Thomas Harriot on Combinations

Maclean, Ian

Revue d'histoire des mathématiques, Tome 11 (2005) no. 1, pp. 57-88.

Thomas Harriot (1560 ?-1621) est célèbre pour ses travaux novateurs tant dans le domaine de l'algèbre que de la philosophie naturelle. Dans cet article, on se propose d'examiner sa pensée sur les combinaisons dans trois contextes ; celui du langage (les anagrammes), celui de la philosophie naturelle (les atomes) et celui de la théorie des nombres. On considérera cette pensée dans le cadre de trois débats historiographiques, à savoir : 1) si ou non, il existe deux mentalités opposées au seuil de la modernité, à savoir l'occulte et la scientifique ; 2) si à cette époque les « sciences mathématiques » sont distinctes de la philosophie naturelle ; et 3) si cette philosophie comprend, au-delà d'une étude de la nature elle-même, celle des attributs du créateur de la nature. Du cas Harriot, on concluera que ce mathématicien est capable d'une pensée mathématique fort abstraite, libérée de l'idéologie sociale, religieuse et politique de son temps (sans que ce contrat s'étende à ce qu'il a à dire sur l'alchimie, ou sur les problèmes des mathématiques appliquées, comme celui de la longitude), et qu'il est capable, comme bien de ses contemporains, de compartimenter son esprit de façon à s'engager mentalement selon des modes fort divers dans les différents domaines de son univers intellectuel.

Thomas Harriot (1560?-1621) is known today as an innovative mathematician and a natural philosopher with wide intellectual horizons. This paper will look at his interest in combinations in three contexts: language (anagrams), natural philosophy (the question of atomism) and mathematics (number theory), in order to assess where to situate him in respect of three current historiographical debates: 1) whether there existed in the late Renaissance two opposed mentalities, the occult and the scientific; 2) whether all mathematical science was clearly demarcated from natural philosophy at that time; and 3) whether all enquiry into nature (including that pursued through mathematics) entailed a consideration of the attributes of God Himself. The paper argues from the case of Harriot that as a man capable of highly abstract mathematical thought, his work on combinations of all kinds is scarcely marked at all by the social, political and religious context from which it arose (which is not to say that his work on alchemy or on practical mathematics is unmarked in the same way), and that he, like many of his contemporaries, was capable of compartmentalising his mind, and of according different modes and degrees of intellectual commitment to different areas of his mental universe.

[1] P. Apian, Eyn New Unnd wolgegründte underweysung aller Kauffmannß Rechnung, 1527

[2] Aristotle, Opera omnia, Guillaume Duval, 1619

[3] W. J. Ashworth, Natural history and the emblematic world view, in: Reappraisals of the Scientific Revolution (David C.LindbergRobert S.Westman ed.), Cambridge University Press, Cambridge, 1990, p. 303-332

[4] F. Bacon, The Advancement of Learning, in: The Oxford Francis Bacon (MichaelKiernan ed.), IV, Oxford University Press, Oxford, 2000, p. 303-332

[5] R. Barnes, Prophecy and Gnosis: Apocalypticism in the Wake of the Lutheran Reformation, Stanford University Press, 1988

[6] G. Batho, Thomas Harriot and the Northumberland household, in: Thomas Harriot: an Elizabethan Man of Science (RobertFox ed.), Ashgate, Aldershot, 1999, p. 28-47

[7] G. Batho, Thomas Harriot's Manuscripts, in: Thomas Harriot: an Elizabethan Man of Science (RobertFox ed.), Ashgate, Aldershot, 1999, p. 286-98

[8] J. Bennett, “The Mechanics' philosophy and the mechanical philosophy”, History of Science 24 (1986), p. 1-28

[9] J. Bennett, Instruments, mathematics, and natural knowledge: Thomas Harriot's place on the map of learning, in: Thomas Harriot: an Elizabethan Man of Science (RobertFox ed.), Ashgate, Aldershot, 1999, p. 137-52

[10] L. Bianchi, Censure et liberté intellectuelle à l'Université de Paris (xiiie-xiv siècles), Belles Lettres, 1999

