Mathematics Sent Across the Channel and the Atlantic: British Mathematical Contributions to European and American Scientific Journals, 1835–1900

Summary

This paper will consider the range of British participation in mathematics internationally during the nineteenth century through an analysis of British mathematical contributions to scientific journals outside of Britain. Viewing scientific papers contained in journals as significant indicators of research, we consider scientists who authored or read and responded in print to papers in a given area within a given group of journals to constitute a publication community. The extent of publication by British mathematicians in these journals can help characterize the role of foreign publication in nineteenth-century British mathematics. Moreover, the isolation of educational, societal, and personal circumstances which motivated British mathematicians to present their work to foreign journals highlights limited but concentrated groups of mathematicians committed to developing and strengthening international mathematical ties with Britain.

Notes

²Charles William, Viscount Milton, Report of the British Association for the Advancement of Science, 1 (1831), 15–17 (16).

¹Britain, in this paper, is taken to be England, Scotland, Wales, and Ireland, that is, the United Kingdom as defined by the Act of Union in 1801 and what ‘was governed from London during the 19th century’ [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, 455–497 (456)].

³Duncan Gregory, ‘Preface’, Cambridge Mathematical Journal, 1 (1837), 1–2 (1).

⁴James Joseph Sylvester, et al., ‘Address to the Reader’, Quarterly Journal of Pure and Applied Mathe-matics, 1 (1855), i–ii.

⁵Roy M. Macleod, ‘Macmillan and the Scientists’, Nature 224 (1969), 428.

⁶Arthur Jack Meadows, Communication in Science (London, 1974), 67; and Bernard Houghton, Scientific Periodicals: Their Historical Development, Characteristics and Control (London, 1975), 12.

⁷James Whitbread Lee Glaisher, ‘Mathematical Journals’, Nature, 22 (1880), 73–75 (73).

⁸Hélène Gispert, ‘'Une comparaison des journaux français et italiens dans les années 1860–1875’, in L'Europe mathématique: Histoires, Mythes, Identités, edited Catherine Goldstein, Jeremy Gray, and Jim Ritter (Paris, 1996), 389–406 (399). ‘Les journaux, principaux vecteurs du savoir mathématique, ont en effet un rôle central dans la diffusion, donc dans l’évolution, du mouvement des mathématiques’.

⁹Derek de Solla Price, ‘Toward a Model for Science Indicators’, in Toward a Metric of Science: The Advent of Science Indicators, edited by Yehuda Elkana et al. (New York, 1978), 69–95 (80). For other views from sociologists of science about the use of journals to trace the development of science, see Henry G. Small, ‘The Lives of a Scientific Paper’, in Selectivity in Information Systems: Survival of the Fittest, ed. by Kenneth S. Warren (New York, 1985), 83–97; Daryl E Chubin and Soumyo D. Moitra, ‘Content Analysis of References: Adjunct or Alternative to Citation Counting?’, Social Studies of Science, 5 (1975), 423–41; and David Edge, ‘Quantitative Measures of Communication in Science: A Critical Review’, History of Science, 17 (1979), 102–134.

¹⁰It would be reasonable to include within a publication community those who merely read the publications produced. Unfortunately, the mere readers—as opposed to actual writers—in the publication community are illusive and have not been considered here.

¹¹Sloan Evans Despeaux, ‘International Mathematical Contributions to British Scientific Journals: 1800–1900’, in Mathematics Unbound: The Evolution of an International Mathematical Community, 1800–1945, edited by Karen Hunger Parshall and Adrian C. Rice (Providence, RI, 2002), 61–87, and Sloan Evans Despeaux, ‘The Development of a Publication Community: Nineteenth-Century Mathematics in British Scientific Journals’ (unpublished doctoral dissertation, University of Virginia, 2002; abstract in Dissertation Abstracts International, 63 (2002), 09B, 4201).

¹²This journal sample consists of all nineteenth-century volumes of British Association for the Advancement of Science Report (1831–present); Cambridge Philosophical Society Transactions (1822–1928); Manchester Literary and Philosophical Society Memoirs (1785–1887), Proceedings (1857–1887), Memoirs and Proceedings (1888–present); Royal Irish Academy Transactions (1787–1907); Royal Societyof Edinburgh Transactions (1783–present); Royal Society of London Philosophical Transactions (1665–present), Proceedings (1832–present); Edinburgh Mathematical Society Proceedings (1883–present); London Mathematical Society Proceedings (1865–present); Royal Astronomical Society Memoirs (1822–present), Monthly Notices (1827–present); Philosophical Magazine (1798–present); Cambridge Mathematical Journal (1837–1845); Cambridge and Dublin Mathematical Journal (1846–1854); Leybourne's Mathematical Repository (1806–1835); Oxford, Cambridge, and Dublin Messenger of Mathematics (1862–1871); Messenger of Mathematics (1871–1929); and Quarterly Journal of Pure and Applied Mathematics (1855–1927). For more on the methodology and other details behind this prosopography, see Despeaux, ‘Development’, 196–240.

