Robert woodhouse and the evolution of cambridge mathematics

History of Science

Volumn 44, Issue 1, 2006, Pages 69-93

  Phillips, Christopher ^a  

^a HARVARD UNIVERSITY   (United States)

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EID: 33644948467     PISSN: 00732753     EISSN: None     Source Type: Journal    
DOI: 10.1177/007327530604400104     Document Type: Review

References (172)

1. This assessment is found in nearly all general histories of mathematics, but even in writings of historians sympathetic to the period, e.g. J. M. Dubbey, The mathematical work of Charles Babbage (Cambridge, 1978), 10 and
2. M. Fisch. "'The emergency which has arrived': The problematic history of nineteenth century British algebra - a programmatic outline", The British journal for the history of science, xxvii (1994), 247-76, p. 250.
3. Recently, N. Guicciardini in "Dot-age: Newton's mathematical legacy in the eighteenth century", Early science and medicine. ix (2004), 218-56
4. and The development of Newtonian calculus in Britain 1700-1800 (Cambridge, 1989) has forwarded a more nuanced and complex view of the British mathematical tradition.
5. His legacy in these areas has been well documented. For his influence on logic and algebra (especially on Augustus De Morgan), see M. Panteki, "The mathematical background of George Boole's Mathematical Analysis of Logic (1847)", in A Boole anthology: Recent and classical studies in the logic of George Boole, ed. by J. Gasser (Dordrecht, 2000),
6. and M. Panteki, idem, "French 'logique' and British 'logic': On the origins of Augustus De Morgan's early logical enquiries, 1805-1835", Historia mathematica, xxx (2003), 278-340;
7. D. Sherry. "The logic of impossible quantities", Studies in history and philosophy of science, xxii (1991), 37-62;
8. H. Pycior, "George Peacock and the British origins of symbolic algebra". Historia mathematica, viii (1981), 23-45;
9. H. Pycior, "Internalism, externalism and beyond: 19th century British algebra". Historia mathematica, xi (1984), 424-41;
10. J. L. Richards, "The art and science of British algebra: A study in the perception of mathematical truth", Historia mathematica, vii (1980), 343-65:
11. H. Becher, "Woodhouse, Babbage, Peacock, and modern algebra". Historia mathematica, vii (1980), 389-400;
12. and J. M. Dubbey, "Babbage. Peacock, and modern algebra", Historia mathematica, iv (1977), 295-302.
13. For the later influence on analysis, see E. Koppelman, "The calculus of operations and the rise of abstract algebra", Archive for the history of exact sciences, viii (1971), 155-242;
14. P. Enros, "The Analytical Society: Mathematics at Cambridge University in the early 19th century", Ph.D. dissertation, University of Toronto, 1979;
15. P. Enros, idem, "Cambridge University and the adaptation of analytics in the early nineteenth-century England", in Social history of nineteenth century mathematics, ed. by Herbert Mehrtens, H. J. M. Bos and Ivo Schneider (Boston. 1981), 135-47;
16. P. Enros, idem , " Analytical Society (1812-1813): Precursor of the renewal of Cambridge mathematics", Historia mathematica. x (1983), 24-47;
17. M. Panteki, "William Wallace and the introduction of Continental calculus to Britain", Historia mathematica, xiv (1987), 119-32;
18. J. M. Dubbey, "Introduction of the differential notation to Great Britain". Annals of science, xix (1963), 37-48; and
19. J. M. Dubbey, idem, "Robert Woodhouse and the establishment of a mathematical basis for the calculus", M.Sc. dissertation, University of London, 1964. For his brief tenure as Lucasian professor,
20. see S. Schaffer, "Paper and brass: The Lucasian Professorship 1820-39, " in A history of Cambridge University's Lucasian Professors of Mathematics, ed. by K. C. Knox and R. Noakes (Cambridge, 2003), 241-93.