[11] T. Billon, Le bon ange de la France, rapportant soixante-deux anagrammes en forme de presages, voeux et benedictions, le tout heureusement tiré sans addition, diminution ou mutation de lettres du tres-fortuné, tres-grand et tres-auguste nom de Louis XIII de Bourbon, roi de France et de Navarre; ensemble de tres-haute et tres-illustre princesse Anne d'Austrie, infante d'Espagne, sur l'heureux mariage de Leurs Majestés, 1613

[12] T. Billon, Sibylla gallica, anagrammaticis magna praedicens oraculis, idque duabus et ultra centuriis, stylo partim soluto, partim versificato, nulla mutata, dempta vel addita littera, in gratiam christianissmi principis Lodovici XIII Galliae et Navarrae regis felicissimi, necnon Annae Mauritiae de Austria, reginae, 1616, 2nd ed. 1624

[13] G. Cardano, Opera omnia, Charles Spon, 1663, 10 vols

[14] W. Cheeke, Anagrammata, et chron-anagrammata regia, 1613

[15] M. Clagett, Oresme and the Medieval Geometry of Qualities and Motions, University of Wisconsin Press, 1968 | MR

[16] S. Clucas, Corpuscular matter theory and the Northumberland circle, in: Medieval and Early Modern Corpuscular Matter Theories (Christoph H.LüthyJohn E.MurdochWilliam R.Newman ed.), Brill, Leiden, 2001, p. 181-201

[17] S. Clucas, Thomas Harriot and the field of knowledge in the English Renaissance, in: Thomas Harriot: an Elizabethan Man of Science (RobertFox ed.), Ashgate, Aldershot, 1999, p. 93-136

[18] B. P. Copenhaver, Astrology and magic, in: The Cambridge History of Renaissance Philosophy (Charles B.SchmittQuentinSkinnerEckhardKessler ed.), Cambridge University Press, Cambridge, 1988, p. 264-300

[19] A. Cunningham, “The identity of natural philosophy: a response to Edward Grant”, Early Science and Medicine 5 (2000), p. 259-78, 299-300

[20] M. Curl, The Anagram Book, Robert Hale, 1982

[21] J. Dan, The Kabbalah of Johannes Reuchlin and its historical significance, in: The Christian Kabbalah (JosephDan ed.), Harvard College Library, Cambridge, Mass., 1997, p. 55-96

[22] A. Edwards, Pascal's Arithmetical Triangle, Charles Griffin/Oxford University Press, 1987 | MR | Zbl

[23] R. French & A. Cunningham, Before Science: the Invention of the Friars' Natural Philosophy, Scholar, 1996

[24] M. Folkerts, E. Knobloch & K. Reich, Mass, Zahl und Gewicht: Mathematik als Schlüssel zu Weltverständnis und Weltbeherrschung, Harrassowitz, 2001 | Zbl

[25] J. Fleming, Graffiti and the Writing Arts of Early Modern England, Reaktion, 2001

[26] H. Gatti, The natural philosophy of Thomas Harriot, in: Thomas Harriot: an Elizabethan man of Science (RobertFox ed.), Ashgate, Aldershot, 1999, p. 64-92

[27] C. Gilly, “Zwischen Erfahrung und Spekulation: Theodor Zwinger und die religiöse und kulturelle Krise seiner Zeit”, Baseler Zeitschrift für Geschichte und Altertumskunde 79 (1979), p. 125-223

[28] E. Grant, “God and natural philosophy: the late Middle Ages and Sir Isaac Newton”, Early Science and Medicine 5 (2000), p. 279-98

[29] T. Harriot, British Library Additional Manuscripts 6782, 6785, 6786, 6787, 6789,

[30] J. Henry, “Thomas Harriot and atomism: a reappraisal”, History of Science 20 (1982), p. 267-96

[31] J. Hucher, De prognosi, 1602

[32] J. Jacquot, Harriot, Hill, Warner and the new philosophy, in: Thomas Harriot. Renaissance Scientist (J.W.Shirley ed.), Clarendon Press, Oxford, 1974, p. 137-8 | MR

