Exploration mathematics: The rhetoric of discovery and the rise of infinitesimal methods

Configurations

Volumn 9, Issue 1, 2001, Pages 1-35

  Alexander, Amir R ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

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EID: 33751414155     PISSN: 10631801     EISSN: 10806520     Source Type: Journal    
DOI: 10.1353/con.2001.0001     Document Type: Article

References (115)

1. Francis Bacon, The New Organon and Related Writings (New York: Liberal Arts Press, 1960), p. 81. For a discussion of Bacon's use of geographical exploration as a model for his reform of knowledge,
2. see \Vayne Franklin, Discoverers, Explorers, Settlers (Chicago: University of Chicago Press, 1979), pp. 7-10.
3. See, for example, Paula Findlen, "II nuovo Colombo: Conoscenza e ignoto nell' Europa del Rinascimento," in La rappresentazione dell'altro nei tor/ del Riimscimento, ed. Sergio Zatti (Lucca: Pazzini Fazzi, 1998), pp. 219-244.
4. William Eamon, Science and thé Secrets of Nature (Princeton: Princeton University Press, 1994);
5. Anthony Grafton, Ne\v Worlds, Ancient Texts (Cambridge, Mass.: Harvard University Press, 1992);
6. Anthony Pagden, "Receding Horizons," in idem, European Encounters with the Ne\v World (New Haven, Conn.: Yale University Press, 1993); Findlen, "II nuovo Colombo" (above, n.2).
7. Mathematicians were not, of course, a homogeneous group. In the introduction to his biography of Pierre de Fermât, Michael Mahoney identifies no fewer than six different types of professions and activities that could be defined as "mathematics" in this period. The traditional views expressed here are most representative of the field he labels "classical mathematics." The subsequent reform of mathematics arose both from within this group, among whom Mahoney includes Galileo and Cavalieri, and from the "applied mathematicians," who included Stevin and Hariot. See Michael S. Mahoney, The Mathematical Career of Pierre de Fermât (Princeton: Princeton University Press, 1973), chap. 1.
8. As many early modem mathematicians insisted, this view of mathematics as a selfcontained deductive system was not necessarily shared by ancient mathematicians. The ancients, they claimed, and Archimedes in particular, had in fact possessed a secret method of discovery; unfortunately they erased all traces of their path of discovery in their official presentations, leaving their successors to endlessly repeat their results and with no clue as to how to proceed. See, for instance, William Oughtred's views, as expressed in his Claris inatheinaticae (London, 1632) and discussed below. John Wallis, in letter 16 of his Commercium epistoliaim (Oxford, 1658), similarly complained that Archimedes hid his "method of discovery" from his successors. François Viète in Ail artein analyticein isagoge (Tours, 1591) expressed the opinion that his "analytic art" was a revival of a lost ancient practice, and Descartes expressed a similar view in rule 4 of his Rules for the Direction of the Mind of 1629.
9. On the "Quaestio de certitudine mathematicarum" see Paplo Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford: Oxford University Press, 1996), chap. 1;
10. Peter Dear, Discipline and Experience: The Mathematical Way in the Scientific Revolution (Chicago: University of Chicago Press, 1995), chap. 2.
11. Christopher Clavius, "In disciplinas mathematicas prolegomena," in idem, Opera inatlieinaticci, vol. 1 (Mainz, 1611), p. 3, quoted in Dear, Discipline and Experience (above, n. 6), p. 40. It should be noted that Clavius himself did not always insist on this account of the nature of mathematics. More than twenty years after the "Prolegomena," in the introduction to his Algebra of 1609, he presented a very different view of this mathematical field: the purpose of algebra, he wrote, is to extract certainty from sense knowledge and to "explore a secret quantity"; he then went on to compare algebra to a hunting dog that tenaciously tracks down its prey until it lays it at its master's feet.