21. There is little extant biographical information, mainly located in J. Venn, Biographical history of Conville and Caius College, 1349-1897: containing a list of all known members of the college from the foundation to the present time, with biographical notes (Cambridge, 1897), 119-20,
22. and A. De Morgan, "Robert Woodhouse", Penny cyclopaedia, xxvii (1843), 526-7.
23. George Morganardently supported the revolution and was likely an early source of French mathematics for Woodhouse. See D. O. Thomas. "Morgan, George Cadogan", in Oxford dictionary of national biography (Oxford, 2004).
24. We know Woodhouse was Morgan's pupil from a letter of Morgan's uncle, Richard Price. See W. B. Peach and D. O. Thomas, Correspondence of Richard Price, iii (Durham, 1994), 178. I am indebted to Simon Schaffer for this point.
25. Enros, in "The Analytical Society" (ref. 2) references B. C. Nangle, The Monthly Review, 2nd series, 1790-1815; indexes of contributors and articles (Oxford, 1955), which I also used to determine authorship of Monthly Review articles.
26. By culminating the story in Lagrange, Woodhouse not only dismisses Maclaurin and other fluxionists but also Britons such as Landen and Waring who used algebraic techniques yet in Woodhouse's view remained vulnerable to Berkeley's criticisms. Berkeley argued that the method of fluxions was not geometrically rigorous, in part because it required a notion of infinitesimals, which Berkeley did not believe existed.
27. (Cambridge, 1803), p. xxv.
28. Richards also makes a useful distinction between generalization and abstraction in Woodhouse's work: J. L. Richards. "Rigor and clarity: Foundations of mathematics in France and England. 1800-1840". Science in context, iv (1991), 279-319, pp. 311-13.
29. See Dubbey, Robert Woodhouse and the establishment of a mathematical basis for the calculus (ref. 2) for a mathematical analysis of the text. The idea that Newton relied primarily on geometry is a late eighteenth-century construct, but as this was generally how Woodhouse and other Cambridge men interpreted Newton. I will continue to associate the two.
30. This was hardly a novel theory and was deeply related to Enlightenment philosophy. See Sherry. "The logic of impossible quantities" (ref. 2), 47-54.
31. Woodhouse, Principles (ref. 6), p. ii.
32. See W. W. Rouse Ball, The origin and history of the Mathematical Tripos (Cambridge, 1880),
33. and J. Gascoigne. "Mathematics and meritocracy: The emergence of the Cambridge Mathematical Tripos", Social studies of science, xiv (1984), 547-84 regarding the Mathematical Tripos.
34. A. Warwick, Masters of theory: Cambridge and the rise of mathematical physics (Chicago. 2003), 89-94. The term 'coach' here is anachronistic as it became common only after Woodhouse's death; nonetheless it referred to the same practice of private tutoring.
35. Of course it was not completely ignored; for some of the response, see Enros, "The Analytical Society" (ref. 2). 88-89.
36. Historians have shown how Woodhouse's arguments in the Principles:, largely developed in his Monthly Review articles, proved suggestive and influential for later developments in algebra and analysis (see ref. 2).
37. C. Brooke, History of Gonville and Caius College (Woodbridge. 1985), 186.
38. See for example, R. Woodhouse, "Review of Vince's Complete System of Astronomy", Monthly Review, xxvii (1798), 121-31, p. 131.
39. R. Woodhouse. Treatise on isoperimetrical problems and the calculus of variations (Cambridge. 1810).
40. G. Peacock, "Report on the recent progress and present state of certain branches of analysis", Third report of the British Association for the Advancement of Science (1834), 185-352. p. 295.
41. R. Woodhouse, Treatise on plane and spherical trigonometry, 1st edn (Cambridge, 1809), pp. vii. 14, 99.
42. R. Woodhouse, Treatise on plane and spherical trigonometry, 1st edn, Ibid., 93.
43. R. Woodhouse, Treatise on plane and spherical trigonometry, 1st edn, Ibid., 117ff.
44. R. Woodhouse, Treatise on plane and spherical trigonometry, 1st edn, Ibid., 180.
45. Peacock, "Report" (ref. 15), 296.
46. Whewell letter to Herschel on 1 November 1818, in I. Todhunter, William Whewell, D.D., Masterof Trinity College, Cambridge: An account of his writings; with selections from his literary and scientific correspondence, ii (London, 1876), 30.