[33] L. Jardine, Experientia literata or Novum Organum, in: Francis Bacon's Legacy of Texts (William A.Sessions ed.), Amer. Math. Soc. Press, New York, 1990, p. 47-67

[34] N. Jardine & M. Frasca-Spada, “Splendours and miseries of the science wars”, Studies in the History and Philosophy of Science 28 (1997), p. 219-35

[35] D. Kahn, L'interpétation alchimique de la Genèse chez Joseph Du Chesne dans le contexte de ses doctrines alchimiques et cosmologiques, in: Scientiae et artes (BarbaraMahlmann-Bauer ed.), Harrassowitz, Wiesbaden, 2004, p. 641-92

[36] Z. Kaluza, “Le De universali reali de Jean de Maisonneuve et les Epicurei litterales”, Freiburger Zeitschrift für Philosophie et Theologie 35 (1986), p. 465-516

[37] Z. Kaluza, Les querelles doctrinales à Paris: nominalistes et réalistes aux confins du xive et du xve siècles, P. Lubrina, 1988

[38] J. Kepler, Dissertatio cum nuncio sidereo nuper apud mortales misso a Galilaeo Galilaeo, 1610

[39] S. Kusukawa, ‘Aspectio divinorum operum': Melanchthon and astronogy for Lutheran medics, in: Medecine and the Reformation (Ole PeterGrellAndrewCunningham ed.), Routledge, London & New York, 1993, p. 33-56

[40] G. Le Blanc, Libellus de ratione anagrammatismi, 1586

[41] A. Libavius, Examen novae philosophiae, quae veteri abrogandae opponitur, 1615

[42] C. Lüthy, J. E. Murdoch & W. R. Newman, Medieval and Early Modern Corpuscular Matter Theories, Brill, 2001

[43] D. C. Lindberg & R. L. Numbers, When Science and Christianity Meet, Chicago University Press, 2003

[44] C. Lohr, Possibility and reality in Suárez's Disputationes metaphysicae, in: Res et verba in the Renaissance (EckhardKesslerIanMaclean ed.), Harrassowitz, Wiesbaden, 2002, p. 273- 86

[45] I. Maclean, White crows, greying hair and eyelashes: problems for natural historians in the reception of Aristotle's logic and biology from Pomponazzi to Bacon, in: Historia: Empiricism and Erudition in Early Modern Europe Cambridge (GiannaPomataNancySiraisi ed.), MIT Press, Cambridge Mass., forthcoming

[46] I. Maclean, Logic, Signs and Nature in the Renaissance: the Case of Learned Medicine, Cambridge University Press, 2001

[47] I. Maclean, Cardano on the immortality of the soul, in: Cardano e la tradizione dei saperi (GuidoCanzianiMarialuisaBaldi ed.), Frano Angeli, Milan, 2004, p. 191-208

[48] I. Maclean, Interpretation and Meaning in the Renaissance: the Case of Law, Cambridge University Press, 1992

[49] S. Mandelbrote, The religion of Thomas Harriot, in: Thomas Harriot: an Elizabethan Man of Science (RobertFox ed.), Ashgate, Aldershot, 1999, p. 246-79

[50] A. Marr, “‘Curious and useful buildings'; the mathematical model of Sir Clement Edmondes”, Bodleian Library Record 18 (2003), p. 108-50

[51] J. Martini, Exercitationum metaphysicarum libri duo, 1608

[52] G. Mathieu-Castellani, “Nombre, lettre, figure à la Renaissance”, Revue des sciences humaines 179 (1980), p. 1-6

[53] P. Melanchthon, Axiomata physica, MS Vatican Pal. Lat., 1038

[54] A. E. Moyer, “The demise of the quadrivium and the beginning of the Scientific Revolution: Boethius in the sixteenth century”, Intellectual News 10 (2002), p. 69-77

[55] J. E. Murdoch, Album of Science: Antiquity and the Middle Ages, Scribner, 1984

[56] B. Nance, Turquet de Mayerne as Baroque Physician: the Art of Medical Portraiture, Rodolpi, 2001

