Duncan F. Gregory, William Walton and the development of British algebra: ‘algebraical geometry’, ‘geometrical algebra’, abstraction

ABSTRACT

This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on ‘algebraical geometry’ and ‘geometrical algebra’ in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the ‘abstract algebra’ and ‘abstract geometry’ of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to ‘algebraical geometry’ and ‘geometrical algebra’ of the second generation of ‘scientific’ symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s–1840s.

Acknowledgments

I would like to thank David Miller for his careful reading of the manuscript and for pointing out numerous improvements and corrections, Adrian Rice and Jeremy Gray for their valuable comments on an earlier version and Lorraine Daston for hosting me in Department II at the Max Planck Institute for the History of Science (Berlin) as I revised this paper. My gratitude goes to Gerard de Vries for his continual support.

Funding

For the stay in Department II at the Max Planck Institute for the History of Science (Berlin) I received a grant from the Duitsland Instituut Amsterdam (DIA).

Notes

¹ See Menachem Fisch, ‘”The emergency which has arrived”: the problematic history of nineteenth-century British algebra’, The British Journal for the History of Science, 27 (1994), 247–76.

² See, for example, Harvey E. Becher, ‘Woodhouse, Babbage, Peacock, and modern algebra’, Historia Mathematica, 7 (1980), 389–400. John M. Dubbey, ‘Babbage, Peacock and modern algebra’, Historia Mathematica, 4 (1977), 295–302. Philip C. Enros, ‘Cambridge University and the adoption of analytics in early nineteenth-century England’, in Social History of Nineteenth Century Mathematics, edited by Herbert Mehrtens, Henk J.M. Bos and Ivo Schneider (Boston: Birkhäuser, 1981), pp. 135–47. Philip C. Enros, ‘The Analytical Society (1812–1813): precursors of the renewal of Cambridge mathematics’, Historia Mathematica, 10 (1983), 24–47. M.V. Wilkes, ‘Herschel, Peacock, Babbage and the development of the Cambridge curriculum’, Notes and Records of the Royal Society of London, 44 (1990), 205–19.

³ George Peacock, A Treatise on Algebra (Cambridge: J. & J.J. Deighton, 1830). For accounts of the work of Peacock see, Menachem Fisch, ‘The making of Peacock’s Treatise on Algebra: a case of creative indecision’, Archive for History of Exact Sciences, 54 (1999), 137–79. Kevin Lambert, ‘A natural history of mathematics. George Peacock and the making of English algebra’, Isis, 104 (2013), 278–302. Helena M. Pycior, ‘George Peacock and the British origins of symbolical algebra’, Historia Mathematica, 8 (1981), 23–45.

⁴ The distinction between the first and second generation of reformers of British mathematics was first introduced in Crosbie Smith and Norton Wise, Energy & Empire. A Biographical Study of Lord Kelvin (Cambridge: Cambridge University Press, 1989), chapter 6. A critical discussion of their characterization of the second generation is found in Lukas M. Verburgt, ‘Duncan Farquharson Gregory and Robert Leslie Ellis: second generation reformers of British mathematics’, under review. Given that the complex task of determining the theoretical relationship between the contributions of, for example, Gregory, De Morgan and Hamilton to algebra is beyond the scope of this paper, the distinction between three groups within the second generation is here presented without any detailed justification. The plausibility of the distinction will, hopefully, become apparent in what follows.

⁵ Gregory's contributions to symbolical algebra are discussed in Patricia R. Allaire and Robert E. Bradley, ‘Symbolical algebra as a foundation for calculus: D.F. Gregory's contribution’, Historia Mathematica, 29 (2002), 295–426. Sloan E. Despeaux, ‘”Very full of symbols”: Duncan F. Gregory, the calculus of operations, and the Cambridge Mathematical Journal’, in Episodes in the History of Modern Algebra (1800–1950), edited by Jeremy J. Gray and K.H. Parshall (London: American and London Mathematical Societies, 2007), pp. 49–72.

⁶ For accounts of the De Morgan’s contributions to algebra see Helena M. Pycior, ‘Augustus De Morgan’s algebraic work: the three stages’, Isis, 74 (1983), 211–26. Joan L. Richards, ‘Augustus De Morgan, the history of mathematics, and the foundations of algebra’, Isis, 78 (1987), 6–30.

⁷ Boole did not write on (the ‘formalization’, in the modern sense of the term, of) algebra, but expressed his views on the nature of algebra and mathematics in his logical works, namely George Boole, The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning (Cambridge: Macmillan, Barclay & Macmillan, 1847). George Boole, An Investigation of the Laws of Thought on Which Are Founded the Mathematical Theories of Logic and Probabilities (London: Walton and Maberly, 1854). See, for example, also Theodor Hailperin, ‘Boole’s algebra isn’t Boolean algebra’, in A Boole Anthology: Recent and Classical Studies in the Logic of George Boole, edited by James Gasser (Dordrecht: Kluwer Academic Publishers, 2000), pp. 61–78.

⁸ Given the argumentation of this paper, this second group of the second generation of reformers of British mathematics will not be discussed.

