Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (57 page)

“If there is taken a series of quantities”
: Wallis,
Arithmetica infinitorum
, prop. 2, from Stedall, trans.,
The Arithmetic of Infinitesimals
, p. 14. Wallis includes an additional step demonstrating that what is true of the series 0, 1, 2, 3 … is also true of any arithmetic series beginning with 0.

“a very good Method of
Investigation
”
: Wallis’s discussion of induction is in John Wallis,
A Treatise of Algebra, Both Historical and Practical
(London: John Playford, 1685), p. 306.

Wallis moved on to do the same for more complex series
: Wallis,
Arithmetica infinitorum
, prop. 19, from Stedall, trans.,
The Arithmetic of Infinitesimals
, p. 26.

“quantities that are as squares of arithmetic proportionals”
: Wallis,
Arithmetica infinitorum
, prop. 21, from Stedall, trans.,
The Arithmetic of Infinitesimals
, p. 27.

“quantities that are as cubes of arithmetic proportionals”
: Wallis,
Arithmetica infinitorum
, prop. 41, from Stedall, trans.,
The Arithmetic of Infinitesimals
, p. 40.

must be true for all powers m of natural numbers
: Wallis,
Arithmetica infinitorum
, prop. 44, from Stedall, trans.,
The Arithmetic of Infinitesimals
, p. 42.

he was engaged in a lively debate with Wallis
: Wallis published the entire exchange as
Commercium epistolicum de quaestionibus quibusdam mathematicis nuper habitum
(Oxford: A. Lichfield, 1658). In addition to Wallis and Fermat, it included letters from Sir Kenelm Digby, Lord Brouncker, Bernard Frénicle de Bessy, and Frans van Schooten. Fermat’s critique of the
Arithmetica infinitorum
is contained mostly in “Epistola XIII,” a letter from Fermat to Lord Brouncker, written in French, that was forwarded to Wallis. Fermat’s contributions to the exchange are also published in Paul Tannery and Charles Henry, eds.,
Oeuvres de Fermat
, vols. 2 and 3 (Paris: Gauthier-Villars et Fils, 1894 and 1896). “Epistola XIII” of Wallis’s
Commercium epistolicum
is printed here as letter LXXXV in 2:347–53.

“But his method of demonstration”
: Fermat to Digby, August 15, 1657, Epistola XII on p. 21 of the
Commercium epistolicum
. Also letter LXXXIV in Tannery and Henry, eds.,
Oeuvres de Fermat
, 2:343.

“one must settle for nothing less than a demonstration”
: “Epistola XIII,” on pp. 27–28 of the
Commercium epistolicum
. Also letter LXXXV in Tannery and Henry, eds.,
Oeuvres de Fermat
, 2:352.

was fully answered in Cavalieri’s books
: Wallis’s assertion that his method is derived from Cavalieri is first stated in the dedication to the
Arithmetica infinitorum
, from Stedall, trans.,
The Arithmetic of Infinitesimals
, pp. 1–2.

“was already done to his hand by Cavallerius”
: Wallis,
Treatise of Algebra
, p. 305. The equivalence of Cavalieri’s method of indivisibles and the method of exhaustion is discussed on p. 280, and the composition of lines, surfaces, and solids on p. 285.

“You may find this work new”
: Wallis, dedication to
Arithmetica infinitorum
, from Stedall, trans.,
The Arithmetic of Infinitesimals
, p. 1.

“If any think them less valuable”
: Wallis,
A Treatise of Algebra
, p. 298. The claim that induction needs no additional demonstration is on p. 306.

“
Euclide
was wont to be so pedantick”
: Quoted from Wallis,
A Treatise of Algebra
, p. 306.

“[M]ost mathematicians that I have seen”
: Quoted ibid., p. 308.

“a conclusive argument”
: Wallis makes the argument that the truth of a demonstration is based on the agreement of “most men” in Wallis,
A Treatise of Algebra
, pp. 307–308.

Elenchus geometriae Hobbianae: John Wallis,
Elenchus geometriae Hobbianae
(Oxford: H. Hall for John Crooke, 1655).

Decameron physiologicum: Thomas Hobbes,
Decameron physiologicum
(London: John Crooke for William Crooke, 1678).

