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‘The emergency which has arrived’: the problematic history of nineteenth-century British algebra – a programmatic outline

Published online by Cambridge University Press:  05 January 2009

Menachem Fisch
Menachem Fisch
Affiliation:
The Cohn Institute for the History and Philosophy of Science and Ideas, Tel Aviv University, Ramat Aviv 69978, Israel.
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Extract

More than any other aspect of the Second Scientific Revolution, the remarkable revitalization or British mathematics and mathematical physics during the first half of the nineteenth century is perhaps the most deserving of the name. While the newly constituted sciences of biology and geology were undergoing their first revolution, as it were, the reform of British mathematics was truly and self-consciously the story of a second coming of age. ‘Discovered by Fermat, cocinnated and rendered analytical by Newton, and enriched by Leibniz with a powerful and comprehensive notation’, wrote the young John Herschel and Charles Babbage of the calculus in 1813, ‘as if the soil of this country [was] unfavourable to its cultivation, it soon drooped and almost faded into neglect; and we now have to re-import the exotic, with nearly a century of foreign improvement, and to render it once more indigenous among us’.

Type
Research Article
Information
The British Journal for the History of Science , Volume 27 , Issue 3 , September 1994 , pp. 247 - 276
Copyright
Copyright © British Society for the History of Science 1994

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References

This paper was initially written while in residence as Fellow of the Institute for Advanced Study Berlin, during the academic year 1991–92. An early and abbreviated version was read at the Institut fiir Wissenschaftsgeschichte History of Science colloquium in Gottingen University in June 1992.1 am grateful to Joan Richards, Ivor Grattan-Guinness, Yehuda Elkana, John Brooke, William Ashworth and Janet Browne, who read early drafts and offered valuable criticisms.

1 Herschel, J. and Babbage, C. (eds.), Memoirs of the Analytical Society 1813, Cambridge, 1813, p. iv.Google Scholar

2 For informed and comparative studies of scientific education in Cambridge and Scotland see Wilson, D. B., ‘The educational matrix: physics education at early-Victorian Cambridge, Edinburgh and Glasgow Universities’, in Wranglers and Physicists: Studies on Cambridge Mathematical Physics in the Nineteenth Century (ed. Harman, P. M.), Manchester, 1985, 12–48Google Scholar, and Smith, C. and Wise, M. N., Energy and Empire: A Biographical Study of Lord Kelvin, Cambridge, 1989, especially ch. 2.Google Scholar For mathematics and mathematical physics in Ireland see Hankins, Thomas L., Sir William Rowan Hamilton, Baltimore, 1980, ch. 1Google Scholar, and Energy and Empire, ch. 1.

3 The Analytical Society was not the first to criticize the sorry state of British mathematics at the time. Its campaign, however, was the first to make a difference. On earlier protesters and critics, notably John Playfair and Robert Woodhouse, and references to the relevant secondary literature, see Fisch, M., William Whewell Philosopher of Science, Oxford, 1991, 25Google Scholar, especially footnotes 13 and 14.

4 Guicciardini, N., The Development of the Newtonian Calculus tn Britain 1700–1800, Cambridge, 1989.CrossRefGoogle Scholar

5 Enros, P., ‘Cambridge University and the adoption of analytics in early nineteenth-century England’, in Social History of Nineteenth-Century Mathematics (ed. Mehrtens, H. et al. ), Boston, 1981, 135–48CrossRefGoogle Scholar, and ‘The Analytical Society 1812–13’, Historia Mathematica (1983), 10, 24–47Google Scholar; Becher, H.W., ‘William Whewell and Cambridge mathematics’, Historical Studies in the Physical Sciences (1980), 11, 1–48CrossRefGoogle Scholar, and ‘Woodhouse, Babbage, Peacock, and modern algebra’, Historia Mathematica (1980), 7, 389–400Google Scholar; Schweber, S. S., Aspects of the Life and Thought of Sir John Frederick Herschel, New York, 1981, 54–67Google Scholar; Fisch, , op. cit. (3), § 2.221–3.Google Scholar

6 Koppelman, E., ‘The calculus of operations and the rise of abstract algebra’, Archives for History of Exact Science (1971–1972), 8, 155–242CrossRefGoogle Scholar; Richards, J., ‘The art and science of British algebra: a study in the perception of mathematical truth’, Historia Mathematica (1980), 7, 343–65CrossRefGoogle Scholar, ‘Augustus De Morgan, the history of mathematics, and the foundations of algebra’, Isis (1987), 78, 7–30Google Scholar, ‘Rigor and clarity: foundations of mathematics in France and England, 1800–1840’, Science in Context (1991), 4 (2), 297–319Google Scholar, and ‘God, truth and mathematics in nineteenth-century England’, in The Invention of Physical Science: Intersections of Mathematics, Theology and Natural Philosophy Since the Seventeenth Century – Essays in Honor of Erwin N. Hiebert (ed. Nye, Mary Jo et al. ), Kluwer, 1992, 51–78CrossRefGoogle Scholar; Pycior, H. M., ‘George Peacock and the British origins of symbolical algebra’, Historia Mathematica (1981), 8, 23–45CrossRefGoogle Scholar, ‘Early criticism of the symbolical approach to algebra’, Historia Mathematica (1982), 9, 392–412Google Scholar, ‘Augustus De Morgan's algebraic work: the three stages’, Isis (1983), 74, 211–26Google Scholar, and ‘Internalism, externalism, and beyond: 19th century British algebra’, Historia Mathematica (1984), 11, 424–41.Google Scholar

7 In the preface to William Whewell: A Composite Portrait, Oxford, 1991Google Scholar, Simon Schaffer and I pointed to ‘the profound significance of Whewell's engagement with the mathematics of his time’ as a key area for future research. While the present paper seeks to outline a broader project, the latter part of it may be viewed as endeavouring to lay the foundations for such an undertaking.

