Abstract
The category of “military mathematical practitioners” consists of those active soldiers and engineers who consciously broadcast their use of mathematical methods to achieve their goals in warfare. These are but a subset of mathematical practitioners more broadly, and they existed on a continuum from the practical to the theoretical, with each demonstrating a mix of the two. In this military case, I investigate the concerns in gunnery and fortification of Thomas Harriot, William Bourne, Thomas Digges, and Edmund Parker—an early-modern scientist, noted author on the mathematical arts, military administrator and author, and a polymath soldier and gunner, respectively—each of whom adopted a certain “mathematical posture” to distinguish themselves in these pursuits. Framed by the work of E.G.R. Taylor, Edgar Zilsel, and Erving Goffman, the examination of how mathematics were actually used by these military mathematical practitioners (which should not be conflated with their actual utility, which is shown here to be often quite lacking) demonstrates the relationship, often a gulf, between theory and practice in one area of the mathematics in later sixteenth-century England. The context, audience, method of development, instruments, and mode of presentation (print vs. manuscript vs. rhetoric) of the mathematical methods applied to warfare also provide evidence of how mathematics was both used and understood as useful in this period to build a self-image of competence and professionalism.
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Notes
- 1.
Carew to Mountjoy, 28 May 1602 [J.S. Brewer and William Bullen (eds.), Calendar of the Carew Manuscripts Preserved in the Archiepiscopal Library at Lambeth [hereafter, CCML], (London: Longmans, Green, Reader, & Dyer, 1867–73), vol. 4 (1601–1603), 239 (no. 241)].
- 2.
H.J. Todd, A Catalogue of the Archiepiscopal Manuscripts in the Library at Lambeth Palace (London: Law and Gilbert, 1812), 123, referring to London, Lambeth Palace Library, ms 615, fol. 594; Mountjoy to Carew, 3 May 1602 [CCML, vol. 4 (1601–1603), 233–4 (no. 234)]. See also C. Falls, Elizabeth’s Irish Wars (London: Methuen, 1950), 324–28.
- 3.
Carew to Mountjoy, 1 June 1602 [CCML, vol. 4 (1601–1603), 242 (no. 242)].
- 4.
Thomas Stafford, Pacata Hibernia, Ireland Appeased and Reduced (London: A[ugustine] M[athewes], 1633], 45 quoted in W.A. McComish, “The Survival of the Irish Castle in an Age of Cannon,” The Irish Sword 9 (1969): 16–21 at 18.
- 5.
See Mountjoy to Carew, 9 June and 29 July 1602 [CCML, vol. 4 (1601–1603), 245 (no. 248) and 285 (no. 274)].
- 6.
London, Lambeth Palace, ms 280 [hereafter, simply “Parker, Notebook”].
- 7.
E.G.R. Taylor, The Mathematical Practitioners of Tudor and Stuart England (Cambridge: Institute of Navigation, 1954).
- 8.
See Edgar Zilsel, The Social Origins of Modern Science, ed., Diederick Raven et al., Boston Studies in the Philosophy of Science 200 (Dordrecht: Springer, 2000); A.C. Keller, “Zilsel, the Artisans, and the Idea of Progress in the Renaissance,” Journal of the History of Ideas 11.2 (1950): 235–240; A. Rupert Hall, “The Scholar and the Craftsman in the Scientific Revolution,” in Critical Problems in the History of Science, ed. Marshall Clagett (Madison: University of Wisconsin Press, 1959), 3–23; and most recently, Pamela O. Long, Artisan/Practitioners and the Rise of the New Sciences, 1400–1600 (Corvallis, Ore.: Oregon State University Press, 2011).
- 9.
See Robin Elaine Rider, “Early Modern Mathematics in Print,” in Non-Verbal Communication in Science Prior to 1900, ed. Renato G. Mazzolini (Firenze: L.S. Olschki, 1993), 91–113; Katie Taylor, “Reconstructing Vernacular Mathematics: the Case of Thomas Hood’s Sector,” Early Science and Medicine 18.1-2 (2013): 153–179; and Kathryn James, “Reading Numbers in Early Modern England,” BSHM Bulletin 26.1 (2011): 1–16.
- 10.
