neal-rhetoric

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Hist. Sci., xxxvii (1999)

THE RHETORIC OF UTILITY: AVOIDING OCCULTASSOCIATIONS FOR MATHEMATICS THROUGHPROFITABILITY AND PLEASURE

Katherine NealUniversity of Sydney

If rhetoric is the art of persuasion, then mathematics may seem to be its antithesis. Thisis believed, not because mathematics does not persuade, but rather because it seem-ingly needs no art to perform its persuasion. The matter does it all; the manner needonly let the matter speak for itself.1

Recent work by R. S. Westman, Mario Biagioli, J. A. Bennett, Peter Dear, AlanGabbey and others has shown how Renaissance mathematicians tried to assert theirintellectual authority and raise their status among other intellectuals.2 With the ex-ception of Biagioli’s wide-ranging survey of the different kinds of mathematicalpractitioners and their different levels of success promoting themselves, historicalanalysis has tended to focus on the ways in which mathematicians were able todevelop their “art” to show its importance for “scientia”. By attempting to demon-strate how their techniques could lead to certain knowledge, mathematicians effec-tively challenged the increasingly beleaguered natural philosophers. It would benaïve to assume, however, that even the most dazzling mathematical results couldbe sufficient in themselves to persuade Renaissance intellectuals to think of math-ematics as anything more than what amounted to a bag of tricks. Accordingly, math-ematicians took active steps to persuade their contemporaries of the importance ofmathematics. The aim of this paper is to show how rhetoric played an importantrole in the mathematicians’ strategies.

It would no doubt be possible to provide a Biagioli-like typology of all the rhe-torical strategies used by different mathematicians throughout Europe, but this pa-per restricts itself to a consideration of English mathematicians, and focuses uponthe rhetorical claim that mathematics can be useful in a wide range of practicaleveryday pursuits. The development of a rhetoric of utility in the Scientific Revolu-tion is, of course, a familiar story but its vigorous use in mathematics textbooks hashardly been noticed. Another aspect of this rhetoric, which this paper seeks to bringout, is its use in dispelling the common belief that mathematics was an occult, evena demonological, pursuit. Although, as we shall see, the occultist associations ofmathematics during this period constituted an important aspect of its public per-ception, this has hardly been discussed in the historical literature. The post-Refor-mation period, with its religious conflicts and witch crazes, made it all the moreexpedient to demonstrate one’s religious orthodoxy or, at least, to demonstrate that

152 · KATHERINE NEAL

one was not unorthodox. If mathematics was to be regarded as a useful and advan-tageous pursuit, it would have to be purged of its occult associations.

The common perception of mathematics in early modern England was that itwas not only a difficult and tedious pursuit but also closely associated with occultand illicit practices. It became very important for mathematical practitioners, par-ticularly those who worked in areas concerned with the pragmatic interests of crafts-men, to break away from occultist tradition. One method for accomplishing thisseparation was to utilize a rhetoric of utility, a technique of persuasion that centredupon stressing the practical uses of mathematics, as one of their tactics to publicizetheir conception of mathematics, as well as to defuse hostility and obtain patronsand students.3 The practical, vernacular mathematics texts used their extended ti-tles and epistles “to the reader”, as well as their instruments and the problems them-selves, to portray mathematics as vital to such useful activities as astronomy,navigation, surveying, gunnery, architecture and mensuration. These sections, to-gether with the type of problems solved in the texts, attempted to persuade thereaders that mathematics could be both profitable and pleasurable. The practition-ers constructed their arguments and their problems so that the utility of mathemat-ics could be used to defuse their contemporaries’ indifference to and hostility towardstheir discipline; they used this rhetoric in order to sell their texts, and as a techniqueto persuade their audience that mathematics was worth supporting and studying.

Mathematics in this period was, however, a diverse set of practices, and manytypes of persuasive techniques were developed.4 This paper will concentrate uponthe rhetoric of a certain type of practitioner, made prominent in the work of E. G. R.Taylor: the practitioners working in the vernacular and often urban tradition ofEnglish practical mathematics, who were usually associated with the use of instru-ments for observation, measurement and calculation.5 Other forms of mathematicalpractice, such as certain aspects of astronomy and more theoretically oriented texts,called upon different sorts of persuasive techniques. Moreover, I shall also exploreto what extent the possibilities for developing mathematical practice were shapedby the local setting. Did local demands and restrictions make certain types of bothmathematical and persuasive techniques more viable than others?

Two of the classic texts that cover this period are E. G. R. Taylor’s Mathematicalpractitioners of Tudor and Stuart England (1954) and D. W. Waters’s The art ofnavigation in England in Elizabethan and early Stuart times (1966). These arefoundational works, and they both mention the emphasis on utility found in practi-cal mathematics texts, but they do not consider why the rhetoric of utility wasconsidered so important by these authors. It is as though Taylor and Waters, takingit for granted that mathematics is useful, saw this talk of utility merely as a matterof fact. Additionally, although Taylor occasionally remarks upon mathematics’ oc-cult associations, she never discussed this in any detail, and failed to see the linkbetween such contemporary attitudes and the need of practitioners to utilize a rhetoricof utility. So, although both the low level of mathematical activity and its occultassociations in the second half of the sixteenth century and the beginning of the

THE RHETORIC OF UTILITY · 153

seventeenth are acknowledged, no corrective persuasive strategies were recognizedto be necessary by Taylor and Waters. Indeed, in both Taylor’s and Waters’s worksthere is an implicit but obvious judgement that the triumph of mathematical practi-tioners was inevitable.