12. See Christopher Clavius, "Proemium de algebrae praestantia," in Algebra Cliristoplwri Clavli Bambergensis (Geneva, 1609), p. 1. (I thank the anonymous reviewer for this reference.) This imagery is, of course, much closer to the exploration rhetoric than Clavius's programmatic statement, which takes Euclidean deduction as its model for proper mathematical procedure. Significantly, Clavius uses this imagery to describe algebra, another emerging branch of the new mathematics. As we shall see, the rhetoric of exploration was used in connection with various aspects of the new mathematics, ranging from mathematical notation to analysis.
13. Quoted in James A. Lattis, Behveen Copernicus and Galileo: Christoph Clavius and tlie Collapse of Ptolemaic Astronomy (Chicago: University of Chicago Press, 1994), p. 35.
14. Clavius's position is representative of the group Mahoney refers to as "classical mathematicians." As Mahoney makes clear, other groups of mathematical practitioners were at the time pursuing more practical uses for mathematics, while making far more modest claims (if any) as to the status of mathematical knowledge: Mahoney, Fermat (above n. 4), chap. 1. On contemporary views of mathematics see Mancosu, Philosophy of Mathematics (above, n. 6);
15. Dear, Discipline and Experience (above, n. 6), esp. chap. 2;
16. Mordechai Feingold, The Mathematicians' Apprenticeship: Science and Society in England, 1560-1640 (Cambridge: Cambridge University Press, 1984); Mahoney, Fermât, chap. 1.
17. This critique of mathematics was leveled most specifically against the practice that Mahoney refers to as "classical mathematics."
18. Francis Bacon, "Of the Dignity and Advancement of Learning" (translation of De aiigmentis scieiitiannn), chap. 6, in The Works of Francis Bacon, ed. James Spedding, Robert Leslie Ellis, and Douglas Denon Heath (London: Longman, 1860), vol. 4, p. 370.
19. Giordano Bruno, Ash \VednesdaySiipper, trans. Stanley L. Jaki (The Hague: Mouton, 1975), p. 55 (first published as Cam de le Ceneri [London, 1584]).
20. On Boyle's suspicion of mathematical studies, see Steven Shapin, A Social History of Truth (Chicago: University of Chicago Press, 1994), chap. 7;
21. Steven Shapin, idem, "Robert Boyle on Mathematics: Reality, Representation, and Experimental Practices," Science in Context 2 (1988): 23-58. On the rhetoric of discovery,
22. see Eamon, Science (above, n. 3), chap. 8 (esp. his discussion of Glanvill, p. 273);
23. Findlen, "II nuovo Colombo" (above, n. 2).
24. Many early modern scientists did, of course, embrace mathematics as a powerful tool in the investigation of nature. My claim is that the attempt to adapt mathematics to the emerging experimental sciences was a highly problematic venture: the attempt to fit it into an experimental mold fundamentally changed the nature of the field.
25. "In these centuries," Paolo Rossi has noted, "there was continuous discussion with insistence that bordered on monotony about a logic of discovery conceived as a venatio, a hunt-as an attempt to penetrate territories never known or explored before" (Paolo Rossi, Philosophy, Technology, and the Arts in the Early Modern Em, trans. Salvator Attanasio [New York: Harper and Row, 1970], p. 42). Rossi is here combining the metaphor of science as a hunt with that of science as geographical exploration. Eamon has explored the close connection between the two imageries in chap. 8 of his Science and the Secrets ofNaùire (above, n. 3), and in
26. William Eamon, "Science as a Hunt," Physis, n.s., 31:2 (1994): 393-432.
27. It should be emphasized that the view of mathematicians as explorers did not exclude admiration for the ancients, but most often went hand in hand with it. See n. 5 above.
28. 1 am not claiming that the rhetoric of exploration was used by the promoters of infinitesimals to the exclusion of all others. Just as it would be silly to claim that only followers of the strict Baconian program utilized this language in natural philosophy, it would be equally wrong to argue that the theme of "exploration" was monopolized exclusively by infinitesimalists. As we shall see, the theme was used in different ways by different authors.
29. Simon Stevin, Tlw Principal Works of Simon Stevin, 6 vols., éd. D. J. Struik, vol. 2a (Amsterdam: Swets and Zeitlinger, 1958), p. 137.
