Political pluralism in England was matched by an unprecedented degree of religious tolerance. The Puritans of the early seventeenth century were not a tolerant lot. They viewed themselves as God’s elect, and their zeal to impose their beliefs and morals on the broader population played no small part in launching the crisis of 1640 and the civil war that followed. But the tragedy of the civil war, broadly attributed to the clash of competing dogmas, gave rise to a more forgiving attitude toward religious truth. Many of the Anglican bishops in the aftermath of the Interregnum, and nearly all the leaders of the newly founded Royal Society, advocated a latitudinarian, rather than dogmatic, approach to religion. Instead of insisting on a strict religious doctrine and excluding anyone who did not fully subscribe to it, they advocated broad latitude in doctrinal matters, acknowledging that ultimate truth was something to search for, not a given. A range of different beliefs was welcomed within the Anglican fold, as long as those beliefs agreed on certain fundamental tenets, such as the Trinity and the supremacy of the king (rather than the Pope).
First advocated for the Church of England itself, religious pluralism was soon extended beyond the Church’s confines. The 1689 Act of Toleration, coming fast on the heels of the Glorious Revolution, granted freedom from persecution to nonconforming Protestant sects such as Presbyterians, Quakers, and Unitarians. Though restricted (until 1828) from many spheres of public life, and from the universities of Oxford and Cambridge, the dissenters were nevertheless secure from state interference and prospered both economically and intellectually. They formed their own churches and their own academies, which were often far more advanced in their teachings than the plodding Anglican universities. Catholicism was not as easily tolerated, however. Both detested and feared, it was associated with the danger of foreign intervention and papal claims of supremacy, and with the claims of the deposed Stuarts. But even though they suffered systematic discrimination, English Catholics were largely left in peace until their official emancipation in 1829.
Political and religious pluralism went hand in hand with scientific, intellectual, and economic openness. The Royal Society, along with the French Academy of Sciences, soon became the leading scientific academy in Europe, and English science set the standard for all Europe. In the realm of letters, England became a locus of public philosophical and political debates in which luminaries such as John Locke, Jonathan Swift, and Edmund Burke took opposing, brilliantly argued, sides. Political liberalization also made possible economic liberalization and private entrepreneurship on an unprecedented scale. The accumulation of capital and the growing size of workshops made it profitable to invest in new technologies, particularly the steam engine. As a result, by the late eighteenth century, England became the first industrialized country in the world, pulling far ahead of its continental rivals and leaving them scrambling to keep up.
Whether the continuum is made up of infinitesimals seems like the quaintest of questions, and it is hard for us to fathom the passions it unleashed. But when the struggle raged in the seventeenth century, the combatants on both sides believed that the answer could shape every facet of life in the modern world that was then coming into being. They were right: when the dust cleared, the champions of infinitesimals had won, their enemies defeated. And the world was never the same again.
DRAMATIS PERSONAE
THE “INFINITESIMALISTS”
Luca Valerio (1553–1618): A mathematician and friend of Galileo who made important contributions to infinitesimal methods. However, when Galileo clashed with the Jesuits in 1616 Valerio sided against him, and was fiercely denounced by his former friends. He died in disgrace shortly thereafter.
Galileo Galilei (1564–1642): The most celebrated scientist of his day. He was persecuted by the Jesuits for his advocacy of Copernicanism, which led to his trial and downfall. Galileo made use of infinitesimals in his work, and supported and encouraged a generation of young mathematicians to develop the concept. Even after his condemnation, he was still the undisputed leader of the Italian infinitesimalists.
Gregory St. Vincent (1584–1667): A Jesuit mathematician who developed a novel method for calculating the volumes of geometrical figures that involved infinite division. His Jesuit superiors considered the method too close to infinitesimals, and forbade him from publishing his work.
Bonaventura Cavalieri (1598–1647): Galileo’s disciple, later professor of mathematics at the University of Bologna, and a member of the order of the Jesuats. His books Geometria indivisibilibus (1635) and Exercitationes geometricae sex (1647) became the standard works on the new mathematics, which he called “the method of indivisibles.”
