Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Alexander, Amir

  ACKNOWLEDGMENTS

  The roots of this book go far back, to my first year as a graduate student at Stanford, when I wrote a paper arguing that infinitesimals were politically subversive in seventeenth-century Europe. In the following years my research interests carried me elsewhere, first to the maritime culture of early modern exploration, and then to the “romantic turn” in mathematics in the early nineteenth century. But I never forgot that early insight, and never doubted that I would one day tell the story. It took longer than I thought, but I finally did. And because I have been thinking about this topic for more than two decades, the list of those I have consulted and whose comments helped shape this book is a long one.

  I would like to thank Timothy Lenoir, Peter Galison, and Moti Feingold, who commented on that paper years ago, as well as Douglas Jesseph, whose detailed critiques spurred me to refine and improve the argument. I spent hours talking about these issues with Christophe Lecuyer, Jutta Sperling, Phillip Thurtle, Josh Feinstein, and Patricia Mázon, my graduate school peers at the time. In later years my colleagues at UCLA were my sounding board, and I thank Margaret (Peg) Jacob, Mary Terrall, Ted Porter, Norton Wise, Soraya de Chadarevian, and Sharon Traweek for their insights and friendship. Carla Rita Palmerino kindly gave me access to her notes from the Jesuit Archives, and Ugo Baldini helped guide me through the maze of Jesuit sources.

  Steven Vanden Broecke became a good friend during a quarter of shared office space, and contributed penetrating comments and a deep knowledge of the early modern world. Conversations with Joan Richards and Arkady Plotnitsky helped shape my thinking on mathematics and broader culture, and Mario Biagioli and Massimo Mazzotti deepened my understanding of early modern Italy and the place of mathematics in its society. Reviel Netz’s “Mathematics as Literature / Mathematics as Text” workshop gave me an opportunity to test-run some of these ideas before a lively and well-informed group, and I benefited greatly from his thoughtful suggestions. Doron Zeilberger, Michael Harris, and Jordan Ellenberg have been generous with mathematical advice, and Siegfried Zielinski has been an example of intellectual open-mindedness. Apostolos Doxiadis, in both his writings and his public outreach, showed me that mathematics, when beautifully presented, has a broad, devoted, and enthusiastic audience.

  Amanda Moon of Farrar, Straus and Giroux shepherded the book through all its stages, from acquisition to publication, always providing incisive and helpful advice. Her colleagues Debra Helfand, Delia Casa, Jenna Dolan, Debra Fried, and Jennie Cohen worked diligently on all aspects of the book from copyediting to proofreading to production, turning a bland-looking electronic file into an elegantly written and beautiful final product. Dan Gerstle read every word of an early draft and made many suggestions, and Laird Gallagher brought a sharp editorial eye to later versions of the text. Both unquestionably made this a better book. Lisa Adams of the Garamond Agency was with this project from its earliest conception to its fruition, and I can truthfully say that Infinitesimal would never have come to be without her advice, support, and professionalism. My childhood friend Daniel Baraz has been a constant presence in my life despite living on the other side of the world. His friendship helped sustain me throughout the process.

  To Bonnie, my love: thank you for being the best wife any man could wish for. Your intelligence and support are in every page of this book. My children were with me throughout the planning, writing, and production process, but they are now embarking on their own life adventures away from home. I will miss their daily presence and companionship, as well as their energy, intelligence, and creativity, and our long talks about everything from football to the Iliad to the art of writing. Jordan and Ella, wherever your paths may lead, my love will always follow.