Henry J. Webb, Elizabethan Military Science: The Books and the Practice (Madison: University of Wisconsin Press, 1965), 145. Taylor gets closer, saying more circumspectly that “heights and distances, maps and plans, were decidedly relevant in warfare carried on by artillery” (Mathematical Practitioners, 8), but she, too, allows Cold War conceptions of scientific warfare to color her later analysis.
- 11.
John Dee, The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara (London: Iohn Daye, 1570), sig. d.i.v and ‘Groundplat’. Dee rather typically for the Praeface failed to develop exactly how guns were mathematical as part of menandrie and they and their projectiles were interestingly not part of his conception of the study of motion.
- 12.
In general, see Webb, Elizabethan Military Science, and on this last area, see William T.Lynch, “Surveying and the Cromwellian Reconquest of Ireland,” in Instrumental in War: Science, Research, and Instruments Between Knowledge and the World, ed. Steven A. Walton, History of Warfare 28 (Leiden, 2005), 47–84.
- 13.
For the later story, see Brett D. Steele, “Muskets and Pendulums: Benjamin Robins, Leonhard Euler, and the Ballistics Revolution,” Technology and Culture 35.2 (1994): 348–82; Janis Langins, Conserving the Enlightenment: French Military Engineering from Vauban to the Revolution (Cambridge, Mass.: MIT Press, 2004); and Ken Alder, Engineering the Revolution: Arms and Enlightenment in France, 1763–1815 (Princeton: Princeton University Press, 1997).
- 14.
For the metaphor of the “clockwork siege” ascribed to the later seventeenth-century French siege engineer Vauban , see Jamel Ostwald, “Like Clockwork? Clausewitzian Friction and the Scientific Siege in the Age of Vauban,” in Instrumental in War, ed. Walton, 85–117.
- 15.
Matthias Schemmel, “Thomas Harriot as an English Galileo: the Force of Shared Knowledge in Early Modern Mechanics,” Bulletin of the Society for Renaissance Studies 21.1 (2003): 1–10 and ibid., The English Galileo: Thomas Harriot’s Work on Motion as an Example of Preclassical Mechanics (Dordrecht: Springer, 2008). See also Matteo Valleriani, Galileo Engineer (Dordrecht: Springer, 2010); and Jürgen Renn and Matteo Valleriani, “Galileo and the Challenge of the Arsenal,” Nuncius 16.2 (2001): 481–503.
- 16.
Erving Goffman , The Presentation of Self in Everyday Life (Garden City, NY: Doubleday, 1959), 18: “Social life is described as a multi-staged drama in which people act out different roles in different social arenas depending on the nature of the situation, their particular roles in it, and the makeup of the audience.”
- 17.
See G.L’E. Turner, “Bourne, William (c.1535–1582),” in Oxford Dictionary of National Biography, 3rd edition (Oxford: Oxford University Press, 2004) [hereafter ‘Oxford DNB’] and the older but still useful, E.G.R. Taylor, “William Bourne: A Chapter in Tudor Geography,” The Geographical Journal 72.4 (1928): 329–339. Turner claims without clear evidence that Bourne learned gunnery from Winter.
- 18.
The canonical Maurice J.D. Cockle, A Bibliography of Military Books up to 1642, 2nd edition (London: Holland Press, 1957), no. 35 and Webb, Elizabethan Military Science, both missed the first edition. The unique 1578 copy, held at the Royal Artillery Institution (STC 3419.7), is set with different type than the 1587 copies (italic vs. black letter) and has hand-drawn or pasted-in illustrations, although the layout and catch-words agree in both editions. The Stationer’s Company transcript records the Arte as licensed to Henry Bynneman on 22 July 1578 and Borne noted in his An Almanacke and Prognostication for x. yeeres (1581), that his “booke called the Art of Shooting in great Ordenaunce” was already in print [E.G.R. Taylor (ed.), A Regiment for the Sea and Other Writings on Navigation by William Bourne, Hakluyt Society 2nd ser. 121 (London: Cambridge University Press, 1963), 328]. Further, John Dee had a copy of the 1578 edition in his library, possibly a gift from Bourne himself [J. Roberts and A.G. Watson (eds.), John Dee’s Library Catalogue (London: Bibliographical Society, 1990), 37].
- 19.
London, British Library, ms Sloane 3651. See also Taylor, Regiment for the Sea, 441–2. The 1587 edition is virtually identical to the 1578 edition, and they are expanded from the manuscript versions, which also provided some of the substance for Bourne’s Inventions and Devices and his Treasure for Travellers (both also 1578).