47. As a more senior fellow than Woodhouse at Caius, Vince's support was most likely needed to be appointed a senior fellow and was likely given as he had already supported Woodhouse's bid to join the Royal Society in 1802; see Royal Society Archives. EC/1802/08.
48. R. Woodhouse, An elementary treatise on astronomy: Physical astronomy, 1st edn, ii (Cambridge, 1818);
49. R. Woodhouse, "Review of Lagrange's Leçons sur le Calcul des Fonctions", Monthly review, xlix (1806), 486-98, p. 489.
50. See Warwick. Masters of theory (ref. 10), 49-113.
51. Richards, "Rigor and clarity" (ref. 6), 307-9.
52. Becher, "Woodhouse, Babbage, Peacock, and modern algebra" (ref. 2), 397;
53. H. Becher, "Radicals, Whigs, and conservatives: The middle and lower classes in the analytical revolution at Cambridge in the age of aristocracy". The British journal for the history of science, xxviii (1995), 405-26, p. 406.
54. Many historians have focused on the 'success' of the Analytical Society and/or the failure of Woodhouse in this context: see W. W. Rouse Ball. A history of the study of mathematics at Cambridge (Cambridge, 1889), 120-2;
55. Enros. "The Analytical Society" (ref. 2), 215;
56. C. Smith and M. N. Wise, Energy and empire: A biographical study of Lord Kelvin (Cambridge, 1989), 151;
57. M. V. Wilkes, "Herschel, Peacock, Babbage, and the development of the Cambridge curriculum". Notes and records of the Royal Society, xliv (1990), 205-19, p. 207:
58. Fisch, "The emergency which has arrived" (ref. 1), 247;
59. and W. Ashworth, "Memory, efficiency, and symbolic analysis: Charles Babbage, John Herschel, and the industrial mind", Isis, lxxxvii (1996), 629-53, pp. 632-6.
60. Enros, "The Analytical Society" (ref. 2), 103.
61. S. Schweber (ed.), Aspects of the life and thought of Sir John Frederick Herschel. i (New York, 1981), 57-59 has earlier dates for the first meetings, but on either account they were held in Bromhead's rooms.
62. Peacock. "Report" (ref. 15). 295;
63. C. Babbage. Passages from the life of a philosopher (London, 1864), 26.
64. Dubbey. The mathematical work of Charles Babbage (ref. 1), 62; Royal Society archives. EC 1816/16. Bromhead also credits Woodhouse with introducing students to Lagrange and other foreign expositors of the calculus,
65. see E. Bromhead. "Differential calculus". Supplement to the Encyclopedia Britannica. iii (1824), 568-72. p. 568.
66. Enros, "The Analytical Society" (ref. 2), 158.
67. Letter of 1 January 1814, quoted in ibid., 119-20.
68. A. Society, Memoirs (Cambridge. 1813), 33.
69. S. Lacroix, Elementary treatise on the differential and integral calculus, transl, by Charles Babbage, John Herschel, and George Peacock (Cambridge, 1816);
70. G. Peacock, A collection of examples of the applications of the differential and integral calculus (Cambridge, 1820).
71. Richards, "Rigor and clarity" (ref. 6), 311-13,
72. and Becher, "Woodhouse. Babbage, Peacock, and modern algebra" (ref. 2), 396.
73. As for the Society, there is some disagreement regarding the formal concluding dates; see M. Fisch, "The making of Peacock's Treatise on Algebra: A case of creative indecision", Archives for the history of exact science, liv (1999), 137-79, p. 137.
74. See Enros, "The Analytical Society" (ref. 2), 102, 215. Fisch, in "Peacock's Treatise" (ref. 3), emphasizes Peacock's pedagogical focus, although he was increasingly distanced from the other members and this focus should be more associated with him than with the Analytical Society.
