The Rhetoric of Utility: Avoiding Occult Associations for Mathematics through Profitability and Pleasure

Davis Philip Hersh Reubin , “Rhetoric and mathematics”, in Nelson J. S. Megill A. McCloskey D. N. (eds), The rhetoric of the human sciences: Language and argument in scholarship and public affairs (Madison, 1987), 53–68.

Westman Robert S. , “The astronomer's role in the sixteenth century: A preliminary study”, History of science, xviii (1980), 105–47; Biagioli Mario , “The social status of Italian mathematicians, 1450–1600”, History of science, xxvii (1989), 41–95; Bennett J. A. , “The challenge of practical mathematics”, in Pumfrey Stephen Rossi Paolo L. Slawinski Maurice (eds), Science, culture and popular belief in Renaissance Europe (Manchester, 1991), 176–90; idem, “The mechanics' philosophy and the mechanical philosophy”, History of science, xxiv (1986), 1–28; Dear Peter , Discipline and experience: The mathematical way in the Scientific Revolution (Chicago and London, 1995); Gabbey Alan , “The case of mechanics: One revolution or many?”, in Lindberg David C. Westman Robert S. (eds), Reappraisals of the Scientific Revolution (Cambridge, 1990), 493–528; idem, “Between ars and philosophia naturalis: Reflections on the historiography of early modern mechanics”, in Field J. V. James Frank A. J. L. (eds), Renaissance and revolution (Cambridge, 1993), 133–46; Jardine Nicholas , The birth of history and philosophy of science: Kepler's “A defence of Tycho against Ursus” with essays on its provenance and significance (Cambridge, 1984); idem, “Epistemology of the sciences”, in Schmitt Charles B. (eds), The Cambridge history of Renaissance philosophy (Cambridge, 1988), 685–711; Johnston Stephen , “Mathematical practitioners and instruments in Elizabethan England”, Annals of science, xlviii (1991), 319–44; idem, “The identity of the mathematical practitioner in 16^th-century England”, in Hantsche Irmgard (ed.), Der ‘mathematicus’: Zur Entwicklung und Bedeutung einer neuen Berufsgruppe in der Zeit Gerhard Mercators (Bochum, 1996), 93–120; Willmoth Frances , Sir Jonas Moore: Practical mathematics and Restoration science (Woodbridge, Suffolk, 1993); Laird W. R. , “Patronage of mechanics and theories of impact in sixteenth century Italy”, in Moran Bruce (ed.), Patronage and institutions: Science, technology and medicine at the European court, 1500–1750 (Woodbridge, Suffolk, 1991), 51–66.

Johnston , op. cit. (ref. 2, 1991), focuses on how mathematical practitioners used instruments as another tool to “negotiate the character and status of the mathematicalls”.

Johnston , op. cit. (ref. 2, 1996), provides an important summary of diversity of mathematical practice.

Taylor E. G. R. , The mathematical practitioners of Tudor and Stuart England (3rd edn, Cambridge, 1968). See also Waters David , The art of navigation in England in Elizabethan and early Stuart times (London, 1958).

Bennett , op. cit. (ref. 2, 1991), and op. cit. (ref. 2, 1986), 11.

Johnston , op. cit. (ref. 2, 1991).

Grattan-Guinness Ivor (ed.), Companion encyclopedia of the history and philosophy of the mathematical sciences (2 vols, London and New York, 1994).

Henry John , “Magic and science in the sixteenth and seventeenth centuries”, in Olby R. C. Cantor G. N. Christie J. R. R. Hodge M. J. S. (eds), Companion to the history of modern science (London and New York, 1990).

Zetterberg Peter , “The mistaking of ‘the mathematicks’ for magic in Tudor and Stuart England”, Sixteenth century journal, xi (1980), 83–97, p. 83; see also Eamon William , “Technology as magic in the late Middle Ages and the Renaissance”, Janus, lxx (1983), 171–212, and Yates Frances A. , “The Hermetic tradition in Renaissance science”, in Singleton Charles S. (ed.), Ideas and ideals in the North European Renaissance: Collected essays (3 vols, Baltimore, 1967), iii, 227–46, p. 230.

