Dot-age: Newton's mathematical legacy in the eighteenth century

Early Science and Medicine

Volumn 9, Issue 3, 2004, Pages 218-256

  Guicciardini, Niccolò ^a  

^a UNIVERSITY OF SIENA   (Italy)

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EID: 33644936141     PISSN: 13837427     EISSN: 15733823     Source Type: Journal    
DOI: 10.1163/1573382042176272     Document Type: Article

References (145)

1. Peter and Ruth Wallis, Newton and Newtoniana 1672-1975: A Bibliography (London, 1977).
2. For an introduction to this field see Scott Mandelbrote, "Newton and Eighteenth-Century Christianity," in The Cambridge Companion to Newton, ed. I. Bernard Cohen and George E. Smith (Cambridge, 2002), 409-430.
3. See, for instance, Robert E. Schofield, "An Evolutionary Taxonomy of Eighteenth-Century Newtonianisms," Studies in Eighteenth Century Culture 7 (1978), 175-92,
4. and Simon Schaffer, "Newtonianism," in Companion to the History of Modern Science, ed. R. C. Olby, G. N. Cantor, J. R. R. Christie, and M. J. S. Hodge (London, 1990), 610-26.
5. Cohen and Smith, Companion to Newton, 20.
6. A. Rupert Hall, "Correcting the Principia," Osiris 13 (1958), 291-326, on 301.
7. Morris Kline, Mathematical Thought from Ancient to Modern Times (New York, 1972), 392.
8. Charles Babbage, Passages from the Life of a Philosopher (London, 1864), 29.
9. See Philip C. Enros, "The Analytical Society (1812-1813): Precursor of the Renewal of Cambridge Mathematics," Historia Mathematica 10 (1983), 24-47;
10. Harvey Becher, "Radicals, Whigs, and Conservatives: The Middle and Lower Classes in the Analytical Revolution at Cambridge in the Age of Aristocracy," British Journal for the History of Science 28 (1995), 405-26,
11. Maria Panteki, "William Wallace and the Introduction of Continental Calculus to Britain: a Letter to George Peacock," Historia Mathematica 14 (1987), 119-32.
12. Joan Richards, review of Niccolò Guicciardini, The Development of Newtonian Calculus in Britain, 1700-1800 (Cambridge, 1989),
13. Isis 83 (1992), 328-9, on 328.
14. Wlilliam Whewell, History of the Inductive Sciences (Cambridge, 1837), 167.
15. Colin Maclaurin, A Treatise of Fluxions, in Two Books (Edinburgh, 1742), 2nd edn. (London, 1801).
16. Florian Cajori, A History of the Conceptions of Limits and Fluxions in Great Britain from Newton to Woodhouse (Chicago and London, 1919), 188.
17. Judith V. Grabiner, "Was Newton's Calculus a Dead End? The Continental Influence of Maclaurin's Treatise of Fluxions," American Mathematical Monthly 104 (1997), 393-410, on 393.
18. See also Judith V. Grabiner, "Maclaurin and Newton: the Newtonian Style and the Authority of Mathematics," in Science and Medicine in the Scottish Enlightenment, ed. C. W. J. Withers and P. Wood (London, 2002), 143-171.
19. One of the best works on the Analyst is Douglas M. Jesseph, Berkeley's Philosophy of Mathematics (Chicago, 1993).
20. Grabiner is successful in showing how the needs for geometrical rigorization pursued in Book 1 were not mathematically fruitless: on the contrary, geometrical rigor was a source of problem-solving power which was deployed in Book 2.
21. Grabiner, "Was Newton's Calculus a Dead End?," 394-5.
22. The critical edition can easily be found in Isaac Newton, The Mathematical Papers of Isaac Newton, ed. D. T. Whiteside, 8 vols. (Cambridge, 1967-1981), 2: 206-47.
23. It is by reading the second Latin edition of Descartes' Géométrie, translated by Frans van Schooten and enriched by commentaries by van Schooten himself and other Dutch mathematicians, that Newton learned how the study of curved lines could be carried out in algebraic terms. René Descartes, Geometria a Renato des Cartes, 2nd Latin edition, translation by Frans van Schooten (Amsterdam, 1659-1661).
24. La Géométrie is one of the three essays appended to Descartes' Discours de la méthode (Leiden, 1637).
25. The original text accompanied by an English translation is accessible as The Geometry of René Descartes with a Facsimile of the First Edition, ed. D. E. Smith and M. L. Latham (New York, 1954).