Evangelista Torricelli (1608–47): Galileo’s disciple and ultimately his successor in Florence. An ardent infinitesimalist, and far less concerned with technical rigor than Cavalieri, he was famous for his powerful and creative techniques, which involved calculating the “width” and “thickness” of infinitesimals. His Opera geometrica of 1644 was widely read by mathematicians across Europe, and Wallis in particular modeled his Arithmetica infinitorum on Torricelli’s work. Among Torricelli’s most surprising results was his success in calculating the volume of a solid of infinite length.
John Wallis (1616–1703): An ardent Parliamentarian and Puritan divine in the early years of the Interregnum, Wallis served as secretary to the Westminster Assembly of Divines. From the mid-1640s he was a regular participant in the private meetings that would later lead to the establishment of the Royal Society of London, and in 1649 he was appointed Savilian Professor of Geometry at the University of Oxford. Wallis made his name in mathematics as a leading infinitesimalist, and in politics as a relative pragmatist and moderate, in line with his fellows at the Royal Society. He was engaged in a decades-long war with Hobbes over his mathematics and his authoritarian politics.
Stefano degli Angeli (1623–97): A friend and disciple of Cavalieri, professor of mathematics at the University of Padua, and a member of the order of the Jesuats. In the 1650s and ’60s he was the last public voice in Italy defending infinitesimals and openly denouncing the Jesuits. But when the Jesuats were unceremoniously dissolved by the Pope in 1668, Angeli finally desisted, and never published on infinitesimals again.
THE “ANTI-INFINITESIMALISTS”
Christopher Clavius (1538–1612): Professor of mathematics at the Jesuit Collegio Romano and founder of the Jesuit mathematical tradition. Clavius insisted on a geometrical approach, which he valued for its orderliness, rigorous deductive method, and absolutely true results. He hoped to apply this methodology to all fields of knowledge, and was not interested in mathematical innovation. Clavius did not address infinitesimals directly, since they were hardly used by mathematicians during most of his career, but he was the author of the core principles of Jesuit mathematics, which led directly to the war on infinitesimals.
Paul Guldin (1577–1643): Leading Jesuit mathematician, charged with discrediting infinitesimals. He attacked Cavalieri’s method in his De centro gravitatis of 1641.
Mario Bettini (1584–1657): Mathematician who became the leading Jesuit critic of the infinitely small after Guldin’s death. He ridiculed infinitesimals in his collections of mathematical curiosities, the Apiaria universae philosophiae of 1642 and the Aerarium philosophiae mathematicae of 1648.
Thomas Hobbes (1588–1679): The author of Leviathan and advocate of an absolutist authoritarian state, Hobbes considered himself a mathematician as well. He believed that his philosophy was founded on mathematical principles, and that it was therefore as certain as a geometrical demonstration. The decrees of the Leviathan, he believed, will be as incontestable as geometrical proofs.
André Tacquet (1612–60): Leading Jesuit mathematician, also charged with discrediting infinitesimals. He denounced infinitesimal mathematics in Cylindricorum et annularium, published in 1651, but accepted their limited use as heuristic devices. He was subsequently directed by his superiors to refrain from publishing original work, and to focus exclusively on wr
iting textbooks. He did so.
JESUITS
Ignatius of Loyola (1491–1556): A Spanish nobleman and soldier from the Basque region, who experienced a religious awakening after being wounded in the Battle of Pamplona in 1521. With ten devoted followers he founded the Society of Jesus, which was officially recognized by Pope Paul III in 1540. Under his leadership the Jesuits became the most dynamic order of the Church and the most effective in battling the Reformation. By the time of Ignatius’s death, the Society had grown to one thousand members and several dozen schools and colleges, and was still expanding rapidly.
Benito Pereira (1536–1610): Clavius’s nemesis at the Collegio Romano, who insisted that mathematics did not qualify as a science. He was also the first Jesuit to directly condemn infinitesimals, although the context was not mathematics, but rather a commentary on Aristotle.
Claudio Acquaviva (1543–1615): Superior general of the Jesuits from 1581 to 1615, Acquaviva established the office of the Revisors General, and supported the early campaign against infinitesimals.