- 20.
Bourne, Arte of Shooting in Great Ordnaunce (1578; London: [Thomas Dawson] for Thomas Woodcocke, 1587), 40 and ms Sloane 3651, fol.25r.
- 21.
Bourne, Arte of Shooting, 86 and ms Sloane 3651, fol. 36v.
- 22.
The distinction is much like that between Leonardo , whose letter to the Sforzas enumerated what practical works he could personally make for them (canals, engines, painting), versus Galileo’s Le operazioni del compasso geometrico et militare (Padua, 1606), which advertises to purchasers what could be done with his military compass.
- 23.
Thomas Digges, Alaæ seu scalæ mathematicæ (London: Thomas Marsh, 1573), on astronomy; Leonard and Thomas Digges, A Geometrical Practise, named Pantometria (London: Henrie Bynneman, 1571), on plane and solid geometry and reissued in 1591 with a new section of artillery definitions; and Leonard and Thomas Digges, An Arithmeticall Militare Treatise, named Stratioticos (London: Henrie Bynneman, 1579), expanded and reissued in 1590.
- 24.
Robert Norton, Of the Art of Great Artillery (London: Edw. Allde for Iohn Tap, 1624), sig. A2r-v. See in general, Stephen Johnston, “Digges, Thomas (c.1546–1595),” in Oxford DNB. Of particular relevance here is A.R. Hall, Ballistics in the Seventeenth Century: a Study in the Relations of Science and War with Reference Principally to England (Cambridge: Cambridge University Press, 1952), 43–49, which generally overestimates Digges’ importance to the field; Stephen Johnston, “Making Mathematical Practice: Gentlemen, Practitioners and Artisans in Elizabethan England,” Ph.D. dissertation, Cambridge University, 1994; Johnston, “Like Father, like Son? John Dee, Thomas Digges and the Identity of the Mathematician,” in John Dee: Interdisciplinary Studies in English Renaissance Thought, ed. Stephen Clucas, International Archives of the History of Ideas 193 (Dordrecht: Springer, 2006), 65–84. Eric H. Ash, ”A Perfect and an Absolute Work: Expertise, Authority, and the Rebuilding of Dover Harbor, 1579–1583,” Technology and Culture 41.2 (2000): 239–268, notes Digges’ extensive service in this particular aspect of his life, but understandably does not examine the military connections or his foreign service.
- 25.
Daniel Santbech, “De Artificio Eiaculandi Sphaeras Tormentarias” in his Problematum astronomicorum et geometricorum sectiones septem (Basel: Henrichum Petri et Petrum Pernam, 1561); Girolamo Ruscelli, Precetti della militia moderna… tutta l’arte del Bombardiero (Venice: Marchiò Sessa, 1568); Nicolò Tartaglia, Nova Scientia (Venice: Stephano da Sabio, 1537) and Quesiti et Invenzioni Diverse (Venice: Venturino Ruffinelli, 1546 and 1554) which were later epitomized by Cyprian Lucar as Three Bookes of Colloquies Concerning the Arte of Shooting in Great and Small Peeces of Artillerie (London: Thomas Dawson for Iohn Harrison, 1588) and A Treatise Named Lucarsolace (London: Richard Field for Iohn Harrison, 1590). See also Matteo Valleriani, Metallurgy, Ballistics and Epistemic Instruments the Nova scientia of Nicolò Tartaglia (Berlin: Edition Open Access, 2013) and Raffaele Pisano and Danilo Capecchi, Tartaglia’s Science of Weights and Mechanics in the Sixteenth Century: Selections from Quesiti et inventioni diverse: Books VII–VIII, History of Mechanism and Machine Science 28 (Dordrecht: Springer, 2016), 39–86 on ballistics and fortificaiton.
- 26.
Stratioticos (1590), 349–60; quote at 361.
- 27.
Pantometria (1591), title page and Stratioticos (1579), sig. [&.iv], respectively, italics in the original. Digges died in 1595, but Thomas Smith returned to Digges’ questions in his The Arte of Gunnerie… by Arithmeticke Skill to be Accomplished (London: Richard Field for William Ponsonby, 1600); he had little success in furthering answers, focusing instead on less complex things that could, as his subtitle announced, “by arithmeticke skill … be accomplished.”
- 28.