26. See Henk J. M. Bos, Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction (NewYork, Berlin, Heidelberg, 2001) which is the best guide to the Géométrie.
27. John Wallis's Arithmetica infinitorum, sive nova methodus inquirendi in curvilineorum quadraturam, aliaque difficiliora matheseos problemata was printed in 1655, but was published one year afterwards in Operum mathematicorum, 2 vols. (Oxford, 1656), 2: 1-199.
28. Newton, Mathematical Papers, 2: 240-1. Translation from the Latin by Whiteside.
29. In 1715 Newton wrote: "But when he [Newton] is not demonstrating but only investigating a Proposition, for making dispatch he supposes the moment o to be infinitely little, & forbears to write it down & uses all manner of approximations wch he conceives will produce no error in the conclusion." Newton, Mathematical Papers, 8: 572. Infinitesimals belonged, in his opinion, to the heuristic tools at the disposal of a mathematician, but were to be avoided in rigorous demonstrations.
30. On the tensions on methodological issues between Newtonians and the naturalists at the Royal Society, see Mordechai Feingold, "Mathematicians and Naturalists: Sir Isaac Newton and the Royal Society," in Isaac Newton's Natural Philosophy, ed. J. Z. Buchwald and I. B. Cohen (Cambridge (Mass), London, 2001), 77-102.
31. On the role of mathematics in Newton's methodology and quest for certainty, see Alan E. Shapiro, Fits, Passions, and Paroxysms: Physics, Method, and Chemistry and Newton's Theories of Colored Bodies and Fits of Easy Reflection (Cambridge, 1993), 12-40.
32. The critical edition of the De methodis can be found in Newton, Mathematical Papers, 3: 32-354.
33. Ibid., 283, 331. Translation from the Latin by D. T. Whiteside.
34. Newton, Mathematical Papers, 4: 420-84.
35. Helena M. Pycior, Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra through the Commentaries on Newton's Universal Arithmetick (Cambridge, 1997), 135-66.
36. One of the texts where Newton distinguishes the two methods is the following: "The synthetic method of fluxions occurs widespread in the following treatise [Principia], and I have set its elements in the first eleven lemmas of the first book and in Lemma II of the second. Specimens of the analytical method occur in Proposition XLV and the Scholium to Proposition XCIII of Book 1, and in Propositions X and XIV of Book 2. It is, furthermore, described in the scholium to lemma II of Book 2. And from their composed demonstrations, also, the analysis by which the propositions were found out can be learnt by going backwards." Newton, Mathematical Papers, 8: 455-7.
37. Newton, Mathematical Papers, 4: 277.
38. Translation from the Latin in Richard S. Westfall, Never at Rest: a Biography of Isaac Newton (Cambridge, 1980), 379.
39. On Newton's works on Pappus and Apollonius see Newton, Mathematical Papers, 4: 217-405; 7: 248-400.
40. A. Rupert Hall, Philosophers at War: the Quarrel between Newton and Leibniz (Cambridge, 1980).
41. Excerpts from the 1676 epistolae to Leibniz were printed in John Wallis, A Treatise of Algebra: both Historical and Practical (London, 1685), 318-20, 330-3, 338-47;
42. the prime theorem on quadratures in John Wallis, Opera mathematica, 3 vols. (Oxford, 1693-99), 2: 390-6; and the full text of the 1676 epistolae to Leibniz in ibid., 3: 622-9, 634-45 .
43. David Gregory, Exercitatio geometrica de dimensione figurarum (Edinburgh, 1684).
44. See Newton, Mathematical Papers, 7: 3ff for Whiteside's accurate reconstruction of this episode
45. Isaac Newton, Tractatus de quadratura curvaurum printed as an appendix in Opticks or, a Treatise of the Reflexions, Refractions, Inflexions and Colours of Light. also two Treatises of the Species and Magnitude of Curvilinear Figures (London, 1704).
46. On Newton's hidden use of fluxions in the Principia see Niccolò Guicciardini, Reading the Principia: the Debate on Newton's Mathematical Methods for Natural Philosophy from 1687 to 1736 (Cambridge, 1999).
47. See Descartes, The Geometry, 16-7.
48. Isaac Newton, The Principia, Mathematical Principles of Natural Philosophy: a New Translation by I. Bernard Cohen and Anne Whitman, Assisted by Julia Budenz, Preceded by a Guide to Newton's Principia by I. Bernard Cohen (Berkeley, Los Angeles, London, 1999), 485.
