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Practical mathematicians and mathematical practice in later seventeenth-century London

  • PHILIP BEELEY (a1)

Abstract

Mathematical practitioners in seventeenth-century London formed a cohesive knowledge community that intersected closely with instrument-makers, printers and booksellers. Many wrote books for an increasingly numerate metropolitan market on topics covering a wide range of mathematical disciplines, ranging from algebra to arithmetic, from merchants’ accounts to the art of surveying. They were also teachers of mathematics like John Kersey or Euclid Speidell who would use their own rooms or the premises of instrument-makers for instruction. There was a high degree of interdependency even beyond their immediate milieu. Authors would cite not only each other, but also practitioners of other professions, especially those artisans with whom they collaborated closely. Practical mathematical books effectively served as an advertising medium for the increasingly self-conscious members of a new emerging professional class. Contemporaries would talk explicitly of ‘the London mathematicians’ in distinction to their academic counterparts at Oxford or Cambridge. The article takes a closer look at this metropolitan knowledge culture during the second half of the century, considering its locations, its meeting places and the mathematical clubs which helped forge the identity of its practitioners. It discusses their backgrounds, teaching practices and relations to the London book trade, which supplied inexpensive practical mathematical books to a seemingly insatiable public.

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Copyright

Footnotes

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The author should like to thank Rebekah Higgitt and Jim Bennett, together with two anonymous referees, for their helpful comments, which have contributed considerably to improving this article. Additionally, he should like to thank the following persons and institutions for granting permission to reproduce images: the Bodleian Libraries, University of Oxford (Figures 1–3); the History of Science Museum, University of Oxford (Figure 5, image taken by the author); and the British Library Board (Figure 6).

All dates in this article are given in Old Style, i.e. according to the Julian calendar used in England until 1752, with the beginning of the new year falling on Lady Day (25 March).

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References

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1 Newton, John, Cosmographia, or a View of the Terrestrial and Coelestial Globes, London: Thomas Passinger, 1679, sig. A4r.

2 Martindale, Adam, The Country-Survey-Book: or Land-Meters Vade-Mecum, London: A.G. and J.P. for R. Clavel, 1682, preface. Martindale describes himself on the title page as ‘A Friend to Mathematical Learning’.

3 In contrast, a considerable amount of work has been done on the practice of mathematics during the Elizabethan and early Stuart period. See, for example, Zetterberg, J. Peter, ‘The mistaking of “the mathematicks” for magic in Tudor and Stuart England’, Sixteenth Century Journal (1980) 11, pp. 8397; Feingold, Mordechai, The Mathematicians’ Apprenticeship: Science, Universities and Society in England, 1560–1640, Cambridge: Cambridge University Press 1984, pp. 166189; Bennett, Jim, ‘Geometry and surveying in early seventeenth-century England’, Annals of Science (1991) 48, pp. 345354; Hester Higton, ‘Elias Allen and the role of instruments in shaping the mathematical culture of seventeenth-century England’, unpublished PhD thesis, Cambridge, 1996; Johnston, Stephen, ‘Mathematical practitioners and instruments in Elizabethan England’, Annals of Science (1991) 48, pp. 319344; Popper, Nicholas, ‘The English Polydaedali: how Gabriel Harvey read late Tudor London’, Journal of the History of Ideas (2005) 66, pp. 351381; Harkness, Deborah E., The Jewel House: Elizabethan London and the Scientific Revolution, New Haven, CT and London: Yale University Press 2007, pp. 97141; Hill, Katherine, ‘“Juglers or Schollers?” Negotiating the role of a mathematical practitioner’, BJHS (1998) 31, pp. 253274.

4 See Ward, John, Compendium of Algebra, London: printed for the author, 1695, title page: ‘John Ward, Phylomath. Heretofore General Gauger in the Revenue of Excise’.

5 Ward, op. cit. (4), sig. A2v. See also Martindale, op. cit. (2), preface, who describes his aims similarly: ‘I have therefore made my Book so little, that the Price can neither much empty the Pocket, nor the Bulk overfill it. And yet so plain, that I doubt not to be understood by very ordinary Capacities’.

6 Ward, John, Compendium of Algebra, 2nd edn, London: John Taylor, 1698, title page.

7 See Baker, Humphrey, The Well Springe of Sciences, 2nd edn [London: Thomas Purfoote], 1574, to the reader. Baker talks of a proliferation of foreign mathematicians that ‘haue of late painted the corners and postes in euery place within this Citie, with their peeuishe billes, makinge promise, and bearinge men in hande that they coulde teache the summe of that Science in breife Methode and compendious rules, suche as before their arriuall, haue not bene taught within this realme’ (sig. A6v). This work was reprinted numerous times up to 1659.