Even E.G.R. Taylor begins her book noting that “at the opening of the eighteenth century the many technical difficulties inherent in making and using accurate and reliable instruments and apparatus, not to speak of finding correct theoretical formulae, were unsolved and insoluble until some further advance… had been made” (Mathematical Practitioners [note 7, above], 3). The full description of a cannonball’s flight is not a closed analytic function, but must be determined experimentally. The earliest effective mathematical ballistic gunnery handbooks date to the late nineteenth century: James M. Ingalls, Exterior Ballistics (Fort Monroe, Va.: U.S. Artillery School, 1885) and Exterior Ballistics in Plane of Fire (New York: D. Van Nostrand, 1886); Lawrence L. Bruff, Exterior Ballistics, Gun Construction, and U.S. Seacoast Guns (West Point, NY: United States Military Academy Press, 1892).
- 29.
On boundary objects, see Susan Leigh Star, “The Structure of Ill-Structured Solutions: Boundary Objects and Heterogeneous Distributed Problem Solving,” in Distributed Artificial Intelligence, ed. Les Gasser and N. Huhns (London: Pitman, 1989), II: 37–54 and Star and James Griesemer, “Institutional Ecology, ‘Translations’ and Boundary Objects: Amateurs and Professionals in Berkeley’s Museum of Vertebrate Zoology, 1907–39,” Social Studies of Science 19.3 (1989): 387–420 at 393.
- 30.
Talor, Mathematical Practitioners; Horst de la Croix, “The Literature on Fortification in Renaissance Italy,” Technology and Culture 4.1 (1963): 30–50; and Barbara Donagan, “Halcyon Days and the Literature of War: England’s Military Education before 1642,” Past & Present 147 (1995): 65–100.
- 31.
For a parallel sort of unconscious self-presentation by personal writing, see Nicholas Popper, “The English Polydaedali: How Gabriel Harvey Read Late Tudor London,” Journal of the History of Ideas 66.3 (2005): 351–381.
- 32.
J.R. Hale, Renaissance Fortification: Art or Engineering? (London: Thames and Hudson, 1977) provides the best brief introduction. For the early history, see Gianni Perbellini, The Fortress of Nicosia, Prototype of European Renaissance Military Architecture (Nicosia: Anastasios G. Leventis Foundation, 1994) and Pietro C Marani, Disegni di fortificazioni da Leonardo a Michelangelo (Firenze: Cantini edizioni d'arte, 1984).
- 33.
Giacomo Lanteri, Due dialoghi di M. Iacomo de’ Lanteri…: ne i quali s’introduce Messer Girolamo Cantanio… & Messer Francesco Treuisi… ‘a ragionare Del modo di disegnare le piante delle fortezze secondo Euclide; et Del modo di comporre i modelli & torre in disegno le piante delle citt’a (Venetia: Costantini, 1557). Curiously, Lanteri’s teacher of mathematics was Girolamo Cataneo , who broke from strictly symmetric geometrical constructions and allowed for irregularities due to terrain and local conditions in his own Opera nuovo di fortificare (Bresica: Gio. Battista Bozola, 1564); see Horst de la Croix, “Literature on Fortification,” 40–41.
- 34.
See Robert Corneweyle, The Maner of Fortification of Cities, Townes, Castelles and Other Places, 1559 (Richmond, Surrey: Gregg, 1972); Lynn White Jr., “Jacopo Acontio as an Engineer,” American Historical Review 72.2 (1967): 425–444; and my “State Building through Building for the State: Domestic and Foreign Expertise in Tudor Fortifications,” in Expertise and the Early Modern State, ed. Eric Ash, Osiris 25 (2010): 66–84. Aconcio’s “lost” book on fortification, originally written in Italian and possibly translated into Latin, survives in an English translation by Thomas Blundeville (another mathematical practitioner) that was independently rediscovered by Stephen Johnston (Oxford University), by me, and by Paola Giacomoni (Università di Trento); see Paola Giacomoni (ed.), Jacopo Aconcio: Trattato sulle fortificazioni, Istituto Nazionale di Studi sul Rinascimento, Studi e Testi 48 (Firenze: Leo S. Olschki, 2011). My modernized transcription will appear in the journal Fort from the Fortress Study Group in 2017.
- 35.
Others have argued that Harriot did his military work (at least the gunnery) for Walter Raleigh and hence for practical reasons; I argue otherwise in Thomas Harriot’s Ballistics and English Renaissance Warfare, Occasional Paper no. 30 (Durham: Durham Thomas Harriot Seminar, 1999).