49. Critical editions can be found in Newton, Mathematical Papers, 7: 24-129, 588-645.
50. [Isaac Newton], Arithmetica universalis: sive de compositione et resolutione arithmetica liber (Cambridge, 1707).
51. A critical edition can be found in Newton, Mathematical Papers, 5: 54-491.
52. Isaac Newton, Analysis per quantitatum series, fluxiones, ac differentias: cum enumeratione linearum tertii ordinis (London, 1711).
53. Newton, Mathematical Papers, 7: 512n.
54. Newton, Mathematical Papers, Ibid., 2: 242-3.
55. Newton, Mathematical Papers, Ibid., 8: 572.
56. Newton, Mathematical Papers, 2, Ibid., 598.
57. On the significance of anonymous title-pages for Newton's works, see I. Bernard Cohen, "The Case of the Missing Author: the Title Page of Newton's Opticks (1704), with Notes on the Title Page of Huygens's Traité de la lumière," in Isaac Newton's Natural Philosophy, ed. J. Z. Buchwald and I. B. Cohen (Cambridge (Mass), London, 2001), 15-45.
58. David Gregory wrote: "He was forced seemingly to allow it, about 14 months agoe, when he stood for Parliament-man at the University. He has not seen a sheet of it, nor knows he what value is in it, nor how many sheets it will make, nor does he well remember the contents of it. He intends to goe down to Cambridge this summer and see it, and if does not please him to buy up the copyes." Quoted in W. G. Hiscock, David Gregory, Isaac Newton and Their Circle: Extracts from David Gregory's Memoranda, 1677-1708 (Oxford, 1937), 36.
59. On Newton's publication of his mathematical works see my "Isaac Newton and the Publication of His Mathematical Manuscripts," Studies in the History and Philosophy of Science 35 (2004).
60. For a survey and bibliographical information see Guicciardini, Development of Newtonian Calculus.
61. A critical edition can be found in Newton, Mathematical Papers, 8: 244-255.
62. See Ian Tweddle, James Stirling's Methodus Differentialis. An Annotated Translation of Stirling's Text (London, 2003).
63. Roger Cotes, "Logometria," Philosophical Transactions 39 (1714), 5-45.
64. John Machin, "De motu nodorum lunae," in Isaac Newton, Philosophiae naturalis principia mathematica, 3d ed. (London, 1726), 451-454.
65. Thomas Simpson, The Doctrine and Application of Fluxions. Containing (Beside What is Common on the Subject) a Number of New Improvements in the Theory. And the Solution of a Variety of New, and Very Interesting, Problems in Different Branches of the Mathematicks, 2 vols. (London, 1750);
66. William Emerson, The Doctrine of Fluxions: Not Only Explaining the Elements Thereof, But Also Its Application and Use in the Several Parts of Mathematics and Natural Philosophy (London, 1743).
67. Robert Simson, "Pappi Alexandrini propositiones duae generales," Philosophical Transactions 32 (1723), 330-40;
68. Opera quaedam reliqua (Glasgow, 1776).
69. Matthew Stewart, "Pappi Alexandrini collectionum mathematicarum libri quarti propositio quarta generalior facta, cui propositiones aliquot eodem spectantes adjicuntur," in Essays and Observations Physical and Literary, Read before a Society in Edinburgh, and Published by Them, 2 vols (Edinburgh, 1754), 2: 141-172;
70. Propositiones geometricae, more veterum demonstratae, ad geometriam antiquam illustrandam et promovendam idoneae (Edinburgh, 1763). See Ian Tweddle, Simson on Porisms: an Annotated Translation of Robert Simson's Posthumous Treatise on Porisms and Other Items on This Subject (New York, Berlin, Heidelberg, 2000).
71. Michel Chasles, Aperçu historique sur l' origine et le développement des méthodes en géométrie, particulièrement de celles qui se rapportent à la géométrie moderne (Paris, 1837).
72. Patrick Murdoch, Neutoni genesis curvarum per umbras, seu perspectivae universalis elementa; exemplis coni sectionum et linearum tertii ordinis illustrata (London, 1746).
73. George E. Davie, The Democratic Intellect: Scotland and Her Universities in the Nineteenth Century (Edinburgh, 1964).