8 Rothblatt, S., The Revolution of the Dons, Cambridge, 1981Google Scholar (first published by Faber and Faber and Basic Books, 1968).

9 On Woodhouse see Becher, , ‘Woodhouse, Babbage, Peacock, and modern algebra’, op. cit. (5).Google Scholar

10 Babbage, C., Passages from the Life of a Philosopher, London, 1864, 30.Google Scholar

11 The full text of the manifesto is quoted in Schweber, , op. cit. (5), 58–9.Google Scholar

12 De Morgan, A., ‘English science’, The British and Foreign Review (1835), 1, 134–57Google Scholar, quoted in Schweber, , op. cit. (6), 66.Google Scholar Despite the discretion granted to the examiners, Peacock's decision to impose his views on the University in this manner was a bold one, and he encountered considerable, if short-lived opposition. Writing to Herschel in December 1816, Peacock was confidently optimistic: ‘We are proposing college reforms, which will introduce the true faith in Trinity; at least if our wise seniority will adopt them… Trust me, the goldenage of the University is approaching’ (Peacock, to Herschel, , 3 12 1816Google Scholar, Letters and Papers of Sir John Herschel, University Publications of America, reel 13, document 247. Further references to this collection will follow the notation adopted in Michael Crowe's A Guide to the Manuscripts and Microfilms, e.g. RSHS 13.247.). Peacock's optimism, however, was premature. ‘White and Fallows [proved] entirely of the old school’, he wrote a few weeks later, ‘& the influence of their examination was so great as completely to overpower my examination. The introduction of d's into the papers excited much remorse…& I believe that I may consider myself as owing entirely to the success of the Johnians in the examination for my escape from some public proceedings against me’ (Peacock to Herschel, undated, postmark: 1817, RS HS 13.248). Peacock's initiative was publicly criticized in Peacock, D. M.'s A Comparative View of The Principles of the Fluxional and Differential Calculus addressed to the University of Cambridge, Cambridge, 1819.Google Scholar But the opposition eventually died down. For further details see Dubbey, J. M., The Mathematical Work of Charles Babbage, Cambridge, 1978, ch. 3.CrossRefGoogle Scholar By comparison, Robert Woodhouse, who, in the textbooks that he published was as outspoken a critic of Cambridge mathematics as Peacock, never exercised his right as moderator to depart in the questions he set from the customary dot notation. (Woodhouse was moderator for the years 1803, 1804, 1807 and 1808.)

13 Cf. Dubbey, , op. cit. (12), 47.Google Scholar

14 See Fraser, C. G., ‘Joseph Louis Lagrange's algebraic vision of the calculus’, Historia Mathematica (1987), 14, 38–53CrossRefGoogle Scholar, and Grattan-Guinness, I., Convolutions in French Mathematics, 1800–1840, 3 vols., Basel, Boston and Berlin, 1990, i, 127–35.Google Scholar In a section entitled ‘On mathematical Denkweisen’, Grattan-Guinness proposes a tripartite taxonomy of mathematical ‘styles of thinking’ current at the time among French mathematicians. He labels them: the algebraic, the geometrical and the analytical, reserving the rubric ‘analytical’ to denote tracts employing ‘techniques based on, or using, limits’ (Convolutions, i, 55–7). Grattan-Guinness makes it clear, however, that his employment of the terms is not intended to recapitulate the way in which they were understood at the time. (Thus he points out that well into the nineteenth century all mathematicians were called ‘geometers’.) In dealing with the British mathematicians under consideration here, I prefer a different terminology, originally introduced by W. R. Hamilton, and elaborated upon in Section 3 below. Suffice it to note at this stage that in the present paper the terms ‘analyst’ and ‘analytical’ will be employed more liberally (and admittedly less precisely) than Grattan-Guinness, in keeping with their historical usage, namely as roughly denoting non-geometrical. Thus, the Society pronounced itself ‘analytical’ par excellence despite its marked ‘algebraic’ orientation.

15 The definitive study of the history of the fluxional calculus to date is Guicciardini, , op. cit. (4)Google Scholar. For cogent summaries of the disparities between British and continental developments of the calculus at the turn of the nineteenth century see pp. 139–42, and Grattan-Guinness, , op. cit. (14), i, chs. 3 and 4, especially pp. 264–5.Google Scholar

16 Smith, and Wise, , op. cit. (2), 151.Google Scholar

17 For a detailed survey of this literature see Grattan-Guinness, I., ‘Mathematics and mathematical physics from Cambridge, 1815–40: a survey of the achievements and of the French influences’Google Scholar in Harman, (ed.), op. cit. (2), 84–111.Google Scholar