Eamon , “Technology as magic” (ref. 10). See also Molland A. George , “Cornelius Agrippa's mathematical magic”, in Hay Cynthia (ed.), Mathematics from manuscript to print 1300–1600 (Oxford, 1988), 209–19.

Feingold Mordechai , “The occult tradition in the English universities of the Renaissance: A reassessment”, in Vickers Brian (ed.), Occult and scientific mentalities in the Renaissance (Cambridge, 1984), 73–91, p. 79.

For Dee, see Sherman William H. , John Dee: The politics of reading and writing in the English Renaissance (Amherst, 1995); Clulee Nicholas H. , John Dee's natural philosophy: Between science and religion (London, 1988); idem, “At the crossroads of magic and science: John Dee's Archemastrie”, in Vickers (ed.), op. cit. (ref. 12), 57–71; Crossley James (ed.), The autobiographical tracts of Dr. John Dee (Manchester, 1851).

No exhaustive study of the literature is intended. As examples I have selected the prefaces and popular works of the mathematicians such as Recorde Robert Wingate Edmund Tapp John Blundeville Thomas Witt Richard , and sundry non-mathematical works where mathematics is discussed.

Zetterberg , op. cit. (ref. 10); Thomas Keith , Religion and the decline of magic (New York, 1971), 362–3.

Aubrey John , ‘Brief Lives’, chiefly of contemporaries, set down by John Aubrey, between the years 1669 and 1696 , ed. by Clark Andrew (2 vols, Oxford, 1898), i, 27.

Allen left his watch in a chamber of his room when he was visiting Mr John Scudamore in Herefordshire. The maid, when she heard it ticking, concluded it was the devil and threw it into the moat. The watch, however, was attached to a string, and was caught on a branch, confirming its association with the devil.

The famous story alleges that when Dee left Mortlake privately in 1583 to embark for Holland, a mob, believing him to be a magician, broke into his house and destroyed a great part of his furniture and books, and also his mathematical instruments, has been conclusively demolished. See the article on Dee in DNB, xiv, 275 for the original story, and Roberts Julian Watson Andrew G. (eds), John Dee's library catalogue (London, 1990), for a refutation.

Aubrey , op. cit. (ref. 16), 109, 213.

As quoted in Zetterberg , op. cit. (ref. 10), 85.

Aubrey , op. cit. (ref. 16), 108, 153.

Worsop Edward , A discoverie of sundrie errours and faults daily committed by landemeaters (London, 1582), sig. P, sig. E3¹.

Worsop , op. cit. (ref. 22), sig. F2'.

Osborne Francis , Advise to a son (Oxford, 1656), 7–8.

Historical Manuscripts Commission, Various collections (Westminster, 1909), v, 246, as quoted in Feingold , “The occult tradition” (ref. 12), 79.

Agrippa Henry Cornelius , De occulta philosophia II.1 (Opera, i, 53); see also Molland , op. cit. (ref. 11).

Yates , “The Hermetic tradition” (ref. 10), 230.

Feingold , “The occult tradition” (ref. 12), 81.

Quoted in Eamon , “Technology as magic” (ref. 11), 201.

Eamon William , Science and the secrets of nature: Books of secrets in medieval and early modern culture (Princeton, N.J., 1994), 39–44.

Yates , op. cit. (ref. 10), 230.

Molland , op. cit. (ref. 11), 210.

Dunn Richard , “The true place of astrology among the mathematical arts of late Tudor England”, Annals of science, li (1994), 151–63.

Thomas , op. cit. (ref. 15), 358–68.

Eamon , “Technology as magic” (ref. 11), 201; see also Rose Paul , Italian renaissance of mathematics (Droz, 1975), chap. 1.

Thomas , op. cit. (ref. 34), 363.

Kieckhefer Richard , Magic in the Middle Ages (Cambridge, 1989), 47–49.

Thomas , op. cit. (ref. 15), 363.

Ascham Roger , The scholemaster , ed. by Mayor John E. B. (London, 1863; rpr. New York, 1967), 14–15. Ascham's text was printed in 1570, with two more editions in 1571, one edition in 1573, and one edition in 1589.