74. Newton, Correspondence, 7: 329.
75. Maclaurin, Treatise, vii-viii.
76. Maclaurin, Treatise, Ibid., 39.
77. Maclaurin, Treatise, Ibid., 47.
78. Maclaurin, Treatise, Ibid., 2-3.
79. Maclaurin, Treatise, Ibid., 57.
80. Maclaurin, Treatise, Ibid., 575.
81. Newton, Mathematical Papers, 8: 122-3.
82. Maclaurin, Treatise, 579.
83. Maclaurin, Traité des fluxions (Paris, 1749);
84. an abridged French translation appeared as Abregé du calcul integral ou méthode inverse des fluxions (Paris, 1765).
85. Bos, Redefining Geometrical Exactness, 10.
86. The existence of "revolutions" in mathematics is a much debated topic: see Donald Gillies (ed.), Revolutions in Mathematics (Oxford, 1992).
87. I am here using this term rather freely in order to denote a major conceptual change. See also Koffi Maglo, "The Reception of Newton's Gravitational Theory by Huygens, Varignon, and Maupertuis: How Normal Science May Be Revolutionary," Perspectives on Science 11 (2003), 135-169, where Kuhn's notion of "invisible revolution" is applied to the development of eighteenth-century mathematics.
88. See works cited in footnotes below.
89. Craig Fraser, "The Calculus as Algebraic Analysis: Some Observations on Mathematical Analysis in the 18th Century," Archive for the History of the Exact Sciences 39 (1989), 317-35, on 317.
90. quot;Funcio quantitatis ergo variabilis est expressio analytica quomodocunque composita ex illa quantitate variabili et numeris seu quantitatibus constantibus," Leonhard Euler, Introductio in analysin infinitorum, (Lausanne, 1748) (= Opera omnia, 1/8-9), § 4.
91. S. B. Engelsman, Families of Curves and the Origins of Partial Differentiation (Amsterdam, 1984).
92. Clifford A. Truesdell, "Rational Fluid Mechanics, 1687-1765," in L. Euler, Opera omnia (Lausanne, 1954), 2/12: i-cxxv;
93. "The Rational Mechanics of Flexible or Elastic Bodies 1638-1788,", in L. Euler, Opera omnia (Zurich, 1960), 2/11: sect. 2.
94. The discovery of the partial differential calculus was an achievement of several Continental mathematicians. J. L. Greenberg has underlined the importance for this development of a little-known French mathematician, Alexis Fontaine des Bertins, who was active in the 1720s. See J. L. Greenberg, "Alexis Fontaine's Integration of Ordinary Differential Equations and the Origins of the Calculus of Several Variables," Annals of Science 39 (1982), 1-36.
95. On partial differential equations see S. S. Demidov, "The Study of Partial Differential Equations of the First Order in the 18th and 19th Centuries," Archive for the History of the Exact Sciences 26 (1982), 325-50.
96. Craig Fraser, "D'Alembert's Principle: the Original Formulation and Application in Jean d'Alembert's Traité de dynamique (1743)," Centaurus 28 (1985), 31-61, 145-159;
97. "J. L. Lagrange's Changing Approach to the Calculus of Variations," Archive for the History of the Exact Sciences 32 (1985), 151-191;
98. H. H. Goldstine, A History of the Calculus of Variations from the 17th Through the 19th Century (New York, 1980);
99. Ivor Grattan-Guinness, "The Varieties of Mechanics by 1800," Historia Mathematica 17 (1990), 313-38.
100. See note 71.
101. Fraser, "Calculus as Algebraic Analysis," 319n.
102. This is a delicate issue, since it could be contended that Euler never completely abandoned the need for a geometric interpretation. In fact Euler paid careful attention to the geometrical dimension of the differentials which he understood as infinitesimal lines, angles, surfaces, etc., rather than as algebraic symbols.
103. Leonhard Euler, Institutiones calculi differentialis (Berlin, 1755) = Opera omnia, 1/10; Institutiones calculi integralis (St. Petersburg, 1768-70) = Opera omnia, 1/ 11-13.
104. Leonhard Euler, Mechanica, sive, motus scientia analytice exposita (St. Petersburg, 1736) = Opera Omnia, 2/1-2.
105. quot;Sed quod omnibus scriptis, quae sine analysi sunt composita, id potissimum Mechanicis obtingit, ut Lector, etiamsi de veritate eorum, quae proferuntur, convincatur, tamen non satis claram et distinctam eorum cognitionem assequatur, ita ut easdem quaestiones, si tantillum immutentur, proprio marte vix resolvere valeat, nisi ipse in analysin inquirat easdemque propositiones analytica methodo evolvat. Idem omnino mihi, cum Neutoni Principia et Hermanni Phoronomiam perlustrare coepissem, usu venit, ut, quamvis plurium problematum solutiones satis percepisse mihi viderer, tamen parum tantum discrepantia problemata resolvere non potuerim. Illo igitur iam tempore, quantum potui, conatus sum analysin ex synthetica illa methodo elicere easdemque propositiones ad meam utilitatem analytice pertractare, quo negotio insigne cognitionis meae augmentum percepi". Euler, Mechanica, 8.