18 See especially ch. 6.

19 Smith, and Wise, , op. cit. (2), 155.Google Scholar On the Cambridge programme see Morrell, J. B. and Thackray, A., Gentlemen of Science: Early Years of the British Association for the Advancement of Science, Oxford, 1981, 479–84.Google Scholar

20 Smith, and Wise, , op. cit. (2), 178–9.Google Scholar

21 Smith, and Wise, , op. cit. (2), 150.Google Scholar

22 Smith and Wise's broad characterizations, especially of the work of the second generation of reformers, deserve careful scrutiny. Such an appraisal, however, lies beyond the modest scope of the present paper. One anecdote concerning Gregory and Ellis is worth noting in passing, for it seems to challenge Smith and Wise's blanket description of the group's orientation. In a journal entry of 24 June 1840 (Trinity College Library Add.Ms.a.821), Ellis writes of a letter he had received from Gregory who was apparently ‘struck in Whewell's book with the difference between Whewell's present & old views about the laws of motion, & by the curiously close coincidence, by which I [Ellis] too was surprized of some of his present notions on the first law of motion with views I have often expressed, viz. that time cannot be a cause – because time is only the condition of the existence of causes, & asks if I had been speaking to Whewell on the subject. I have not’, Ellis continues, ‘but have often sported the idea in examinations’. And adds: ‘I caught the germ of it, viz. Kant's view of time & space, from Logan ages ago – curious if I made my fortune with Whewell by so casual an advantage’! Gregory and Ellis, in short, attribute to Whewell a Kantian change of heart which they ascribe to Ellis's possible (if indirect) influence. Even when taken at face value, Ellis's entry attests to a view of mechanics rather different from the non-hypothetical instrumentalism ascribed to his generation by Smith and Wise. But the story acquires an intriguing further twist when one realizes that both Gregory and Ellis had misread Whewell twice over. First, I can detect no change of view on Whewell's behalf regarding the a-causal nature of time between his ‘On the nature of the truth of the laws of motion’ (first published in Transactions of the Cambridge Philosophical Society (1834), 5, 149–72) and the corresponding chapter of the Philosophy referred to by Ellis and Gregory. Secondly, and more importantly, contrary to Ellis and Gregory's contention, in neither of them is time denied causal power by Whewell on analytical grounds. On the contrary, he goes to great pains to argue that it is an empirically proven fact that velocity does not change merely as a function of time! Indeed, in both instances, the entire point of Whewell's analysis was to exhibit the intimate (later dubbed ‘antithetical’) meshing of a priori and purely experiential constituents in the formation of scientific knowledge. Ellis, and perhaps also Gregory, seem to have been so deeply committed to Kant as to confuse friend and foe.

23 Whewell, W., ‘Report on the recent progress and present condition of the mathematical theories of electricity, magnetism and heat’, BAAS Report (1835), 5, 1–34Google Scholar, and Smith, and Wise, , op. cit. (2), 163–4.Google Scholar On the other hand, Whewell's ‘Theory of electricity’, published in the Encyclopedia Metropolitana in 1830 (iv, 140–70)Google Scholar, ‘compiled from Poisson's papers’, remained squarely Laplacian in its basic orientation. See Whewell, to Forbes, J. D., 9 06 1831Google Scholar, cited in Todhunter, I., William Whewell, D.D. Master of Trinity College, Cambridge: An Account of his Writings with Selection from his Literary and Scientific Correspondence, 2 vols., London, 1876, ii, 119–20.Google Scholar The dates would seem to confirm Smith and Wise's contention that Whewell underwent a Lagrangian change of heart (in physics that is) between 1831 and 1835, although I seriously doubt it. As I have argued elsewhere, Whewell's mature philosophy of science sits poorly with the instrumentalism associated in Energy and Empire with Fourier's theory of heat. Cf. Fisch, , op. cit. (3), 42–60 and 133–9.Google Scholar

24 For a different, though equally critical assessment of his chapter in Energy and Empire see Grattan-Guinness's review of the book in Mathematical Reviews (1992b), 609–10.Google Scholar

25 Smith, and Wise, , op. cit. (2), 151–68.Google Scholar

26 For interestingly conflicting accounts of the prior convictions that informed the algebraic debate see Bloor, D., ‘Hamilton and Peacock on the essence of algebra’, in Social History of Nineteenth Century Mathematics (ed. Mehrtens, H., Bos, H. and Schneider, I.), Boston, 1981, 202–32CrossRefGoogle Scholar and Richards, , ‘God, truth and mathematics’, op. cit. (6)Google Scholar. Bloor argues that the meta-mathematical debate between the ‘formalists’ (Peacock, De Morgan, Herschel, Babbage and the young Whewell) and the ‘intuitionist' Hamilton owes its origin to political differences. Richards’ finer grained study, by contrast, detects a shared ‘intuitionism’ even among some of the so-called ‘formalists’ that derived, in her opinion, from a broadly shared ‘evangelical’ religiosity.

27 Cf. Fisch, , op. cit. (3), 7–12Google Scholar, and developed further in Fisch, M., ‘Towards a rational theory of progress’, Synthese (1994), 99Google Scholar, forthcoming. For the logic of question and answer see Collingwood, R. G., An Autobiography, Oxford, 1989, 29–3Google Scholar (first published, Oxford, 1939).

28 Guicciardini, , op. cit. (4), 138 and 141–2.Google ScholarGrattan-Guinness, I., ‘Babbage's mathematics in its time’, BJHS (1979), 12, 82–88, on 83CrossRefGoogle Scholar, and op. cit. (17), 84–111 and 95–101.