Seth Ward had traveled to London
: On Ward and Hobbes, see Jesseph,
Squaring the Circle
, p. 50.

“the equal of ‘Leviathan’”
: John Wallis, dedication to John Owen of
Elenchus
, folios A2r, A2v. The translation from the Latin original is from letter 37 in Peter Toon, ed.,
The Correspondence of John Owen
(Cambridge: James Clarke and Co. Ltd., 1970), pp. 86–88.

“I have done that business for which Dr. Wallis receives the wages”
: Thomas Hobbes, Epistle Dedicatory to Henry Lord Pierrepont to
Six Lessons
. See Molesworth, ed.,
The English Works of Thomas Hobbes
, 7:185.

“the most deformed necessary business which you do in your chambers”
: Hobbes,
Six Lessons
, 7:248.

“scab of symbols”
: Ibid., 7:316.

“the pompous ostentation of Lines and Figures”
: Wallis,
A Treatise of Algebra
, p. 298.

“You do shift and wiggle”
: Thomas Hobbes,
STIGMAI
,
or markes of the absurd geometry, rural language, Scottish church-politicks, and barbarisms of John Wallis
(London: Andrew Crooke, 1657), p. 12, quoted in Stedall, trans.,
The Arithmetic of Infinites
, pp. xxix–xxx.

“Here comes the beare to be bayted!”
: The anecdote is included in Aubrey’s biography of Hobbes, “Thomas Hobbes,” p. 340.

“should I undertake to refute his Geometry”
: John Wallis, dedication to John Owen of
Elenchus
, p. 86.

“set such store by geometry”
: John Wallis,
Elenchus
, p. 108, quoted in Jesseph,
Squaring the Circle
, p. 341.

“there is no more to be feared of this Leviathan”
: John Wallis, dedication to John Owen of
Elenchus
, p. 87.

“how little he understands this mathematics”
: Wallis to Huygens, January 11, 1659, quoted in Jesseph,
Squaring the Circle
, p. 70.

“like Beetles from my egestions”
: Hobbes,
Six Lessons
, 7:324.

“I do not wish to change, confirm, or argue”
: Hobbes to Sorbière, 7/17 March, 1664, quoted in Jesseph,
Squaring the Circle
, pp. 272–73.

“Who ever, before you”
: Wallis,
Elenchus
, p. 6, quoted in Jesseph,
Squaring the Circle
, p. 78–79.

“
was not so much to shew a Method of Demonstrating things already known
”: Wallis,
Treatise of Algebra
, p. 305.

“with all the ecclesiastics of England”
: The discussion of the motive for writing
Leviathan
is in Hobbes,
Six Lessons
, 7:335. The letter to Sorbière is quoted in Simon Schaffer, “Wallification: Thomas Hobbes on School Divinity and Experimental Pneumatics,”
Studies in History and Philosophy of Science
19 (1988): 286.

“Egregious logicians and geometricians”
: Hobbes,
Six Lessons
, 7:308.

“Is this the language of geometry?”
: Ibid.

“If you say that by the parallels you mean infinitely little parallelograms”
: Ibid., 7:310.

“it could hardly have been proposed by a sane person”
: Thomas Hobbes,
Lux mathematica
(1672), in William Molesworth, ed.,
Thomae Hobbes Malmesburiensis opera philosophica
, vol. 5 (London: Longman, Brown, Green, and Longmans, 1845), p. 110, quoted and translated in Jesseph,
Squaring the Circle
, p. 182.

“Nor can there be anything infinitely small”
: Hobbes,
Lux mathematica
, 5:109, quoted and translated in Jesseph,
Squaring the Circle
, p. 182.

Epilogue

the dozens of sermons he delivered
: The sermons were collected in John Wallis,
Three Sermons Concerning the Sacred Trinity
(London: Thomas Parkhurst, 1691); and John Wallis,
Theological Discourses and Sermons on Several Occasions
(London: Thomas Parkhurst, 1692).

“a man of most admirable fine parts”
: The quote is by his younger contemporary, the English antiquarian Thomas Hearne. Quoted in “John Wallis,” in Sidney Lee, ed.,
Dictionary of National Biography
, vol. 49 (London: Smith, Elder, and Co., 1899), p. 144.

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