For Ascham's career as a mathematical lecturer, see Rose P. L. , “Erasmians and mathematics at Cambridge in the early sixteenth century”, Sixteenth century journal, viii, supplement (1977), 47–59, p. 56. Ascham also advised against studying mathematics too intensely in a 1564 letter to the Earl of Leicester, stating “I think you did yourself injury in changing Tully's wisdom with Euclid's pricks and lines” (in Giles J. A. (ed.), The whole works of Roger Ascham (3 vols, London, 1864–65), ii, 103).

Shapin Stephen , A social history of truth: Civility and science in seventeenth-century England (Chicago, 1994); Dear Peter , “Totius in verba: Rhetoric and authority in the early Royal Society”, Isis, lxxvi (1985), 145–61. See also Pera Marello Shea William (eds), Persuading science (Canton, Mass., 1991) and Dear Peter (ed.) The literary structure of scientific argument: Historical studies (Philadelphia, 1991).

See Holmes Frederic L. , “Argument and narrative in scientific writing”, in Pera Shea (eds), Persuading science (ref. 41), 164–94, for an exploration of the method of obtaining consensus used by the Academicians.

Shapin , op. cit. (ref. 41), 337. See also Shapin , “Robert Boyle and mathematics: Reality, representation, and experimental practice”, Science in context, ii (1988), 23–58.

Dear , “Totius in verba” (ref. 41), 159.

Cantor Geoffrey , “The rhetoric of experiment”, in Gooding David Pinch Trevor Schaffer Simon (eds), The uses of experiment: Studies in the natural sciences (Cambridge, 1989), 159–80. Cantor discusses the rhetoric of experimental reports. See also Hunter's Michael Science and society in Restoration England (Cambridge, 1981), especially chap. 5, for a discussion of the role of utility in Restoration science.

Kitcher Philip , “Persuasion”, in Pera Shea (eds), Persuading science (ref. 41), 3–27.

Johnston , op. cit. (ref. 4), 108.

Shapin , A social history of truth (ref. 41), 222, 236.

Taylor , op. cit. (ref. 5), 205.

Wingate Edmund , Arithmetique made easie, in tvvo bookes (London, 1630), sig. A^r.

Ibid.

Tapp John , The path-way to knowledge; Containing the whole art of arithmeticke … (London, 1613).

Taylor , op. cit. (ref. 5), 173. Waters, op. cit. (ref. 5), 569.

Blundevile M. , M Blundevile His exercises, containing sixe treatises … (London, 1594), sig. A3^r; Worsop , A discoverie of sundrie errours … (London, 1582), sig. A2^r-A2^r.

Braudel Fernand , The wheels of commerce: Civilization and capitalism 15th-18th century, transl. by Reynolds Siân (2 vols, New York, 1986), ii, 448.

For more details on the problems of audience, see Johnston , “Mathematical practitioners” (ref. 3), 342.

Davis Ralph , The rise of the english shipping industry in the 17^th and 18^th centuries (2nd edn, Newton Abbot, 1972); Brenner R. , “The social basis of English commercial expansion 1550–1650”, Journal of economic history, xxxii (1972), 361–441.

Brenner Robert , Merchants and revolution: Commercial change, political conflict, and London's overseas traders, 1550–1653 (Cambridge, 1993), 5.

Brenner , op. cit. (ref. 58), 12–14.

Taylor , op. cit. (ref. 5), 33, 171.

Waters , op. cit. (ref. 5), 542–3.

Waters , op. cit. (ref. 5), 558.

Davis , op. cit. (ref. 57), 125.

Jardine Lisa , Francis Bacon: Discovery and the art of discourse (Cambridge, 1974), 69–71.

See Turner's A. J. “Mathematical instruments and the education of gentlemen”, Annals of science, xxx (1973), 51–88 for a detailed explanation of why these skills became necessary.

Elyot Thomas Sir , The boke named the Governour (ed. from 1st edn of 1531 by Henry Herbert Stephen Croft; 2 vols, New York, 1968, reprint of 1883 edn), 28.