The more recent work on practical mathematics of J. A. Bennett has also brieflydiscussed the promotion of mathematics on the basis of its utility.6 The focus of hiswork, however, is on establishing the practical mathematical sciences as a plausi-ble candidate for a source for the mechanical philosophy. Likewise, although StephenJohnston has noted the use of utility in putting “the mathematicalls” in a positivelight, generally his work concentrates on describing the roles of instruments inpromoting practitioners’ roles.7 Johnston’s work on the identity of mathematicalpractitioners in sixteenth-century England briefly discusses both the rhetoric ofutility and mathematics’ associations with magic, but again it seems true to say thatthese issues have never been explored in any detail.

The association of mathematics with magic is particularly under-researched. Inthe Companion encyclopaedia of the history and philosophy of the mathematicalsciences, for instance, there are no entries in the index for either magic or the oc-cult.8 Additionally, scholars such as John Henry have noted that “mathematicalmagic” has not yet received sufficient scholarly analysis, and its role is, conse-quently, difficult to assess with confidence.9 Although Frances Yates provocativelyenquired if mathematics was “for Bacon, too much associated with magic and themiddle world of the stars” to be emphasized in his method, she did not delve intothe association between mathematics and magic in any detail. Peter Zetterburg in-vestigates the assumption of the “vulgar” that the gadgets connected with“mathematicks” were the “result of a dangerous confederacy with spirits and de-mons”.10 He attributes this association to “gossip, misunderstanding, and fable”, aswell as to the extravagant claims of the mathematicians concerning the power oftheir devices. This extremely suggestive study, however, is primarily concernedwith automata and mechanical devices, and although it makes a promising begin-ning, it leaves out a variety of pertinent factors. William Eamon’s very interestingexamination of the association of the mechanical arts with magic also examinesautomata; however, it mainly concentrates on the relationship between magic andtechnology.11 Additionally, Mordechai Feingold has explored the role of the occulttradition in the English universities in general. His broad-ranging study at leastpointed out that “For the upper classes these studies [mathematics and the occultsciences] carried dangerous connotations”.12 Otherwise, the association betweenmathematics and magic is confined mainly to extremely well known cases such asJohn Dee.13

To understand the development and utilization of claims of mathematics’ utilityas a persuasive device, we must first explore why such a technique was needed.Therefore, I shall discuss mathematics’ dubious reputation and some of the reasonswhy mathematics was believed to be associated with occult practices. After inves-tigating the types of complaints made against mathematics, I shall examine the


factors that made it possible to create a new vision of mathematical practice. Sec-ondly, an examination of the early modern English, vernacular mathematical litera-ture will show how the rhetoric of utility was used in the attempts to negotiate a rolefor mathematical practitioners in early modern England.14 Not only the text, but theproblems themselves were shaped by the need to show that mathematics could beboth profitable and pleasurable. Yet one should keep in mind that although thisrhetoric played an important part in creating a social space for mathematical prac-tice, it should not be accepted uncritically as an indication of the actual “practical-ity” of the mathematicians’ work, or necessarily as an indication of strong linkswith craftsmen. I will investigate, in particular, the alliances that practitioners as-pired to create with navigation and exploration in order to examine if there wassometimes a gap between their proclamations and performances. Finally, I shallbriefly discuss factors other than the rhetoric of utility that might have aided indissociating mathematics from magic. Although the rhetoric of utility was an im-portant factor in persuading practitioners’ audiences that mathematics was a profit-able, instead of a dangerous, activity, other issues, such as natural magic’s changingrelationship with natural philosophy, should also be taken into account.

A RUINOUS REPUTATION

The utilization of a persuasive technique for supporting the value of mathematicalpractice was vital, due to mathematics’ dubious reputation. One of the difficultiesthat practitioners needed to overcome in the period 1550–1650 was that mathemat-ics was often linked to magic, and the use of it was sometimes taken as a sign thata practitioner was in league with spirits and demons.15 For instance, John Aubrey,in his discussion of Thomas Allen (1542–1632), one of the best mathematiciansand astrologers of that time, says “In those dark times astrologer, mathematician,and conjurer, were accounted the same things; and the vulgar did verely beleevehim to be a conjurer”.16 Allen’s servant apparently encouraged this point of view bytelling stories of meeting “the spirits coming up his [Allen’s] stairs like bees”, andthe story of a maid mistaking Allen’s watch for a devil and throwing it into the moatis well known.17 Indeed, Aubrey’s Brief lives is full of such stories; he tells us that“the children dreaded [John Dee] because he was accounted a conjurer”,18 and thatthe country people had similar beliefs about William Oughtred (1574–1660).19

Moreover, Aubrey makes the earlier perception of mathematics clear when he re-lates that at the beginning of this period authorities “burned Mathematical bookesfor Conjuring bookes”.20

Mathematics, in Aubrey’s work, also appears to be a dangerous course of study,or an occupation fit only for those who had no other options. One of Oughtred’sstudents, a certain Mr Austin, studied mathematics so much “... that he becamemad, fell a laughing, and so dyed”, while Sir Charles Cavendish (1591–1654) isdescribed as “a little, weake, crooked man” whom “nature [had not] adapted for thecourt nor camp” and who therefore took up mathematics.21