  INDEX

  The index that appears in the print version of this title does not match the pages in your e-book. Please use the search function on your e-reading device to search for terms of interest. For your reference, the terms that appear in the print index are listed below.

  absolute infinity

  absolute rule of kings

  Académie des Sciences

  Academy

  Academy of the Desirous

  Accademia dei Lincei

  Achilles and the Tortoise

  Acquaviva, Claudio; indivisibles opposed by; innovation disdained by; Revisors created by

  Acta Eruditorum (Leibniz)

  Act of Toleration (1689)

  Adolphus, Gustavus

  Aerarium philosophiae mathematicae (Bettini)

  aerodynamics

  Against the Murderous, Thieving Hordes of the Peasants (Luther)

  air currents

  Alberti, Leon Battista

  Albrecht of Hohenzollern

  alchemy

  Alcibiades, Albert

  Aldobrandini, Cardinal

  Alexander VI, Pope

  algebra

  Algebra (Clavius)

  algebraic numbers

  “all the lines”

  “all the planes”

  Alvarado

  Ammannati, Bartolommeo

  Anabaptists

  analysis

  Angeli, Stefano degli

  Anglican Church, see Church of England

  Antwerp Cathedral

  Apiaria universae philosophiae mathematicae (Bettini)

  Apollonius

  Apostolic Clerics of St. Jerome, see Jesuats

  Archimedes

  Archimedian spiral

  architecture

  Aristotelian physics

  Aristotle; indivisibles opposed by

  Aristotle’s wheel

  arithmetic

  Arithmetica infinitorum (Wallis)

  Arriaga, Rodrigo de

  Arti, Duke of

  Arundel, Earl of

  Assayer, The (Galileo)

  Assembly of Divines

  astrolabe

  Astronomia nova (Kepler)

  astronomy

  atoms

  Aubrey, John

  Augsburg

  Austria

  authority

  axiom of Archimedes

  Babbage, Charles

  Bacon, Francis

  Bacon, Roger

  Baldi, Bernardino

  Bamberg

  Barberini, Antonio

  Barberini, Francesco

  Barberini, Maffeo (Pope Urban VIII)

  barometers

  Barrow, Isaac

  Basilica of Santa Croce

  Bavaria

  Beaugrand, Jean

  Belgium

  Bellarmine, Robert; death of; heliocentrism edict of; papal supremacy defended by

  Benedictines

  Berkeley, George, infinitesimals mocked by

  Bernini, Gian Lorenzo

  Bernoulli, Daniel

  Bettini, Mario

  Biberus, Nithard

  Bible; science vs.

  Biblioteca Nazionale Centrale

  Bibliotheca selecta (Possevino)

  Bidermann, Jacob

  black holes

  Black Robes

  black swans

  blood, circulation of

  Bockelson, Jan

  Bohemia

  Boleyn, Anne

  Bologna, University of

  Bolognese Senate

  Bolsheviks

  Borgia, Cesare

  Borgia, Francis, Duke of Gandia

  Borgia, Gaspar

  Borromeo, Federico

  Boscovich, Roger

  Bosse, Abraham

  Botticelli, Sandro

  Boulliau, Ismael

  Boulonois, E. de

  Bourbons

  Boyle, Robert

  Brahe, Tycho

  Bramante, Donato

  Brazil

  Breitenfeld, Battle of

  Briggs, Henry

  Brouncker, Lord

  Brunelleschi, Filippo

  Bucer, Martin

  Burke, Edmund

  business models

  calculus; creation of; infinitesim
al

  calendar reform

  Calvin, John

  Calvinism

  Cambridge University

  Campion, Edmund

  Canisius, Peter

  Canons Regular

  Cappon, Ignace

  Carafa, Vincenzo

  Cardano, Gerolamo

  Cartesian philosophy

  Castelli, Benedetto

  Castel Sant’Angelo

  Catherine of Aragon

  Catherine of Siena

  Catholic Church, Catholicism; Bible requiring interpretation in; on good works; hierarchy of; Hobbes’s criticism of; infinitesimals and; Luther’s opposition to; Protestantism vs.; suppressed in England

  Cavalieri, Bonaventura; Angeli as disciple of; background of; Bettini’s dispute with; on composition of continuum; conservatism of; death of; Guldin’s dispute with; indivisibles supported by; Jesuats joined by; Jesuits’ attack on; mathematics taught by; method of indivisibles created by; paradoxes as seen by; parallelogram proof of; Tacquet’s dispute with; Torricelli vs.