⁹ The origin and content of the work of Hamilton are analyzed in, for example, Thomas L. Hankins, Sir William Rowan Hamilton (Baltimore: John Hopkins University Press, 1980/2004). Thomas L. Hankins, ‘Algebra as pure time: William Rowan Hamilton and the foundations of algebra’, in Motion and Time, Space and Matter, edited by Peter K. Machamer and Robert G. Turnbull (Columbus: Ohio State University Press, 1976), pp. 327–59. Jerold Mathews, ‘William Rowan Hamilton's paper of 1837 on the arithmetization of analysis, Archive for History of Exact Sciences, 19 (1978), 177–200. Peter Ohrstrom, ‘Hamilton's view of algebra as the science of pure time and his revision of this view’, Historia Mathematica, 12 (1985), 45–55. Edward .T. Whittaker, ‘The sequence of ideas in the discovery of quaternions’, Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, 50 (1944/1945), 93–8.

¹⁰ The authoritative account of the life and works of this ‘forgotten mathematician’ is Tony Crilly, Arthur Cayley: Mathematician Laureate of the Victorian Age (Baltimore: John Hopkins University, 2005). See also Tony Crilly, ‘The young Arthur Cayley’, Notes and Records. The Royal Society Journal of the History of Science, 52 (1998), 267–82.

¹¹ See, for example, Eric Temple Bell, The Development of Mathematics (New York: McGraw-Hill, 1954), p. 180. Lubos Novy, Origins of Modern Algebra. Translated by Jaroslave Tauer (Leiden: Noordhoff, 1973), p. 199.

¹² See, for example, Daniel A. Clock, A New British Concept of Algebra, 1825-1850 (Madison: University of Wisconsin, 1964).

¹³ Joan L. Richards, ‘The art and the science of British algebra: a study in the perception of mathematical truth’, Historia Mathematica, 7 (1980), 343–65 (p. 345).

¹⁴ Catarina Dutilh Novaes, ‘The different ways in which logic is (said to be) formal’, History and Philosophy of Logic, 32 (2011), 303–32. See, for example, Leo Corry, ‘The empiricist roots of Hilbert’s axiomatic approach’, in Proof Theory. History and Philosophical Significance, edited by Vincent F. Hendricks, Stig Andur Jørgensen and Klaus Frovin (Dordrecht: Kluwer Academic Publishers, 2000), pp. 35–54, for a problematization of the ‘received view’ of the formalism of modern mathematics.

¹⁵ Smith & Wise, Energy (note 4), p. 171, my emphasis.

¹⁶ Verburgt, ‘Gregory and Ellis’ (note 4) argues that there is also a significant problem with the characterization of the differences between the first generation and the first group of the second generation underlying these accounts.

¹⁷ John M. Dubbey, The Mathematical Work of Charles Babbage (Cambridge: Cambridge University Press, 1978) established that Peacock knew of Babbage’s ‘Essays on the philosophy of analysis’ of 1821. Fisch, ‘Creative indecision’ (note 3) shows not only that the portrayal of Peacock’s Treatise on Algebra of 1830 as an application of (Lagrangian) formalist views, to which he was committed during the 1810s–1820s, to algebra is mistaken, but also that ‘Babbage’s construal of pure analysis differed significantly from the system [of algebra] Peacock would eventually propose’. Fisch, ‘Creative indecision’ (note 3), p. 155.

¹⁸ Hamilton famously dismissed both the ‘philological’ view of algebra ‘as a system of rules or else as a system of expressions’ and the ‘practical’ view of algebra ‘as an art or as a language’ and. If these two viewpoints correspond to, on the one hand, the first generation (Peacock, but, thus, especially Babbage) and the first group of the second generation and, on the other hand, the second group of the second generation, Hamilton himself approached algebra as ‘a system of truths’.

¹⁹ Smith & Wise, Energy (note 4), p. 171.

²⁰ This is meant as an implicit reference to the more general problems connected to those accounts of the history of modern mathematics and logic that equate ‘modern’ with an increase in ‘abstractness’ and ‘formality’ and define ‘abstract’ and ‘formal’ in terms of its being part of the process of ‘modernization’. Compare Herbert Mehrtens, Moderne, Sprache, Mathematik: Eine Geschichte des Streits um die Grundlagen der Disziplin und des Subjekts formaler Systeme (Berlin: Surhkamp, 1990) and Jeremy Gray, Plato’s Ghost. The Modernist Transformation of Mathematics (Princeton & Oxford: Princeton University Press, 2008) to, for example, Dennis E. Hesseling, The Reception of Brouwer’s Intuitionism in the 1920s (Basel: Birkhäuser, 1991/2012). The ‘debate’ between the symbolical algebraists and the abstract algebraists does, indeed, foreshadow the complexities of the famous formalist-intuitionist debate of almost a century later – as Israel Kleiner has suggested. Israel Kleiner, A History of Abstract Algebra (Boston: Birkhäuser, 2007), 151.

²¹ Smith & Wise, Energy (note 4), p. 171.

²² For a thorough analysis of the traditional relationship between arithmetic, geometry and algebra see, for example, Henk J.M. Bos, Redefining Geometrical Exactness. Descartes’ Transformation of the Early Modern Concept of Construction (New York: Springer-Verlag, 2001), chapter 6.