¹³For more details about higher education in Britain during the nineteenth century, see Sloan Evans Despeaux, ‘Launching Mathematical Research without a Formal Mandate: The Role of University Affiliated Journals in Britain, 1837–1870’, Historia Mathematica, 34 (2007), 89–106.

¹⁴Other degrees in science, divinity, medicine, and law, were also obtained by some of the members of the prosopography. The doctorate, firmly established in the German and American educational systems during the nineteenth century, did not reach Britain until 1917, and at least 13 of the 14 PhD holders in this prosopography received the degree in Germany (the origin of one PhD has not been determined). For more details, see Despeaux, ‘Development’, 204–11.

¹⁵This sample of societies is the Royal Society of London, the Royal Society of Edinburgh, the Royal Irish Academy, the London Mathematical Society, the Edinburgh Mathematical Society, the Royal Astronomical Society, the Physical Society, the Cambridge Philosophical Society, the Manchester Literary and Philosophical Society, and the Association for the Improvement of Geometrical Teaching.

¹⁶The Annali di matematica pura ed applicata replaced the Annali di scienze matematiche e fisiche, founded in 1850 by Barnaba Tortolini. This journal takeover occurred in the midst of Italian political unification and was led by Enrico Betti, Francesco Brioschi, and Angelo Genocchi.

¹⁷For more on these journals, see Wolfgang Eccarius, ‘August Leopold Crelle als Herausgeber wis-senschaftlicher Fachzeitschriften’, Annals of Science, 33 (1976), 229–61; Silvina Duvina, ‘Le Journal de mathématiques pures et appliquées sous la férule de J. Liouville (1836–1874)’, Sciences et techniques en perspective, 28 (1994), 179–217; Laura Martini, ‘The Politics of Unification: Barnaba Tortolini and the Publication of Research Mathematics in Italy, 1850-1865’, in Il sogno di Galois: scritti di storia della matematica dedicati a Laura Toti Rigatelli per il suo 60° compleanno, ed. by R. Franci, P. Pagli, and A. Simi (Siena, 2003), 171–198; and June E. Barrow-Green, ‘Gösta Mittag-Leffler and the Foundation and Administration of A cta Mathematica’, Mathematics Unbound: The Evolution of an International Mathematical Community, 1800–1945, ed. by Karen Hunger Parshall and Adrian C.Rice (Providence, RI, 2002), 138–64.

¹⁸The American Journal of Mathematics published thirty-three articles from British mathematicians from the first sample, the Nouvelles annales published at least twenty-seven such articles, and the Mathematische Annalen published twenty-four such articles. The French journal, Comptes rendus hebdomadaires des séances de l'Académie des Sciences, contained 128 articles from British mathematicians from the first sample, but is not included here. It has general science format, while the other journals in the sample are devoted solely to mathematics; this challenging format makes the Comptes rendus beyond the scope of the present study. For more on these journals, see Karen Hunger Parshall and David E. Rowe, The Emergence of the American Mathematical Research Community, 1876–1900: J.J. Sylvester, Felix Klein, and E.H. Moore, (Providence, RI, 1994), 88–94; Hélène Gispert, ‘Le Milieu mathématique fran,cais et ses journaux en France et en Europe (1870–1914)’, in Messengers of Mathematics: European Mathematical Journals 1810–1939, ed. by Elena Aussejo and Mariano Hormigon (Madrid, 1993), 133–198; and Heinrich Behnke, ‘Rückblick auf die Geschichte der Mathematischen Annalen’, Mathematische Annalen, 200 (1973), I–VII.

¹⁹These four mathematicians are discussed in detail below.

²⁰The following account of Hirst's life is based on J. Helen Gardner and Robin J. Wilson, “Thomas Archer Hirst—Mathematician Xtravagant Parts I–VI,” American Mathematical Monthly, 100 (1993), 435–41, 531–38, 619–25, 723–31, 827–34, and 907–15.

