Until recently, historians of mathematics usually agreed in refusing to consider the numerous geometrical publications of Thomas Hobbes as a contribution to the development of mathematics in the seventeenth century. From time to time, one could find statements that although Hobbes did not find new theorems he undoubtedly had profound insights into the logical foundations of mathematics, but these occasional remarks did not encourage historians to go deeper into Hobbes's mathematical thought. In the end, the general conclusion was that Hobbes's preoccupation with squaring the circle, doubling the cube (starting when the philosopher was more than forty years of age), and challenging Euclid's definitions were better ignored, at least in the history of science. In particular, his controversy with the Savilian professors Seth Ward and John Wallis was seen as a ‘deplorable affair’, liable only to damage the reputation of the protagonists.
The author wishes to thank Michael Hunter for his valuable comments and suggestions on earlier drafts of this essay and to acknowledge the support provided by the Gustav Kettel-Stiftung, Königswinter.
As far as possible, the works of Thomas Hobbes are quoted from the first editions; in addition, the locations in the editions of Sir William Molesworth, London, 1839-1845, English Works, 11 volumes, and Opera Latina, 5 volumes (subsequently quoted as: EW, respectively OL) are indicated. The Elements of Law, Natural and Politic, are quoted from the edition of Ferdinand Tonnies (2nd edn, London, 1969; subsequently quoted as: Elements).
1 For a comprehensive survey on Hobbes and the historians of mathematics see Schuhmann, Karl, ‘Geometrie und Philosophie bei Thomas Hobbes’, Philosophisches Jahrbuch (1985), 92, 161–77, especially 171f, and Breidert, Wolfgang, ‘Les mathematiques et la méthode mathématique chez Hobbes’, Revue Internationale de Philosophie (1979) 33, 415–31.
2 Cf. De Morgan, August, A Budget of Paradoxes, 2nd edn (ed. Smith, David E.), 2 vols., n.p., 1915 (reprinted Freeport, New York, 1969), i, 110; Hoffmann, J. C. V., ‘Zur Geschichte der Mathcmatik (Der englische Philosoph Hobbes als Mathematiker)’, Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht (1902), 32, 262–7, especially 265; Boyer, Carl B., The History of the Calculus and Its Conceptual Development, 2nd edn, New York, 1949 (reprinted 1959 etc.), 175–8.
3 Scott, Joseph Frederick, The Mathematical Work of John Wallis, London, 1938 (reprinted New York, 1981), 166–72, quoted from 166. See also Robertson, George Croom, Hobbes, Edinburgh, 1886 (reprinted, New York, 1971), 160–85, especially 162.
4 Shapin, Steven and Schaffer, Simon, Leviathan and the Air-Pump, Princeton, 1985, especially chapter 3, ‘Seeing double: Hobbes's politics of plenism before 1660’, 80–109. Mancosu, Paolo and Vailati, Ezio, ‘Torricelli's infinitely long solid and its philosophical reception in the seventeenth century’, Isis (1991), 82, 50–70; Giorello, Giulio, ‘Pratica geometrica e immagine della matematica in Thomas Hobbes’, in Napoli, Andrea (ed.), Hobbes oggi, Milano, 1990, 215–44.
5 The continuation of the discussion between Hobbes and Wallis will be investigated in a sequel to this paper.
6 Ward, Seth, A Philosophicall Essay Towards an Eviction of the Being and Attributes of God, The Immortality of the Souls of Men, The Truth and Authority of Scripture, Oxford, 1652, ‘To the Reader’, A3ff (subsequently quoted as: Essay).
7 See Hunter, Michael, ‘Science and heterodoxy: an early modern problem reconsidered’, in Lindberg, David C. and Westman, Robert S. (eds.), Reappraisals of the Scientific Revolution, Cambridge, 1990, 437–60.
8 Blackbourne, Richard, ‘Vitae Hobbianae auctarium’ in Thomae Hobbes Angli Malmesburiensis philosophi vita, London, 1681, 21–221, reprinted in OL, i, pp. xxii–lxxx; cf. p. lxxiii; EW, vii, p. 334, 339.
9 Robertson, , op. cit. (3), 169. Metzger, Hans-Dieter, Thomas Hobbes and die Englische Revolution 1640–1660, Stuttgart-Bad Cannstatt, 1991, 188, follows Robertson but makes the important observation that Ward's remarks in the Essay (op. cit. 6) already contain the main elements of later critiques of the Leviathan.
10 Mintz, Samuel I., The Hunting of Leviathan, Cambridge, 1962, 157.
11 Ibid., 55.
12 Shapiro, Barbara J., Probability and Certainty in Seventeenth-Century England, Princeton, 1983, 83f; see also Hunter, , op. cit. (7), 442.