²³ ‘Abstractionism’ is the Aristotelian-Scholastic doctrine which holds that ‘mathematical objects are constructed out of our ordinary experience but constructed in such a way that mathematics does not depend on specific features of the sensible world’. Douglas M. Jesseph, Berkeley’s Philosophy of Mathematics (Chicago & London: The University of Chicago Press), p. 10. The doctrine was rejected by Berkeley, Kant and many other idealist philosophers and mathematicians; see, in this context, Margaret Atherton, ‘Berkeley’s anti-abstractionism’, in Essays on the Philosophy of George Berkeley, edited by Ernest Sosa (Dordrecht: Kluwer Academic Publishers, 1986), pp. 45–60. Carl J. Posy, ‘Kant’s mathematical realism’, The Monist, 67 (1984), 115–34. George Elder Davie has made a convincing argument for the presence of this doctrine – popular, as it was, among eighteenth-century Scottish mathematicians such as Gregory's teacher, William Wallace (1768–1843) – in the contributions of Gregory to symbolical algebra. George Elder Davie, The Democratic Intellect: Scotland and Her Universities in the Nineteenth Century (Edinburgh: Edinburgh University Press, 1961), part 2. The fact that ‘abstractionism’, in mathematics, was closely related to Newtonian ‘fluxional method’ – which had been popularized, among others, by the Scottish geometer Robert Simson (1687–1768) – seems to support this argument. For the content and influence of Newtonian fluxions see, for example, Florian Cajori, A History of the Conceptions of Limits and Fluxions in Great Britain from Newton to Woodhouse (London: Open Court, 1919). Philip Kitcher, ‘Fluxions, limits, and infinite littlenesse. A study of Newton’s presentation of the calculus’, Isis, 64 (1973), 33–49. For its presence in Scotland see Judith V. Grabiner, ‘Was Newton’s calculus a dead end? The continental influence of Maclaurin’s Treatise of Fluxions’, The American Mathematical Monthly, 104 (1997), 393–410. These considerations will briefly be reflected upon in section 4.

²⁴ The Kantian nature of Hamilton's work on algebra has been extensively studied – see, for example, Anthony T. Winterbourne, ‘Algebra and pure time: Hamilton's affinity with Kant’, Historia Mathematica, 9 (1982), 195–200. In his seminal Sir William Rowan Hamilton, Thomas L. Hankins has expressed his belief that Hamilton was able to establish abstract, rather than symbolical, algebra because of his ‘constructivism’. Hankins, Hamilton (note 9), 352. The fact that Hamilton and Cayley adhered, at least to a certain degree, to Kantian epistemology – see Mathews, ‘Hamilton on analysis’ (note 9), 192 – is briefly discussed in the section with concluding remarks.

²⁵ William Rowan Hamilton, ‘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, 17 (1837), 293—422 (p. 295).

²⁶ For this observation see Fisch, ‘Creative indecision’ (note 3), p. 156, p. 145.

²⁷ In chronological order: Wooster Woodruff Beman, ‘A chapter in the history of mathematics’, Proceedings of the American Association for the Advancement of Science, 46 (1897), 33–50. G. Windred, ‘The history of the theory of imaginary and complex quantities’, Mathematical Gazette, 14 (1929), 533–41. Ernest Nagel, ‘Impossible numbers: a chapter in the history of logic’, Studies in the History of Ideas, 3 (1935), 429–74. Phillip S. Jones, ‘Complex numbers: an example of recurring themes in the development of mathematics’, Mathematics Teacher, 47 (1954), 106–44. Adrian Rice, ‘Inexplicable? The status of complex numbers in Britain, 1750-1850. In Around Caspar Wessel and the Geometric Representation of Complex Numbers. Proceedings of the Wessel Symposium at the Royal Danish Academy of Sciences and Letters Copenhagen, August 11-15 1998, edited by Jesper Lützen (Copenhagen: Det Kongelige Danske Videnskabernes Selskab, 2001), 147–80.

²⁸ Rice, ‘Inexplicable?’ (note 27) is a valuable exception.

²⁹ See, for example, Nagel, ‘Impossible numbers’ (note 27). In her seminal paper, Elaine Koppelman has not only demonstrated the importance of the so-called ‘calculus of operations’ for the formation of the abstract view of algebra, but also suggested that this calculus ‘clearly was not the only relevant factor. One that should be mentioned, and which deserves further study, is the influence of new ideas in geometry’. Elaine Koppelman, ‘The calculus of operations and the rise of abstract algebra’, Archive for History of Exact Sciences, 8 (1971), 155–242 (p. 238). She referred to a 1939 paper of Ernest Nagel as a starting-point for this study. Ernest Nagel, ‘The formation of modern conceptions of formal logic in the development of geometry’, Osiris, 7 (1939), 142–223. It may be remarked that both papers ‘are marred by their treatment of the British works as steps toward abstract algebra in an unacceptably modern sense’. Smith & Wise, Energy (note 4), p. 171, f. 53.

³⁰ Robert Woodhouse, ‘On the necessary truth of certain conclusions obtained by means of imaginary expressions’, Philosophical Transactions of the Royal Society of London, 92 (1802), 89–119. On Woodhouse see Becher, ‘Woodhouse’ (note 2). Dubbey, ‘Babbage and Peacock’ (note 2). Christopher Phillips, ‘Robert Woodhouse and the evolution of Cambridge mathematics’, History of Science, 44 (2006), 1–25.

³¹ Robert Woodhouse, The Principles of Analytical Calculation (Cambridge: Cambridge University Press, 1803). Harvey Becher, ‘William Whewell and Cambridge mathematics’, Historical Studies in the Physical Sciences, 11 (1980), 1–48 (p. 7).

³² John O’Neill, ‘Formalism, Hamilton and complex numbers’, Studies in History and Philosophy of Science, 17 (1986), 351–72 (p. 357).