²¹Silvanus P. Thompson, The Life of William Thomson, Baron Kelvin of Largs, vol. 1 (London, 1910), 113–120. For more on Thomson's relationship with Liouville and the international contributions to the French mathematician's journal, see Jesper Lützen, ‘International Participation in Liouville's Journal de mathématiques pures et appliquées’, in Mathematics Unbound, 89–104 and Jesper Lützen, Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics (New York, 1990), pp. 135–46. Soon after returning from his semester-long sojourn, Thomson agreed to take over the editorship of the Cambridge Mathematical Journal, which he rechristened the Cambridge and Dublin Mathematical Journal. With international experiences fresh in his memory, Thomson tried to widen the contributorship of his newly acquired journal. For more on Thomson's role in this journal, see Crilly, ‘The Cambridge Mathematical Journal and its descendants’.

²²Lützen, ‘International Participation’, 92.

²³Laura Martini, ‘Political and Mathematical Unification: Algebraic Research in Italy, 1850–1914’ (un-published doctoral dissertation, University of Virginia, 2006; abstract in Dissertation Abstracts International, 67 (2006), 05B (2006), 2592), 186.

²⁴The only foreign article of Cambridge fellow, Arthur Thacker and George Green's three foreign contributions were made to Crelle's Journal during the 1850s. Trinity College, Dublin fellow, Charles Graves, and William Donkin, Savilian Professor of Astronomy at Oxford, made two foreign contributions each between 1850 and 1854.

²⁵For more on Sylvester's efforts to establish and sustain the American Journal, see Karen Hunger Parshall, James Joseph Sylvester: Jewish Mathematician in a Victorian World (Baltimore MD, 2006), 239–48.

²⁶James Joseph Sylvester to Arthur Cayley, 24 Nov. 1877, in Tony Crilly, Arthur Cayley, Mathematician Laureate of the Victorian Age (Baltimore 2006), 348–49.

²⁷Arthur Cayley, ‘Desiderata and Suggestions’, ‘No. 1 The Theory of Groups’, ‘No. 2 The Theory of Groups, Graphical Representation’, ‘No. 3 The Newton–Fourier Imaginary Problem’, and ‘No. 4 Mechanical Construction of Conformable Figures’, American Journal of Mathematics, 1 (1878), 50–52; 174–76; and 2(1879), 97, 186.

²⁸Cayley, ‘Desiderata and Suggestions’, 52.

²⁹Sylvester also printed Cayley's work in the intra-university publication, the Johns Hopkins University Circular. Sylvester wrote to Cayley that ‘I shall try and cull out of your letters some additional matter for the J[ohns] H[opkins] Circular. I am glad (and so is [Daniel Coit] Gilman [first President of Hopkins]) that you approve of the development which it has undergone’. James Joseph Sylvester to Arthur Cayley, 16 March 1883, in Parshall Life and Work, 221.

³⁰James Joseph Sylvester to Daniel Coit Gilman, 7 September 1878, in Parshall, Life and Work, 191.

³¹Out of the 385 contributions, only four appeared in German, one in Italian, and two in Latin.

³²These contributions are not limited to those in the eight journals from which the study obtained the sample of British mathematicians. Besides the eight initial journals, these articles were printed in Les Mondes, Revue hebdomadaire des sciences et de leurs applications aux arts et à l'industrie (Paris), Société philomatique de Paris, Bulletin des sciences (Paris), La revue scientifique de la France et de l’étranger (Paris), Annali di scienze matematiche e fisiche (Rome), R. Accademia dei Lincei, Memorie (Rome), Giornale di matematiche (Naples), Deutsche chemische Gesellschaft (Berlin), Annalen der Physik und Chemie (Leipzig), Mathésis (Gand), Van Nostrand's Engineering Magazine (New York), Transactions of the Royal Society of New South Wales (Sydney), and Transactions of the Victoria Royal Society (Melbourne). This list indicates the range of foreign publications to which British mathematicians contributed.

^{33 33}James Joseph Sylvester, ‘Sur une extension de la théorie des equations algébriques’, Comptes rendus 58 (1864): 689–91 (689); also in The Collected Mathematical Papers of James Joseph Sylvester, ed. Henry Frederick Baker, 4 vols, (Cambridge, 1908; reprint ed., New York 1973), 2: 361–62 (361). ‘Quelques recherches que j'ai faites tout récemment sur la régle donnée sans d′emonstration par Newton … pour trouver une limite inférieure au nombre de racines imaginaires d'une equation’.