106. Jacob Hermann, Phoronomia, sive de viribus et motibus corporum solidorum et fluidorum libri duo (Amsterdam, 1716).
107. On Hermann, see Silvia Mazzone and Clara S. Roero, Jacob Hermann and the Diffusion of the Leibnizian Calculus in Italy (Florence, 1997).
108. Euler, Mechanica, 38-9.
109. Fraser "Calculus as Algebraic Analysis," 319.
110. John Playfair (1748-1819) is today best remembered as a geologist. In fact from 1797 he embarked on the project of systematizing and commenting on James Hutton's theory of the Earth. However, Playfair also occupies an important place in the history of eighteenth-century British mathematics, since he was one of the first to bring Continental methods to the attention of British mathematicians. See Amy Ackerberg-Hastings, "Analysis and Synthesis in John Playfair's Elements of Geometry," British Journal for the History of Science 35 (2002), 43-72.
111. John Playfair, "Traité de Mechanique Celeste," The Edinburgh Review 22 (1808), 249-284.
112. John Playfair, "Traité de Mechanique Celeste," The Edinburgh Review 22 (1808), Ibid., 279-280.
113. That is: Maclaurin's Treatise and Simpson, Doctrine and Application (see note 52).
114. Ibid., 281.
115. Similar criticisms can be found in an anonymous review of Sylverstre F. Lacroix's three-volume treatise on calculus. This review is discussed in Andrew Warwick, Masters of Theory: Cambridge and the Rise of Mathematical Physics (Chicago and London, 2003), 66.
116. It can be reasonably contended, as P. Kitcher maintains, that a shared view on the nature and role of mathematics stands behind the mathematical practices accepted by a specific community, and that changes in such meta-mathematical views are intertwined with large-scale changes in other components specific to the mathematician's culture and society. See Philip Kitcher, The Nature of Mathematical Knowledge (New York and Oxford, 1983), 191.
117. See J. B. Shank's paper in this issue. Feingold, "Mathematicians and Naturalists"; John Gascoigne, Cambridge in the Age of Enlightenment: Science, Religion, and Politics from the Restoration to the French Revolution (Cambridge, 1992);
118. Andrew Warwick, Masters of Theory;
119. Mary Terrall, "Metaphysics, Mathematics, and the Gendering of Science in Eighteenth-Century France," in The Sciences in Enlightened Europe, ed. W. Clark, J. Golinski, and S. Schaffer (Chicago and London, 1999), 246-271;
120. Michel Blay, La naissance de la mécanique analytique: la science du mouvement au tournant des XVIIe et XVIIIe siècles (Paris, 1992).
121. Just to name a few of these mathematicians: Nicholas and Daniel Bernoulli as well as Jacob Hermann stayed for a certain period in St. Petersburg; Euler divided his life between Berlin and St. Petersburg; Lagrange was in Berlin (1766-1787) and then in Paris; all the great Frenchmen, including Clairaut, d'Alembert, and Laplace, were prominent members of the French Academy.
122. It is interesting to note that at the beginning of the eighteenth century the differential and integral calculus - a theory whose status was still much in dispute - was introduced and defended at these academies by philosophers such as Nicolas Malebranche, who from 1699 fought a battle in favor of Leibniz's calculus, by Leibniz himself and by Wolff in Berlin and St. Petersburg. These philosophers - as Shank argues in his discussion of Malebranche in his contribution to this fascicle - were deeply involved in mathematical research and assigned a high place to mathematics in their philosophical systems. This helped create a conceptual as well as a social space for mathematicians at Continental academies. It would of course be a crude mistake to see the eighteenth-century Continental academies as dominated by Malebranche, and by the so-called Leibniz-Wolffian philosophy. Quite to the contrary, both the French siècle des lumières and the German Aufklärung soon fractured into many strands, which showed considerable independence from such philosophical roots. However, the Continental academies continued to guarantee mathematicians a high social status, in accordance with the desiderata of these eminent founding fathers.