³³ Peacock referred to the geometrical interpretation of imaginary quantities put forward by Adrien-Quentin (Abbé) Buée (1748–1826) and John Warren (1796–1852) in the preface to his Treatise on Algebra. ‘The first attempt which I can find of an interpretation of the meaning of such quantities was given by M. Buée [in] a Memoir which contains some original views on the use and signification of the signs of Algebra, though presented in a very vague and unscientific form […] At a much later period […] the work of Mr. Warren […] appeared […] Mr. Warren has completely succeeded in giving an interpretation of the roots of unity, when attached to symbols which denote lines in Geometry, or any quantities which such lines may represent: in doing so however he has adhered strictly to the practice of all writers on Algebra, in making the interpretation govern the results and not the results the interpretation’. Peacock, Treatise (note 3), xxvii–xxviii.

³⁴ See Peacock, Treatise (note 3), xxx–xxxi (p. xxxi).

³⁵ Peacock seems to implicitly refer to this textbook when he spoke of ‘my system of Algebraic Geometry’ in the preface of the Treatise on Algebra. Peacock, Treatise (note 3), p. xxxiv. Although it does not mention its exact date of publication, the following passage from the Report of the Council to the Thirty-Ninth Annual General Meeting of the Astronomical Society is worth quoting: ‘[I]n 1819 Mr. Peacock was Moderator [of the Cambridge Tripos]. All the chief actors in producing [the change of the curriculum] have lived to see their work fully done, and their country in full communication with all the world after more than a century of nearly complete exclusion. Mr. Peacock subsequently published an anonymous Syllabus of Trigonometry and Algebraic Geometry […] In 1826 he published in the Encyclopedia Metropolitana his historical article on Arithmetic [and in] 1830 appeared the first of his two works on Algebra’. Royal Astronomical Society, ‘Report of the Council to the Thirty-Ninth Annual General Meeting of the Astronomical Society’ (1859), p. 126. This historical fact suggests that the theoretical application of symbolical algebra not ‘back to’ arithmetical algebra, but to geometry in volume II of the second version of the Treatise of 1845 could not have been such ‘a major step away from the Treatise of fifteen years earlier’. Fisch, ‘Emergency’ (note 1), p. 174.

³⁶ O’Neill, ‘Formalism’ (note 32), p.358.

³⁷ Helena M. Pycior, Symbols, Impossible Numbers, and Geometric Entanglements. British Algebra Through the Commentaries on Newton’s Universal Arithmetick (Cambridge & New York: Cambridge University Press, 1997), p. 315.

³⁸ Joan L. Richards, Mathematical Visions. The Pursuit of Geometry in Victorian England (Boston: Academic Press Inc., 1988), p. 51.

³⁹ This much will be established in section 4.1 – in which their contributions to ‘algebraical geometry’ are scrutinized. See also Richards, Mathematical Vision (note 38), chapter 1. Joan L. Richards, ‘Projective geometry and mathematical progress in mid-Victorian Britain’, Studies in History and Philosophy of Science Part A, 17 (1986), 297–325. For accounts of the ambivalent reception of higher-dimensional space see, for example, June Barrow-Green and Jeremy Gray, ‘Geometry at Cambridge, 1863-1940’, Historia Mathematica, 33 (2006), 315–56. Amirouche Moktefi, ‘Geometry. The Euclidean debate’, in Mathematics in Victorian Britain, edited by Raymond Flood, Adrian Rice and Robin Wilson (Oxford & New York: Oxford University Press, 2011), pp. 321–38. Joan L. Richards, ‘The reception of a mathematical theory: non-Euclidean geometry in England 1868-1883’, in Natural Order: Historical Studies of Scientific Culture, edited by Barry Barnes and Steven Shapin (Beverly Hills: Sage, 1979), pp. 143–66.

⁴⁰ Duncan Farquharson Gregory, ‘On the elementary principles of the application of algebraical symbols to geometry’, The Cambridge Mathematical Journal, 2 (1939), 1–9 (p. 9).

⁴¹ Hamilton's work on algebra is briefly discussed in section 5.2.

⁴² This distinction will be explained in the introduction to section 4.

⁴³ Fisch, ‘Emergency’ (note 1), p. 254.

⁴⁴ On Gregory see Robert Leslie Ellis, ‘Biographical memoir of Duncan Farquharson Gregory’, in The Mathematical Writings of Duncan Farquharson Gregory, edited by William Walton (Cambridge: Deighton, Bell and Co., 1865), pp. xi–xxiv. John Archibald Venn, Alumni Cantabrigienses. A Biographical List of All Known Students, Graduates and Holders of Office at the University of Cambridge, From the Earliest Times to 1900. Part II. From 1752 to 1900. Volume III. Gabb-Justamond (Cambridge: Cambridge University Press, 1947), p. 141.

⁴⁵ The history of the CMJ and its role within the British mathematical community is described in Tony Crilly, ‘The Cambridge Mathematical Journal and its descendants: the linchpin of a research community in the early and mid-Victorian age’, Historia Mathematica, 31 (2004), 455–97.

⁴⁶ The following two accounts provide detailed descriptions of these events. Alex D.D. Craik, Mr. Hopkins’ Men. Cambridge Reform and British Mathematics in the 19^th Century (London: Springer-Verlag, 2007), pp. 101–2, 230–2. Davie, Democratic Intellect (note 23), chapter 7.

⁴⁷ Ellis, ‘Biographical memoir’ (note 44), xxii.