75. Enros, "The Analytical Society" (ref. 2). 179-200.
76. J. Herschel, "Review of Somerville's Mechanism of the Heavens". Quarterly Review, xlvii (1832), 537-59, p. 547.
77. K. C. Knox. "The negative side of nothing: Edward Waring, Isaac Milner and Newtonian values." in From Newton to Hawking, ed. by Knox and Noakes (ref. 2), 205-40, p. 236.
78. See the index to J. M. F. Wright. Solutions of the Cambridge problems: From 1800 to 1820 (London, 1825),
79. also cited in Becher, "Woodhouse. Babbage, Peacock, and modern algebra" (ref. 2). 393.
80. Of course, Wright was not listing the books that were useful for past examinations. but rather those that contemporary (1825) students would find useful for answering the older questions. Nonetheless, it indicates how pervasive Woodhouse's book had become by the 1820s as preparation for the Tripos.
81. G. Airy, Autobiography of Sir George Biddell Airy, ed. by Wilfred Airy (Cambridge. 1896), 33.
82. Ibid., 20. 'Senate-House' referred to the location of the Tripos examination.
83. Ibid., 26;
84. Macaulay quoted in S. F. Cannon. Science in culture: The early Victorian period (New York, 1978), 31.
85. Rouse Ball, Mathematics at Cambridge (ref. 27), 119-20.
86. See Becher. "Radicals, Whigs, conservatives" (ref. 26), 413. who cites Babbage Correspondence. fol. 26. letter of 4 February 1814.
87. Anonymous, Cambridge problems: being a collection of the printed questions proposed to the candidates for the degree of bachelor of arts at the general examinations from 1801 to 1820 inclusive (Cambridge, 1821), 268-75.
88. Although this might be making too much of the wording of the question, it seems significant that these were the only times algebra and the calculus were explicitly connected in a question. For the problems,
89. see ibid., 145. 265.
90. Becher, "Radicals, Whigs, conservatives" (ref. 26), 414.
91. Warwick, Masters of theory (ref. 10), 51.
92. H. Becher. "William Whewell and Cambridge mathematics". Historical studies in the physical sciences, xi (1980), 1-48, pp. 9-10;
93. Schaffer, "Paper and brass" (ref. 2), 261.
94. See also H. Becher. "Woodhouse, Robert," in Oxford dictionary of national biography (ref. 3).
95. In Todhunter, William Whewell (ref. 21), 30.
96. A typical critic was A. Browne. A short view of the first principles of the differential calculus (Cambridge, 1824), p. ix:
97. Woodhouse made this same point in R. Woodhouse. "Review of Lacroix's Traité des Differences et des Series". Monthly review. xxxvi (1801), 498-501, p. 500
98. and R. Woodhouse. "Review of Bonnycastle's Treatise on Plane and Spherical Trigonometry". Monthly Review, liii (1807), 279-85. p. 285.
99. This criticism was closely related to the debate over whether students would be turned into mindless calculating machines through the new analysis. See Schaffer, "Paper and brass" (ref. 2);
100. W. Ashworth, "The calculating eye: Baily, Herschel. Babbage and the business of astronomy", The British journal for the history of science, xxvii (1994), 409 -41;
101. W. Ashworth, idem, "Memory, efficiency, and symbolic analysis" (ref. 27);
102. and I. Grattan-Guinness, "Charles Babbage as an algorithmic thinker", IEEE Annals of the history of computing, xiv/3 (1992), 34-48.
103. Anonymous, "Review of Woodhouse's Physical Astronomy", Edinburgh review, xxxi (1819), 375-94. p. 394, likely authored by William Brougham.
104. M. Crosland and C. Smith, "Transmission of physics from France to Britain: 1800-1840", Historical studies in the physical sciences, ix (1978). 1-61, p. 11.
105. R. Woodhouse, "Review of Vince's Complete System of Astronomy, vol. 2", Monthly Review, xxxv (1801), 72-82, p. 82.
106. R. Woodhouse, Monthly Review, Ibid., 80.
107. Crosland and Smith. "Transmission of physics" (ref. 55), 5-7.
108. Certainly others were interested in the promotion of Continental mathematics, and these are well highlighted by Crosland and Smith and include Playfair in Scotland and Lloyd in Ireland.