Similarly, the association between magic and mathematics can be seen in Adiscoverie of sundrie errours (1582) when the author, Edward Worsop, suggestedthat this was merely the result of a Roman Catholic conspiracy to undermine theusefulness of mathematics. The Catholics, according to Worsop, had “fedde thepeople with scumme and drosse ... they brought in superstition and idolatrie: so instead of the pure Mathematicall knowledges, they used coniurations, sorceries, in-vocations of spirits, enchauntments, and other unlawfull practices, under the namesof Divinatorie and Judiciall Astriligie”.22 But this, of course, was merely “the abus-ing and contemning of the Mathematicalls”. For Edward Worsop, it seems clearthat the association of mathematics with astrology was sufficient to link it to occultpractices, but to make his point more forcefully he exploited the Protestant beliefthat many of the doctrines of the Roman Catholic Church were based upon super-stition and the illicit use of occult notions. Having suggested that the commonimage of mathematics was the result of the Papists’ desire to suppress useful knowl-edge, Worsop immediately went on to utilize the rhetoric of utility by dwelling onthe usefulness of mathematics to navigation, makers of almanacs, business, and, ofcourse, surveying to persuade his readers that “if a man have a Mathematicall head,& mathematicall art, that man is to be reputed a most excellent and most necessariemember of the comman weale”.23 Thus one way legitimization could be pursuedwas by excluding the dangerous and undesirable, while simultaneously emphasiz-ing usefulness and pleasure.

Francis Osborne, in his Advice to a son (1656), also shows us not only the earlierattitude towards mathematics, but that the rhetoric of utility was making progresstowards replacing the more hostile point of view. He makes it clear that no study isworthwhile unless it will lead to profit, and that mathematics is such a useful skill.He also, however, describes the depth of the older feelings against mathematics,stating “my memory reacheth the time, when the Generality of People thought hermost usefull branches; spels, and her Professers, Limbs of the Devill”, and he addsthat when Oxford created a chair of mathematics, “Not a few of our then foolishGentry, refusing to send their sons thither, lest they should be smutted by the BlackArts”.24 The accuracy of Osborne’s recollection is confirmed in a letter of James,Lord Ogilvy to his grandson, written in 1605, in which he worries about youngscholars at the university becoming involved with “magick” and “necromancy”which are “the greatest sins against God that can be ...”.25

MATHEMATICS AND MAGIC

There are a variety of reasons for the association of mathematics with the occult.Importantly, the “magicians” themselves claimed there was a close connection. Forinstance, Henry Cornelius Agrippa, in his De occulta philosophia (1533), stated:

The mathematical disciples are so necessary and cognate to magic that, if any-one should profess the later without the former, he would wander totally fromthe path and obtain the least desired result.26


It appears that Agrippa believed that when a magician was learned in natural phi-losophy and mathematics, particularly in arithmetic, music, geometry, optics, as-tronomy and mechanics, he could do “marvellous things”.27 In England, there wereenough mathematicians reputed to be connected with occult practices to keep theassociation alive in the public’s mind. There are, of course, the famous examples ofJohn Dee and Thomas Allen, but such figures as Gabriel Harvey and John Fletcher,a fellow of Caius College, were also linked both to mathematics and occult stud-ies.28 There are also literary examples that show the cultural strength of these ties;the most famous was Dr Faustus, of course, said to be “the most famous name of allthe Mathematicks that lived in his time”, but Friar Roger Bacon was almost as wellrenowned, as can be seen in “The Honourable History of Friar Bacon and FriarBungay”, a popular play by Robert Greene (1594, reprinted in 1599, 1630 and1655).29

Another factor in mathematics’ association with occult practices is, as previ-ously mentioned, its ties to astrology. Arabic scholars had earlier provided Euro-peans with “a comprehensive and highly organized system of occult science” inwhich astrology played an important role.30 For instance, Agrippa believed thatmathematical magic belonged to “the middle celestial world of the stars”.31 Indeed,he felt that from “abstract, mathematical and celestial things we receive celestialvirtues” that might lead to predictions of the future.32 In particular, astrology wasbelieved to be derived from arithmetic and geometry.33 The Church was alwaysopposed to judicial astrology because it held that the idea of astral determinationwas incompatible with Christian doctrines of free will and moral autonomy, althougha number of Anglican clergymen were not above branding judicial astronomy as a“Popish” practice.34 Many educated lay-people, on the other hand, regarded judicialastrology with “profound suspicion”,35 because of the ease with which it could beexploited by charlatans. These negative connotations were bound to impinge uponmathematicians, many of whom did earn money through astrological consultations.

Perhaps one of the major problems confronting the mathematical practitionerwho wished to show the practical utility of mathematics was the very fact that theworkings of automata and other mechanical devices were all too often attributed tonecromancy and demonology. Even university-trained intellectuals in the early mod-ern period tended to think of machines as devices that operated by occult means. Inthe Scholastic tradition natural objects which had the ability to act on other passiveobjects were held to operate either by virtue of their manifest qualities (hotness,coldness, dryness, or wetness), or by virtue of non-manifest, occult qualities (suchas magnetism or other qualities which could not be seen to derive from, or be re-ducible to, the four manifest qualities). By analogy, mechanical contrivances whichwere capable of performing marvellous feats were also held to work by occultmeans. It is this background that John Wilkins had in mind when he called hispopular explanation of the workings of simple machines, Mathematicall magick(1648). Neither Wilkins, nor any other educated man, believed that the magic ofmachinery depended upon the inner machinations of demons, but it seems that


there were plenty who did appear to believe this. In the popular consciousness, amathematical wizard who could design and build a machine capable of performingastonishing feats must have enlisted the aid of demons.