Mutio Vitelleschi (1563–1645): Superior general of the Jesuits from 1615 to 1645, during their period of eclipse (1623–31) and their return to power in Rome. He presided over the launch of the final campaign against infinitesimals, and wrote to the provinces to forbid the doctrine.
Jacob Bidermann (1578–1639): Leader of the Revisors General in 1632, when the Jesuits renewed their assault on infinitesimals.
Vincenzo Carafa (1585–1649): Superior general of the Jesuits from 1646 to 1649. Enforced the ban on infinitesimals, and humiliated Pallavicino by compelling him to retract his views. He wrote to his underlings to remain vigilant against infinitesimals, and began the process to add infinitesimals to a list of permanently banned doctrines.
Rodrigo de Arriaga (1592–1667): Leading Jesuit philosopher. In 1632 he published his Cursus philosophicus, which surprisingly concludes that infinitesimals are plausible. By the time the book was published, however, the Jesuits were back in power in Rome and determined to quash the support for infinitesimals. Superior General Carafa declared that there would be no more Arriagas.
Pietro Sforza Pallavicino (1607–67): A marchese (marquis) by birth, and an ardent Galilean in his youth, Pallavicino was exiled from Rome following Galileo’s fall. He returned and, sensing which way the wind was blowing, became a Jesuit, and ultimately a cardinal. Still attached to Galilean views, he taught at the Collegio Romano that infinitesimals were plausible. In 1649 he was denounced in a letter by the superior general, and forced to publicly retract his views.
THE ROYAL SOCIETY
Sir Francis Bacon (1561–1626): English jurist, philosopher, statesman, and Lord Chancellor to James I from 1618 to 1621. Although not a scientist himself, Bacon is considered one of the leading figures in the scientific revolution for his advocacy of empiricism in the study of nature. In a series of influential essays and books, Bacon argued that the proper way to investigate nature was through systematic observation and experimentation, rather than through a priori reasoning or mathematics. Long after his death, Bacon became the unofficial patron saint of the Royal Society, which championed his empiricist approach.
Henry Oldenburg (1619–77): German by birth, Oldenburg settled in London in the 1650s and became a key figure in intellectual and scientific circles, famous for his wide network of correspondents. Along with Robert Boyle, John Wallis, and others he was a leading founder of the Royal Society and served as its first secretary. In this capacity he guided the Society through its difficult early years, and established it as the leading scientific academy in Europe, known for its empiricist bent.
Robert Boyle (1627–91): A founder of modern chemistry, Boyle was the most distinguished and admired scientist among the early members of the Royal Society. Boyle advocated a humble empiricism as the correct approach to the study of nature, believing this would be to the benefit of both religion and the state.
Thomas Sprat (1635–1713): The leading propagandist of the early Royal Society. In 1667 he published the History of the Royal Society of London, which laid out the Society’s scientific principles as well as its political goals. Sprat argued that the experimental study of nature not only increases human knowledge, but also promotes civic and religious harmony. In 1665 Sprat was the author of a sharp satirical response to Sorbière’s account of his visit to England.
RULERS
Charles V (1500–58): Holy Roman Emperor from 1519 and king of Spain (as Charles I) from 1516 until his abdication in 1556. With domains stretching from eastern Europe to Peru, he was the titular ruler of one of the greatest empires in history, though his hold on his territories was often tenuous. Viewing himself as the sword of the Church, he confronted Luther in 1521 at the Diet (general assembly) of Worms, and issued an edict outlawing Luther and his teachings. He spent the rest of his reign trying unsuccessfully to eradicate Protestantism from his lands.
Gustavus Adolphus (1594–1632): King of Sweden from 1611, widely considered one of the greatest military innovators of all time. In June 1630 he landed with his army in northern Germany in support of the hard-pressed Protestant princes in the Thirty Years’ War. Over the two years that followed, he proceeded to defeat the Catholic armies of the Holy Roman Empire in a string of battles, dramatically changing the balance of power in Europe. The Swedish threat also changed the political calculus in Rome, ending a period of Galilean ascendancy and handing power back to the Jesuits. Gustavus died in the Battle of Lützen while leading a cavalry charge against imperial forces.