109. Crosland and Smith. "Transmission of physics" (ref. 55), 19-20.
110. Crosland and Smith. "Transmission of physics", Ibid., 51-56.
111. Crosland and Smith. "Transmission of physics", Ibid., 13.
112. Airy owned and annotated copies of vol. i of the 1st edn of R. Woodhouse, An elementary treatise on astronomy (Cambridge. 1812).
113. and of Woodhouse, Physical astronomy (ref. 23). He made appropriate changes when the new edition of Woodhouse was printed in 1822.
114. Airy. Autobiography (ref. 42), 29-30;
115. Whewell, letter to Herschel in Todhunter, William Whewell (ref. 21), 30.
116. Woodhouse's Physical astronomy
117. was the first to incorporate Laplace and would be almost the only nearly complete treatise on the subject for two decades more. See Panteki, "French 'logique'" (ref. 2), 300.
118. Schweber (ed.), Herschel (ref. 28), 64. Woodhouse's influence upon Airy is discussed later in this section.
119. Woodhouse, Physical astronomy (ref. 23), p. Ivii.
120. Woodhouse, Elementary astronomy. 1st edn (ref. 63), p. xiv.
121. Although the title page of R. Woodhouse, An elementary treatise on astronomy, 2nd edn (Cambridge, 1822) indicates a publishing date of 1821, the preface was not written until 1822.
122. Woodhouse, Physical astronomy (ref. 23), p. xiii.
123. Woodhouse, Elementary astronomy, 2nd edn (ref. 66), pp. xvii-xviii.
124. R. Woodhouse, "Review of Laplace's Traité de Mécanique Céleste, part 1", Monthly review, xxxi (1800), 493-505, p. 472;
125. R. Woodhouse, idem, "Review of Laplace's Traité de Mécanique Céleste, part 2", Monthly review, xxxii (1800), 478-85, p. 484.
126. As a continued demonstration of the link between the younger generation and Woodhouse, Peacock was influential in securing the observatory of which Woodhouse would become the first supervisor.
127. Woodhouse, Elementary astronomy, 2nd edn (ref. 66), pp. xix-xxii.
128. For instance, the method of the variation of constants presented by Woodhouse made its way repeatedly into problems. See Panteki, "French 'logique'" (ref. 2), 300. Panteki also discusses Woodhouse's work on the three-body problem.
129. Cf. Becher, "William Whewell and Cambridge Mathematics" (ref. 51). 10.
130. Woodhouse, Physical astronomy (ref. 23), p. Iviii.
131. Analysis additionally provided the assurance that the system was stable, with overtones of desirable political and social stability; see Smith and Wise, Energy and empire (ref. 27), 153.
132. Woodhouse, Elementary astronomy, 2nd edn (ref. 66), p. xxii.
133. R. Woodhouse, "Some account of the transit instrument made by Mr. Dolland. and lately put up at the Cambridge Observatory", Philosophical transactions of the Royal Society of London, cxv (1825), 418-28;
134. R. Woodhouse, idem, "On the transit instrument of the Cambridge Observatory; being a supplement to a former paper", Philosophical transactions of the Royal Society of London, cxvi (1826), 75-76;
135. and R. Woodhouse, idem, "On the derangements of certain transit instruments by the effects of temperature", Philosophical transactions of the Royal Society of London, cxvii (1827), 144-58.
136. R. Woodhouse, "Review of Wood's Optics and Vince's Astronomy", Monthly review, xxxiv (1801). 239-45, p. 244,
137. and R. Woodhouse, idem, "Review of Wood's Mechanics and Vince's Hydrostatics". Monthly review, xxviii (1799), 313-23, p. 323.
138. Anonymous, "Notices", London times, 28 February 1823, 3G. Despite being the Plumian Professor, Woodhouse
139. probably gained prestige through his marriage into the family of architect William Wilkens. Whilst his wife's death was noted in the London Times,
140. the same was not true of Woodhouse's; see Anonymous. "Notices", London Times, 7 April 1826, 4B.
