“The quadrature of the circle”: The account is based on Jesseph, Squaring the Circle, pp. 22–26; and Archimedes, “Measurement of a Circle,” chap. 6 in E. J. Dijketerhuis, Archimedes (Princeton, NJ: Princeton University Press, 1987), pp. 222–23.
the three classical problems were simply insoluble: On views on the solubility of the quadrature of the circle, see Jesseph, Squaring the Circle, pp. 25–26.
“that which has no parts”: For Hobbes’s discussion of Euclid’s definitions, see Six Lessons, 7:201.
“If the magnitude of a body which is moved”: Hobbes, De corpore, 2.8.11, reprinted in Latin in Thomas Hobbes, Opera philosophica (London: John Bohn, 1839), 1:98–99, more commonly known as Hobbes’s Opera Latina. The passage is translated in Jesseph, Squaring the Circle, pp. 76–77.
Points have a size, lines have a width: On Hobbes’s view that points have size and lines have width in order to construct geometrical bodies, see Hobbes, Six Lessons, p. 318.
“conatus”: For Hobbes’s discussion of his concepts “conatus” and “impetus,” see De corpore, 3.15.2, and Opera Latina, 1:177–78, both translated in Jesseph, Squaring the Circle, pp. 102–103.
“instead of saying that a line is long”: Sorbière, A Voyage to England, p. 94.
“Every demonstration is flawed”: Hobbes, De principiis, chap. 12, quoted in Jesseph, Squaring the Circle, p. 135.
“speake loudest in his praise”: Ward published this anonymously in [Seth Ward], Vindiciae academiarum (Oxford: L. Litchfield, 1654), p. 57, quoted in Jesseph, Squaring the Circle, p. 126.
It was a trap, and Hobbes knew it: On the beginnings of the war between Hobbes and Wallis, and Ward’s role in it, see Jesseph, Squaring the Circle, p. 126.
“problematically”: The disclaimer is in Hobbes, De corpore (1655), p. 181. Translation from Latin in Jesseph, Squaring the Circle, p. 128. The titles of the chapters are in De corpore, p. 171.
were all gleefully exposed: See John Wallis, Elenchus geometriae Hobbianae (Oxford: H. Hall for John Crooke, 1655).
“Solutions of Problems that have hitherto remained insoluble”: Sorbière, A Voyage to England, p. 94.
The attempted proofs of the quadrature: For a modern exposition of two of these proofs, see Jesseph, “Two of Hobbes’s Quadratures from De corpore, Part 3, Chapter 20,” in Jesseph, Squaring the Circle, pp. 368–76.
squaring of the circle is impossible: For a fuller discussion of the impossibility of squaring the circle, see Jesseph, Squaring the Circle, pp. 22–28.
“solved some most difficult problems”: Hobbes’s list of accomplishments is included in Aubrey, “Thomas Hobbes,” pp. 400–401. Translated in Jesseph, Squaring the Circle, pp. 3–4.
8. Who Was John Wallis?
“we Tast and See it to be so”: John Wallis, Truth Tried (London: Richard Bishop for Samuel Gellibrand, 1643), pp. 60–61.
“Beside his constant preaching”: The account of Wallis’s childhood and the quotes are from John Wallis, “Autobiography,” in Christoph J. Scriba, “The Autobiography of John Wallis, F.R.S.,” Notes and Records of the Royal Society of London 25, no. 1 (June 1970): 21–23.
“Mr Holbech was very kind to me”: Ibid., p. 25.
“scarce bred any man that was loyall to his Prince”: The quote is from The Autobiography of Sir John Bramston, quoted in Vivienne Larminie, “Holbeach [Holbech], Martin,” in Oxford Dictionary of National Biography (Oxford: Oxford University Press, 2004–12).
“were scarce looked upon as Accademical studies”: Wallis, “Autobiography,” p. 27.
“a pleasing Diversion at spare hours”: Ibid.
“Knowledge is no Burthen”: Ibid., p. 29.
the trial of Archbishop William Laud: Agnes Mary Clerke, “Wallis, John (1616–1703),” Dictionary of National Biography, vol. 59 (1899).
A Serious and Faithful Representation: The pamphlet’s full title was A Serious and Faithful Representation of the Judgements of Ministers of the Gospel within the Province of London, Contained in a Letter from Them to the General and His Counsel of War, Delivered to His Excellency by Some of the Subscribers, January 18, 1649 (printed in Edinburgh, 1703).
“now to be my serious study”: Wallis, “Autobiography,” p. 40.
two mathematical treatises: Wallis’s two treatises were De sectionibus conicis (Oxford: Leon Lichfield, 1655) and Arithmetica infinitorum (Oxford: Leon Lichfield, 1656), both included in Wallis, Opera mathematicorum (Oxford: Leonard Lichfield, 1656–57), vol. 2.
Presbyterian referred to respectable clergymen: Wallis wrote, “When they were called Presbyterians it was not in the sense of Anti-Episcopal, but Anti-Independents.” See Wallis, “Autobiography,” p. 35.
custos archivorum: On Wallis’s election as keeper of the archives and Stubbes’s opposition, see Christoph J. Scriba, “John Wallis,” in Gillispie, ed., Dictionary of Scientific Biography. On Stubbe and Hobbes, see Jesseph, Squaring the Circle, p. 12. On the demonization of Hobbes, see Mintz, “Thomas Hobbes,” in Gillispie, ed., Dictionary of Scientific Biography.
he be burned at the stake: John Aubrey, “Thomas Hobbes,” p. 339nc.
“Their first purpose”: Thomas Sprat, History of the Royal Society of London (London: T.R., 1667), p. 53.
“such a candid and unpassionate company as that was”: Ibid., pp. 55–56.