8 See Cuttica, Cesare, ‘Sir Francis Kynaston: the importance of the “nation” for a 17th-century English Royalist’, History of European Ideas (2006) 32, pp. 139161; Webster, Charles, The Great Instauration: Science, Medicine and Reform 1626–1660, London: Duckworth 1975, p. 218; Lee, Patricia-Ann, ‘Some English academies: an experiment in the education of Renaissance gentlemen’, History of Education Quarterly (1970) 10, pp. 273286, esp. 279–280.

9 Speidell, John, A Geometricall Extraction, or a Compendious Collection of the Chiefe and choyse Problemes, London: for Edward Allde, 1616, sig. A4r: ‘Having for these tenne yeares space, bene a professor of the Mathematickes in this Cittie, during which time, I have instructed many Gentlemen and others (in Arithmeticke, Geometrie, Astronomy)’. On Speidell see Aubrey, John, Brief Lives with an Apparatus for the Lives of Our English Mathematical Writers, ed. Bennett, Kate, 2 vols., continuously paginated, Oxford: Oxford University Press, 2015, pp. 752, 17161717.

10 Speidell, op. cit. (9), sig. A4r.

11 Speidell, op. cit. (9), sig. A4v. On Thompson and Allen see Taylor, Eva Germaine Rimington, The Mathematical Practitioners of Tudor & Stuart England, Cambridge: at the University Press, 1968, pp. 6062; Higton, Hester, ‘Portrait of an instrument-maker: Wenceslaus Hollar's engraving of Elias Allen’, BJHS (2004) 37, pp. 147166, esp. 154–157.

12 Speidell, John, An Arithmeticall Extraction or a collection of diuers questions with their answers, London: Elizabeth Allde, 1628, sig. A4r.

13 See Kersey, John, The Elements of that Mathematical Art commonly Called Algebra, two parts, London: William Godbid for Thomas Passinger 1673–1674, sig. b1r.

14 See Wingate, Edmund, The Construction and Vse of the Line of Proportion, London: John Dawson, 1628. See also Feingold, Mordechai, ‘Gresham College and London practitioners: the nature of the English mathematical community’, in Ames-Lewis, Francis (ed.), Sir Thomas Gresham and Gresham College: Studies in the Intellectual History of London in the Sixteenth and Seventeenth Centuries, Aldershot: Ashgate, 1999, pp. 174188, esp. 182–183; Bennett, Jim, ‘Early modern mathematical instruments’, Isis (2011) 102, pp. 697705, esp. 701–702.

15 Wingate, Edmund, Mr Wingate's Arithmetick, containing a perfect method for the knowledge and practice of Common Arithmetick, 3rd edn, ed. Kersey, John, London: for Philemon Stephens, 1658, sig. A3v–A4r.

16 Kersey, op. cit. (13), sig. b3r. The encouragement could only have resulted from the 1650 edition of the Arithmetique. In the 1658 edition Kersey purposely left out his chapter on algebra, because already at that time he intended ‘to frame a familiar Introduction to that mysterious Art, in a distinct Treatise’. See Wingate, op. cit. (15), sig. A5v.

17 Wingate, Edmund, Arithmetique made easie, or, a perfect Methode for the true knowledge and practice of Natural Arithmetique, 2nd edn, ed. Kersey, John, London: J. Flesher for Philemon Stephens, 1650, pp. 462464.

18 High-quality paper had to be imported to England throughout the seventeenth century, making it the most substantial element in the capital costs of publishing a book. See Mandelbrote, Giles, ‘Workplaces and living spaces: London book trade inventories of the late seventeenth century’, in Myers, Robin, Harris, Michael and Mandelbrote, Giles (eds.), The London Book Trade: Topographies of Print in the Metropolis from the Seventeenth Century, London: The British Library, 2003, pp. 2143, 36.

19 Kersey, op. cit. (13), sig. b3r. See Beeley, Philip, ‘To the publike advancement: John Collins and the promotion of mathematical knowledge in Restoration England’, BSHM Bulletin (2017) 32, pp. 6174.

20 Collins provides the most complete account of his early life himself in his preface to An Introduction to Merchants-Accompts, 3rd edn, London: William Godbid for Robert Horne, 1674, sig. B1r–B2r. See also Taylor, op. cit. (11), p. 94.