30. Ibid., p. 392. Stevin's characterization of decimal notation as a "discovery" comparable to finding an "unknown island" may seem surprising to a modern reader used to regarding it as a notational technique, rather than as an uncovering of something "out there" in the world. The important point for the argument is that Stevin himself viewed it as a "discovery" and described its invention in terms of a geographical voyage of exploration.
31. 1 thank Peter Dear for this point.
32. The movement here is again very similar to what was occurring in natural philosophy. It is often noted that practicing artisans possessed a long-standing empirical tradition of knowledge, which was separate from the official learned tradition and ignored by it; Bacon and his fellow reformers drew upon this tradition and transformed it into legitimate scholarly knowledge. In the same manner, it is less than surprising to find the view of mathematics as a voyage of exploration voiced by a practicing engineer, before it is legitimized by Galileo, Torricelli, Wallis, and their colleagues.
33. See, e.g., Galileo Galilei, Dialogue Concerning the Two Chief World Systems, trans. Stillman Drake (Berkeley: University of California Press, 1967), pp. 186-188 (orig. pub. Florence, 1632).
34. In the Two Ne\v Sciences of 1638, for example, Galileo noted: "conclusions that are true may seem improbable at first glance, and yet when only some small thing is pointed out, they cast off their concealing cloaks and, thus naked and simple, gladly show off their secrets" (Galileo Galilei, Two New Sciences, ed. and trans. Stillman Drake [Madison: University of Wisconsin Press, 1974], p. 14).
35. Galileo, Two World Systems (above, n. 22), p. 104.
36. Galileo to Torricelli, September 27, 1641, letter 15 in Evangelista Torricelli, Opère dl Evangelista Tonicelli, 4 vols., éd. Gino Loria and Giuseppe Vassura (Faenza: Montanari, 1919), vol. 3, p. 60.
37. In identifying a narrative of exploration and discovery I look for three standard elements. First, great riches and marvels must be posited in a hidden land; second, the land must be protected by natural barriers, such as mountains, forests, or mists; and third, clear passageways to the land are then opened by the intrepid explorer. This simple story was used by explorers from Columbus and Magellan to Frobisher and Raleigh. All the basic elements are present in the correspondence of Galileo, Cavalieri, and Torricelli, although they are not always present together. For more on the narrative of exploration, see Amir Alexander, "The Imperialist Space of Elizabethan Mathematics," Studies in the History and Philosophy of Science 26:4 (1995): 559-591.
38. Evangelista Torricelli, Opère scelle dl Evangelista Tonicelli, ed. Lanfranco Belloni (Turin: Unione Tipografico Editrice Torinese, 1975), p. 624.
39. Ibid., p. 383.
40. Ibid., p. 382.
41. I thank Peter Dear and Michael Mahoney for pointing out the classical reference in the "royal road" metaphor. Significantly, Torricelli uses Euclid's imagery to point out Euclid's error: there is, he claims, a clear, open "royal" road to mathematics, and it is Cavalieri's method of indivisibles.
42. Cavalieri to Torricelli, August 20, 1641, letter no. 12 in Opère di Tonicelli (above, n. 25), vol. 3, p. 57.
43. Cavalieri to Torricelli, May 12, 1643, letter no. 53 in ibid., vol. 3, p. 123.
44. Ibid., vol. 3, letters no. 14, 20, 36, 39, 74, 82.
45. The trope of hidden gems of knowledge and secret natural marvels is an old one, with roots in both antiquity and the Middle Ages. As William Eamon demonstrates, however, it was used differently in different periods. In antiquity it connoted esoteric knowledge, which could be discovered only through supernatural means or revelation, and should therefore be jealously guarded by the few initiates. In medieval scholasticism it referred to problems that were very difficult or simply unknowable; such prob-lems could not be solved through direct demonstration, and were therefore excluded from the realm of proper "scientia." Finally, in the early modern period it came to connote the unobservable "inner workings" of nature, which were to be exposed and brought to light. It is only this last sense that suggests that an exploration of nature is desirable. This is the sense in which Galileo, Cavalieri, and their colleagues are, using the trope, while applying it to the study of mathematics. See Eamon, Science (above, n. 3) generally, and the "Conclusion" for a summary.