⁴⁸ Duncan Farquharson Gregory and William Walton, Treatise on the Application of Algebra to Solid Geometry (Cambridge: Deightons, 1845). Duncan Farquharson Gregory, The Mathematical Writings of Duncan Farquharson Gregory, edited by William Walton (Cambridge: Deighton, Bell and Co., 1865). Duncan Farquharson Gregory, Examples of the Processes of the Differential and Integral Calculus, edited by William Walton (Cambridge: Deighton, Bell and Co., 1841).

⁴⁹ On Walton see John Archibald Venn, Alumni Cantabrigienses. A Biographical List of All Known Students, Graduates and Holders of Office at the University of Cambridge, From the Earliest Times to 1900. Part II. From 1752 to 1900. Volume VI. Square-Zupitza (Cambridge: Cambridge University Press, 1954), p. 338.

⁵⁰ William Walton, A Collection of Problems in Illustration of the Principles of Theoretical Mechanics (Cambridge: Deighton, Bell and Co., 1855). William Walton, A Collection of Problems in Illustration of the Principles of Elementary Mechanics (Cambridge: Deighton, Bell and Co., 1958). William Walton, Problems in Illustration of the Principles of Coordinate Geometry (Cambridge: Deighton, Bell and Co., 1851). William Walton, A Treatise on the Differential Calculus (Cambridge: Deighton, Bell and Co., 1846). William Walton, A Collection of Problems in Illustration of the Principles of Theoretical Hydrostatics and Hydrodynamics (Cambridge: Deighton, Bell and Co., 1847). William Walton, Elementary Problems in Statics and Dynamics. Designed for Candidates for Honours, First Three Days (Cambridge: Deighton, Bell and Co., publication year unknown). William Walton, A Collection of Elementary Problems in Hydrostatics and Optics, Designed for the Use of Those Candidates for Mathematical Honors, Who Are Preparing for the First Three Days of the Senate-House Examination (Cambridge: Deighton, Bell and Co., publication year unknown).

⁵¹ William Walton and Charles Frederick Mackenzie, Solutions of the Problems and Riders Proposed in the Senate-House Examination for 1854 (Cambridge: Macmillan and Co., 1854). William Walton and M. Champion, Solutions of the Problems and Riders Proposed in the Senate-House Examination for 1857 (Cambridge: Macmillan and Co., 1857).

⁵² See Crilly, ‘The Cambridge Mathematical Journal’ (note 45), p. 464; Duncan Farquharson Gregory, ‘Preface’, Cambridge Mathematical Journal, 1 (1837), 1–9 (p. 1).

⁵³ Crilly, ‘The Cambridge Mathematical Journal’ (note 45), p. 465.

⁵⁴ Crilly, ‘The Cambridge Mathematical Journal’ (note 45), p. 468.

⁵⁵ See Koppelman, ‘The calculus of operations’ (note 29).

⁵⁶ Despeaux, ‘”Very full of symbols”’ (note 5), p. 53.

⁵⁷ Niccolò Guicciardini, The Development of the Newtonian Calculus in Britain 1700-1800 (Cambridge: Cambridge University Press, 1989), p. 138.

⁵⁸ As Fisch, ‘Creative indecision’ (note 3) shows, the members turned away for different reasons and in strikingly different ways.

⁵⁹ Koppelman, ‘The calculus of operations’ (note 29), pp. 210–11.

⁶⁰ Despeaux, ‘”Very full of symbols”’ (note 5), 56.

⁶¹ William Thomson, ‘Presidential address to the British Association, Edinburgh, 1871’, in Popular Lectures and Addresses. Three Volumes (Cambridge & New York: Cambridge University Press, 2011), 132–206 (p. 139).

⁶² See Despeaux, ‘”Very full of symbols”’ (note 5), pp. 54–67.

⁶³ See Smith & Wise, Energy (note 4), p. 150.

⁶⁴ Enros, ‘Analytical society’ (note 2), p. 28.

⁶⁵ Davie, Democratic Intellect (note 23), p. 154.

⁶⁶ Compare Fisch, ‘Creative indecision’ (note 3). Allaire & Bradley, ‘D.F. Gregory's contribution’ (note 5). John M. Dubbey, ‘The introduction of the differential notation to Great Britain’, Annals of Science, 19 (1963), pp. 37–48. For example, Peacock had dismissed the method of fluxions not only for ‘introducing extraneous ideas (geometrical and mechanical) into the study of purely algebraic[al] problems’, but also because ‘the differential notation is ‘equally convenient for representing both operation and quantity’. Koppelman, ‘The calculus of operations’ (note 29), p. 153.

⁶⁷ See Davie, Democratic Intellect (note 23), pp. 161–2. Smith & Wise, Energy (note 4), pp. 184–5.

⁶⁸ Allaire and Bradley observe that Examples ‘functioned equally well whether the student’s understanding of first principles was algebraic in flavor [or] used limits […] Indeed, someone who had learned calculus using the Newtonian doctrine of fluxions would have benefited from Gregory's text’. Allaire & Bradley, ‘D.F. Gregory's contribution’ (note 5), p. 396.

⁶⁹ Craig G. Fraser, ‘Calculus as algebraic analysis: some observations on mathematical analysis in the 18^th century’, Archive for History of Exact Sciences, 39 (1989), pp. 317–35. For a detailed account of Cauchy’s approach to the calculus see, for example, Judith V. Grabiner, The Origins of Cauchy’s Rigorous Calculus (Cambridge, MA: The MIT Press, 1981).