21 Collins, op. cit. (20), sig. B1v.

22 Collins, op. cit. (20), sig. B2r.

23 Collins put the case for Dary in his letter to John Frederick, president of Christ's Hospital, of 24 June 1673, Cambridge University Library MS Add. 9597/13/5, f. 82r–av; Rigaud, Stephen Jordan (ed.), Correspondence of Scientific Men of the Seventeenth Century, 2 vols., Oxford: Oxford University Press 1841, vol. 21, pp. 204206. See also The Diary of Robert Hooke, 1672–1680, ed. Robinson, Henry W. and Adams, Walter, London: Taylor & Francis, 1935, pp. 39, 48; Willmoth, Frances, Sir Jonas Moore: Practical Mathematics and Restoration Science, Woodbridge: Boydell Press, 1993, p. 196. On Dary see Taylor, op. cit. (11), p. 94.

24 On Pepys's involvement with the Mathematical School at Christ's Hospital see Iliffe, Rob, ‘Pepys and the New Science’, in Lincoln, Margarette (ed.), Samuel Pepys: Plague, Fire, Revolution, London: Thames & Hudson, 2015, pp. 196203; Willmoth, op. cit. (23), pp. 195–207.

25 Mayne, John, Socius Mercatorius: or the Merchant's Companion, London: W[illiam] G[odbid] for N[athanial] Crouch, 1674, p. [208].

26 Mayne, op. cit. (25), sig. A3r–v.

27 See Mayne, John, Arithmetick: Vulgar, Decimal, & Algebraical, London: for J.A., 1675, title page. The tract itself is contained on pp. 145–206 and bears a different title: Stereometry: or, A New and the most Practical Way of Gauging Tunns in the form of a Prismoid & Cylindroid, London: William Godbid for Nathanial Crouch, 1673.

28 Playford, John, Vade mecum, or the Necessary Companion, 2nd edn, London: A[rthur] G[odbid] and J[ohn] P[layford] for T. Passinger, 1680. The Companion for Excise-Men, paginated separately, is bound in at the end of the Vade mecum. The combination of the two texts was facilitated by the fact that they were produced by the same printers.

29 Willmoth, op. cit. (23), p. 147.

30 John Collins to James Gregory, 1671, Cambridge University Library MS Add. 9597/13/6, f. 113r–114v; Rigaud, op. cit. (23), pp. 195–201, 198.

31 See, for example, Moore, Jonas, Arithmetick in Four Books, 3rd edn, London: R.H. for Obadiah Balgrave, 1688, epistle dedicatory to James, Duke of York, sig. A4v: ‘To you, therefore, Illustrious Sir, (whose Word next to His Sacred Majesty, can only Patronize and Advance the Mathematicks and Mathematicians) I … Dedicate these my Labours’. See also Willmoth, op. cit. (23), pp. 130, 134, 151.

32 Dary, Michael, The Complete Gauger. In two Parts. Theoretical and Practical, London: for Robert Horne and Nathanael Ponder, 1678, title page.

33 Mayne, op. cit. (25), pp. 148, 153, 191–192, 198. Like many other late Elizabethan mathematicians, Oughtred, who had studied at King's College, Cambridge, combined the theoretical study of mathematics with the construction of mathematical and astronomical instruments. See Feingold, op. cit. (3), p. 81; Bennett, op. cit. (14), p. 702.

34 Mayne, op. cit. (27), p. 111: ‘That excellent Accomptant Mr. Collins, in a Sheet printed Anno 1665. hath taught a more exact way of Equation’.

35 Speidell, John, An Arithmeticall Extraction: or, a collection of eight hundred Questions with their Answers, 2nd edn, ed. Speidell, Euclid, London: H.C. for Philip Lea, 1686, sig. A2r–A3r.

36 Speidell, Euclid, Logarithmotechnia: or, the Making of Numbers called Logarithms, London: Henry Clark for the author, 1688, sig. A2v. In this tract, Speidell sought to present logarithms in an easy yet certain way and to adapt geometrical figures to them. He was attacked harshly by Hooke in his diary, who claimed that he had plagiarized Nicholas Mercator's (c.1620–1687) eponymous book ‘but understand not what he writes’. See Gunter, Robert Theodore, Early Science in Oxford, vol. 10, Oxford: for the author, 1935, p. 103.

37 Speidell, op. cit. (36), pp. 1–3.

38 Mayne, op. cit. (27), p. 80.

39 Speidell, op. cit. (36), p. [51].

40 See John Collins to Francis Vernon, 7 February 1670/1671, Cambridge University Library MS Add. 9597/13/5, f. 70r–71v; Rigaud, op. cit. (23), vol. 1, pp. 139–141, esp. 139; and Collins to Vernon, 14 December 1671, Cambridge University Library MS Add. 9597/13/5, f. 64r–65v; Rigaud, op. cit. (23), vol. 1, pp. 176–179, esp. 177.