46. On the new experimental sciences as a systematic search for hidden secrets, see ibid., chap. 8.
47. Alexander, "Imperialist Space" (above, n. 26).
48. John Napier, A Description of the Admirable Table of Logarithms, trans. Edward Wright (London, 1616), n.p. The notion of a "key to mathematics," used also by William Oughtred, evokes the related theme of mathematics as a secret code in need of deciphering. This theme is undoubtedly similar to the "exploration" trope, especially because it views the mathematician as one who brings to light hidden secrets.
49. Ibid., n.p.
50. Stephen J. Rigaud, Correspondence of Scientific Men of the Seventeenth Century, vol. 1 (Oxford: Oxford University Press, 1841), letter 24, pp. 65-66.
51. On Baconianism in England, see Charles Webster, The Great Instauration: Science, Medicine, and Reform, 1626-1660 (London: Duckworth, 1975);
52. Christopher Hill, Intellectual Origins of the English Revolution (Oxford: Clarendon Press, 1965).
53. Wallis published the entire correspondence related to his dispute with Fermât: see John Wallis, ed., Commerchnn epistolicum de quaestionibus quibiisdam mathematicis niiper habitiim (Oxford, 1658).
54. Not all mathematicians, however, accepted infinitesimals even then. Descartes, after experimenting with them in his early years, had abandoned them by the time he published the Geometry in 1638. The Jesuit order went even further, and repeatedly banned the teaching of infinitesimals in its schools: see Egidio Festa, "Quelques aspects de la controversie sur les indivisibles," in Geometric! e atomismo nella scuola Galileana, ed. Massimo Bucciantini and Maurizio Torrini (Florence: Olschki, 1992), pp. 193-207.
55. See Wallis's account of the work of Cavalieri, Harlot, and Oughtred in John Wallis, A Treatise on Algebra, Both Historical and Practical (London: Playford, 1685).
56. Peter Dear has recently emphasized the role of mathematics in the Royal Society's original program, described by John Wilkins as "Physico-Mathematicall-Experimentall learning"; there is no doubt, however, that most of the work done under the auspices of the Royal Society in its early years was nonmathematical. See Dear, Discipline and Experience (above, n. 6), esp. p. 2 and chap. 8.
57. See John Wallis, Truth Tried, or, Animadversions on a Treatise Published by the Right Honorable Robert Lord Brooke, Entitled the Nature of Truth, printed by Richard Bishop for Samuel Gellibrand (London, 1642).
58. On Wallis's preference for sensible over abstract truths, see ibid., p. 60. The method of induction is used throughout his works, but is explicitly defended in his Treatise of Algebra (above, n. 43), pp. 306-308.
59. John Wallis, Opera mathematica, vol. 1 (Oxford, 1699), p. 491.
60. On Bacon's pursuit, capture, and examination of nature through interrogation and torture, see Carolyn Merchant, The Death of Nature: Women, Ecology, and the Scientific Revolution (San Francisco: Harper and Row, 1980), chap. 7.
61. Wallis, Opera inatheinatica, vol. 1, p. 491.
62. Bacon uses the hunt metaphor in De sapientia vctenun, in Works (above, n. 11), vol. 6, p. 7l3;
63. andinDeaiigmentisscientanimS.2, in Works, 1: 633, trans. 4:421.
64. See the discussion in Eamon, Science (above, n. 3), p. 283. On the image of judicial interrogation in Bacon,
65. see Merchant, Death of Nature (above, n. 48), esp. chap. 7;