⁷⁰ Ivor Grattan-Guinness wrote that ‘Cauchy resurrected the Newtonian approach, for Cauchy’s derived function is not Lagrange’s algebraically defined object [but] a Newton-style fluxion’. Ivor Grattan-Guinness, ‘Babbage’s mathematics in its time’, The British Journal for the History of Science, 12 (1979), 82–8 (p. 83). See also Giovanni Ferraro, ‘Analytical symbols and geometrical figures in eighteenth-century analysis’, Studies in History and Philosophy of Science, 32 (2001), 535–55,

⁷¹ Allaire & Bradley, ‘The contribution of D.F. Gregory’ (note 5), p. 397.

⁷² See Davie, Democratic Intellect (note 23), p. 164.

⁷³ Koppelman, ‘The calculus of operations’ (note 29), p. 176.

⁷⁴ For example, in his 1852 plea for the inclusion of ‘new geometry’ in the Cambridge curriculum, Ellis noted that it ‘seems to be little studied in the University yet the method of which it makes so much use, namely, the generation and transformation of figures by ideal motion, is more natural and philosophical than the (so to speak) rigid geometry to which our attention has been confined’. Robert Leslie Ellis, ‘Evidence on mathematical studies and examinations. Answers from Robert Leslie Ellis, Esq., M.A., late Fellow of Trinity College’, in Cambridge University Commission. Report of Her Majesty’s Commissioners Appointed to Inquire Into the State, Discipline, Studies and Revenues of the University and Colleges of Cambridge; Together with the Evidence, and an Appendix (London: W. Clowes and Sons, 1852), pp. 222–6 (p. 224), my emphasis. See also Richards, ‘Projective geometry’ (note 39).

⁷⁵ Hamilton wrote that Newton ‘whose revolutionary work in the higher parts of both pure and applied Algebra was founded mainly on the notion of fluxion which involves the notion of time’. Hamilton, ‘Theory of conjugate functions’ (note 25), pp. 5–6.

⁷⁶ George Peacock, ‘Report on the recent progress and present state of certain branches of analysis’, in Report of the Third Meeting of the British Association for the Advancement of Science (London: John Murray, 1834), pp. 185–352.

⁷⁷ Fisch, ‘Creative indecision’ (note 3), p. 140.

⁷⁸ Pycior, ‘George Peacock’ (note 3), pp. 27–8.

⁷⁹ John Playfair, ‘On the arithmetic of impossible quantities’, Philosophical Transactions of the Royal Society of London, 68 (1778), 318–43. William Greenfield, ‘On the use of negative quantities in the solution of algebraic equations’, Transactions of the Royal Society of Edinburgh, 1 (1778), 99–107. Adrien-Quentin de Bueé, ‘Mémoire sur les quantités imaginaires’, Philosophical Transactions of the Royal Society of London, 96 (1806), 23–88; Woodhouse, ‘On the necessary truth’ (note 30).

⁸⁰ Woodhouse, ‘On the necessary truth’ (note 30), p. 90, p. 93.

⁸¹ Peacock, ‘Report’ (note 76), pp. 189–92.

⁸² Fisch, ‘Creative indecision’ (note 3), p. 162.

⁸³ Peacock, Treatise (note 3), p. vi.

⁸⁴ Peacock, ‘Report’ (note 76), pp. 190–1, my emphasis.

⁸⁵ Fisch, ‘Creative indecision’ (note 3), 162. Silvestre François Lacroix, An Elementary Treatise on the Differential and Integral Calculus. Translated From the French. With an Appendix and Notes (Cambridge: Deighton and Sons, 1816).

⁸⁶ Fisch, ‘Creative indecision’ (note 3), pp. 162–3.

⁸⁷ Fisch, ‘Creative indecision’ (note 3), p. 165.

⁸⁸ Peacock, ‘Report’ (note 76), pp. 194–5.

⁸⁹ Fisch, ‘Creative indecision’ (note 3), p. 166.

⁹⁰ Fisch, ‘Creative indecision’ (note 3), p. 167.

⁹¹ Fisch, ‘Creative indecision’ (note 3), pp. 167–8.

⁹² Fisch, ‘Creative indecision’ (note 3), p. 140.

⁹³ Duncan Farquharson Gregory, ‘On the real nature of symbolical algebra’, Transactions of the Royal Society of Edinburgh, 14 (1840), 208–16 (p. 208).

⁹⁴ Gregory, ‘Real nature’ (note 93), p. 208.

⁹⁵ Gregory, ‘Real nature’ (note 93), pp. 208–9.

⁹⁶ The following account draws on, and quotes from Gregory, ‘Real nature’ (note 93), pp. 209–15.

⁹⁷ Gregory, ‘Real nature’ (note 93), p. 208.

⁹⁸ Allaire & Bradley, ‘D.F. Gregory's contribution’ (note 5), p. 407.

⁹⁹ Allaire & Bradley, ‘D.F. Gregory's contribution’ (note 5), p. 407.

¹⁰⁰ Davie, Democratic Intellect (note 23), p. 164.

¹⁰¹ Duncan Farquharson Gregory, ‘On the solution of linear differential equations with constant coefficients’, Cambridge Mathematical Journal, 1 (1838), 22–32 (p. 30).

¹⁰² Gregory, ‘Linear differential equations’ (note 101), p. 30.

¹⁰³ Duncan Farquharson Gregory, ‘On the impossible logarithms of quantities’, Cambridge Mathematical Journal, 1 (1839), 226–34.