A final reason for the association of mathematics with magic was the obvioususe of mysterious symbols and diagrams in mathematical textbooks. Keith Thomashas pointed out that there was a “widespread conviction that anything mysteriousmight have a diabolical origin”,36 and, as Richard Kieckhefer has noted, magic wasintimately bound up with writing, especially mysterious writing, like rune inscrip-tions, or the squiggles of mathematics.37 Many texts included at least geometricdrawings which the uninitiated might take for conjuring devices. Several of theearly works on arithmetic and algebra, ranging from Robert Recorde’s The whet-stone of witte (1557) to John Tapp’s The path-way to knowledge (1613), included“cossike” notation, an early form of algebraic symbolism, which might have lookedmagical to the uninformed eye. Famously, reformers in the reign of Edward VIdestroyed mathematical manuscripts at Oxford because they believed them to beconjuring books. Additionally, as late as 1644 when sequestrators seized the papersof Walter Warner, they were said to be “much troubled at the sight of so manycrosses and circles in the superstitious algebra and that black art geometry”.38 Otherbooks, such as John Blagrave’s The mathematical iewel (1585), as Zetterburg haspointed out, included texts of marvellous-looking instruments coupled with ex-travagant claims of the instruments’ capabilities. Thus many visual aspects of math-ematical texts might have encouraged suspicions.

Not all criticisms of mathematics, however, were associated with magic. RogerAscham, in The scholmaster (1570), supplies a different sort of reason to avoidstudying mathematics, saying

Some wittes, moderate enough by nature, be many times marde by over mochstudie and use of some sciences, namelie, Musicke, Arithmetick, and Geometrie.Theis sciences, as they sharpen mens wittes over moch, so they change mensmaners over sure, if they be not moderately mingled, and wiselie applied tosum good use of life. Marke all Mathematicall heades, which be onely andwholy bent to these sciences, how solitarie they be themselves, how unfit tolive with others, and how unapte to serve in the world.39

Thus, even when mathematics was not directly linked to magic, studying it toointensely was deemed to be dangerous. Ascham here clearly identifies mathematicsas a potentially morally dangerous pastime. Following such studies could lead tolosing one’s sense of civic duty, and did nothing to further the commonwealth.Civic humanism encouraged forms of knowledge that could take place in the worldand be expressed as active service; a solitary life was rejected. Moreover, Aschamdid not reach this position out of ignorance: he was the Cambridge mathematicallecturer from 1539 to 1541.40 Yet although this quotation is not, in itself, encourag-ing, it does suggest a possible route that mathematical practitioners might take inorder to reverse the generally unfavourable impression of their arts. If the


practitioners could show that their arts made men more fit to serve the common-wealth, that their arts were practical, profitable, and pleasurable, they might hopeto overcome both the indifference to, and the hostility towards, mathematics.

FORMULATING PERSUASIVE STRATEGIES

Steven Shapin and Peter Dear have provided detailed accounts of how the burgeon-ing seventeenth-century English experimental community found rhetorical tech-niques for supporting their statements when the use of ancient authority wasdisavowed as a viable strategy.41 They argue that highly detailed narrative accountsof experiments and observations were used as a means of obtaining authority forexperimental assertions.42 Mathematics, on the other hand, was viewed by someexperimentalists, such as Robert Boyle, as being relatively inaccessible due to itstechnical vocabulary and special techniques; its incorporation into experimentalactivities, it was feared, might restrict the size of the practising experimental com-munity.43 Additionally, Dear points out that the authority for the mathematical con-tributions to the Philosophical transactions “seems to have lain somewhat in doubt,since mathematical forms of argument and demonstration could not be fitted read-ily into the usual form of presentation”.44 The early modern English mathematicalpractitioners, however, share a common goal with experimental scientists of thisperiod: they were trying to persuade their audience of the validity of their activities.45

In many circumstances, persuading others requires no great rhetorical creativity,but merely the ability to anticipate how the reader can be soothed and reassured asto the correctness of one’s argument. Persuasion requires a sense of the backgroundof the readers and an anticipation of the responses that they are likely to make.46

Two factors of the background of possible early modern readers will be considered.First, the beginnings of a movement towards a “plain style” of language, coupledwith claims that the texts would not be too difficult to be useful. Second, and moreimportantly, the developing belief that the primary end of knowledge, and its mainjustification, was the benefit of man’s earthly estate.

A “PLAIN STYLE”

Mathematical practitioners became proponents of a language of ease and accessi-bility. A common theme often drawn on in prefaces and title pages was the easinessof mathematical practice. By stressing facility, humanists’ edicts against obscurityand deliberate difficulty could be circumvented.47 Perhaps this plainness of stylewas also partially an attempt to show that they did not deal in the obscure obfuscationsof the magicians. Magical texts were notoriously mysterious and difficult to under-stand. Additionally, speaking plainly had, in this period, its own persuasive effects. Itcan be thought of as a rhetoric that denies that it is rhetoric.48 This was a tactic some-times employed by the mathematical practitioners. Edmond Wingate (1596–1656),educated at Queen’s College, Oxford and a follower of the lectures at Gresham