66. Evelyn Fox Keller, Reflections on Gender and Science (New Haven: Yale University Press, 1985), chaps. 2, 3.
67. These metaphors have often been treated interchangeably, both in the seventeenth century and in modern scholarship. Bacon, as has been noted, uses all three tropes to argue his case for empiricism; Wallis, in the passage quoted, clearly uses the hunt and interrogation metaphors interchangeably; and see n. 15, above. A related but somewhat different theme was the notion of mathematics as an art for breaking secret codes, which gains added significance from the fact that many mathematicians, most notably François Viele and John Wallis, were indeed professional code-breakers. There are, it seems to me, two versions of this trope: One, used by Viele and Descartes (among others), views mathemalics as a tool for deciphering hidden secrels in the outside world. This version uses mathemalics in new ways, but does not touch the core underslanding of whal malhemalics is; it is used to refer, most often, to algebra. On Ihis see Peter Pesic, "Secrets, Symbols, and Systems: Parallels between Cryptanalysis and Algebra, 1580-1700," Isis 88 (1997): 674-692. The other version, used by Edward Wrighl ("who for malhemalics found Ihe key") and William Oughlred ("The Key lo Malhemalics"), implies lhat malhematics itself withholds secrets, and lhal Ihe role of Ihe malhemalician is lo uncover Ihem; Ihis lasl version is very close lo Ihe "exploration" trope.
68. This, of course, was precisely the view of early modern reformers of knowledge on the role of the natural philosopher. On early modern science as an attempt to systematically uncover the hidden secrets of nature, see Eamon, Science (above, n. 3), chap. 8, "Science as Veimtio."
69. Joseph Glanvill, The Vanity of Dogmatizing (London, 1661), p. 178.
70. As I noted before, the case in mixed mathematics is not much different from that in pure mathematics. In astronomy, for example, one may describe the observed positions of the planets through the use of rigorous and deductive geometry. This in no way alters the fundamental nature of geometry as rigorous and deductive.
71. Cavalieri to Galileo, letter 3889 in Galileo Galilei, Le opère di Galileo Galilei (Florence: Edizione Nazionale, 1929-39), vol. 18, p. 67; Cavalieri was quoting from ode 3 of Horace's Carmiim.
72. Torricelli, Opère scelle (above, n. 27), p. 382. From a modern perspective, the method of indivisibles may seem like a technique for solving problems rather than a "discovery" out there in the world. As the language here suggests, however, and as 1 will argue at length later on, Cavalieri, Torricelli, Harlot, and their colleagues viewed the method very differently: for them it revealed something fundamental about the true nature of mathematical objects.
73. Cavalieri to Torricelli, March 10, 1643, letter no. 47 in Opère de Torricelli (above, n. 25), vol. 3, p. 114.
74. Bonaventura Cavalieri, Exerdtationes geoinetricae sex (Bologna: Montij, 1647).
75. While this conveys the general argument of the proof, certain details have been omitted in the interest of simplicity. Unlike Torricelli and other "indivisibilist" mathematicians, Cavalieri was a cautious practitioner of his method. He insists, for example that a single line ("régula") traverse both triangles simultaneously, and that the mathematical relation between the figures is valid only if it holds between every two lines that the régula intersects at the same moment. Furthermore, he never claims that the lines actually comprise a plane figure (as did Torricelli), but only that if a certain relation holds between "all the lines" of one figure and "all the lines" of another, then the same relationship holds between their surface areas. See François de Gandt, "Naissance et métamorphose d'une théorie mathématique: La géométrie des indivisibles en Italie," in Sciences et techniques en perspective, vol. 9 (Nantes: Université de Nantes, 1984-85), pp. 179-229;
76. François de Gandt, idem, "Les indivisibles de Torricelli," in L'oeime de Torricelli: Science Galiléenne et nouvelle géométrie, éd. idem (Nice: Université de Nice, 1987), pp. 147-206;
77. François de Gandt, idem, "Cavalieri's Indivisibles and Euclid's Canons," in Revolution and Continuity: Essays in the History and Pliiiosopliy of Early Modem Science, éd. Peter Barker and Roger Ariew (Washington, D.C.: Catholic University of America Press, 1991), pp. 157-182;
78. Kristi Andersen, "Cavalieri's Method of Indivisibles," Archive for History of Exact Sciences 31:4 (1985): 293-367.
79. Cavalieri was a cautious mathematician, much concerned with preserving the logical consistency of his system. He carefully avoided making any programmatic statements about the composition of the continuum, which would have opened him to criticism based on the ancient paradoxes of Zeno and the problem of incommensurability. He did, however, have a clear idea of what the internal structure of geometrical figures was like. His basic intuition about the structure of the continuum is made clear in the opening passages of his Exercitationes geoinetricae sex, where he writes: "It is therefore evident that the plane figures should be conceived by us in the same manner as cloths are made up of parallel threads. And solids are in fact like books, which are composed of parallel pages" (Exercitationes geometricae sex [above, n. 58], p. 3). Cavalieri's method of indivisibles was based on this fundamental intuition of the composition of the continuum.