41 Speidell, op. cit. (35), sig. A5r–v: ‘I have communicated to my loving Friend Mr. Reeve Williams, Professor of Mathematicks in London, who hath lately done into English out of the French, D'Chales Euclid, and performed the same well … and delightful to the Readers thereof; this having Uses subjoined to each Proposition, which was not to any English one before’. See also de Chales, Claude-François Milliet (ed.), The Elements of Euclid, Explained and Demonstrated in a New and most easie Method (tr. Williams, Reeve), London: for Philip Lea, 1685.

42 de Chales, Claude-François Milliet (ed.), The Elements of Euclid Rxplain'd, in a new, but most easie method (tr. Hallifax, William (?)), Oxford: Leonard Lichfield, 1685.

43 Newton, John, The Art of Practical Gauging, London: for Dixy Page, 1669, sig. A3r–v.

44 Collins, John, Geometricall Dyalling Performed by a Line of Chords onely, or by the Plain Scale, London: Thomas Johnson for Francis Cossinet, 1659, sig. A3r–v. See Feingold, op. cit. (14), p, 184, who points out that Gresham professors were akin to their counterparts in the universities in their perception of the relationship between theory and practice. In their writings they had in mind an informed readership and not the ‘vulgar’.

45 Collins, op. cit. (44), pp. 1–2.

46 Collins, op. cit. (44), sig. A4v. See Eagleton, Catherine and Jardine, Boris, ‘Collections and projections: Henry Sutton's paper instruments’, Journal of the History of Collections (2005) 17, pp. 113.

47 Collins, John, The Sector on a Quadrant, London: London: J.M. for George Hurlock et al. , 1659, preface. Despite its length, the book was completed before the instrument, leading to a certain divergence between the two set out on a page of errata (sig. a4v). See Eagleton and Jardine, op. cit. (46), p. 4.

48 John Collins to John Wallis, 28 February 1665/1666, in Beeley, Philip and Scriba, Christoph J. (eds.), The Correspondence of John Wallis (1616–1703), 4 vols., Oxford University Press 2003–2014, vol. 2, pp. 191194.

49 John Collins to John Pell, 9 April 1667, Cambridge University Library MS Add. 9597/13/5, f. 89v–90r; Rigaud, op. cit. (23), vol. 1, pp. 125–129, 125. Similarly, Elias Allen's workshop in the Strand was a general meeting place for members of London's mathematical community and served also as a post office for letters exchanged between scholars. See Higton, op. cit. (11), p. 155.

50 British Library Add. MS 4279, f. 273r. See Webster, op. cit. (8), p. 91. The intention of the meeting on this occasion was evidently to conduct astronomical observations. Moorfields also served at this time as a book market. See Mandelbrote, op. cit. (18), pp. 30–31. On Thompson and Stirrup see the advertisement at the end of Stirrup, Thomas, Horometria: or the Compleat Diallist, 2nd edn, London: R. & W. Leybourne for Thomas Pierrepont, 1652, where it is stated, ‘All the worke of this Book is performed either Geometrically or Instrumentally … if any be desirous to have either Scale, Sector, Quadrant, or any other Mathematicall Instrument whatsoever, they may be furnished by Master Anthony Thompson in Hosier lane neer Smithfield’ (sig. a2v).

51 John Collins to John Wallis, 21 March 1670/1671, in Beeley and Scriba, op. cit. (48), vol. 3, pp. 435–439, 437 (apparatus).

52 See Bennett, op. cit. (14), p. 703. Biagioli, Mario, ‘The social status of Italian mathematicians, 1450–1600’, History of Science (1989) 27, pp. 4195, 43, points out that in some Italian cities like Florence abacists and land surveyors had their own guilds, while sometimes they were grouped together with masons or other elementary-level teachers.

53 See Crawforth, Michael A., ‘Instrument makers in the London guilds’, Annals of Science (1987) 44, pp. 319377, esp. 328–329. Thus Henry Sutton and William Sutton both belonged to the Guild of Joiners, while Elias Allen attached himself to the Clockmakers’ Company soon after its creation in 1631, eventually becoming the master of this guild. See Eagleton and Jardine, op. cit. (46), pp. 2–3; Higton, op. cit. (3), pp. 74–77; and Higton, op. cit. (11), pp. 155–156.

54 British Library Add. MS 4398, f. 147r. On Seller see Taylor, op. cit. (11), pp. 108–111.

55 Waller, Richard (ed.), The Posthumous Works of Robert Hooke, London: Samuel Smith and Benjamin Walford, 1705, p. 457.

56 Hearing from him of Collins's illness, Wallis wrote to John Aubrey on 17 September 1683, Oxford, Bodleian Library MS Aubrey 13, f. 243r–v: ‘The good character you give him, I concur with you in it: And own the progress of Mathematick Learning to owe much to his Industry therein’.