123. Terrall, "Metaphysics," 256.
124. E. Brian, La mesure de l'état: administrateurs et géomètres au XVIIIe siècle (Paris, 1994).
125. Feingold, "Mathematicians and Naturalists," 78.
126. Feingold opposes the idea that the distinction between the two competing groups can be defined in terms of "scientists" versus "amateurs." He warns us that "by the early eighteenth century the cleavage between a group comprised primarily of mathematicians, astronomers, and physicists, on the one hand, and naturalists, physicians, and general scholars, on the other, was indicative of taste, not competence." Ibid., 94.
127. Albert G. Howson, A History of Mathematics Education in England (Cambridge, 1982), 66.
128. The mutineers included Francis Maseres, a prolific writer on logarithms, James Glenie and John Landen, two proponents of new algebraic foundations of calculus, Thomas Hornsby, Savilian Professor of Astronomy at Oxford, and George Atwood, today remembered for the Atwood-machine. See John L. Heilbron, "A Mathematicians' Mutiny, with Morals," in World Changes: Thomas Kuhn and the Nature of Science, ed. P. Horwich (Cambridge (Mass)/London, 1993), 81-129, on 89;
129. and John Gascoigne, Joseph Banks and the English Enlightenment: Useful Knowledge and Polite Culture (Cambridge, 1994), 11-13.
130. June Barrow-Green, "'A Correction to the Spirit of Too Exclusively Pure Mathematics': Robert Smith (1689-1768) and his Prizes at Cambridge University," Annals of Science 56 (1999), 271-316;
131. John Gascoigne, "Mathematics and Meritocracy: The Emergence of the Cambridge Mathematical Tripos," Social Studies of Science 14 (1984), 547-84,
132. reprinted as chapter 5 in John Gascoigne, Science, Politics, and Universities in Europe, 1600-1800 (Ashgate, 1998);
133. Warwick, Masters of Theory.
134. Gascoigne, Cambridge in the Age of Enlightenment, 273.
135. Gascoigne, Cambridge in the Age of Enlightenment, 273, Ibid.
136. Warwick, Masters of Theory, 66.
137. Terrall, "Metaphysics," 247.
138. Terrall, "Metaphysics," Ibid., 267-9.
139. This in part explains why George Berkeley's attack against the cogency of the new analysis in The Analyst (1734) was acutely felt in Britain, while it was almost ignored on the Continent.
140. Terrall, "Metaphysics," 256-7.
141. See Jean-Charles Borda's skepticism on the possibility of applying contemporary hydrodynamics to engineering in George E. Smith, "The Newtonian Style in Book II of the Principia," in Isaac Newton's Natural Philosophy, ed. J. Z. Buchwald and I. B. Cohen (Cambridge, Mass., London, 2001), 249-298, on 284.
142. Thomas Kuhn, "Mathematical versus Experimental Traditions in the Development of Physical Science," in The Essential Tension: Selected Readings in Scientific Tradition and Change (Chicago, 1977), 31-65.
143. The studies of John Gascoigne quoted above remain fundamental for understanding the strong link between mathematics, natural philosophy and theology in eighteenth-century England.
144. Late eighteenth-century British mathematicians were at their best when refraining from investigative mathematics and applying themselves to down-toearth enterprises with laborious problems more like bookkeeping (such as Henry Cavendish's theory of electricity and Charles Hutton's calculation of gravity at Mount Schehallien): their mathematics had to prove its utility.
145. I refer the reader to Warwick, Masters of Theory, an excellent book that I was able to read only when this paper was almost completed. In studying the reform of mathematics which occurred at the beginning of the nineteenth century - a reform in which people such as Woodhouse, Peacock, Babbage and Herschel introduced Continental algebraic methods into the Cambridge curriculum - Warwick shows in detail how difficult it was to exchange mathematical "craft" skills across different local cultures. We tend to forget that what is given pride of place in the pages of twentieth-century histories of eighteenth-century mathematics was the work of a very small and privileged group of mathematicians, who shared specific skills, sensibilities, and values. Warwick documents the time and efforts that were necessary in order to trade such a mathematical culture across the Channel: those who worked outside the elitist protected space of the academies, and who did not participate in an exchange of information, which often took place through oral communication and in manuscript form, had to overcome considerable difficulties in order to acquaint themselves with the Eulerian calculus. As Warwick shows, the reform was brought about not only by overcoming local prejudice, but especially by transferring skills through new teaching methods based on a close master-pupil relationship.