¹⁰⁴ Duncan Farquharson Gregory, ‘On a difficulty in the theory of algebra’, Cambridge Mathematical Journal, 3 (1843), 153–9.

¹⁰⁵ Duncan Farquharson Gregory, ‘On the elementary principles of the application of algebraical symbols to geometry’, Cambridge Mathematical Journal, 2 (1839), 1–9 (p. 1).

¹⁰⁶ Gregory, ‘On a difficulty’ (note 104), p. 153.

¹⁰⁷ Koppelman, ‘The calculus of operations’ (note 29), p. 216.

¹⁰⁸ Gregory, ‘On a difficulty’ (note 104), p. 154.

¹⁰⁹ Gregory, ‘On a difficulty’ (note 104), pp. 154–5. Robert Murphy, ‘First memoir on the theory of analytical operations’, Philosophical Transactions of the Royal Society of London, 127 (1836), 179–210.

¹¹⁰ Gregory, ‘On a difficulty’ (note 104), p. 156.

¹¹¹ Gregory, ‘On a difficulty’ (note 104), p. 157.

¹¹² Gregory, ‘On a difficulty’ (note 104), p. 155, my emphases.

¹¹³ Duncan Farquharson Gregory, ‘On the existence of branches of curves in several planes,’ Cambridge Mathematical Journal, 1 (1839), 259–66.

¹¹⁴ See, for example, William Walton, ‘On the general interpretation of equations between two variables in algebraic geometry’, Cambridge Mathematical Journal, 2 (1840), 103–13.

¹¹⁵ Augustus De Morgan, ‘On the signs + and – in geometry (continued), and on the interpretation of the equation of a curve’, Cambridge and Dublin Mathematical Journal, 7 (1852), 242–51 (p. 243).

¹¹⁶ See Richards, Mathematical Visions (note 38), 53–4.

¹¹⁷ Duncan Farquharson Gregory, ‘On some elementary principles in the application of algebra to geometry’, Cambridge Mathematical Journal, 1 (1838), 74–7.

¹¹⁸ See Davie, Democratic Intellect (note 23), p. 154, p. 161, p. 166. Smith & Wise, Energy (note 4), p. 184. The Newtonian character of Gregory's application of the calculus of operations to geometry must be sought in his treatment of the relation between process and operation or, more specific, of lines as being generated by moving points. Newton himself employed geometrical fluxional analysis precisely to treat curves as arising from the motion of a point and to formulate them in terms of quantities changing with time (‘fluents’). See note 23 for references to accounts of Newton’s mathematical method.

¹¹⁹ Gregory, ‘On some elementary principles’ (note 117), p. 75.

¹²⁰ Gregory, ‘On some elementary principles’ (note 117), pp. 81–2.

¹²¹ Gregory, ‘On some elementary principles’ (note 117), pp. 74–5.

¹²² Gregory, ‘On the elementary principles’ (note 105), p. 1.

¹²³ See Davie, Democratic Intellect (note 23), p. 161.

¹²⁴ Gregory, ‘On the elementary principles’ (note 105), p. 2.

¹²⁵ Gregory, ‘On the elementary principles’ (note 105), p. 5.

¹²⁶ The following draws on Gregory, ‘On the elementary principles’ (note 105), pp. 5–7.

¹²⁷ See Despeaux, ‘”Very full of symbols”’ (note 5), pp. 60–7. Smith & Wise, Energy (note 4), pp. 151–68.

¹²⁸ Gregory, ‘On the elementary principles’ (note 105), p. 2.

¹²⁹ Gregory, ‘On the elementary principles’ (note 105), p. 5.

¹³⁰ Gregory, ‘On the elementary principles’ (note 105), p. 9.

¹³¹ Gregory, ‘On the existence of branches’ (note 113), p. 139, my emphasis.

¹³² Gregory & Walton, Treatise on Solid Geometry (note 48), p. 177. Buée, ‘Mémoire’ (note 79).

¹³³ Gregory & Walton, Treatise on Solid Geometry (note 48), p. 175.

¹³⁴ Gregory & Walton, Treatise on Solid Geometry (note 48), p. 176.

¹³⁵ Gregory, ‘On the existence of branches’ (note 113), p. 257.

¹³⁶ Gregory, ‘On the existence of branches’ (note 113), p. 259.

¹³⁷ Gregory, ‘Impossible logarithms’ (note 103), p. 229.

¹³⁸ Richards, Mathematical Visions (note 38), p. 51.

¹³⁹ Gregory & Walton, Treatise on Solid Geometry (note 48), p. 177.

¹⁴⁰ Gregory, ‘On the existence of branches’ (note 113), p. 262.

¹⁴¹ Gregory, ‘On the existence of branches’ (note 113), pp. 265–6, my emphasis.

¹⁴² Crilly, ‘The Cambridge Mathematical Journal’ (note 45), p. 468.

¹⁴³ William Walton, ‘On the doctrine of impossibles in algebraic geometry’, Cambridge Mathematical Journal, 7 (1852), 234–42 (p. 235).

¹⁴⁴ George Salmon, A Treatise on the Higher Plane Curves. Intended as a Sequel to A Treatise on Conic Sections (Dublin: Hodges and Smith, 1852), p. 302.

¹⁴⁵ Salmon, Treatise on Higher Plane Curves (note 144), p. 303.

¹⁴⁶ In his articles Walton mostly referred to Gregory, ‘Linear differential equations’ (note 101). Gregory, ‘On the existence of branches’ (note 113). Gregory, ‘On the elementary principles’ (note 105).

