famous Jesuits of the age. But it may have had something to do with the fact that Clavius spent much of that time campaigning inside the Society for an unpopular cause: raising the status of mathematics in the Jesuit hierarchy of knowledge and improving the teaching of it in the order’s schools. Some Jesuits, such as Benito Pereira, who was Clavius’s colleague at the Collegio Romano, vigorously opposed him. Nevertheless, by the time Clavius joined the ranks of the “professed” Jesuits in 1575, he was well on his way to winning the fight.

  Clavius spent only a year in Rome after being admitted to the Society as a novice before he was sent off to the Jesuit house in Coimbra, Portugal. Unlike the secluded monasteries of traditional orders such as the Benedictines, such “houses” (or “residences”) were located in the heart of the city or town. There, the local Jesuits lived as a tight-knit community under an appointed superior, and emerged daily to conduct their activities in the broader community. Little is known about the four years Clavius spent in Coimbra in his late teens and early twenties, but they were undoubtedly formative ones. The city was famous in those days as the seat of an ancient university, whose most celebrated resident was Pedro Nuñez, one of the great mathematicians and astronomers of the age. There is no direct evidence that Clavius studied under Nuñez, but the mathematician Bernardino Baldi (1553–1617), who wrote a short biography of Clavius, does mention that the two knew each other. To be sure, given the young German’s interests and the small size of the University of Coimbra, it is hard to imagine that Clavius and Nuñez did not meet. But, for the most part, according to Baldi, Clavius was self-taught, gaining his knowledge of mathematics through his own careful study of classical mathematical texts.

  When Clavius was recalled to Rome in 1560, it was to continue his study of philosophy and theology, and to teach mathematics. In 1563 he was lecturing on mathematics at the Collegio Romano, and around 1565, when thirty years old, he became a professor of mathematics, a position that he would hold more or less continuously until his death forty-seven years later. Up to this point, Clavius’s career was respectable but hardly remarkable. Although recognized by his superiors for his mathematical abilities, he was nevertheless just a young faculty member toiling in obscurity among colleagues who did not much respect his field of expertise. Even years later he was still fighting for the right of the mathematics professor to take part in public ceremonies and disputations along with his colleagues, a complaint that suggests that this was not usually done. And despite holding a chair at the Society’s flagship college, he was excluded for years from the ranks of the “professed,” which tells us all we need to know about his status in the rigid hierarchy of the order.

  But sometime between 1572 and 1575, more than a decade after his return from the provinces, Clavius’s career took a dramatic turn. The newly elected Pope Gregory XIII assembled a distinguished commission to deal with an issue that had troubled the Church for centuries: calendar reform. As technical adviser to the commission the Pope selected the young Jesuit professor of the Collegio Romano who was making a name for himself as an expert on mathematical and astronomical matters. The appointment was unquestionably a great honor for Clavius, putting him at the center of one of the most ambitious projects the Church had undertaken. It also made him the official representative of the Jesuits in a high-ranking panel of the Church, whose recommendations would be known to all and would be scrutinized by scholars across Europe. Placed in such a visible position, Clavius was expected to bring honor and distinction to the Society, and enhance its prestige in the papal court. It was a difficult and even risky proposition for a young and rather obscure professor of mathematics. But Clavius and his cause had been waiting for just such an opportunity.

  ORDERING THE UNIVERSE

  The problem the commission was called to address had been in the making for more than twelve hundred years. Back in the year 325 CE the Council of Nicea had determined that Easter should be celebrated on the first full moon after the vernal equinox, which, according to the council, fell on March 21. Unfortunately the Julian calendar that was used at the time did not quite match the true length of the solar year—the time that it takes the sun to return to the exact same spot in the sky. Whereas the Julian year is 365 days and 6 hours, the true solar year is almost exactly 11 minutes shorter. Such a minuscule discrepancy does not matter from one year to the next, or even over a person’s lifetime, but an error of 11 minutes repeated more than 1,200 times does add up. By the 1570s the date of the vernal equinox had slipped to March 11, and the date of Easter, the most important feast in the Christian calendar, had slipped with it. If nothing were done to correct the problem, the error would continue to grow, and Easter continue to slip. The lunar calendar, meanwhile, which is used to calculate when a full moon will occur, had a comparable problem, slipping one day every 310 years. By the sixteenth century the full moon would appear four days after the date predicted by the calendar.