- 36.
His fortification pages are London, British Library, ms Add. 6788, fol. 55–65. They bear a similarity to Samuel Marolois’ , Opera mathematica, ou Oeuvres mathématiques traictans de géométrie, perspective, architecture et fortification (Hagæ-Comitis: Henrici Hondii, 1613–14), so is either a late foray by Harriot (†1621), or his work prefigures Marlois.
- 37.
“Rules touching Great Ordnance” in Parker, Notebook (note 6, above), esp. fol. 32v–36.
- 38.
Wolff-Michael Roth, “Where is the Context in Contextual Word Problems?” Cognition and Instruction 14.4 (1996) 487–527.
- 39.
Richard Norwood, Fortification or Architecture Military (London: Tho. Cotes for Andrew Crooke, 1639), p. 59.
- 40.
Erving Goffman, Frame Analysis: An Essay on the Organization of Experience (New York: Harper and Row, 1974), 39.
- 41.
See for example, Digges and Digges, Stratioticos (note 23, above), ch. 1–9, pp. 1–52; Smith, The Arte of Gunnery (note 27, above), 1–8; and Robert Norton, The Gunner Shewing the Whole Practise of Artillery (London: A.M. for Humphrey Robinson, 1628), 1–30, which sets out theorems for artillery practice.
- 42.
Smith, Arte of Gunnery, 46. “Randon” is the farthest range of a shot, and describes great speed, force, or violence [OED, s.v. “Random 1a”] and hence distance. The modern meaning of random as haphazard or without exactness [OED, s.v. “Random 3”] seems to derive from the tendency of things that rush headlong with great speed and violence to loose accuracy.
- 43.
Smith, Arte of Gunnery, p. 35. “Saker” and “cannon” are proper names for different sizes of ordnance, of nominally 3½ and 7 or 8 inches bore diameter, respectively.
- 44.
The didactic style of the manuscript suggest use in the academies; see Steven A. Walton, “The Bishopsgate Artillery Garden and the First English Ordnance School,” Journal of the Ordnance Society 15 (2003): 41–51, and “Proto-Scientific Revolution or Cookbook Science? Early Gunnery Manuals in the Craft Treatise Tradition,” Ricardo Cordoba (ed.), Craft Treatises and Handbooks: The Dissemination of Technical Knowledge in the Middle Ages, De Diversis Artibus 91 (Turnhout: Brepols, 2013), 221–236.
- 45.
Oxford, Bodleian Library, ms Ashmole 343 [hereafter, “Secrets of Gunmen”], fol. 128r, emphasis added. Although the manuscript is an early seventeenth-century copy (suggestive of durability of interest in such topics), the material is clearly from the mid- to late sixteenth century.
- 46.
Digges, Stratioticos, 66.
- 47.
E.g., Susan Rose, “Mathematics and the Art of Navigation: the Advance of Scientific Seamanship in Elizabethan England,” Transactions of the Royal Historical Society 14 (2004): 175–184.
- 48.
“Secrets of Gunmen,” fol. 128r, emphasis added.
- 49.
Carew to Mountjoy, 1 June 1602 [CCML (note 1, above), vol. 4 (1601–1603), 242 (no. 242)].
- 50.
Parker, Notebook (note 6, above), fol. 19 and 25v.
- 51.
Parker, Notebook, fol. 83.
- 52.
Lucar, Three Bookes of Colloquies (note 25, above), 43–5 (bk. I, colloquy 23), which is apparently from Tartaglia’s Quesiti, as it is not in the Nuova Scienza (see Valleriani, note 25, above).
- 53.
Bourne, Arte of Shooting, 8–9.
- 54.
The following is from Parker, Notebook, fol. 11.
- 55.
Parker, Notebook, fol. 29.
- 56.
It may not be a signature as much as an ascription of invention as it is in a clear italic hand while the majority of the ms is a rather scrawling secretary hand.
- 57.
Parker, Notebook, fol. 10.
- 58.
See Steven A. Walton, “Mathematical Instruments and the Creation of the Scientific Military Gentleman,” in Instrumental in War, ed. Walton (note 12, above) 17–46.
- 59.
Harrison also published other mathematical works, such as Robert Recorde , The Pathewaie to Knowledge Containyng the First Principles of Geometrie (London: J. Kingston for Ihon Harrison, 1574), which relied heavily on illustrations.
