Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (51 page)

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.

T
HE
R
OYAL
S
OCIETY

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.

R
ULERS

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.

P
OPES

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 implemented 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.

O
THER
R
EFORMERS,
R
EVOLUTIONARIES, AND
C
OURTIERS

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.

Gerrard Winstanley (1609–76):
Leader of the Diggers, who began digging up the land in St. Georges Hill, Surrey, in 1649. Winstanley and his followers believed that land was common property and that all men had the right to cultivate it. Their activities alarmed the local property owners, who managed to dislodge them through legal actions and violent attacks. Fear of the Diggers and other radical groups pushed the propertied classes to overcome their differences, leading to the restoration of the monarchy in 1660.

Samuel Sorbière (1615–70):
French courtier, physician, and man of letters, as well as a friend and admirer of Thomas Hobbes. From 1663 to 1664 Sorbière visited England, spending much of the time as a guest of the Royal Society. His later account of the visit deeply offended his former hosts, especially his glorification of Hobbes and his ridicule of Wallis. This elicited a strong rebuttal from Thomas Sprat, and also ended Sorbière’s career at the French court.

TIME LINE

Sixth century BCE:
Pythagoras and his followers declare that “all is number,” meaning that everything in the world can be described by whole numbers or their ratio.

Fifth century BCE:
Democritus of Abdera uses infinitesimals to calculate the volume of cones and cylinders.

Fifth century BCE:
The Pythagorean Hippasus of Metapontum discovers incommensurability (i.e., irrational numbers). It follows that different magnitudes are not composed of distinct tiny atoms, or infinitesimals. After his discovery, Hippasus is mysteriously lost at sea, possibly drowned at the hands of his Pythagorean brethren.

Fifth century BCE:
Zeno of Elea proposes several paradoxes showing that infinitesimals lead to logical contradictions. Thereafter infinitesimals are shunned by ancient mathematicians.

300 BCE:
Euclid publishes his highly influential treatise on geometry,
The Elements
, which carefully avoids infinitesimals. It serves as a model for the style and practice of mathematics for nearly two thousand years.

Ca. 250 BCE:
Archimedes of Syracuse bucks the trend and experiments with infinitesimals. He arrives at remarkable new results regarding the areas and volumes enclosed by geometrical figures.

1517:
Martin Luther launches the Reformation by nailing a copy of his Ninety-Five Theses to the door of the Castle Church in Wittenberg. The ensuing struggles between Catholics and Protestants continue for two centuries.

1540:
Ignatius of Loyola founds the Society of Jesus, popularly known as the Jesuits, dedicated to reviving Catholic doctrines and restoring Church authority.

1544:
A Latin translation of the works of Archimedes is published in Basel, making his study of infinitesimals widely available to scholars for the first time.

1560:
Christopher Clavius begins teaching at the Jesuit Collegio Romano. He founds the Jesuit mathematical tradition on the bedrock of Euclidean geometry.

Late sixteenth to early seventeenth century: Revival of interest in infinitesimals among European mathematicians.

1601–15:
The Jesuit “Revisors General,” responsible for ruling on doctrines, produce a string of denunciations of infinitesimals.

1616:
The Jesuits clash with Galileo over his advocacy of Copernicanism, but also for his use of infinitesimals. Galileo tones down his rhetoric, but bides his time for a chance to reopen the debates.

1616:
The mathematician Luca Valerio sides with the Jesuits against his friend Galileo. He dies in disgrace soon after.

1618:
Outbreak of the Thirty Years’ War, pitting Catholics against Protestants.

1623:
Galileo’s friend Maffeo Barberini becomes Pope Urban VIII and sides openly with Galileo and his followers.

1623–31:
A golden “liberal age” in Rome. Galileans ascendant.

1625–27:
The Jesuit mathematician Gregory St. Vincent is prohibited by his superiors from publishing a work considered too close to infinitesimals.

1628:
Thomas Hobbes encounters a geometrical proof for the first time while on a European tour.

1629:
Bonaventura Cavalieri is appointed professor of mathematics at the University of Bologna.

1630s:
Evangelista Torricelli develops his infinitesimal methods, but publishes nothing.

1631:
The Protestant King Gustavus Adolphus of Sweden defeats the armies of the Holy Roman Emperor in the Battle of Breitenfeld during the Thirty Years’ War. His victory alters the European balance of power.

1631:
Under pressure from traditionalists, Urban renounces his liberal policies and restores the Jesuits to favor. End of Galilean ascendancy.

1632:
The Jesuit Revisors General issue the most comprehensive condemnation of infinitesimals to date. Similar decrees follow in subsequent years.

1632:
The Jesuit general superior, Mutio Vitelleschi, writes to the provinces to denounce infinitesimals.

1632–33:
Galileo is charged with heresy, tried by the Inquisition, and condemned to spend the rest of his life under house arrest, which he does in his villa in Arcetri, outside Florence.

1635:
Cavalieri publishes
Geometria indivisibilibus
, which becomes the standard work on infinitesimals across Europe.

1637:
Galileo’s
Dialogues Concerning Two New Sciences
is published in Leiden, in the Netherlands. The book discusses infinitesimals at length and praises Cavalieri as a “new Archimedes.”