29 Cf. Bell, E. T., The Development of Mathematics, 2nd edn, New York, 1945, 287–90Google Scholar, Fraser, C. G., ‘The calculus as algebraic analysis in the 18th century’, Archive for History of Exact Sciences (1989), 39 (4), 317–35Google Scholar, and Grattan-Guinness, , op. cit. (14), i, ch. 3, especially §3.2.4.Google Scholar

30 Koppelman, , op. cit. (6).Google Scholar

31 Recent studies, however, cast serious doubt on Grattan-Guinness and Guicciardini's contention that Cauchy's contributions to the calculus were passed over in England simply as a result of Lagrangian bias. By 1830 or so, the only English Lagrangians of note seem to have been Peacock and Herschel. Peacock rejected Cauchy's procedures explicitly, deeming his attempt ‘to conciliate the direct consideration of infinitesimals with the purely algebraical views of the principles of this calculus, which Lagrange first securely established … altogether inconsistent with the spirit and principles of symbolical algebra’, cautioning that Cauchy's suggestions ‘would necessarily bring us back again to that tedious multiplication of cases which characterized the infancy of the science’ (‘Report on the recent progress and present state of certain branches of analysis’, Report on the Third Meeting of the British Association for the Advancement of Science, London, 1834, 185–352, at 247–8)Google Scholar. Herschel's allegiance to Lagrange and apparent disregard of Cauchy's revision of the calculus is equally evident (if implicit) in his long entry ‘Mathematics’ in The Edinburgh Encyclopedia, 1832, xii, 434–59.Google Scholar Here Lagrange's revision of the calculus is hailed as ‘the greatest revolution which the nature of a science like mathematics could admit in the principles of its most extensive branch’ (p. 451). Cauchy, by contrast, is not even mentioned. While one might agree with Guicciardini and Grattan-Guinness that Peacock and Herschel's disregard of Cauchy indeed derived from their sympathy towards Lagrange, the same cannot be said of those who had by then abandoned Lagrange's version of the calculus. Whewell and De Morgan, who in the 1830s strongly spoke up for a return to a limit-based calculus, also seem not to have attributed any importance to Cauchy's work. One explanation might be that in reacting to the stark algebraic formalism advocated by Herschel and Peacock, Whewell and De Morgan had tended to cast Cauchy among the formalists and to view his epsilon-delta procedures as offering little if any to the conceptual clarification of limit, continuity or convergence. (See Fisch, , op. cit. (3), 53–6Google Scholar and Richards, , ‘God, truth and mathematics’, op. cit. (6), 64.Google Scholar) A somewhat different approach, and interestingly opposed to that offered by Guicciardini and Grattan-Guinness, has been proposed by Robert Fox and Smith and Wise. In their view Cauchy's reputation in England derived primarily from his physics which was (justly) perceived and eventually rejected for being Laplacian – i.e. reductionist and hypothetical. In other words the failure of many English mathematicians to appreciate Cauchy's reform of the calculus is attributed, by them, not to them having been out of touch with French developments but to having followed them too closely. See Fox, R., ‘The rise and fall of Laplacian physics’, Historical Studies in the Physical Sciences (1974), 4, 89–136CrossRefGoogle Scholar, and Smith, and Wise, , op. cit. (2), 164–5.Google Scholar

32 Although far less concerned than Peacock, Whewell or De Morgan with foundational studies proper, a similar point may be made regarding James Thomson's Introduction to the Differential and Integral Calculus. In the first edition of 1831 Lagrange's algebraic approach was adopted, but, possibly as a result of the standard set by De Morgan's The Differential and Integral Calculus, London, 1842, it was abandoned in the second edition of 1848 in favour of the limit approach. Here, too, Cauchy's epsilon–delta procedures are conspicuously absent. See Smith, and Wise, , op. cit. (2), 35, 151 and 170.Google Scholar

33 Duhem, P., The Aim and Structure of Physical Theory, New York, 1981, 55–104.Google Scholar

34 Hamilton, W. R., ‘Theory of conjugate functions, or algebraic couples; with a preliminary and elementary essay on algebra as the science of pure time’, Transactions of the Royal Irish Academy (1837), 17, 293–422Google Scholar, republished in Halberstam, H. and Ingram, R. E. (eds.), The Mathematical Papers of Sir William Rowan Hamilton, Cambridge, 1967, iii, 3–96Google Scholar (pagination according to the latter).

35 Hamilton's debt to Coleridge in this respect mainly concerns the latter's much-laboured distinction (late of Kant) between Reason and Understanding. In Coleridge's opinion, the former alone constitutes the ‘scientific faculty’ in being ‘the power by which we become possessed of Principle … and of Ideas’ prior and independently of the senses (The Friend, No. 7, 28 09 1809Google Scholar, in The Collected Works of Samuel Taylor Coleridge (ed. Rooke, B. E.), London, 1969, iv (pt II), 104Google Scholar; see also Aids to Reflection, 9th edn, London, 1861, 167–204)Google Scholar. Interestingly, Coleridge notices the similarity between ‘Reason of contemplation’ (as opposed to understanding) and sensual perception. ‘Reason’, he writes, ‘is much nearer to Sense than to Understanding: For Reason … is a direct aspect of truth, an inward beholding, having a similar relation to the intelligible or spiritual, as Sense has to the material or phenomenal’ (Aids to Reflection, 174)Google Scholar. The science of mathematics, obtained, according to Coleridge, by the contemplation of ‘pure sense’ (p. 183) pertains therefore quite explicitly to a notion of truth by correspondence in the realm of ideas akin to its experimental counterpart in the realm of fact. For Hamilton's debt to Coleridge see Hankins, T. L., op. cit. (2), 255–7Google Scholar and Bloor, , op. cit. (26), 209–20.Google Scholar