- 60.
Smith, The Arte of Gunnery (note 27, above), facing 58.
- 61.
Kurt Petersen, Det Militære Målesystem: Kaliberstokken Og Dens Udvikling Fra 1540 Til 1850 (Lyngby, Denmark: Polyteknisk Forlag, 2005); A. Konstam, “A Gunner’s Rule from the ‘Bronze Bell’ Wreck, Tal-y-Bont,” Journal of the Ordnance Society 1 (1989): 23–26; Ruth R. Brown, “Comment on The Tal-y-Bont Gunner’s Rule,” Journal of the Ordnance Society 2 (1990): 71–2; Jeremy N. Green, “Further Information on Gunner’s Rules or Tally Sticks,” Journal of the Ordnance Society 2 (1990): 25–32; Winifred Glover, “The Spanish Armada Wrecks of Ireland,” in Excavating Ships of War, ed. Mensun Bound (Oswestry, Shrops.: Nelson, 1998), 51–63; David S. Weaver, “The English Gunner’s Caliper,” Arms Collecting 33 (1995): 111–25; Colin Martin, “De-Particularizing the Particular: Approaches to the Investigation of Well-Documented Post-Medieval Shipwrecks,” World Archaeology 32 (2001): 383–99; Alex Hildred, “The Material Culture of the Mary Rose (1545) as Fighting Vessel: The Uses of Wood,” in Artefacts from Wrecks: Dated Assemblages from the Late Middle Ages to the Industrial Revolution, ed. Mark Redknap (Oxford: Oxbow Books, 1997), 51–72; and Alexzandra Hildred, Weapons of Warre: The Armaments of the Mary Rose (Portsmouth: Mary Rose Trust, 2011), 392–407.
- 62.
Incidentally, it also suggests that gunners need not have been able to do the calculations themselves. Michael Korey, The Geometry of Power: Mathematical Instruments and Princely Mechanics around 1600 (Munich: Deutscher Kunstverlag, 2007), 19, puts it nicely by noting that a table or instrument is “less than a calculator than… an instrument for avoiding calculation,” emphasis original.
- 63.
London, Society of Antiquaries, ms 94 [hereafter, “Wright, Notes”]. As early as the late sixteenth century, gunners took the invention of instruments as a both signs of accomplishment as well as keys to employment and advance: consider the Radio Latino invented by Latino Orsini that could be used for gunnery, fortification, and surveying; the quadrant of Johann Carl , Zeugmaster and engineer of Nurenberg; or the instrument described by Thomas Bedwell in his Aurea Regula Coss, Nova Geometrica (“The Golden Algebra, a New Geometry”). For all of these, see Walton, “Mathematical Instruments and the Creation of the Scientific Military Gentleman”.
- 64.
R.S. Perinbanayagam, “How to do Self with Things,” in Beyond Goffman: Studies on Communication, Institution, and Social Interaction, ed. S.H. Riggins (Berlin: Mouton de Gruyter, 1990), 315–340.
- 65.
S.H. Riggins, “The Power of Things: the Role of Domestic Objects in the Presentation of Self,” in Riggins (ed.), Beyond Goffman, 347–49. They also then became occupational objects, which include both objects that credential for the common gunners (using wooden gunners’ rules and gauges), as well denote privilege and prestige for the princes (with their complex, compound gilt instruments in their Kunstkammern).
- 66.
A. J. Turner, “Mathematical Instruments and the Education of Gentlemen,” Annals of Science 30 (1973): 51–88; Gerard L’E. Turner, Elizabethan Instrument Makers: The Origins of the London Trade in Precision Instrument Makers (New York: Oxford University Press, 2001); and especially Jim Bennett, “Early Modern Mathematical Instruments” Isis 102.4 (2011): 697–705. Instruments would later become foundational for the mathematics itself: see for example, Hester Higton, “Does Using an Instrument Make You Mathematical? Mathematical Practitioners of the 17th Century,” Endeavour 25.1 (2001): 18–22.
- 67.
Richard W. Stewart, The English Ordnance Office, 1585–1625. A Case Study in Bureaucracy (Woodbridge: Boydell, 1996).
- 68.