Now, with his new companions: On Wallis’s involvement with the “Invisible College” during the Interregnum, and of the group’s diverse fields of interest, see Wallis, “Autobiography,” pp. 39–40.
“continued such meetings in Oxford”: Wallis, “Autobiography,” p. 40.
Philosophical Transactions: The other candidate for “first scientific journal” is Le journal des scavans of the French Academy of Sciences, whose first issue appeared two months before Philosophical Transactions.
“gives us room to differ, without animosity”: Sprat, History of the Royal Society, p. 56.
“they work and think in company”: Ibid., p. 427.use their experience to reconstitute the entire body politic: For more on the early Royal Society and its mission to recast English political life and prevent a return to the disastrous dogmatism of the Interregnum, see Shapin and Schaffer, Leviathan and the Air-Pump; Margaret C. Jacob, The Newtonians and the English Revolution 1689–1720 (New York: Gordon and Breach, 1990), first published in 1976; James R. Jacob, Robert Boyle and the English Revolution (New York: Burt Franklin and Co., 1977); Barbara J. Shapiro, Probability and Certainty in Seventeenth Century England (Princeton, NJ: Princeton University Press, 1983); Steven Shapin, A Social History of Truth: Civility and Science in Seventeenth-Century England (Chicago: University of Chicago Press, 1995).
Cartesian philosophy: The most concise summary of Descartes’s philosophy is contained in his eminently readable Discourse on the Method, first published anonymously in Leiden in 1637 as Discours de la méthode.
“more imperious, and impatient of contradiction”: Sprat’s views on the dangers of dogmatism can be found in Sprat, History of the Royal Society, p. 33.
“slowness of the increase of knowledge amongst men”: Ibid., p. 428.
“The reason of men’s contemning all Jurisdiction and Power”: Ibid., p. 430.
“the most fruitful parent of Sedition is Pride”: Ibid., p. 428–29.
a public statement of the goals and purpose of the Royal Society: On Sprat and the grandees of the Royal Society, see the “Introduction” in Jackson I. Cope and Harold Whitmore Jones, eds., The History of the Royal Society by Thomas Sprat (St. Louis, MO: Washington University Studies, 1958), esp. pp. xiii–xiv.
dogmatism leads to sedition: In Sprat’s words, “it gives them fearless confidence in their own judgments, it leads them from contending in sport to opposition in earnest … in the State as well as in the Schools.” See Sprat, History of the Royal Society, p. 429.
“the influence of experiments is Obedience to the Civil Government”: Sprat, History of the Royal Society, p. 427.
“that teaches men humility�
�: On the beneficial effects of experimentalism, see ibid., p. 429.
“one great man”: For the Royal Society’s idolization of Bacon, see ibid., p. 35.
Francis Bacon, Lord Chancellor to James I: Bacon’s major works include The Advancement of Learning (1605), Novum organum (1620), and New Atlantis (1627).
True knowledge, worthy of: True knowledge, in the Aristotelian scheme, was referred to as scientia, and required absolute certainty based on logical reasoning and ancient authority.
“the daintiness and pride of mathematicians”: The quote is from Francis Bacon, “Of the Dignity and Advancement of Learning,” book 3, chap. 6, in James Spedding, ed., The Philosophical Works of Francis Bacon, vol. 4 (London: Longman and Co., 1861), p. 370.
It was a conundrum that left the Society with an ambivalence toward mathematics: On the early Royal Society’s ambivalence toward mathematics, and particularly Robert Boyle’s suspicion of the field, see Shapin, A Social History of Truth, chap. 7.
9. Mathematics for a New World
“It hath been my endeavour”: John Wallis, “Autobiography,” in Scriba, “The Autobiography of John Wallis, F.R.S.,” p. 42.
“On Conic Sections”: John Wallis, De sectionibus conicis, nova methodo expositis, tractatus (Oxford: Leon Lichfield, 1655). On the publication of this treatise, along with the Arithmetica infinitorum, see Jacqueline Stedall, trans., The Arithmetic of Infinitesimals, John Wallis, 1656 (New York: Springer-Verlag, 2004), p. xvii.
“any plane is made up, so to speak, of infinite parallel lines”: Wallis, De sectionibus conicis, prop. 1, in Wallis, Opera mathematica (Oxford: Theatro Sheldoniana, 1695), p. 297.
“the area of the triangle is equal to the base times half the altitude”: Wallis, De sectionibus conicis, prop. 3, in Opera mathematica, p. 299.
When criticized by Fermat: Fermat’s criticisms are included in a wide-ranging correspondence of Wallis’s Arithmetica infinitorum, which Wallis published in 1658 under the title Commercium epistolicum. Fermat’s letters were published in French translation in volumes 2 and 3 of Paul Tannery and Charles Henri, eds., Oeuvres de Fermat (Paris: Gauthiers-Villars et Fils, 1894–96).
“Mathematical entities exist”: The quote is from Wallis, Mathesis universalis (Oxford: Leon Lichfield, 1657), chapter 3; reprinted in Wallis, Opera Mathematica (Oxford: Sheldonian Theatre, 1695), p.21.
“it seems not in the power of the Will to reject”: Ibid., pp. 60–61.
“a general proposition may become known by induction”: Wallis, Arithmetica infinitorum (Oxford: Leon Lichfield, 1656), p. 1, prop. 1. Translation is from Stedall, trans., The Arithmetic of Infinitesimals, p. 13.
study minute creatures under a microscope: Hooke’s startling enlarged images of common insects and microbes invisible to the naked eye were published in Robert Hooke, Micrographia or Some Physiological Descriptions of Minute Bodies Made by Magnifying Glasses (London: John Allestry, 1667).
“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.
&n
bsp; “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.
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Page 38