13 Bernhardt, Jean and Tricaud, François, ‘Thomas Hobbes: Biographie’, in Schobinger, Jean Pierre (ed.), Die Philosophie des 17. Jahrhunderts. Band 3: England (= Grundriss der Geschichte der Philosophie, begründet von Friedrich Ueberweg. Völlig neu bearbeitete Auflage), Basel, 1988, 121; Schaffer, Simon, ‘Wallifaction: Thomas Hobbes on school divinity and experimental pneumatics’, Studies in the History and Philosophy of Science (1988), 19, 285; Giorello, Giulio, op. cit. (4), 229.
14 Vindiciae academiarum Containing, Some briefe Animadversions upon Mr. Websters Book, Stiled, The Examination of Academies. Together with an Appendix concerning what M. Hobbs, and M. Dell have published on this Argument, Oxford, 1654; reprinted in: Debus, Allen G., Science and Education in the Seventeenth Century. The Webster–Ward Debate, London and New York, 1970, 193–259. (Subsequently quoted as: Vindiciae.)
15 Although Reik, Miriam M. (The Golden Lands of Thomas Hobbes, Detroit 1977, 83 n. 12) stresses the point that a considerable part of the clergy had been prejudiced against Hobbes already by the publication of De cive, recent research confirms that the Leviathan was to bring about the turning point in the Anglicans' attitude to Hobbes: see Tuck, Richard, ‘The “Christian Atheism” of Thomas Hobbes’, in Hunter, Michael and Wootton, David (eds.), Atheism from the Reformation to the Enlightenment, Oxford, 1992, 111–30.
16 See Hobbes, Thomas, Leviathan, London, 1651, especially Part IV, ‘Of the kingdome of darknesse’, 333–88 (EW, iii, 603–700); Ward referred to these reproaches in the Vindiciae, op. cit. (14), 54. See also Tuck, Richard, ‘Introduction’ to Hobbes, Thomas, Leviathan, Cambridge, 1991, pp. xxii–xxv.
17 See Shapiro, , op. cit. (12), 83f.
18 We should bear in mind that the Savilian chairs of geometry and of astronomy did not exist before 1619. The unsettled theological position of the Experimental Philosophy is delineated in chapter 7 of Shapin, and Schaffer, , op. cit. (4), 283–331. See also Malcolm, Noel, ‘Hobbes and the Royal Society’, in Rogers, G. A. J. and Ryan, Alan (eds.), Perspectives on Thomas Hobbes, Oxford, 1988, 43–66, especially 57f.
19 Essay, ‘To the Reader’, op. cit. (6), A3.
20 Hobbes, Thomas, Humane Nature: or, The Fundamental Elements of Policie, London, 1650 (cf. EW, iv, 1–76; Elements, 1–69; subsequently quoted as: Humane Nature). For the question of the authorship of Ward see Tönnies, Ferdinand, Studien zur Philosophie und Gesellschaftslehre im 17. Jahrhundert, Stuttgart-Bad Cannstatt, 1975, 39f; see also EW, vii, 336.
21 See Pope, Walter, The Life of the Right Reverend Father in God Seth, Lord Bishop of Salisbury, And Chancellor of the Most Noble Order of the Garter. With a Brief Account of Bishop Wilkins, Dr. Isaac Barrow, Mr. Lawrence Rooke, Dr. Turbervile, And others, London, 1697, and Fletcher, J. M. J., ‘Seth Ward, Bishop of Salisbury, 1667–1689’, Wiltshire Archaeological and Natural History Magazine (1940), 49, 1–16, especially 2–5. But see à Wood, Anthony, Athenae Oxonienses (ed. Bliss, Ph.), 3rd edn, 4 vols., London, 1813 (reprinted, 1967), iv, 246–52.
22 Metzger, , op. cit. (9), 89–103, 121–30, 146–57.
23 Ibid., 156f. See also Tuck, , op. cit. (16), p. xxv.
24 So the English translation in the Seven Philosophical Problems (EW, vii, 5f). (Quae cum ita sint, lectores meos monitos hic vellem, ne malevolorum convitiis temere credentes aliter de me quam aequum est sentire velint: nec vitio vertant, si contra hostes tuos pugnans, et quaecunque potui tela corripiens, gladio uno usus sum ancipite. OL, iv, p. 303).
25 See Dzelzainis, Martin, ‘Edward Hyde and Thomas Hobbes's Elements of Law, Natural and Politic’, The Historical Journal (1989), 32, 303–17, especially 305f; see also Metzger, , op. cit. (9), 93f.
26 See The Nicholas Papers. Correspondence of Sir Edward Nicholas, Secretary of State (ed. Warner, George F.), Oxford, 1886, reprinted, New York and London, 1965, i, 286f.
27 Malcolm, , op. cit. (18), 51–4.
28 See Tuck, Richard, ‘Hobbes and Descartes’, and Malcolm, , op. cit. (18), 11–42, especially 11ff and 51f; see also Beaulieu, Armand, ‘Les relations de Hobbes et de Mersenne’, in Yves Charles, Zarka and Bernhardt, Jean (eds.) Thomas Hobbes: Philosophie première, théorie de la science et politique, Paris, 1990, 81–90.