141. Enros, "The Analytical Society" (ref. 2), 244.
142. B. Powell. "Review of Woodhouse's Elementary Treatise on Astronomy". British critic, xx (1823),143-56, pp. 145-56.
143. See also P. Corsi, Science and religion: Baden Powell and the Anglican debate, 1800-1860 (Cambridge, 1987), 37ff.
144. De Morgan. "Robert Woodhouse" (ref. 3), 527. The archival evidence unfortunately does not indicate where or how Woodhouse learned to be an "expert practical astronomer" before 1812, although he certainly had the resources to undertake such a task.
145. Anonymous. "Review of Woodhouse's Physical Astronomy" (ref. 54), 377.
146. O.15.47/381 in Whewell Papers. Wren Library, Trinity College.
147. On 'progressive' v. 'permanent', see R. Yeo, Defining science: William Whewell, natural knowledge, and public debate in early Victorian Britain (Cambridge, 1993), 209-30.
148. See Panteki, "French 'logique'" (ref. 2), 313-17,
149. and Becher, "William Whewell and Cambridge mathematics" (ref. 51), 10-19.
150. Airy's copies still remain in the Wren Library.
151. Airy. Autobiography (ref. 42), 77-78.
152. Letter 47, 11 February 1828, Airy Papers, Wren Library.
153. Letter 46, 27 January 1828, Airy Papers, Wren Library.
154. Letter 48, 19 February 1828, Airy Papers, Wren Library.
155. G. Airy, Mathematical tracts on physical astronomy, the figure of the Earth, precession and nutation, and the calculus of variations (Cambridge, 1826), p. iv. See
156. Panteki, "Boole's Mathematical Analysis of Logic" (ref. 2), 169-73,
157. and Panteki, idem. "Relationships between algebra, differential equations and logic in England, 1800-1860", Ph.D., Council for National Academic Awards, 1991,
158. §1.3 for more on Airy's use of the Earth-figure equation, and Panteki, idem. "French 'logique'" (ref. 2), 301 for Airy's movement toward Whewell.
159. G. Airy, Mathematical tracts on physical astronomy, the figure of the Earth, precession and nutation, and the calculus of variations, 2nd edn (Cambridge. 1831), p. iv. Woodhouse's work was reduced to two brief citations in this edition.
160. Panteki. "French 'logique'" (ref. 2), 316.
161. Airy. Autobiography (ref. 42), 29-30. Airy's autobiographical notes were never published during his lifetime.
162. G. Airy, "Report on the progress of astronomy during the present century". First report of the British Association for the Advancement of Science (1833), 125-89, pp. 185-6.
163. Airy, Autobiography (ref. 42), 37-38.
164. Schuffer, "Paper and brass" (ref. 2), 246-76.
165. See also Schaffer. "Astronomers mark time: Discipline and the personal equation". Science in context, ii (1988), 115-45,
166. and A. Chapman, "George Biddell Airy (1801-1892): A centenary commemoration", Notes and records of the Royal Society, xlvi (1992), 103-10, pp. 107-8.
167. Woodhouse's early articles could provide a fruitful link, for instance, between the narratives of I. Grattan-Guinness, "French calcul and English fluxions around 1800: Some comparisons and contrasts", in Science and imagination in XVIIIth century British culture, ed. by S. Rossi (Milan, 1987), 215-30:
168. I. Grattan-Guinness, idem, Convolutions in French mathematics, 1800-1840 (Boston, 1990):
169. and Guicciardini, Newtonian calculus (ref. 1).
170. In explaining the lack of extant images of Woodhouse, a note in the picture album of Lucasian Professors in the Wren Library suggests that Woodhouse was unwilling to have any likenesses of him produced, apparently because of a pock-marked face.
171. Guicciardini. Newtonian calculus (ref. 1), 136.
172. Schaffer, "Paper and brass" (ref. 2), 255.