Practice geometrical entrenchments, for example, seem to have been dug in the Artillery Garden outside Bishopsgate in London during the Civil War; see Steven A. Walton, “The Tower Gunners and the Artillery Company in the Artillery Garden before 1630,” Journal of the Ordnance Society 18 (2006): 53–66 at 58.
- 69.
Mario Biagioli, “The Social Status of Italian Mathematicians, 1450–1600,” History of Science 27 (1989): 41–95 and more recently, Cesare S Maffioli, “A Fruitful Exchange/Conflict: Engineers and Mathematicians in Early Modern Italy,” Annals of Science 70.2 (2013): 197–228.
- 70.
For a related study of the rise of architecture, see Anthony Gerbino and Stephen Johnston, Compass and Rule: Architecture as Mathematical Practice in England 1500–1750 (New Haven, Conn.: Yale University Press 2009).
- 71.
Fluid mechanics of projectile flight would continue to dog natural philosophers until well past the mid-eighteenth century; see Hall, Ballistics in the Seventeenth Century (note 24, above); John F. Guilmartin, Jr., “Ballistics in the Black Powder Era,” in British Naval Armaments, ed. Robert D. Smith (London: Trustees of the Royal Armouries, 1989), 73–98; and Steele, “Muskets and Pendulums” (note 13, above), and see note 28, above.
- 72.
This has been contested in modern scholarship, but it is just such scholarship that showed how forcefully seventeenth-century natural philosophers argued for the objectivity of their “facts” at the time. See Mary Poovey, A History of the Modern Fact: Problems of Knowledge in the Sciences of Wealth and Society (Chicago: University of Chicago Press, 1998) and Steven Shapin and Simon Schaffer, Leviathan and the Air-Pump: Hobbes, Boyle, and the Experimental Life (Princeton: Princeton University Press, 1985) and the literature that has flowed from their work.
- 73.
The useful term “artifact physics” is from Christopher R. Hoffman and Marcia-Anne Dobres, “Conclusion: Making Material Culture, Making Culture Material,” in The Social Dynamics of Technology: Practice, Politics, and World Views, ed. Dobres and Hoffman (Washington, DC: Smithsonian Institution Press, 1999), 209–22 at 216. See also Davis Baird, Thing Knowledge: a Philosophy of Scientific Instruments (Berkeley: University of California Press, 2004).
- 74.
See Bryan Pfaffenerger, “Worlds in the Making: Technological Activities and the Construction of Intersubjective Meanings,” in The Social Dynamics of Technology, 147–164, as well as the conclusion to that volume, esp. 213–215ƒƒ.
- 75.
Goffman, Presentation of Self (note 16, above), 13, emphasis original.
- 76.
Marcia-Anne Dobres, referring to the work of Judith McGraw and Ruth Schwartz-Cowan , among others, in her, “Technology’s Links and Chaînes: the Processural Unfolding of Technique and Technician,” in The Social Dynamics of Technology, 129. This claim was made for differences of gender, but the same argument works in reference to difference in ability based upon access to and fluency with mathematics. While Dobres is interested in the chaîne opératoire methodology of studying the transformation of raw materials into products and the meanings engendered along the way, her more general point is exactly what I am arguing here for the military mathematical practitioners: “while undertaking productive activities, individuals create and localize personal and group identities, making statements about themselves that are ‘read’ by others with whom they are interacting. Technical acts can thus be treated as a medium for defining, negotiating, and expressing personhood” (129, emphasis in original).
- 77.
The terms “matters of concern” and “matters of fact” are from Bruno Latour , “Why has Critique Run out of Steam? From Matters of Fact to Matters of Concern,” Critical Inquiry 30 (2004): 225–48, esp. 246 (and reiterated in his “From Realpolitik to Ding Politik, or How to Make Things Public,” in Making Things Public: Atmospheres of Democracy, ed. Bruno Latour and Peter Weidbel ([Cambridge, Mass,: MIT Press, 2015], 14–41), although I have inverted his assessment of current critique.
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Walton, S.A. (2017). Technologies of Pow(d)er: Military Mathematical Practitioners’ Strategies and Self-Presentation. In: Cormack, L., Walton, S., Schuster, J. (eds) Mathematical Practitioners and the Transformation of Natural Knowledge in Early Modern Europe. Studies in History and Philosophy of Science, vol 45. Springer, Cham. https://doi.org/10.1007/978-3-319-49430-2_5
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DOI: https://doi.org/10.1007/978-3-319-49430-2_5
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