  Cavaliers

  Cavendish, Charles

  Cavendish, Margaret

  Cavendish, William

  Cavendish family

  centers of gravity

  Centrobaryca (Guldin)

  Cesarini, Virginio

  Cesi, Federico

  Charlemagne

  Charles I, King of England

  Charles II, King of England; restoration of

  Charles V, Emperor

  chemistry

  China

  Christina of Lorraine

  Church of England; creation of; indivisibles opposed by; reform of

  Church of St. Ignatius

  Ciampoli, Giovanni

  circle, area of

  civil engineering

  Civil War, English; causes of; as context of Leviathan; threat of anarchy in

  Clarendon, Earl of

  Clavis mathematicae (Oughtred)

  Clavius, Christopher; calendar reformed by; death of; Galileo’s astronomical discoveries accepted by; geometry seen as central by; Hobbes vs.; Jesuit reverence for; lack of originality of; mathematical academy created by; mathematical curriculum of; mathematics seen as key to religious disputes by; mathematics seen as stable by; mathematics taught by; received into Society of Jesus; textbooks written by

  Clement VII, Pope

  Clement VIII, Pope

  Clement IX, Pope

  codes

  cohesion

  College of Cardinals

  Collegio Romano; founding of; Galileo honored by; Galileo’s dispute with; indivisibles banned by; mathematics academy created at; Pallavicino’s defense of dissertation in

  Cologne

  Colombini, John

  comets

  Commandino, Federico

  compass

  conatus

  Concis (Apollonius)

  Constitutions (Ignatius of Loyola)

  constructive proofs

  continuum: Cavalieri on; Galileo and; indivisibles in; paradoxes of, see paradoxes; physical reality and

  contradictions, see paradoxes

  Copernicus, Nicolaus, heliocentric theory of

  cossists

  Council of Constance

  Council of Nicea

  Council of Trent

  Counter-Reformation

  Crick, Francis

  Cromwell, Oliver

  Cromwell, Richard

  curing wounds at a distance

  Cursus philosophicus (Arriaga)

  Cylindrica et annularia

  Cylindricorum et annularium (Tacquet)

  d’Alembert, Jean

  Darwin, Charles

  d’Aviso, Urbano

  Decameron physiologicum (Hobbes)

  De centro gravitatis (Guldin)

  De centro gravitatis (Valerio)

  De cive (Hobbes)

  deconstruction

  De corpore (Hobbes)

  De Corpore Politico, or the Elements of the Law (Hobbes)

  deduction

  deductive proofs

  De homine (Hobbes)

  De infinitis parabolis (Angeli)

  de la Chaise, François

  della Rovere family

  del Monte, Guidobaldo

  de Lugo, Giovanni

  Democritus

  de Nemore, Jordanus

  Denmark

  De revolutionibus (Copernicus)

  Descartes, René; infinitesimals disdained by

  De sectionibus conicis (Wallis)

  de Thou, Jacques-Auguste

  Dialogue on the Two Chief World Systems (Galileo)

  Dialogues Concerning Two New Sciences (Galileo)

  Diet of Worms

  differential calculus

  Digby, Kenelm

  Diggers

  Discourse on Floating Bodies (Galileo)

  Discourses and Mathematical Demonstrations Relating to Two New Sciences (Galileo)

  Disputationes metaphysicae (Suárez)

  divine grace

  divine omnipotence

  divine right of kings

  DNA

  Dominicans

  Donatello

  doubling of cube

  Drake, Francis

  ductus plani in planum

  Duperron, Jacques Davy

  Dutch Republic

  Easter

  Eck, Johann

  Edward VI, King of England

  Einstein, Albert

  electricity

  electric motors

  electrodynamics

  electronics

  Elementorum philosophiae sectio tertia de cive (Hobbes)

  Elements, The (Euclid); Clavius’s edition of

  Elenchus geometriae Hobbianae (Wallis)

  Elizabeth I, Queen of England

  engineering

  England; Catholicism suppressed in; Dutch war with; manners in

  Enlightenment

  Enthusiasts

  Estates-General

  Eucharist

  Euclid

  Euclid (Clavius)