80. Eamon, Science (above, n. 3), p. 270.
81. See Enrico Giusti, Bonaventura Cavalier! and the Theory of Indivisibles (Cremona: Edizioni Cremonese, 1980).
82. Guldin's critique of indivisibles is contained in Paul Guldin, De centra gravitatis, 4 vols. (Vienna, 1635-41), more commonly known as Centrobarycae. (See, for example, Book 2, p. 3 [1639], and Book 4, p. 341 [1641].)
83. For the full discussion, see Galileo, Tïvo Ne\v Sciences (above, n. 23), pp. 28-33.
84. Tïvo Ne\v Sciences (above, n. Ibid., pp. 35, 36, 38.
85. Tïvo Ne\v Sciences (above, n. Ibid., p. 51.
86. Tïvo Ne\v Sciences (above, n. Ibid., p. 54.
87. Tïvo Ne\v Sciences (above, n. Ibid., p. 39.
88. On the respective approaches of Galileo, Cavalieri, and Torricelli to the method of indivisibles, see de Gandt, "Naissance" (above, n. 59).
89. For a fuller account of Tbrricelli's lists of paradoxes, see de Gandt, "Les indivisibles" (above, n. 59).
90. Ibid., p. 182.
91. am here following de Gandt's argument on the nature of Torricelli's mathematics in ibid., esp. p. 181.
92. Quoted in L. Belloni, "Tbrricelli et son époque (le triumvirat des élevés de Castelli: Maggiotti, Nardi, et Torricelli)" in L'Oeum de TonicelH (above, n. 59), p. 31. Nardi is referring to Archimedes' celebrated "method of exhaustion," which proved results on areas and volumes of geometrical objects deductively and without the use of infinitesimals. Essentially, this method proved that the area/volume of an object could not be greater or smaller than a given figure, because that would lead to a logical contradiction. Nardi (and many others) pointed out that while this method proved the result with certainty, it gave no hint as to how the result was originally arrived at: see above, n. 5. Many suspected that Archimedes had in fact used infinitesimals as a method of discovery, before erasing all trace of them in his final proofs; their suspicions, of course, proved accurate with the discovery of Archimedes' "Method" by Heiberg at the turn of the twentieth century.
93. See, e.g., Bucciantini and Torrini, Geonietria e atoinisino (above, n. 42).
94. The biographical information is from Stevin, Principal Works (above, n. 18) vol. 1, pp. 3-14.
95. Ibid., p. 230. Also discussed in Carl B. Boyer, The History of the Calculus and Its Conceptual De\'elopinent (New York: Dover, 1959), p. 100.
96. Archimedes was the forerunner of infinitesimal techniques, including the "weighing" of geometrical figures, and was acknowledged as such by sixteenth- and seventeenthcentury mathematicians. In his formal presentations, however, he substituted these methods with the rigorous and deductive proofs, most famous of which was the "method of exhaustion."
97. William Oughtred, preface to The Key of Mathematics (London: Thomas Harper, 1647), n.p.; this English translation closely follows the Latin of the original Clavis matliematicae of 1632.
98. preface to The Key of Mathematics (Ibid., n.p.
99. Oughtred reiterates the same view of the purpose of mathematical studies in an introductory poem to his book entitled Mathematicall Recreations (London: William Leake, 1653), where he writes: "Here questions of ARITHMETICK are wrought / And hidden secrets unto light are brought" (n.p.).
100. On Oughtred see J. F. Scott, "Oughtred, William," in Dictionary of Scientific Biography 10: 254-255;
101. Florian Cajori, William Outfitted, a Great Seventeenth-Century Teacher of Mathematics (Chicago: Open Court, 1916).
102. Rigaud, Correspondence of Scientific Men (above, n. 39), pp. 65-66.
103. The visual imagery fits neatly with the exploration motif. Unlike the bookish scholars who remain in their studies, the explorer views and obsems the wonders of unknown lands, and then presents them to those who stayed behind.