^{34 34}James Joseph Sylvester ‘Algebraical Researches, Containing a Disquisition on Newton's Rule for the Discovery of Imaginary Roots, and an Allied Rule Applicable to a Particular Class of Equations, together with a Complete Invariantive Determination of the Character of the Roots of the General Equations of the Fifth Degree, &c.’, Philosophical Transactions of the Royal Society of London, 154 (1864), 579-666; also in Collected Works 2, 376–79; ‘On an Elementary Proof and Generalization of Sir Isaac Newton's Hitherto Undemonstrated Rule for the Discovery of Imaginary Roots’, Proceedings of the London Mathematical Society 1 (1865–1866): 1–16; also in Collected Works, 2, 498-516; and ‘Observations sur un article de M. Poulain’, Les Mondes 11 (1866): 435–37; also in Collected Works, 2,514–16.

^{35 35}Karen Hunger Parshall and Eugene Seneta, ‘Building an International Reputation: The Case of J.J. Sylvester (1814-1897)’, American Mathematical Monthly, 104 (1997), 210–22 (217).

³⁶Charles Graves, William Roberts, and William MacCullagh, all at Trinity, made at least seven duplicate publications in this study's sample.

³⁷Patrick S. Cross, ‘The Organization of Science in Dublin from 1785 to 1835: The Men and Their Institutions’ (unpublished PhD dissertntion, University of Oklahoma, 1996), 142.

³⁸Charles Graves, ‘Elementary Geometrical Proof of Joachimstal's Theorem’, Proceedings of the Royal Irish Academy 5 (1853), 70–71, Journal für die reine und angewandte Mathematik 42 (1851), 279, and Nouvelles annales de mathématiques, 11 (1852), 322–23.

³⁹Also not included in the sample were British propositions and solutions of problems from the textitNou-velles annales de mathématiques, although over twenty such instances were observed.

⁴⁰William Spottiswoode, ‘Elementary Theorems relating to Determinants’, Journal für die reine und ange-wandte Mathematik, 51 (1856), 209–71, 328–81 (209).

⁴¹For more detailed information, see Despeaux, ‘Development’, 297–323.

⁴²Lützen, ‘International Participation’, 91.

⁴³Lützen, ‘International Participation’, 92.

⁴⁴Eccarius, 238.

⁴⁵I thank Dr. Martini for sharing these data with me. For more on her discussion of the Annali, see Martini, ‘Political and Mathematical Unification’, 164–223.

⁴⁶Barrow-Green, l56

⁴⁷Barrow-Green, 157. For more on the role of analysis in Britain, see Adrian Rice and Robin J. Wilson,'The rise of British analysis in the early 20th century: the role of G. H. Hardy and the London Mathematical Society', Historia Mathematica 30 (2003), 173–94.

⁴⁸Reinhard Siegmund-Schultze, ‘The Emancipation of Mathematical Research Publishing in the United States from German Dominance (1878–1945)’, Historia Mathematica, 24 (1997), 135–66 (138).

⁴⁹John Venn, Alumni Cantabrigienses; A Biographical List of All Known Students, Graduates and Holders of Office at the University of Cambridge (Cambridge, 1922–1954); Joseph Foster, Alumni Oxonienses: The Members of the University of Oxford, 1715–1886: Their Parentage, Birthplace, and Year of Birth, with a Record of Their Degrees (Oxford 1887–1888); George Dames Burtchaeli, Thomas Ulick Sadleir, (ed.), Alumni Dublinenses (Dublin, 1935); J.C Poggendorff's Biographisch-Literarisches Handwörterbuch zur Geschichte derExactenWissenschaften (Leipzig, 1863–1926 and Sir Leslie Stephen and Sir Sidney Lee (eds), The Dictionary of National Biography (London, 1885–1901).

⁵⁰No educational information has been found for John C. Malet.

⁵¹George Boole's first degree was an LLD, which he received three years after after becoming Professor of Mathematics at Queen's College, Cork. Thomas Archer Hirst's most advanced formal training had been at the Halifax Mechanics Institute before he followed his friend John Tyndall to the University of Marburg for graduate study. Gardner and Wilson, 439. Thomas Turner Wilkinson received no formal higher educational training.

⁵²For all but six cases, the undergraduate training presented in table 5a culminated in the BA degree. Graduate training presented in table 5a included the MA, MD, PhD (two cases), LLD, DD, DSc, or DCL.

⁵³James MacCullagh, ‘Réclamation de priorité relativement a certaines formules pour calculer l'intensité de la lumière’, Comptes rendus, 8 (1839), 961–71.