College, was a tutor to Princess Henrietta in France. Although he was later a land-owner, a Justice of the Peace, and a Member of Parliament, he was also intermit-tently a mathematical practitioner.49 In his Arithmetique made easie (1630), one ofthe most popular arithmetics of his day, he claimed that “Arithmetique needes notthe Logicians arguments, nor the Rhetoricians Eloquence to prove or parswade theusefulnesse thereof to the world, every mans particular occasion, to use it, is suffi-cient to satisfie any man in that point”.50 In other words, he asserted that mathemat-ics did not need rhetoric to persuade people as to its usefulness. Plainness andsimplicity were often claimed as virtues by mathematical texts. Wingate furtherargued that many men who would have liked to learn mathematics despaired, be-cause they had encountered texts that were both “tedious and obscure”. As a resultof these textual difficulties, intricate branches of arithmetic, such as multiplicationand division, “perplex the new Practitioner, that hee takes them to be HerculesPillers, and writes upon them Non plus ultra”.51 His work, on the other hand, wassaid to be easy to use.

Of course, to a certain extent, Wingate might mainly have been trying to increasethe sales of his texts by claiming that they were somehow “easier” than others.Certainly the content of his Arithmetique made easie, which included a typicaldiscussion of number, addition, subtraction, multiplication, division, and roots, to-gether with the usual explanation of the rule of three, the golden rule, rule of fel-lowship, and rule of false position, was very similar to previous arithmetics exceptfor its inclusion of logarithms. The very popularity of logarithms upon their intro-duction, however, should remind us how difficult multiplication, division, and theextraction of roots were seen to be in this period.

The practitioners claimed that mathematics’ difficulties could be overcome witha plain but clear style. John Tapp, whose work centred mainly around navigation, inthe extended title of his The path-way to knovvledge (1613), stated that his bookwas “Digested into a plaine and easie methode by way of Dialogue, for the betterunderstanding of the learners thereof ”.52 If one compares Tapp’s work to Recorde’sThe whetstone of witte (1557), the contents, mainly numeration, addition, subtrac-tion, multiplication, division, coupled with the usual rules of practice and a briefexplanation of “cossike” numbers, is almost identical. In some ways, however, it is“easier” as it leaves aside Recorde’s discussions about the nature of number en-tirely, bypassing Recorde’s concern with the question of how exactly fractions and“surds”, or irrationals, should be understood if number means an entity built upfrom units. Thomas Blundeville (fl. 1560–1602) was a landed gentlemen who wasa mathematical tutor in the households of Sir Nicholas Bacon and Justice Windham.His work was oriented towards the instruction of young gentlemen, especially inthe use of maps, globes, instruments and navigation.53 Blundeville declared thatone of his texts contained “a verie easie Arithmeticke so plainlie written as any manof mean capacite may easilie learn the same without the help of any teacher”; whileWorsop, in his A discoverie of sundrie errours ... (1582), stated: “I, a simple manamong common people, have set forth this discourse to their behoofe, by the playnest


waies I could devise, and for their easiest understanding ... I have thought good bya plaine and popular discourse, to laie open unto the understanding of every reason-able man....”54 Thus one tactic to persuade the intended audience of the value ofmathematics was to claim that it was written in a plain style that made it easy tolearn. Magic, of course, was traditionally written in an obscurantist way to ensurethat only the adept could understand it. By claiming that their style ensured thateveryone could understand their texts, writers were dissociating themselves frommagic. The strategy of emphasizing the utility of mathematical practice was, how-ever, much more widely used than the stressing of plainness of style.

THE ENGLISH CONTEXT

To understand the appeal of the argument based on utility, and why the practition-ers believed this argument might be able to persuade their intended audience thatmathematics was a profitable activity that had no dangerous associations, we mustexamine the broader context. In the period around 1500 there was not a great de-mand for mathematical skills. England was, in some respects, a “backward” coun-try, without a powerful navy, with a largely rural population, and with only woolproduction and the strong cloth industry as sources of wealth.55 Although in thesector of foreign trade large English companies were created from around 1553, math-ematicians would have had little luck attempting to gain wider public acceptance, orstudents prepared to pay for tuition, by persuading the largely illiterate rural popula-tion or the labouring poor; neither group needed their skills. Increased involvement innavigation and exploration contributed to a greater demand for mathematical prac-tice. Nevertheless, the audience for their texts was restricted, in many respects, to theliterate urban classes of merchants and master craftsmen and to gentlemen.56

Several factors contributed to the expansion of England’s shipping and commer-cial enterprises. Although in 1560 there was only approximately 50,000 tons ofmerchant shipping of every kind, concern with promoting the fisheries and the growthof the coal trade began the build-up of smaller ships. Moreover, the disruptions inAntwerp in the 1560s and 1570s curtailed the English traditional dependence uponEuropean middlemen, and increased the number of English ships in Mediterraneanwaters.57 Additionally, the cloth export crisis in the 1550s created a need for newmarkets, but there was also a growing interest in searching for spices and gold fromthe East. Revisionist accounts that are sceptical about the cloth export crisis asleading to a search for new cloth markets still emphasize the importance of therising new trades based on imports extending from Morocco, Russia, Persia, andGuinea to Turkey.58 For instance, the company for the discovery of the northeastpassage of 1553, which evolved into the Muscovy Company of 1555, a drivingforce in the eastward expansionist movement, aimed to open up a route to the spicesand gold of the Far East that would be free from Portuguese interference.59 Theburgeoning of foreign trade and increasingly longer voyages eventually led to ademand for mariners who were skilled in navigational instruments and practical


mathematics. It was not, however, a speedy process. Before 1572, Stephen Bor-ough had made unsuccessful attempts to establish the position of chief pilot as wellas the training of pilots on the Spanish model;60 and Richard Hakluyt had called forthe establishment of a navigational lectureship in London in 1582.61 Sir ThomasSmith seems to have sponsored lectures in navigation at his own expense, for theEast India Company.62 It was not, however, until 1673 that a government-sponsoredmathematical navigational school was founded at Christ’s Hospital. This schoolwas originally founded for purposes relating more closely to war than trade, al-though it appears that in 1631 there was a short-lived stipend given for a math-ematical lecture on navigation “to his Majesty’s servants”.63 Other factors, however,besides avoiding occult associations, expanding trade, and exploration, encouragedthe expansion of certain types of mathematical practice.