36 See note 40 below.

37 Whewell, I should add immediately, would have balked at this description. Not only did his conception of a science resemble that of Hamilton, but he did his best to present higher algebra as a bona fide science. However, as shall be argued below, his rhetoric notwithstanding, Whewell chose to follow Peacock too closely for his own philosophy to apply, and ended up abandoning algebra altogether.

38 For the sake of brevity I shall adopt Hamilton's terminology. Following Hamilton's usage, the terms ‘practical’, ‘philological’ and ‘theoretical’ will be taken in what follows to denote ‘applied’ or ‘instrumental’, ‘formal’ and ‘pursued as a system of mathematical truths’ respectively.

39 For a fuller elaboration of such a means–ends analysis of problems and problem-solving see Fisch, , op. cit. (27).Google Scholar

40 Lamenting the fact that Peacock was unable to attend the Dublin meeting of the British Association in August, Hamilton wrote to him in July 1835 as follows: Besides the pleasure which I expected from even merely seeing you again, I wished to talk to you about Algebra. I have taken some pains to understand your views, & almost persuaded myself that I do … but you have perhaps perceived enough of the turn of my mind in metaphysics to expect that I should rather admire them than adopt them. So far, indeed, have I gone in the opposite direction, that wishing lately to publish some sketch of my own views on the subject, I found myself forced back on an old thought of mine, & have nearly printed an essay on Algebra as the Science of Pure Time. The introductory remarks, however, {without expressly mentioning you *),} recognize the existence (I might say the necessity) of a Philological School of Algebra, in which school you perhaps, have been the most bold and consistent teacher [Hamilton to Peacock, 23 July 1835, Trinity College Library, Cambridge, Add.ms.b.4944; the phrase in curly brackets was inserted as an afterthought].

41 Lacroix, S. F., An Elementary Treatise on the Differential and Integral Calculus, Translated from the French with an Appendix and Notes, Cambridge, 1816.Google Scholar Herschel and Babbage's joint preface to the Memoirs of the Analytical Society, by contrast, focuses exclusively on the notational, rather than the conceptual merits of Lagrangian analysis (namely, the ‘accurate simplicity of its language’, ‘the symmetry’ and ‘the conciseness of its notation’). See Herschel, and Babbage, , op. cit. (1), pp. i–ii.Google Scholar

42 As far as I can tell at present, Babbage at the time was single-mindedly a-theoretical (in Hamilton's sense of the term). Herschel's initial attitude to the foundations of mathematics is unclear. In later years, as we shall see, he carefully steered wide of foundational problems, and remained largely uncommitted. But even at this early stage he seems to have remained aloof. Thus, in a section of his Appendix to the Society's Lacroix, entitled ‘On the connection between the differential calculus and that of differences’ (pp. 539ff) the former is introduced liberally as ‘considered in the light in which Leibnitz presented it, or as depending on the theory of limits’ without a further word of reservation.

43 The Treatise was a textbook and an elementary one at that. The ‘Report’ (Peacock, op. cit. (31)), on the other hand, was written for a far better informed readership. Naturally, the latter presents a more substantial and polemical version of Peacock's position. And since there exist no detectable changes of his basic view of algebra between the two works, we shall refer in what follows mostly to the ‘Report’.

44 Nagel, Ernest, ‘“Impossible numbers”: a chapter in the history of modern logic’, in Studies in the History of Ideas (edited by the department of philosophy of Columbia University), Columbia, 1935, iii, 426–74, on 454.Google Scholar See also Pycior, H., ‘George Peacock and the British Origins of Symbolical Algebra’, op. cit. (6), especially 27–33.Google Scholar

45 A position argued for forcefully, for example in Frend, William's Principle of Algebra, London, 1796.Google Scholar

46 E.g. Nagel, , op. cit. (44), 448–55Google Scholar, Bell, , op. cit. (29), 180–1Google Scholar, Clock, D. A., ‘A New British Concept of Algebra: 1825–1850’, Ph.D. dissertation, University of Wisconsin, 1964, 4–34Google Scholar, Novy, L., Origins of Modern Algebra, Leyden, 1973, 189–94Google Scholar, and Koppelman, , op. cit. (6), 215–17.Google Scholar

47 See note 6 above.

48 Richards, , ‘The art and science of British algebra’, op. cit. (6), 345–6.Google Scholar

49 See especially Richards, , ‘God, truth and mathematics’, op. cit. (6).Google Scholar

50 He appears to have intended to do so but never did. In both the opening paragraph of the ‘Report’ (Peacock, , op. cit. (31), 185)Google Scholar and in the preface to the Treatise (2nd edn, 1845, i, p. iii)Google Scholar he expresses hope of producing in the future similar studies of ‘the more important departments of analysis, with the view of presenting their principles in such a form, as may make them component parts of one uniform and connected system’. Hopefully further studies of Peacock's papers will reveal whether his decision to discontinue his ambitious project was due to more than the call ‘of higher duties’.