29 See Controversiae de verâ circuli mensurâ anno MDCXLIV exortae, inter Christianum Severini, Longomontanum, Cimbrum, Superiorum Mathematum in Regiâ Danorum Academiâ Havniensi, professorem publicum et loannem Pellium, Coritano-regnum, Anglum, Matheseos, in illustri Amstelodamiensium Gymnasia, professorem publicum, pars prima, Amstelodami, 1647, 46. Three years later, Robert Boyle had informed Samuel Hartlib of the results of (probably anatomical) studies of his; Hartlib communicated the letter to Hobbes who welcomed Boyle's results. See Hartlib, to Boyle, , The Works of the Honourable Robert Boyle (ed. Birch, Thomas), London, 1772, vi, 77; see also Skinner, Quentin, ‘Thomas Hobbes and the nature of the early Royal Society’, The Historical Journal (1969), 12, 232.
30 See Huygens, Christiaan, Oeuvres complètes, La Haye, 1888ff, i, 176f. Hobbes also has spoken favourably of this tract to Christiaan's brother Lodewijck. See Huygens, Lodewijck, The English Journal 1651–1652 (ed. and tr. Bachrach, A. G. H. and Collmer, R. G.), Leiden, 1982, 74f and 218.
31 Cf. EW, vii, 336–41, especially 337; see also Wallis, John, Elenchus geometriae Hobbianae. Sive, geometricorum, quae in ipsius ‘Elementis Philosophiae’, à Thoma Hobbes Malmesburiensi proferuntur, refutatio, Oxford, 1655, 113–19, especially 116f.
32 See EW, ii, 340.
33 Vindiciae, op. cit. (14), 6f and 53f.
34 Essay, ‘To the Reader’, op. cit. (6), A3.
35 Vindiciae, op. cit. (14) 57f.
36 Malcolm, , op. cit. (18), 54, observes that the question of the universities became suddenly topical in 1654 when their abolition was proposed in the Barebones Parliament.
37 Johnston, David, ‘Hobbes's Mortalism’, History of Political Thought (1989), 10, 647–63, and Metzger, , op. cit. (9), 241–9, especially 244f, point to the fact that Hobbes took up the concept of the mortalism of the soul only after De cive (1642) and published his view for the first time in the Leviathan, at a moment when mortalism – after a period of liberal discussion – had been finally rejected by the presbyterian and the anglican clergy; see also Tuck, , op. cit. (15), 128.
38 Essay, op. cit. (6), 14–17.
39 In his subsequent discussion of the same topic with Hobbes, Wallis was quite delicately to maintain a standpoint diametrically opposed to Ward's: see Wallis, John, ‘An answer to four papers of Mr Hobs, lately published in the months of August, and this present September, 1671’, Philosophical Transactions (18 09 1671), 75, 2241–50, especially 2242f.
40 Wallis will present a counter-example in his dispute with Hobbes: an infinitely produced line, starting or ending at a fixed point (ibid.).
41 Cf. North, John D., ‘One truth or more?’, in Unguru, Sabetai (ed.), Physics, Cosmology, and Astronomy, 1300–1700: Tension and Accommodation, Dordrecht, 1991, 253–93, especially 277–83.
42 See OL, i, 334ff, For a profound investigation of Hobbes's related discussion of Thomas White's arguments for a creation of the world see Pacchi, Arrigo, ‘Hobbes and the problem of God’, Perspectives on Thomas Hobbes, op. cit. (18), 171–87.
43 Ward, Seth, In Thomae Hobbii philosophiam exercitatio epistolica, Oxford, 1656, 116.
44 See OL, i, 88.
45 ‘For the effects we acknowledge naturally, do include a Power of their producing, before they were produced; and that Power presupposeth something existent that hath such power: And the thing so existing with power to produce, if it were not Eternal, must needs have been produced by somewhat before it, and that again by something else before that, till we come to an Eternal (that is to say the first) Power of all powers, and first Cause of all causes: And this is it which all men conceive by the Name of GOD, implying Eternity, Incomprehensibility, and Omnipotency.’ (Humane Nature, 1321; cf. EW, i, 591., Elements, 53f.) It is perhaps worth noting that, from the beginning of the dispute, Hobbes tried to avoid meetings with Ward; see Malcolm, , op. cit. (18), 53, 59 and Pope, , op. cit. (21), p. 26.
46 Ward, Seth, Against Resistance of Lawful Powers: A Sermon Preached at White-Hall, Novemb. Vth 1661. By Seth Ward, D. D. Chaplain to His Majesty. Published by His Majestie's Command, London, 1661.
47 Malcolm, , op. cit. (18), 57.
48 Ward, , op. cit. (46), 35.
49 Ibid., 2f, 4f, 35.
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