[57] P. D. Nolhac, Ronsard et l'humanisme, Champion, 1921

[58] J. D. North, Thomas Harriot and the first telescopic observations of sunspots, in: Thomas Harriot. Renaissance Scientist (J.W.Shirley ed.), Clarendon Press, Oxford, 1974, p. 137-8 | MR

[59] J. D. North, Stars and atoms, in: Thomas Harriot: an Elizabethan Man of Science (RobertFox ed.), Ashgate, Aldershot, 1999, p. 186-28

[60] I. Pantin, “La représentation des mathématiques chez Jacques Peletier du Mans: Cosmos hiéroglyphique ou ordre rhétorique?”, Rhetorica 20 (2002), p. 375-90

[61] B. Pereira, De communibus omnium rerum naturalium principiis et affectionibus, 1576

[62] S. Pumfrey, “Harriot: a patronage studies analysis”, Bulletin of the Society of Renaissance Studies 21 (2003), p. 11-22

[63] N. Reusnerus, Anagrammatographia [...] accessit Gulielmi Blanci Libellus de ratione anagrammatismi, Elias Reusnerus, 1602

[64] J. J. Roche, Harriot, Oxford, and twentieth-century historiography, in: Thomas Harriot: an Elizabethan Man of Science (RobertFox ed.), Ashgate, Aldershot, 1999, p. 246-79

[65] P. D. Ronsard, Sonnets pour Hélène, Droz/Minard, 1970

[66] E. Rosen, Harriot's science: the intellectual background, in: Thomas Harriot. Renaissance scientist (J.W.Shirley ed.), Clarendon Press, Oxford, 1974, p. 5-18 | MR

[67] S. Sanctorius, Methodi vitandorum errorum, 1630

[68] M. Seltman, Harriot's algebra: reputation and reality, in: Thomas Harriot: an Elizabethan Man of Science (RobertFox ed.), Ashgate, Aldershot, 1999, p. 153-85 | Zbl

[69] E. Serene, Demonstrative science, in: The Cambridge History of Later Medieval Philosophy (A.K. NormanKretzmannJanPinborg ed.), Cambridge University Press, Cambridge, 1982, p. 496-517

[70] J. W. Shirley, Thomas Harriot: a Biography, Clarendon Press, 1983 | MR

[71] M. Stewart, “Libertas philosophandi: from natural to speculative philosophy”, Australian Journal of Politics and History 40 (1994), p. 29-46

[72] M. Stifel, Ein Rechen Büchlin. Vom EndChrist, 1532

[73] M. Stifel, Arithmetica integra, 1544

[74] M. Stifel, Ein sehr wunderbarliche wortrechnung sampt einer merckliche erklerung etlicher zalen Danielis und der Offenbarung Sanct Johannis, 1553

[75] R. Sutton, “The phrase ‘libertas philosophandi'”, Journal of the History of Ideas 14 (1953), p. 310-16

[76] N. Taurellus, Alpes caesae, hoc est, Andreae Caesalpini Itali monstrosa et superba dogmata discussa et excussa, 1597

[77] N. Torporley, “A synopsis of the controversie of Atoms, published as an appendix to Jean Jacquot, Thomas Harriot's reputation for impiety”, Notes and Records of the Royal Society of London 9 (1952), p. 164-87

[78] H. Trevor Roper, Harriot's physician: Théodore de Mayerne, in: Thomas Harriot: an Elizabethan Man of Science (RobertFox ed.), Ashgate, Aldershot, 1999, p. 48-63

[79] A. Vesalius, Corporis humani fabrica, 1543

[80] B. Vickers, Occult and Scientific Mentalities in the Renaissance, Cambridge University Press, 1984

[81] B. D. Vigenère, Traicté des chiffres, ou secretes manieres d'escrire, 1587

[82] J. Wippel, “The condemnations of 1270 and 1277 at Paris”, Journal of Medieval and Renaissance Studies 7 (1977), p. 169-201

[83] F. A. Yates, The Occult Philosophy in the Elizabethan Age, Routledge, 1979

[84] P. Zumthor, “D'une pensée littérale”, Revue des sciences humaines 179 (1980), p. 7-21