¹⁴⁷ William Walton, ‘On the general interpretation of equations between two variables in algebraic geometry’, Cambridge Mathematical Journal, 2 (1840), 103–13 (p. 103).

¹⁴⁸ Walton, ‘On the doctrine of impossibles’ (note 143), 238.

¹⁴⁹ William Walton, ‘On the general theory of multiple points’, Cambridge Mathematical Journal, 2 (1840), 155–67; William Walton, ‘On the existence of possible asymptotes to impossible branches of curves’, Cambridge Mathematical Journal’, Cambridge Mathematical Journal, 2 (1841), 236–9.

¹⁵⁰ Walton, ‘On the general interpretation’ (note 147), p. 104.

¹⁵¹ Salmon, Treatise on Higher Plane Curves (note 144), p. 303.

¹⁵² Walton, ‘On the general theory’ (note 149), p. 165.

¹⁵³ Augustus De Morgan, ‘On the foundation of algebra’, Transactions of the Cambridge Philosophical Society, 7 (1842), 173–87, 287–300; 8 (1842), 139–42, 241–53.

¹⁵⁴ Pycior, ‘De Morgan’s algebraic work’ (note 6), p. 222.

¹⁵⁵ De Morgan, ‘On the foundation’ (note 153), p. 173.

¹⁵⁶ Koppelman, ‘The calculus of operations’ (note 29), p. 220.

¹⁵⁷ George Boole, The Mathematical Analysis of Logic, Being an Essay Towards a Calculus in Deductive Reasoning (Cambridge: Macmillan, Barclay & Macmillan, 1847).

¹⁵⁸ See Charlotte Simmons, ‘William Rowan Hamilton and George Boole’, BSHM Bulletin: Journal of the British Society for the History of Mathematics, 23 (2008), 96–102.

¹⁵⁹ These two views may be said to correspond to the ‘non-scientific’ symbolical algebraists and ‘scientific’ symbolical algebraists, respectively.

¹⁶⁰ Hamilton, ‘Theory of conjugate functions’ (note 25), p. 5.

¹⁶¹ Ohrstrom, ‘Hamilton's view of algebra’ (note 9), p. 46.

¹⁶² William Rowan Hamilton, Lectures on Quaternions (Dublin: Hodges and Smith, 1953), 1–64 (p. 2).

¹⁶³ Hamilton, ‘Theory of conjugate functions’ (note 25).

¹⁶⁴ Hamilton, ‘Theory of conjugate functions’ (note 25), p. 6.

¹⁶⁵ Hamilton, ‘Theory of conjugate functions’ (note 25), p. 7.

¹⁶⁶ Hamilton, ‘Theory of conjugate functions’ (note 25), p. 7.

¹⁶⁷ Hamilton, Lectures (note 162), p. 2, f. 2.

¹⁶⁸ Thomas L. Hankins, ‘Triplets and triads: Sir William Rowan Hamilton on the metaphysics of mathematics’, Isis, 68 (1977), 174–93 (p. 177).

¹⁶⁹ Rice, ‘Inexplicable?’ (note 27), p. 169.

¹⁷⁰ Koppelman, ‘The calculus of operations’ (note 29), p. 228.

¹⁷¹ See, for example, Rice, ‘Inexplicable?’ (note 27), section 7.

¹⁷² Hamilton in Rice, ‘Inexplicable?’ (note 27), p. 23.

¹⁷³ Hankins, ‘Triplets and triads’ (note 168), p. 176, emphasis in original.

¹⁷⁴ Rice, ‘Inexplicable?’ (note 27), p. 173.

¹⁷⁵ Hamilton, ‘Theory of conjugate functions’ (note 25), p. 4.

¹⁷⁶ The three-dimensional properties of quaternions were, indeed, soon to be adopted for use in physics by another mathematician associated with the CMJ, James Clerk Maxwell (1831–1879).

¹⁷⁷ Arthur Cayley, ‘A memoir on abstract geometry. Read December 16, 1869’, Philosophical Transactions of the Royal Society of London, 65 (1870), 51–63 (p. 52).

¹⁷⁸ Richards, Mathematical Visions (note 38), p. 52.

¹⁷⁹ Salmon, Treatise on Higher Plane Curves (note 144), p. 303.

¹⁸⁰ Richards, Mathematical Visions (note 38), p. 52.

¹⁸¹ Richards, Mathematical Visions (note 38), p. 55.

¹⁸² Richards, Mathematical Visions (note 38), p. 55.

¹⁸³ See, for example, Gray, Plato’s Ghost (note 20). Following Mehrtens, Moderne (note 20), Gray distinguishes between ‘moderns’ and ‘countermoderns’. Where the former held that mathematics needs no independent referent to justify the existence of the objects about which it speaks, the latter argued that the existence of mathematical objects cannot be derived solely from their function within a formal system. Gray writes that ‘British algebraists of the first half of the nineteenth century can be seen as quite formal in their study of systems of meaningless symbols’, and then attributes the destruction of this ‘pre-modern’ British view to Hamilton's ‘modernist’ formalism. Gray, Plato’s Ghost (note 20), p. 28. But this leaves unexplained not only the ground of the possibility of referring to different meanings of ‘formal’, but also that and in what sense there is something ‘modern’ about the work of the ‘pre-moderns’ Peacock and Gregory, and ‘countermodern’ about the ‘modern’ work of Hamilton.