  All this was unacceptable: not only was the date of Easter at stake, but the entire religious calendar of feasts and saints’ days, not to mention the seasonal and agricultural calendar, was thrown into disarray. Already in the th
irteenth century the English philosopher Roger Bacon had complained that the calendar was “intolerable to all the wise, horrible to all astronomers, and ridiculed by all computists.” In truth, the very sense of time and its regularities was perturbed for all Christendom, and the Church, guardian of the sacred rhythms of life, was called to take action. Several Church councils, beginning with the Council of Constance (1414–18), tried to address the problem, but nothing came of these efforts. Finally the Council of Trent, which met periodically in the northern Italian town of Trento between 1545 and 1563, ordered that a special commission be convened for the express purpose of reforming the calendar. Around a decade later, the newly elected Pope Gregory XIII finally acted on the council’s decree.

  The task of the commission, of which Clavius was a member, was complex. First it had to determine the exact size of the errors in the Julian and lunar calendars. Then it had to produce new lunar tables that would accurately predict the future phases of the moon. Finally, it had to correct for the cumulative slippage that had already taken place, and propose a new calendar that would prevent the error from recurring. In 1577 the commission sent a “compendium” of proposed changes to leading Catholic scholars, soliciting comments and suggestions. After reviewing and sorting through the many responses, the commission was particularly impressed with the elegant and simple proposals of the Calabrian doctor Aloysius Lilius. In September of 1580, when the commission presented its conclusions to the Pope, it based its recommendations largely on Lilius’s suggestions.

  The first recommendation was for an immediate one-time correction to the calendar that would eliminate ten days. To prevent the problem from reemerging in future centuries, the commission also proposed a permanent adjustment to the Julian calendar: As before, every year that was divisible by 4 would be a leap year, lasting 366 days instead of 365. But unlike the old calendar, years that were divisible by 100 (e.g., 1800, 1900) would be standard 365-day years, with the exception that years that were divisible by 400 would remain leap years. The combined effect would be to reduce the average length of a year by 10 minutes and 48 seconds, effectively synchronizing the calendar year with the solar year. Henceforth the vernal equinox would always fall on March 21. In February 1582, in the the papal bull “Inter gravissimas,” the Pope made it official: Accepting the commission’s recommendations, he decreed that Thursday, October 4, of that year would be followed by Friday, October 15, making 1582 the only 355-day year on record. He also instituted the calendar devised by Clavius and his colleagues, the one still used around the world today. It is known, appropriately, as the Gregorian calendar.

  Throughout this entire process Clavius’s astronomical and mathematical expertise was indispensable. It had been his job to present the most up-to-date astronomical calculations to his less technically adept colleagues on the committee. He also undoubtedly played a leading role in recalculating the phases of the moon and in vetting various scholars’ proposals for calendar reform. Through it all, he proved himself not only an excellent mathematician and astronomer, but also someone who could effectively navigate the intricate politics of the papal court. In later years, when the other members of the commission had returned to their regular occupations, Clavius would emerge as the public spokesman for the new system, publishing a six-hundred-page “explanation” of the new calendar and taking on its vociferous critics. The obscure and underappreciated professor at the Collegio Romano had become a leading mathematician, a spokesperson for the rising mathematical sciences, a “professed” Jesuit, and a public face of the order. He would never look back.

  A MATHEMATICAL VICTORY

  The Gregorian reform of the calendar was a spectacular triumph for the Catholic Church in the dark years of its struggle with the Protestant “heretics.” Here was the Pope exercising his universal authority to correct a problem that had troubled all Christians for more than a millennium. In a display of near-godlike power, the Pope transformed the year, the religious festivals, and the seasons for millions across the globe. A bas-relief on a monument to Pope Gregory XIII in St. Peter’s Basilica in Rome by the sculptor Camillo Rusconi shows members of the calendar reform commission, with Clavius (according to Jesuit tradition) kneeling in the middle. He is presenting the new calendar to the Pope, who is seated on his throne with his arms spread wide, gesturing toward a globe, as if it were once again his and no one else’s. While the Pope’s Protestant enemies certainly counted among their number scholars as learned and accomplished as Clavius and his fellow commission members, no Protestant prince or clergyman could become, as the Pope had, a master of time itself.