51 Smith, and Wise, , op. cit. (2), 180–92.Google Scholar

52 Fisch, , op. cit. (3), 17–57Google Scholar, and ‘A philosopher's coming of age: a study in erotetic intellectual history’, Fisch, and Schaffer, (eds.), op. cit. (7), 30–66.Google Scholar

53 Novy, , op. cit. (46), 189.Google Scholar

54 Both of whom attributed the instrumental success of the calculus to a ‘compensation of errors’. On Berkeley, see Wisdom, O., ‘The compensation of errors in the method of fluxions’, Hermalhena (1941), 57, 111–28Google Scholar, and Grattan-Guinness, I., ‘Berkeley's criticism of the calculus as a study in the theory of limits’, Janus (1970), 56, 215–27.Google Scholar On Lagrange see Boyer, C. B., The History of the Calculus and its Conceptual Development, New York, 1949, 251.Google Scholar

55 Peacock, to Herschel, , 14 11 1816Google Scholar, RS HS 13.246.

56 Peacock, to Herschel, , 12 12 1816Google Scholar, RS HS 13.247.

57 Maseres, F., A Dissertation on the Use of the Negative Sign in Algebra, London, 1758Google Scholar, and Tracts on the Resolution of Affected Algebraick Equations, London, 1800Google Scholar, and Frend, W., The Principles of Algebra, London, 1796.Google Scholar For a discussion of Maseres' and Frend's proposals, the manner in which they wrought havoc in the theory of algebraic equations, and Peacock's later reaction to them, see Pycior, , ‘Peacock and the British origins of symbolic algebra’, op. cit. (6), 27–31.Google Scholar

58 Peacock, , ‘Report’, op. cit. (31), 190–1.Google Scholar Herschel by comparison, in his ‘Mathematics’, seems not to have regarded the problem of negatives and imaginaries a conceptual problem at all. According to Herschel ‘the reality [sic] of the apparently imaginary expressions’ was established, and the ‘mysterious terror in which imaginary quantities had hitherto held mathematicians’ transgressed, precisely when in 1629 Albert Girard ‘first notice[d] the fact [sic] that an equation has as many roots, of one or the other species, as dimensions’, and when John Bernoulli in 1697 laid the foundations for the exponential expression of the trigonometric functions (Herschel, , op. cit. (31), 439 and 446).Google Scholar

59 On De Morgan's insistence on viewing the construction of the conceptual foundations of algebra as an ongoing progressive and historical process, see Richards, , ‘De Morgan and the history of mathematics’, op. cit. (6), 17–29.Google Scholar Richards, wrongly in my opinion, ascribes to Peacock a similar approach to the history of mathematics, although she admits that it is ‘more explicit in Whewell than in Peacock’ (pp. 17–18). If at all, Peacock's theory of algebra was certainly the least historically dependent of the group.

60 On Cauchy's theory of algebra see Peacock, , ‘Report’, op. cit. (31), 192–3.Google Scholar

61 De Morgan's review of the Treatise was published in the Quarterly Journal of Education (1835), 9, 91–110 and 293–311.Google Scholar On the development of De Morgan's view of algebra see Richards, ' seminal ‘De Morgan and the history of mathematics’, op. cit. (6).Google Scholar

62 De Morgan, A., ‘Negative and impossible quantities’, Penny Cyclopedia, supplementary volume (1840), 133–4Google Scholar, quoted and discussed in Richards, , ‘De Morgan’, op. cit. (6) 15–17Google Scholar, italics added. ‘This formulation’, notes Richards, ‘reverses the order Peacock's principle established in algebra. Rather than moving [by arithmetical suggestion] from meaning to abstract forms, the development is from abstract forms to meaning’ (p. 16).

63 De Morgan, A., ‘On the foundations of algebra, no. II’, Transactions of the Cambridge Philosophical Society (1842), 7, 287–300.Google Scholar

64 Hamilton, , op. cit. (34), 295.Google Scholar

65 See note 102 below.

66 The manuscript is in Trinity College Library, Cambridge, Whewell Papers R.18.178. The very last entry is dated 16 December 1833.

67 Cf. Fisch, , op. cit. (3), 27–36.Google Scholar

68 Referring in a footnote to John Playfair's review of Laplace's Mècanique cèléste (Edinburgh Review (1808), 11, 243–84)Google Scholar, Herschel and Babbage write in their joint Preface to the Memoirs of the Analytical Society: ‘it should be recollected, that the Author of the Essay confines his attention entirely to the subject of Analytical dynamics; referring to the discoveries in the integral calculus merely as connected with that subject, and that too very cursorily. Our business is exclusively with the pure Analytics.’, op. cit. (1), p. iiGoogle Scholar, the italics are theirs.