Oliver Cromwell (1599–1658): One of the leading commanders of the Parliamentary New Model Army during the English Civil War, as well as leader of the Puritan Independents (against the Presbyterians). He became lord protector of England, Scotland, and Ireland in 1653. Some believed that Hobbes’s Leviathan was written in support of his authoritarian rule.
Charles I (1600–49): King of England from 1625, his reign was marked by increasing confrontation with Parliament, and ultimately civil war. Following the example of the French kings, Charles tried to establish an absolute monarchy in England but encountered fierce resistance from Parliament, which controlled state revenues. His ill-fated attempt at personal rule led to the crisis of 1640 and the outbreak of civil war between Parliament and the king. Defeated in battle, Charles was captured by Parliamentary forces and executed in 1649.
Charles II (1630–85): The son of Charles I, he grew up in the court in exile, where he was tutored for a time by Hobbes. In 1660 he was recalled to England and reinstated as king by a coalition of former Parliamentarians and Royalists who were concerned about the rise of religious and social radicals. Careful to avoid the fate of his father, he ruled cautiously alongside Parliament. In 1662 he granted a royal charter to a group of natural philosophers who believed that the study of nature held the key to social and political peace. The group became the Royal Society of London.
POPES
Leo X (Pope from 1513 to 1521): A member of the Florentine Medici family, he was a cultured and learned man and a great patron of Renaissance art. But his slow and hesitant response to Luther’s challenge helped turn a local problem in Germany into an existential crisis for the Church.
Paul III (Pope from 1534 to 1549): After ascending to the Papal throne at the height of the Reformation, when the Protestant tide was sweeping all before it, he set in motion the counteroffensive that would restore the fortunes of the Catholic Church. In 1540 he granted Ignatius of Loyola’s request to found a new order called the Society of Jesus, which was destined to play a key role in the Counter-Reformation. In 1545 he convened the Council of Trent, which set the fundamental doctrines of the Catholic Church that are upheld to this day.
Gregory XIII (Pope from 1572 to 1585): A friend and protector of the Society of Jesus, he granted the Jesuits the land and resources to build a permanent home for their leading university, the Collegio Romano. He convened a commission for the reform of the calendar, in which Clavius played a key role, and implemen
ted its recommendations in 1582. The Gregorian calendar, almost universally in use today, is named after him.
Urban VIII (Pope from 1623 to 1644): A friend and protector of Galileo’s before ascending to the Papacy (known then as Cardinal Maffeo Barberini), Urban continued his patronage of Galileo as Pope, leading to a golden “liberal age” in Rome. But in 1632, following the publication of Galileo’s Dialogue on the Copernican system and some unfavorable political developments, Urban turned on Galileo, leading to the latter’s trial and banishment. The Jesuits returned to favor in Rome, and were given a free hand to suppress infinitesimals.
Clement IX (Pope from 1667 to 1669): As part of a short and undistinguished Papacy, Clement IX was responsible for suppressing the order of the Jesuats, of which two leading infinitesimalist mathematicians—Bonaventura Cavalieri and Stefano degli Angeli—were members.
OTHER REFORMERS, REVOLUTIONARIES, AND COURTIERS
Martin Luther (1483–1546): Originally an Augustinian monk and professor of theology at the University of Wittenberg, Luther launched the Reformation in 1517 by posting his Ninety-Five Theses on the door of the city’s Castle Church. By 1621 he had been excommunicated by the Pope and banned by the emperor, but the spread of Protestantism proved irreversible. Other religious reformers soon followed Luther’s lead, establishing their own brands of Protestantism.
Charles Cavendish (1594–1654): Respected mathematician and a member of one of the great aristocratic clans of England, known in the seventeenth century as patrons and practitioners of the arts and sciences. His brother William (1592–1676), the Duke of Newcastle, maintained a laboratory on the grounds of his estate, and William’s wife, Margaret (1623–73), was an acclaimed poet and essayist. The Cavendishes turned their manors of Chatsworth and Welbeck Abbey into thriving intellectual centers, and were lifelong patrons of Hobbes.
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