Elyot , op. cit. (ref. 66), 45–46.

Gilbert Humphrey Sir , “Queene Elizabethes Academy”, in Funivell F. J. (ed.), Queene Elizabethes Academy (by Sir Humphrey Gilbert). A booke of precedence, The ordering of a funerall, &c. Varying versions of the good wife, The wise man, &c. Maxims, Lydgate's order of fools, A poem on heraldry, Occleve on Lords' men &c (London, 1969), 4–5.

Percy Henry , Advice to his son, ed. by Harrison G. B. (London, 1930), 67–72.

Library British , Add MS 27,606 (The design and method of Mr Crow's studies).

Peacham Henry , The compleat gentleman (London, 1622).

Peacham , op. cit. (ref. 71), 72.

Turner , “Mathematical instruments” (ref. 65), 51.

Willan T. S. , The Muscovy merchants of 1555 (Manchester, 1953), 5–10.

Willan , op. cit. (ref. 74), 7.

Andrews Kenneth , Trade, plunder and settlement: Maritime enterprise and the genesis of the British Empire 1480–1630 (Cambridge, 1984), 8.

Andrews , op. cit. (ref. 76), 33, for an excellent review of the rhetoric used by the literature advocating voyages of discovery, new trades, and the colonies.

Recorde Robert , The grounde of artes (London, 1542), sig. Aiv^r.

Recorde , Grounde (ref. 78), sig. Biii^r.

See Johnston , op. cit. (ref. 2, 1996), 112, on this point.

Recorde , Grounde (ref. 78), sig. Biii^r.

Baker Humphrey , The well springe of sciences (London, 1562), sig. Aiiii¹.

Recorde Robert , Pathway of knowledge (London, 1551), Preface sig. i.

Digges Thomas , Stratioticos (London, 1579), sig. Aiij^r.

Johnston Stephen , “Making mathematical practice: Gentlemen, practitioners and artisans in Elizabethan England”, Ph.D.dissertation, Cambridge University, 1994, provides a lucid explanation of the connections between Digges's shift in orientation and a highly nuanced conception of patronage as social credit. By contrast, see also Feingold Mordechai , The mathematicians' apprenticeship: Science, universities and society in England 1560–1640 (Cambridge, 1984), 186, and cf. 206–7.

Blundevile , His exercises (ref. 54), sig. A4¹.

Witt Richard , Arithmeticall qvestions … (London, 1613), sig. A3^r.

Taylor , op. cit. (ref. 5), 337.

Tapp John , The path-way to knowledge (London, 1613), 10–11.

Tapp , op. cit. (ref. 89), 49.

Tapp , op. cit. (ref. 89), 305–6.

Worsop Edward , A discoverie of sundrie errours (London, 1582), sig. A2^1-r.

Recorde Robert , Whetstone of witte (London, 1557), sig. Miir.

Blagrave John , The mathematical iewel (London, 1585), title-page.

See Hill Katherine , ‘“Juglers or Schollers’: Negotiating the role of a mathematical practitioner”, The British journal for the history of science, in press.

Johnston , “Mathematical practitioners” (ref. 3).

Parker John , Books to build an empire: Bibliographical history of English overseas interest to 1620 (Amsterdam, 1965), 55–56.

Recorde , Whetstone of witte (ref. 93), fol. Aiii. See also Willan , The Muscovy merchants of 1555 (ref. 74), 23; Taylor E. G. R. , The haven-finding art: A history of navigation from Odysseus to Captain Cook (London, 1957), 197; Waters , op. cit. (ref. 5), 94–95.

Leonard Digges Thomas , Stratioticos (London, 1579), sig. Aiii³.

As we shall see, however, his attempts were not always greeted with open arms; Digges complained that “In like sort by Masters, Pilotes, and Mariners, I have bene aunswered, that my Demonstrations were pretie devises: But if I had bene in any Sea services, I should finde all these my Inventions meere toyes, and their Rules onely practizeable…”, op. cit. (ref. 99), sig. Aiv^r.