69 Later, during the late 1820s and 1830s, Lagrange's programme for mechanics was gradually adopted, not so much in mechanics proper as in wave-optics and the theory of heat. For a detailed study of the ‘increasing preference for the new works of Fresnel and Fourier over those of Laplace and his disciples’ see Smith, and Wise, , op. cit. (2), 155–68.Google Scholar

70 Cf. Peacock, , ‘Report’, op. cit. (31), 186–8.Google Scholar

71 Published in 1830 as vol. 1 of Dionysius Lardner's Cabinet Cyclopedia. See also his ‘Mechanism of the Heavens’ (review of Somerville, Mary's Mechanism of the Heavens, London, 1832Google Scholar, and Laplace's Mècanique cèléste), Quarterly Review (1832), 47, 537–59Google Scholar, reprinted in Herschel, J. F. W., Essays from the Edinburgh and Quarterly Reviews with Addresses and Other Pieces, London, 1857, 21–62.Google Scholar

72 The two French schools manifested themselves paradigmatically in the mechanics(es) of Laplace and Lagrange and in the physics of Laplace and of their respective followers. The former, represented apart from Laplace by Biot, Navier, Cauchy and Poisson, strove to ground a comprehensive physical account of the world upon a hypothesized ontology of particles and/or forces and/or ‘fluids’ acting at a distance, whereas the latter, represented, apart from Lagrange, by Fresnel, Fourier, Ohm and Ampere, proceeded analytically, striving to ‘save the phenomena’ by means of a set of general equations devoid of ‘any visual, geometrical or mechanical representations or constructions’. For a description of the two schools see Hendry, J., James Maxwell and the Theory of the Electromagnetic Field, Bristol, 1986, 5 and passimGoogle Scholar, and Fisch, M., ‘A physicists' philosopher-James Clerk Maxwell on mathematical physics’, Journal of Statistical Physics (1988), 51, 309–19.CrossRefGoogle Scholar Smith and Wise, wrongly in my opinion, seem to imply that the line of division between the two schools was synchronie rather than diachronic. Thus they write, for instance, that ‘the methods epitomized by Fourier fared much better than those of latter-day followers of Laplace and Lagrange, such as Cauchy’ (op. cit. (2), 162).

73 For an account of the problematic implications of such a position see Fisch, , op. cit. (3), 33–5.Google Scholar

74 Whewell, W., The Philosophy of the Inductive Sciences Founded on their History, Cambridge, 1840, i, 8.Google Scholar

75 Memoirs of the Analytical Society 1813 (op. cit. (1)), was the only official publication of the Analytical Society. The mention of the year in the title indicates that the Society members envisaged successive volumes. None ever appeared.

76 See also Fisch, , op. cit. (3), 36–9.Google Scholar This claim of mine has been dismissingly contested by Becher, H., ‘The Whewell story’, Annals of Science (1992), 49, 377–84.Google Scholar

77 Bromhead to Babbage, British Museum Add 37182 f13, cited in Schweber, , op. cit. (5), 62–3.Google Scholar See also Babbage, to Herschel, , 12 12 1816, RS HS 2.56.Google Scholar

78 The translation appeared in 1816. The Advertisement promised ‘a sequel … containing a collection of examples and results’, the work on which, it promised, ‘is already begun, and it is expected to be ready for publication in the course of a few months’. The three volumes of examples, however, were only published in 1820: Babbage, C., A Collection of Examples of the Solutions of Functional EquationsGoogle Scholar; Peacock, G., A Collection of Examples of the Applications of the Differential and Integral CalculusGoogle Scholar; Herschel, J., A Collection of Examples of the Application of the Calculus of Finite Differences, all by Deighton, Cambridge, 1820.Google Scholar

79 Whewell, to Herschel, , 6 03 1817Google Scholar, cited in Todhunter, , op. cit. (23), ii, 16.Google Scholar

80 See Peacock to Herschel, note 12 above.

81 ‘The true faith’ seems to have been a common catchword among the young dissenters to denote their Lagrangian approach. See for example: Babbage, to Herschel, , 13 06 1813Google Scholar, RS HS 2.15; Babbage, to Herschel, , 8 12 1815, RS HS 2.50Google Scholar; Herschel, to Babbage, , 14 07 1816, RS HS 2.64Google Scholar; Babbage, to Herschel, , 20 07 1816, RS HS 2.65Google Scholar; Peacock, to Herschel, , 3 12 1816, RS HS 13.247Google Scholar; as well as Bromhead to Babbage, note 77 above.

82 Woodhouse, R., An Elementary Treatise on Astronomy, Vol. II Containing Physical Astronomy, Cambridge, 1818.Google Scholar

83 Whewell, to Herschel, , 1 11 1818Google Scholar, cited in Todhunter, , op. cit. (23), ii, 30.Google Scholar

84 Whewell, W., An Elementary Treatise on Mechanics Designed for the Use of Students in the University, Cambridge, 1819.Google Scholar

85 Fisch, , op. cit. (3), 39–56.Google Scholar

86 Herschel's two principal works in this respect are his lengthy essay ‘Mathematics’ of 1832 in The Edinburgh Encyclopedia (op. cit. (31)), and his thoroughly Baconian A Preliminary Discourse, op. cit. (71). In the former, as we have seen, the calculus is presented as having been in the past ‘exhibited, all more or less dependent on the notions of infinity, or, which is still more revolting to analysis, on mechanical conceptions’, but having undergone, owing to Lagrange, ‘the greatest revolution which the nature of a science like mathematics could admit in the principles of its most extensive branch’. While in the latter, natural philosophy is described in traditional Baconian terms, its eventual mathematization all but ignored.