⁵⁴Lützen, Joseph Liouville, 134.

⁵⁵T.D. Spearman, ‘James MacCullagh’, in Science in Ireland 1800–1930: Tradition and Reform, ed. John R. Nudds et al. (Dublinl, 1988), 83–97 (53).

⁵⁶In fact, of all mathematical students during MacCullagh's tenure, twenty became fellows. Spearman, p. 42.

⁵⁷At least twenty-one of the mathematicians in this sample received university fellowships.

⁵⁸While studying law, Sylvester worked as an actuary in the Equity and Law Life Assurance Society, and he probably undertook legal studies in order to advance his actuarial career. Parshall, Life and Work, 2.

⁵⁹Crilly, Mathematician Laureate, 121.

⁶⁰Three in this group, John Couch Adams, William Spottiswoode, and William Thomson, served as presidents of the Royal Society.

⁶¹These LMS presidents were Cayley, James Cockle, Andrew Forsyth, Hirst, Percy MacMahon, Samuel Roberts, Henry Smith, Spottiswoode, and Sylvester.

⁶²These BAAS presidents were Cayley, George Howard Darwin, Andrew Forsyth, Percy MacMahon, and Spottiswoode.

⁶³For more on Sylvester's reaction to his election to the American Academy, see Parshall, James Joseph Sylvester: Jewish Mathematician in a Victorian World, 219.

⁶⁴James Joseph Sylvester to Thomas Archer Hirst, 1 November 1865, in Karen Hunger Parshall, James Joseph Sylvester: Life and Work in Letters (Oxford, 1998), 128. The discovery Sylvester referred to is a method for solving questions entailing conic sections which Chasles presented in ‘Considérations sur la méthode générale exposée dans la séance du 15 février—Différence entre cette méthode et la méthode analytique—Procédés générale de démonstration’, Comptes rendus, 58 (1864), 1167–75. Parshall, Life and Work, 128.

⁶⁵Arthur Cayley to Thomas Archer Hirst, 31 October (no year recorded), LMS Papers, University College London. Chasles, in fact, won the Copley Medal in 1865, and Plücker won the year later.

⁶⁶Adrian C. Rice and Robin J. Wilson, ‘From National to International Society: The London Mathematical Society, 1867-1900’, Historia Mathematica, 25 (1998), 185–217 (187–89).

⁶⁷These 220 articles were recorded in the Royal Society Catalogue and are not limited to this study's journal sample.

⁶⁸Arthur Cayley, ‘Mémoire sur les courbes du troisi‘eme ordre’, Journal de mathématiques pures et ap-pliquées 9 (1844), 285–93.

⁶⁹Crilly, Mathematician Laureate, 106.

⁷⁰Lützen, Joseph Liouville, 134.

⁷¹Joseph Liouville to William Thomson, 29 July 1847, quoted in Crilly, Mathematician Laureate, 126.

⁷²Arthur Cayley, ‘Note sur deux formules données par M.M. Eisenstein et Hesse’, Journal für die reine und angewandte Mathematik, 29 (1845), 54–57 (55). ‘Je me propose de poser les premiers fondemens de cette théorie dans un mémoire qui va paraître dans le prochain No. du “Cambridge Mathematical Journal”’.

⁷³Arthur Cayley, ‘On the Theory of Linear Transformations’, Cambridge Mathematical Journal 4 (1845), 193–209; ‘On Linear Transformations’, Cambridge and Dublin Mathematical Journal 1 (1846), 104–22; and ‘Mémoire sur les hyperdéterminants” Journal für die reine und angewandte Mathematik 30 (1846), 1–37.

⁷⁴Crilly, Mathematician Laureate, 108–109.

^{75 75}In 1854, Cayley's ‘Nouvelles recherches sur les covariants’ appeared in Crelle's Journal and presented the method which would direct his study of invariant theory. While Sylvesterconsidered this method ‘an engine that mightiest instrument of research ever yet invented by the mind of man—a Partial Differential Equation, to define and generate invariantive forms’, Crilly stated that ‘[i]n retrospect, it was the wrong road taken since the Ω-process [a process governed by the hyperdeterminant derivative] held more theoretical potential and was the basis for the future German symbolic calculus developed in the 1860s’. Crilly, Mathematician Laureate, p. 176. Arthur Cayley ‘Nouvelles recherches sur les covariants’, Journal für die reine und angewandte Mathematik, 47 (1854), 109–24.