  Euclidean geometry; as construction of world; deduction in; method of presentation in; pure reasoning in; rigidity of

  Eudoxus of Cnidus

  Euler, Leonhard

  evolution

  Exercitationes geometricae sex (Cavalieri)

  experiments; in math; as standing for tolerance

  Fairfax, Thomas

  faith; and scientific research

  Faithorne, William

  Farnese family

  Favre, Pierre

  Ferdinand II, Duke of Tuscany

  Ferdinand II, Emperor

  Fermat, Pierre de

  Fifth Monarchy Men

  Finland

  fluids

  fluxions

  Fontenelle, Bernard Le Bovier de

  Fordinus

  fortifications

  Foscarini, Paolo

  Fourier, Joseph

  France; English alliance with; Huguenots expelled from; Jesuits banned from teaching in; manners in; Protestants in

  Francisco de Toledo

  Francis I, King of France

  Francis Xavier

  Franck, Sébastian

  Franklin, Benjamin

  Frederick I, King of Denmark

  Frederick the Wise of Saxony

  freedom to philosophize

  French Academy of Sciences

  French Revolution

  French Royal Academy of Sciences

  Galen

  Galileo Galilei; astronomical discoveries of; Barberini’s friendship with; conservatism of; and continuum; death of; denunciation of; fame of; freedom to philosophize advocated by; heliocentrism defended by; Hobbes’s visit to; house arrest and humiliation of; indivisibles accepted by; Jesuits’ tension with; Urban’s discussions and relationship with; Urban’s removal of protection from; Wallis influenced by

  Galvani, Luigi

  Gandia, Duke of

  Gandia, Spain

  Gassendi, Pie
rre

  General Congregation

  generators

  Geneva, Switzerland

  geography

  Geometria indivisibilibus (Cavalieri)

  geometrical curves

  geometry; as embodiment of order; in Plato’s Academy

  Germany

  Germany, Nazi

  Gilbert, William

  Giotto

  Glorious Revolution

  gnomon; semi-

  Goa

  Gonzaga, Charles, Duke of Nevers

  good works, Catholic belief in

  Grassi, Orazio

  gravitation

  Gregorian Calendar

  Gregory XIII, Pope

  Gregory XV, Pope

  Grienberger, Christoph

  Guldin, Paul; Cavalieri disputed by

  gunpowder

  Gustavus I Vasa, King of Sweden

  Habsburgs

  Hall of the Consistory

  Harriot, Thomas

  Harvey, William

  Hawking, Stephen

  heat diffusion

  heliocentrism

  Henri IV, King of France

  Henry VIII, King of England

  hierarchy: of Church; Hobbes’s advocacy of; indivisibles and; of Jesuits; of knowledge; and opposition to indivisibles; papal supremacy in; in religion; of truth

  Hieronymites

  Hippasus

  History (de Thou)

  History of the Royal Society (Sprat)

  Hitler, Adolf

  Hobbes, Francis

  Hobbes, Thomas; as Bacon’s secretary; Clavius vs.; education of; as ejected from ranks of mathematicians; flaws in geometry seen by; Galileo visited by; geometry as model of philosophy of; geometry reformed by; grand tours of; indivisibles interpreted by; indivisibles rejected by; introduction to mathematics of; and knowledge as based on logic; material notions in mathematics of; own philosophy praised by; quadrature of circle and; rigidity of; as Royalist in Civil War; Scholasticism disdained by; Sorbière’s praise for; Wallis’s dispute with

  Hobbes, Thomas (father)

  Holbeach, Martin

  Holy Roman Empire

  homogenes

  Hooke, Robert

  Horace

  horror vacui

  Huguenots

  human nature

  Human Nature (Hobbes)

  Huygens, Christiaan

  hydrodynamics

  Ignatius His Conclave (Donne)

  Ignatius of Loyola; Aquinas as authority of; background of; Clavius received into Society of Jesus by; Jesuits created by; mathematical education valued by; order defended by; sainthood of; and start of Jesuit colleges

  Imperial Cortes

  impetus

  incommensurability

  Independents