87 For similar reasons he also came to reject the use of all non-representative physical magnitudes – most conspicuously virtual displacements and velocities of which he wrote: ‘To invest these geometrical phantoms with any attributes arising from physical laws, is a proceeding altogether arbitrary and illogical’ (The First Principles of Mechanics with Historical and Practical Illustrations, Cambridge, 1832, p. ix, italics added).Google Scholar

88 Herschel, on the other hand, was forced to describe the application of ‘abstract science’ to ‘knowledge of nature’, as he dubbed the mathematical and inductive constituents of modern physics, as a ‘sudden’ and inexplicable ‘transfer of ideas from one remote station in one to an equally remote station in the other’ (A Preliminary Discourse, op. cit. (71), 71–2).Google Scholar

89 The affinity between the views of the two men was neither accidental nor limited to the nature of mathematical truth. They first met in April 1832 in Cambridge and remained in close contact for several years, most significantly during the early 1830s in which they were both laying the foundations for their most important work: Hamilton on optics, algebra and dynamics, and Whewell on the nature of scientific truth and induction. Both men seem to have benefited much from this exchange. ‘He is indeed’, wrote Hamilton of Whewell, ‘a man of very powerful intellect … we have had many little fights, in the most loving spirit, and hope to have many more – not that I own myself to be particularly pugnacious – but I cannot resist the temptation of an argument with a great original like him’ (Hamilton to the Marquess of Northampton, 2 February 1838, cited in Graves, R. P. (ed.), Life of Sir William Rowan Hamilton, New York, 1975, ii, 249–50.)Google Scholar. For a preliminary account and evaluation of the exchange see Hankins, , op. cit. (2), 174–80Google Scholar, and Fisch, , op. cit. (3), 63–7, 93–8.Google Scholar

90 Cf. Whewell, , op. cit. (74), i, 54.Google Scholar

91 Op. cit. (66). Whewell significantly dithers between ‘Pure’ and ‘Formal’ (p. 105).

92 On the facing page Whewell jotted down an alternative title: ‘Of the Intuitive Power with Regard to Extension, or of Inward-Seeing Power’ (p. 106).Google Scholar

93 For a chronology of Whewell's philosophical development see Fisch, , op. cit. (3), 133–8.Google Scholar

94 On Whewell's notion of necessity see Fisch, , op. cit. (3), 154–63Google Scholar, and ‘Antithetical knowledge’ in Fisch, and Schaffer, (eds.), op. cit. (7), 289–309.Google Scholar

95 This, however, leads to other problems, an is perhaps too easy a case since it involves no obvious exceptions, but what of m/n or log n? Is one allowed on the basis of the principle of absolute generality to substitute 0 for n? A decade later De Morgan would raise this problem forcefully with respect to divergent series, and would answer it with a forceful yes! See De Morgan, A., ‘On divergent series and various points of analysis connected with them’, Transactions of the Cambridge Philosophical Society (1844), 8, 182–203Google Scholar, and Richards, , ‘De Morgan’, op. cit. (6), 26–9.Google Scholar

96 By 1842, Peacock had apparently also arrived at the same conclusion. In the preface to the first volume of the second edition of his Treatise the ‘Direct proposition’ is dropped entirely and the ‘principle of the permanence of equivalent forms’ is expressed as follows: It is this adoption of the rule of the operations of arithmetical algebra as the rules for performing the operations which bear the same names in symbolical algebra, which secures the absolute identity of the results in the two sciences as far as they exist in common: or in other words, all the results of arithmetical algebra which are deduced by the application of its rules, and which are general in form, though particular in value, are results likewise of symbolical algebra, where they are general in value as well as in form [Peacock, G., A Treatise on Algebra, Vol. 1.: Arithmetical Algebra, Cambridge, 1842, pp. vi–vii].Google Scholar

97 I am referring to his ‘On the nature of the truth of the laws of motion’ (op. cit. (22)) first published in 1834. On the possible dating of the discovery of his theory of induction see Fisch, , op. cit. (3), 99–100.Google Scholar

98 See especially ‘On the steps of which knowledge consists’, Trinity College Library, Cambridge, Whewell Papers, R.18.178, pp. 14–36. On this phase in Whewell's philosophical development see Schaffer, S., ‘The history and geography of the intellectual world: Whewell's politics of language’Google Scholar, in Fisch, and Schaffer, (eds.), op. cit. (7)Google Scholar, and Fisch, , op. cit. (3), 84–93 and 100–1Google Scholar, especially footnote 2.

99 See in particular his draft essay ‘Of practical skill and of speculative knowledge, of the different nature of pure and of physical science’, Trinity College Library, Cambridge, Whewell Papers, R.18.176. Here the deductive structure of the mathematical sciences is explored with special reference to the nature of truth to which they pertain – namely, truth of their respective Fundamental Ideas. Significantly perhaps, the first entry of this draft is dated 23 December 1833.

100 Whewell, , op. cit. (74), i, 142–3.Google Scholar

101 Hankins, , op. cit. (2), 254.Google Scholar

102 Hamilton's ‘On quaternions: or a new system of imaginaries in algebra’ was published in eighteen instalments in the Philosophical Magazine between 1844 and 1850, and his ‘On symbolical geometry’, in ten instalments in the Cambridge and Dublin Mathematical Journal between 1846 and 1849.

103 See Fisch, , op. cit. (3), 140–2.Google Scholar