⁷⁶Cayley made 32 contributions to the Comptes rendus, 19 to Liouville's Journal, 9 to the Annali, 26 to the American Journal of Mathematics, and 12 to the Johns Hopkins University Circular.

⁷⁷Crilly, Mathematician Laureate, pp. 189–91.

⁷⁸Crilly, Mathematician Laureate, 350.

⁷⁹This study did not count foreign contributions that Sylvester he made while at the Johns Hopkins University during 1876 to 1883. Similarly, it did count the contributions of the German Franz Woepke, an expert in the history of Arabic mathematics, while he resided in Britain from 1861 and 1863. Because of the difficulties inherent in tracking the movements of these mathematicians throughout their lifetimes, this study has most certainly included some British mathematicians’ contributions made abroad and has missed some contributions made by non-British mathematicians visiting Britain.

⁸⁰Parshall and Seneta, 210. This first contribution was ‘Sur une propriété nouvelle de l’équation qui sert a déterminer les inégalités séculaires des planètes’, Nouvelles annales de mathématiques 11 (1852), 434–40. See also, Parshall, James Joseph Sylvester: Jewish Mathematician in a Victorian World, 121.

⁸¹Parshall and Seneta, 213; and Parshall, James Joseph Sylvester: Jewish Mathematician in a Victorian World, 121.

⁸²Lützen, Joseph Liouville, 100.

⁸³James Joseph Sylvester, ‘Sketch of a Memoir on Elimination, Transformation, and Canonical Forms’, Cambridge and Dublin Mathematical Journal 6 (1851), 192–93; also in Collected Mathematical Papers, 1: 189.

⁸⁴James Joseph Sylvester, ‘On a Linear Method of Eliminating between Double, Treble, and Other Systems of Algebraic Equations’, Philosophical Magazine 18 (1841), 425–35; also in Collected Mathematical Papers, 1, 75–85.

⁸⁵Parshall and Seneta, 215–16.

⁸⁶Just two years later, in 1854. Sylvester picked Hesse as a reference for his bid for the professorship of mathematics at the Royal Military Academy at Woolwich. Ibid., 216.

⁸⁷Charles Hermite to J. J. Sylvester, 29 April 1881, in Parshall and Rowe, 123.

⁸⁸Fabian Franklin, ‘Sur le Développement du Produit infini (1 − x)(1 − x2)(1 − x3)(1 − x4)…’, Comptes rendus 82 (1881), 448–50. Parshall and Rowe, 22.

⁸⁹Parshall and Rowe,p. 123. This theorem states that ‘for any positive integer m, the difference between the number of partitions of m into an even number of parts and the number of partitions of m into an odd number of parts equalled (−1)ⁿ if m= and zero otherwise’. The theorem's name comes from the exponents m =  , which for n>0 give the sequence of pentagonal numbers, 1, 5,12, 22, … Ibid., 121–23.

⁹⁰Parshall and Rowe,p. 123. This theorem states that ‘for any positive integer m, the difference between the number of partitions of m into an even number of parts and the number of partitions of m into an odd number of parts equalled (−1)ⁿ if m= and zero otherwise’. The theorem's name comes from the exponents m =  , which for n>0 give the sequence of pentagonal numbers, 123.

⁹¹These fifty articles were recorded in the Royal Society Catalogue and are not limited to this study's journal sample.

⁹²These fifty articles were recorded in the Royal Society Catalogue and are not limited to this study's journal sample, 129

⁹³Of the articles recorded in the Royal Society Catalogue,forty-eight out of fifty-three of William's and twenty-seven out of thirty-sevenof Michael's were made to foreign journals.

⁹⁴Frederic Boase, ed., Modern English Biography (London, 1965), s.v. ‘Roberts, Michael’ and ‘Roberts, William’.

⁹⁵Liouville to M. Roberts, 13 March 1847, quoted in Lützen, Joseph Liouville, 134.

⁹⁶Lützen, Joseph Liouville, 134.

⁹⁷Lützen, Joseph Liouville., 700, 714–15.

⁹⁸Joseph Liouville, ‘Théorèmes de géométrie par M. Michael Roberts’, Journal de mathématiques pures et appliquées, 10 (1845), 466–68 (466). ‘Ces théorèmes très intéressants sont surtout relatifs aux lignes géodésiques et aux lignes de courbure que l'on peut tracer sur la surface d'un ellipsoïde à trois axes inégaux … M. Michael Roberts démontre ces théorèmes d'une manière très simple’.

⁹⁹Lützen, Joseph Liouville, 714–15.