The perceived need for an education that would allow men to take advantage ofnew opportunities, and increase their ability to give service to the state, had beengrowing since Henry VIII’s “reform” of the church. A new class was needed tofulfil the functions of government that had previously been performed by men whohad been drawn mainly from the church.64 Additionally, Henry VIII’s reformationgave the crown a great deal of church land to manage, which increased the need forsurveying. Mathematics, in this period, began to be included in some educationaltexts as a valued gentlemanly recreation and as a practical activity. Although, as wehave seen in Asham’s remarks, overmuch study of mathematics was viewed withsuspicion, and although the danger of being tainted with occult associations per-sisted in some minds, some knowledge of practical mathematics was becomingmore and more important for gentlemen.65 Sir Thomas Elyot, in his The boke namedthe Governour (1531), made it clear that it was a virtue for those who would begovernors to study subjects that would improve the “publike weale”.66 He urgedthat gentlemen learn “geometry, astronomie, and cosmographie”, but that they learnthese things with instruments, charts, and figures; otherwise this pursuit would takeup too much time.67 Humphrey Gilbert’s scheme for an academy to educate thenobility and gentry at Queen Elizabeth’s court also included practical mathematics.There were to be mathematicians who were to read arithmetic, geometry, astronomy,and cosmography; the instruction, however, was to focus on navigation and war-fare, with an emphasis on instruments.68 Additionally, Henry Percy, the ninth Earlof Northumberland, in his Advice to his son (1609), recommended the study ofarithmetic, geometry, astronomy, and “the Art Nautical and Military”.69 Mr Crow’stutor explained to his father that he would at least teach him arithmetic, geometryand the use of globes, and also assist him as much as he was able if the son’s“Genius delights to wade deeper”.70 Later, Henry Peacham, who tutored WilliamHoward (who later became Viscount Stafford), also wrote a text on gentlemanlyconduct titled Compleat gentleman (1622).71 He ranked mathematics with poetry,pictures, and heraldry as a pleasing gentlemanly recreation, and he called math-ematics “this most ingenious and useful Art ... a science of such Importance, thatwithout it, we can hardly eate our bread, lie dry in our beds, buy, sell, or use any


commerce whatsoever”.72 It became increasingly contended that if a gentlemanwanted to maintain his position and engage in the activities traditional to his class,a measure of skill in mathematics was essential.73 Mathematics, however, does notseem to have been valued for its own sake, but only insofar as it could be portrayedas serving the commonwealth and private interests.

Moreover, such ventures as the Muscovy Company’s attempts to find a northerlypassage to the Indies and the East India Company’s charter for voyages beyond theCape of Good Hope and Magellan’s Straits eventually increased the interest inmathematics through the company’s structure and the extensive nature of the voy-ages. The new joint-stock companies were pioneers in the expansion of trade andthey depended upon financial devices that utilized at least basic mathematics.74

Even the slaving voyages required mathematical techniques for redistributing capi-tal.75 These companies were possible targets of mathematical practitioners in theirquest for patronage. For example, Robert Recorde dedicated the first English alge-bra, The whetstone of witte, to the Muscovy Company in November 1557, and hisCastle of knowledge (1556), a treatise on the sphere, was written expressly for theMuscovy Company’s navigators. Increasing trade created a wider audience for therhetoric of utility; the tonnage of shipping doubled between 1570 and 1630, and bythe end of this period the English merchants and ships handled a complex networkof trade.76 The rhetoric used to advocate voyages of discovery, new trades and colo-nies was often phrased in terms of promoting the public good. Indeed, there arestriking similarities between the claims of advocates of overseas expansion and theclaims of the promoters of mathematical practice. Both groups tended to claim thatthey were concerned with the public good, or the common weal, and that theiractivities would increase trade and lead to work for the unemployed.77

THE RHETORIC OF UTILITY

English mathematical practitioners began their attempts to convince their readersof the usefulness of mathematics even in the earliest texts written in the vernacular.Robert Recorde, a student at both Oxford and Cambridge where he studied medi-cine, was a Controller of the Mint, and acted as a Surveyor of Mines and Moneys inIreland. He was also a writer of mathematical textbooks. In his work, The groundeof artes (1542), an elementary arithmetic, he began with the almost metaphysicalproclamation that “nombre ... is the onely thyng (all moste) that separateth mannefrom beastes. He therefore that shall contemne nombre, he declareth hym self asbrutishe as a beaste, and unworthy to be counted in the fellowshyp of menne.”78