104. Wallis, Opera inatlieinatica (above, n. 47), vol. 1, p. 363.
105. It should be noted, though, that Wallis initially received his information on Cavalieri's method through Torricelli's Opera geoinetrica (Florence, 1641). His conception of the method of indivisibles was accordingly much closer to Torricelli's notions than to Cavalieri's original method. See Wallis's introduction to his Aritlunetica infinitonun of 1656 in Wallis, Opera inatlieinatica, vol. 1, p. 357;
106. de Gandt, "Les indivisibles" (above, n. 59), p. 152.
107. Wallis, Opera inatliematica, vol. 1, p. 299.
108. On Fermat's reaction to Wallis, see his letter of August 15, 1657, to Kenelm Digby, "Remarques sur l'arithmétique des infinis du S. J. Wallis," in Ouvres de Fermat, ed. Charles Henry and Paul Tannery (Paris: Gauthier-Villars, 1891-1912), vol. 2, pp. 347-353; and see Mahoney, fermat (above, n. 4), chap. 6.
109. Wallis, Opera matlieinatica (above, n. 47), vol. 1, p. 491.
110. Bernard de Fontenelle, Éléments de la géométrie de l'infini (Paris, 1727), p. ciii; quoted in Michael S. Mahoney, "Infinitesimals and Transcendent Relations: The Mathematics of Motion in the Late Seventeenth Century," in Reappraisals of the Scientific Revolution, ed. David C. Lindberg and Robert S. Westman (Cambridge: Cambridge University Tress, 1990), pp. 461-191, n. 46.
111. For a discussion of science as the search for novelties, see Eamon, Science (above, n. 3), chap. 8. Eamon emphasizes the imagery of the hunt over that of geographical exploration, but his discussion and examples make it clear that the two images capture a very similar vision of the purpose and practice of science.
112. As I argued above, this should be qualified in the case of "mixed mathematics," where "pure mathematics" was joined with natural philosophy. In those cases, of course, all truths could not be said to be derived from the initial assumptions. Nevertheless, as Clavius makes clear in the "Prolegomena," the basic nature of mathematics is not changed by combining it with natural philosophy: it maintains its strict logical and deductive character, and bestows it upon the new field.
113. Significantly, much as French philosophers were more skeptical of empiricism than their English contemporaries, French mathematicians did not, on the whole, employ the rhetoric of exploration, and were far more cautious with the use of indivisibles. Viele did not discuss indivisibles at all, and Descartes famously excluded them from the realm of mathematics. Fermat used infinitesimals skillfully but gingerly, insisting that, unlike VVallis, he was proceeding in a "via ordinaria, legitimea, et Archimedea." On Fermat, see Mahoney, Fermat (above, n. 4), chap. 5; the quote is from a letter from Fermat to Kenelm Digby, published in Wallis's Commercium epistoliaim (above, n. 41). It is interesting to note that Fermat ultimately gave up his insistence on the classical nature of his practice in his later years, when it was clear that the new methods had far surpassed the classical model. It is in this context that we encounter the following passage in an introduction by Antoine de Lalouvère to Fermat's treatise on rectification, which he published as an appendix to his own book in 1660: "hac tempestate in Geometricis inventum et superatum féliciter esse Bonae Spei promontorium illud, unde expedita existât navigatio ad inaccessas ante tetragonismorum praesertim regiones" [At this time in geometry, that Cape of Good Hope was successfully found and surpassed, from which a clear navigation is open to regions that were certainly inaccessible before the quadratures] (Antoine de Lalouvère, Vetemm geometria promota ... [Toulouse, 1660], part 1, appendix 2, preamble to some propositions of Fermat's; reprinted in Oeuvres de fermat [above, n. 86], vol. 1, pp. 199-200, n. 1). (I thank Michael Mahoney for this reference.)
114. Wallis, Treatise of algebra (above, n. 43), p. 305.
115. Clavius, "Proemium de algebrae praestantia" (above, n. 7), p. 1. See also n. 7 above.