57 See the motion passed by the Royal Society on 5 November 1667, as recorded in Birch, Thomas, The History of the Royal Society of London for Improving of Natural Knowledge, 4 vols., London: for A. Millar, 1756–1757, vol. 2, p. 206: ‘Dr. Wilkins moved, that Mr. Collins might be declared exempt from the payment of admission-money and the weekly payments, he having but a small revenue, and being capable and willing to do the society very good service. The council declared him exempt willingly.’ See also Hunter, Michael, Science and Society in Restoration England, Cambridge: Cambridge University Press 1981, pp. 7273.

58 Gregory's election was on 11 June 1668. See Hunter, Michael, The Royal Society and Its Fellows, 1660–1700: The Morphology of an Early Scientific Institution, 2nd edn, London: British Society for the History of Science, 1994, p. 184. Collins was elected eight months previously, on 17 October 1667 (ibid., p. 178). See also Beeley, op. cit. (19), p. 66.

59 Martindale, op. cit. (2), Mr Collins to the Reader.

60 John Collins to Edward Bernard, 16 March 1670/1671, in Beeley and Scriba, op. cit. (48), vol. 3, pp. 431–435, 431: ‘I will not goe about to detaine you with a Discourse to intimate how happy it is for a Man inferior Subselli, and a Non-Academick to have the honour of the Acquaintance with the learned, such as you are’.

61 Thomas Smith to Edward Bernard, 26 October 1676, Oxford, Bodleian Library MS Smith 57, pp. 31–32: ‘Meeting very lately with your Brother Mathematician Mr Collins & acquainting him with those books your Letter mentioned, hee earnestly desired mee to write to you’.

62 See Beeley, Philip, ‘The progress of mathematick learning: John Wallis as historian of mathematics’, in Wardhaugh, Benjamin (ed.), The History of the History of Mathematics, Oxford: Peter Lang, 2012, pp. 930, esp. 11–14.

63 John Collins to James Gregory, early March? 1668, Cambridge University Library MS Add. 9597/13/6, f. 92r–93v; Rigaud, op. cit. (23), vol. 2, pp. 174–179, 175.

64 Although some academically trained mathematicians such as Robert Recorde (c.1512–1558), John Dee (1527–1609) or William Oughtred (1575–1660) covered practical topics such as dialling in their writings, this did not generally translate into positive appreciation of practitioners themselves. Hill, op. cit. (3), pp. 257–260, documents how Oughtred attacked the teaching methods, ability and authority of Richard Delamain as a mathematical practitioner. See further Higton, op. cit. (3), pp. 28–30. Similarly, Ash, Eric H., Power, Knowledge, and Expertise in Elizabethan England, Baltimore and London: Johns Hopkins University Press, 2004, p. 16, has argued that practical mathematicians in the sixteenth century were better off portraying themselves not as experienced practitioners, but as masters of the theoretical principles that underlay that practice. See also Bennett, Jim, ‘Geometry in context in the sixteenth century: the view from the museum’, Early Science and Medicine (2002) 7, pp. 214230, esp. 222–225; Popper, op. cit. (3), p. 371.

65 Gregory, David, ‘Scheme for the teaching of mathematics’, in Tanner, Joseph Robson (ed.), Private Correspondence and Miscellaneous Papers of Samuel Pepys, 1679–1703, London: G. Bell, 1926, pp. 9194.

66 Barrow, Isaac, Lectiones XVIII, Cantabrigiae in Scholis publicis habitae; in quibus opticorum phaenomenωn genuinae rationes investigantur, ac exponuntur, London: W. Godbid, 1669, p. 6.

67 John Collins to John Beale, 20 August 1672, Cambridge University Library MS Add. 9597/13/5, f. 83r–85av; Rigaud, op. cit. (23), pp. 195–204.

The author should like to thank Rebekah Higgitt and Jim Bennett, together with two anonymous referees, for their helpful comments, which have contributed considerably to improving this article. Additionally, he should like to thank the following persons and institutions for granting permission to reproduce images: the Bodleian Libraries, University of Oxford (Figures 1–3); the History of Science Museum, University of Oxford (Figure 5, image taken by the author); and the British Library Board (Figure 6).

All dates in this article are given in Old Style, i.e. according to the Julian calendar used in England until 1752, with the beginning of the new year falling on Lady Day (25 March).

Practical mathematicians and mathematical practice in later seventeenth-century London

  • PHILIP BEELEY (a1)

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