After this attack upon those who would disparage numbers, Recorde went on toinvoke the concept of utility, stating that without numbers men could do almostnothing.79 He provided a long list of activities, including all the contemporary aca-demic subjects, that depend upon numbers, such as astronomy, geometry, music,“physicke”, law, grammar, and divinity. This sort of generalized defence that arguedfor the value of mathematics to the lawyer, the physician, and even theologians


should, perhaps, be seem as more of a rhetorical trope than as a serious attempt tocolonize the territory of what Recorde labelled the “learned professions”. It was astandard defence that was used alongside frequent appeals to the support of ac-knowledged authorities.80 More importantly, he also claimed that numbers are nec-essary to the “Common Weale in times of peace and in due provision and order ofarmies in tyme of battle”.81 By the time he had finished describing numbers’ usefulattributes, his audience might be pardoned for believing that it was their civic dutyto learn arithmetic. Humphrey Baker, in his The well springe of sciences (London,1562), a basic arithmetic that was dedicated to the Merchant Adventurers’ Com-pany, went even further, stating that anyone who did not accept the value of arith-metic was a “foole, and vnfit member, to rule or deale in a Common Welthe”.82 Inthe dedication of Recorde’s Pathway of knowledge (1551), an elementary geometrytext, he presented the case for the usefulness of geometry as well, claiming “Car-penters, karvers, Joyners and masons doo willingly acknowledge that they can workenothyng without reason of geometrie”.83 Both branches of mathematics were pro-claimed to be practical, profitable and vital to the “Common Weales”.

On the other hand, Thomas Digges, in his and his father Leonard’s Stratioticos,first published in 1579, seems to confirm the fears of those who like Ascham be-lieved that mathematical practice might lead to isolation, while simultaneously uti-lizing the rhetoric of utility to rehabilitate its reputation. He stated:

the more subtle parts of these Mathematical Demonstrations did breede in mefor a time a singular delection, yet finding none, or very few, with whome toconferre & communicate those my delites, (& remembering also that gravesentence of Diuine Plato, that we are not borne for our selves, but also for ourParents, Countrie, and Friends).... After I grew to years of riper iudgement, Ihaue wholey bent my self to reduce those Imaginatiue contemplations, to sen-sible Practicall Conclusions: as well thereby to haue some companions of thosemy delectible studies, as also to be able, when time is, to employ these to theseruice of my Prince and Countrie.84

Thus although he had been seduced for a time into Ascham’s stereotype of an iso-lated “mathematical head”, he claims to have seen the light and transformed him-self into a useful member of the commonwealth. Moreover, in Digges’s dedicationto his patron, the Earl of Leicester, he emphasizes his many practical services espe-cially in navigation and military applications.85

Certain sections of the texts, such as the extended titles, epistles “to the reader”,and dedications to a patron, were commonly used to attract the interests of poten-tial readers and to portray mathematics as vital to such useful activities as astronomy,navigation, surveying, gunnery, architecture and mensuration. Aside from the ex-travagant claims about the usefulness of some of the instruments, there was never ahint of anything that might be construed as an occult association. In Blundevile’sExercises (1594), a comprehensive treatise that covered topics ranging overarithmetic, trigonometry (at least in terms of using tables), cosmography, the use of


globes, maps, and a discussion of navigation, he makes it clear in his address to thereader that he hopes to appeal to the English gentlemen, from both court and coun-try, who were interested in travelling by sea and who would therefore like to learnenough mathematics to understand the principles of navigation.86 Other practition-ers, such as Richard Witt, made their intended audience clear by the title of theirwork, for example, “Arithmeticall Qvestions, Touching The Buying or Exchangeof Annuities, Taking of Leases for Fines, or yearly Rent; Purchase of Fee-Simples;Dealing for a Present or Future Possessions; and Other Barggines and Accounts,wherein Allowance for Disbursing or Forbeareance of Money is Intended ...”. InWitt’s address to the reader, he makes it very clear that men could not properlytrade with one another, except with great losses, without using mathematics. Heends this section by commanding his readers to “Perswade thy selfe, Arithmetickeis profitable, not onely to men in their private affaires, but also in the Common-wealth bussinesses: as well in time of warre as peace”.87

Extreme cases of the rhetoric of utility were the texts that were so obviouslypractical that nothing needed to be said. For instance, in Edward Wright’s The de-scription and vse of the sphaere (London, 1613) there was no dedication, preface,or introductory remarks whatsoever. The work begins with a table of contents thatmakes it clear that the text consists of a detailed, technical account of the uses ofthe sphere. Only one picture of the instrument is included. The work begins with adescription of the sphere and globe, and goes on to explain the use of various as-pects such as the zodiac, the meridian, and the tropics. It then explicates proceduressuch as “To rectife the sphere; that is, to sett the sphaere to the latitude of the placefor which you would use it”. This text was recommended by Captain John Smithfor the use of mariners,88 and perhaps Wright felt that the usefulness of the spherewas too obvious to need labouring. Edmund Gunter’s Use of the sector, crosse-staffe, and other instruments (London, 1624) also dispenses with these typical de-vices. Instead, it jumps directly into details, and seems aimed more towards fellowprofessionals, being dedicated in the title to “such as are studious of Mathematicallpractise”, than a general audience. It might have been intended more to familiarizeother mathematical practitioners with these instruments so they could, in turn, ex-plain them to their students. Perhaps, once again, the instruments’ utility is sup-posed to speak for itself.