Taylor , op. cit. (ref. 5), 193.

Andrews , op. cit. (ref. 76), 20.

Tapp , op. cit. (ref. 89), sig. A2^l.

Waters , op. cit. (ref. 5).

Taylor , The haven-finding art (ref. 98).

See Foxe Luke , North-West Fox, or, Fox from the North-West Passage (London, 1635).

James Thomas , The strange and dangerous voyage of Captain Thomas James (London, 1633), sig. A2^v.

Rundall Thomas (ed.), Narratives of voyages towards the north-west in search of a passage to Cathay and India 1496 to 1631 (London, 1869), 205.

Foxe not only managed not to lose a man, but also to sail much further north, and was at least able to discount earlier suggestions as to the possible location of the passage.

Borough William , A discourse of the variation of the cumpas (London, 1581). He suggests that all seamen who wished to excel at their profession should learn arithmetic and geometry, but he urged this course so that they could design their own instruments and not depend upon outside advice.

Wright Edward , Certaine errors in navigation; The voyage of … to the Azores (London, 1599), sig. ll3^r.

Wright , op. cit. (ref. 111), sig. ll3^r-l.

James does not mention very precisely which books he brought with him, stating merely “A Chest full of the best and choicest Mathematicall bookes that could be got for money in England; as likewise Hackkluite Master Purchas Master , and other books of Iournals and Histories”. He did, however, supply a list of instruments, including “glasses, logg-line, meridian-line, plumb-lines, globes, semi-circles, meridian compasses, loade-stone, watch-clocke, a Table, euery day Calculated, correspondent to the Latitude, Master Gunter's Cross-staff, Jacobs Staues, Quadrant, Equilateral Triangle…” (pp. 604–6). Foxe, on the other hand, seems to have brought very few instruments. He mentions a compass, which he said was unreliable (p. 309), a log-line, and a semi-circle.

Barlow William , The navigators svpply (London, 1597) is another example of suggestions offered combined with concern over acceptance by sailors. He relates a story about one of Sir Francis Drake's voyages to show what happens to those who fail to listen to their experts; Drake ended up sailing in a circle, arriving back where he began after 16 days, because he would not listen to his navigator who was aware of the difficulties that ensued due to the variation of the compass (sig. a3^l). Moreover, he realized that some would suggest that his lack of experience made him an unfit guide, but he claimed “And in the minde onely, pure and true Arte, refined from the droße of sensible or experimental knowledge, is to be found” (sig. a5^r-b^r). In the end, he assumed that the more skilful sailors would accept his suggestions.

Some types of mathematical practice, such as astrology, used different sorts of persuasive devices. For instance, Christopher Heydon's A defence of judicial astrology (Cambridge, 1603), written in response to Chamber's John A treatise against judicial astrology (London, 1601), where the main argument seemed to be that astronomy was the highest form of human knowledge in that the “ravishing beautie, constant order, and powerfull efficacy of the celestiall bodyes … lead us to God” (sig. &3^r). Additionally, he claimed that astrology and astronomy were in fact the same thing (sig. A1r), and also maintained that he was only defending ‘proper’ astrologers who “contain themselves within the bounds of naturall Philosophie”.

Clark Stuart , “The rational witchfinder: Conscience, demonological naturalism and popular superstitions”, in Pumfrey Rossi Slawinski (eds), op. cit. (ref. 2), 222–48; idem, “The scientific status of demonology”, in Vickers (ed.), op. cit. (ref. 12), 351–74; and Henry John , “The touch of cold philosophy? …': The fragmentation of Renaissance occultism and the origin of the Enlightenment”, unpublished manuscript. I thank John Henry for letting me read this work.

Henry John , The Scientific Revolution and the origin of modern science (London, 1997), 47.

Rossi Paolo , Francis Bacon: From magic to science (Chicago, 1968); and Henry , opera cit. (refs 9 and 117).

London William , A catalogue of the most vendible books in England, sig. Dd2r-Dd31.

Hill Katherine , “Mathematics as a tool for social change: Educational reform in seventeenth- century England”, The seventeenth century, xii (1997), 23–36.