  Protestants had no choice but to acknowledge the binding power of the Pope’s proclamation, and his unrivaled ability to reorder the universe. Just how troubling this was to them can be glimpsed in Ignatius His Conclave, an anti-Jesuit satire by the English poet and clergyman John Donne dating from 1611. Ignatius, in Donne’s depiction, resides in hell with his associates, Clavius among them. “Our Clavius,” Ignatius proclaims, is to be honored

  for the great pains … which he tooke in the Gregorian Calendar, by which both the peace of the Church and the Civill businesses have been egregiously troubled: nor hath heaven itself escaped his violence, but hath ever since obeied his appointments: so that S. Stephen, John Baptist, & all the rest, which have been commanded to work miracles at certaine appointed daies … do not now attend till the day come, as they were accustomed, but are awaked ten daies sooner, and constrained by him to come downe from heaven to do that businesse.

  The biting satire does not mask the anti-Catholic Donne’s sincere consternation at having to succumb to the Pope’s reordering of both religious and civil time.

  Protestant princes were forced into an unpleasant choice: they could either accept the Gregorian calendar, and thereby implicitly acknowledge the universal authority of the Pope, or they could reject it, and knowingly retain an embarrassingly erroneous calendar. Feeling cornered, they reacted with understandable confusion. Queen Elizabeth I of England at first announced that she would go along with the reform, only to backtrack in the face of opposition from the Church of England. The Gregorian reform would not come to the British Isles until 1752. The Dutch Republic split, with some provinces adopting the reform immediately while others retained the Julian calendar until 1700; and Sweden went back and forth between the two calendars until finally settling on the Gregorian in 1753. Farther east, the Russian Orthodox Church, whose quarrel with the Pope predated Luther’s by seven hundred years, also held on to the Julian calendar, until the Bolsheviks, who could not be suspected of being papal agents, imposed the reformed calendar in 1918. The last European nation to adopt the Gregorian calendar was Greece, in 1923, almost three and a half centuries after Clavius and his associates completed their work. By enacting a calendar that effectively took over the world, the Roman Church exhibited commanding authority, whereas its rivals showed only weakness and confusion, and the inherent limitations of their national churches.

  The calendar reform was precisely the kind of triumph the Society of Jesus was laboring to achieve. Here was a perfect example of the Catholic Church imposing truth, order, and regularity upon an unruly world. Like St. Ignatius in Rubens’s masterpiece, Pope Gregory was bringing the light of universal truth to the people, who had long suffered in darkness and confusion. The reaction to the reforms confirmed this: wherever the Pope’s command was law, order, peace, and truth prevailed; wherever heretics and schismatics ruled, error, confusion, and strife persisted. Nothing could better illustrate the justice of the approach that was at the core of the Jesuit worldview. Here, the Jesuits believed, was a template for the ultimate triumph of the Roman Church.

  The decisive victory of the Roman Church in the matter of the calendar seemed all the more striking when compared to the stalemate that prevailed in other areas of theological dispute. Catholics, for example, believed that the grace of God was bestowed upon sinners only through the holy Church and its sacraments, enacted by an
ordained priest. Protestants, conversely, believed in a “priesthood of all believers,” meaning that God would bestow his grace directly upon them. Catholics believed that Christ was physically present in the bread and wine during the sacrament of the Mass. Protestants believed that Christ was either present everywhere (Luther) or that the Mass was a mere commemoration of his sufferings (Zwingli). Catholics believed that God would take into account a man’s good works in this world in determining whether he would be saved or lost. Protestants believed that only faith and divine grace mattered. Catholics believed that the Bible required interpretation by the hierarchy and the traditions of the Church. Protestants believed that the Bible was a clear guide for righteous behavior, accessible to anyone. And so on. What these arguments had (and still have) in common is that they are entirely inconclusive. From Luther’s day to ours, neither side has given an inch, or has seen any reason to.

  To be sure, advocates on both sides engaged in passionate, and often violent, debate. They published crude caricatures of each other, depicting Luther as the Devil’s emissary, or the Pope as anti-Christ, and disseminated them as broadly as the new technology of the printing press allowed. They published popular pamphlets denouncing each other’s doctrines as heretical, and catechisms detailing the fundamentals of each faith. They authored learned treatises such as Calvin’s Institutes of the Christian Religion or the Jesuit Francisco Suárez’s Disputationes metaphysicae, and they engaged, occasionally, in formal debate, as Luther had with Eck in 1519. But despite the effort, time, and resources invested in these battles, neither side was able to impose its position upon the other. What a contrast between this bog of indecision and the glorious, clear-cut victory afforded the Roman Church by the reform of the calendar! If only the secret of the calendrical triumph could be infused into those other fields, the ultimate victory of Pope and Church would be assured.