UTILITY EMBEDDED IN MATHEMATICAL PRACTICE

The problems themselves were often constructed in what we might now consider a“story problem” fashion in order to present mathematics as useful in everyday prac-tices. John Tapp, in his The path-way to knowledge (1633), used a variety of differ-ent occupations such as merchants and noblemen’s stewards when giving examplesof addition. Although the examples are somewhat contrived, they show a desire todisplay mathematics in action. A merchant, for instance, has received severalpayments from a debtor, and wishes to know the total sum. The steward wishes to


know the total he has spent on household expenses.89 Questions concerning moneywere very popular: “In 345 pounds, how many pence?”90 Even more advanced ques-tions, such as the extraction of roots, were put into what was supposed to be apractical setting and given “real-life” characters who could benefit from mathemati-cal knowledge. Tapp wrote:

A Captain over an Army of men, being to attempt the winning of a Cittie: So itis that by reason of a ditch which compasseth the walles thereof, hee cannotcome neere the walles to place Ladders there....91

The problem, which is to find the proper height of the ladder, involves taking asquare root, and is solved by using a geometrical instrument. In some instances, notonly is a ‘real-life’ problem provided, but also a picture of a person solving itinstrumentally is included. Even without pictures of instruments, however, the moredifficult procedures, such as the taking of square roots, were usually linked to prac-tical activities. In Blundeville’s Exercises (1594), after he describes his methodsfor taking square roots, he immediately shows how this technique may be used inthe “setting of battels”. This is characterized as being an Italian procedure, usableby a “Sargent major” in the field, for the arranging of his armies. It was supposed toinform one of how many ranks, and how many men per rank, were needed in orderto form popular battle formations. Although it is difficult to envisage soldiers paus-ing before the beginning of a battle to compute square roots, these examples pro-vided at least a practical façade. The intention seems to be to provide illustrations,whether they were feasible or not, of mathematics in practice that could be utilizedin a variety of situations.

The questions of exactly how useful practical mathematics was, and who actu-ally read these texts, are both difficult to answer. We should be extremely carefulabout accepting at face value practitioners’ claims concerning mathematics’ utilityand ties to craft practices. Some of the more basic arithmetic and geometric opera-tions taught in the more elementary texts were both undoubtedly useful and some-thing one could learn by oneself through careful reading. Addition, subtraction,multiplication and division could all be useful in keeping household accounts, per-forming monetary conversions, and dividing up profits. Skill at measuring, weigh-ing and finding of distances could similarly be used for many practical applications.There are signs, however, that the practitioners themselves were not convinced thatall students could easily acquire these various techniques by reading texts. Worsop,in his work on surveying, claimed to be a simple man writing for the commonpeople, hinting that texts by such learned scholars as Recorde and Digges would betoo difficult.92. William Bourne, in his The treasure for traveilers (1578), also claimedto be an “unlearned” author, writing for the common people. Practitioners seemedto make distinctions concerning the difficulty level of various texts; although to acertain extent this might have been a strategy to promote their texts at other au-thors’ expense, some texts were clearly simpler than others.

For instance, Recorde’s The whetstone of witte (1557), unlike his other works,


The “Mathematical Iewel”, from the title-page of John Blagrave’s book of that name(London, 1585).

FIG. 1.

was never reprinted. Here, besides using “practical” examples, Recorde claims thatthe extraction of roots “serueth so many waies, in building: in proieaion of plattes,for measuring of ground Timber, or stone: And also in warre, for framyng of battailes,for makying of diuerse engines, and generally for all woorkes of Geometrie andAstronomie”.93 There are, however, two problems with his claim. First, it wouldhave taken an exceptional student, in the days when multiplication and divisionwere viewed as complex operations, to have mastered the art of taking square rootsfrom his explanation alone, without further aid from a teacher. The explanation ofthe process given is purely by rote, in a recipe-like fashion, with no hint of theunderlying reasoning behind it. One could very quickly run into situations where itwould have been unclear just how one should proceed. Second, as already remarked,


it seems highly unlikely that battle captains actually paused before engagements tocompute square roots to form their battalions. In the section dealing with “cossike”numbers, Recorde introduced algebraic equations that were supposed to be an-swering questions concerning bricks, silk, walls, and wills among other things; yetthe fact that no other text devoted mainly to algebra was written for another sev-enty-four years makes it unlikely that he convinced his audience of the utility ofthese techniques.

William London’s A catalogue of the most vendible books in England (1657),gives us some idea of the variety of the mathematical texts sold, if not of who waspurchasing them. The section discussing mathematics, however, is grouped togetherwith “Horsemanship, Faulconry, Merchandize, Limning, Millitary Discipline,Herauldry, Fire-works, Husbandry, &”, making it seem that he was trying to appealto a reasonably well-off audience. The gentry, and the better-off merchants, mighthave purchased “practical” texts for their amusement, even though they were saidto be aimed at a more “vulgar” audience. Indeed, some of the instruments described,such as John Blagrave’s “Mathematical Iewel” (Figure 1), would have been beyondmost people’s means. Interestingly, the list also indicates by its inclusion of magi-cal and astrological texts that mathematics could be construed to have occult asso-ciations as late as 1657.

Not only the choice of problem, but often the instruments themselves, were usedas tools of persuasion. Many texts were dedicated to explicating the making anduse of a particular instrument, and they often contain drawings of the instrumentsin action (Figure 2). Fabulous claims, about what the instrument could accomplishif only it were purchased, served not only to sell the instrument in question but alsoto convey the extraordinary usefulness of mathematical practice. For instance, in1585 Blagrave in his The mathematical iewel claimed that his own instrument couldserve in place of a quadrant, ring, “dyall”, astrolabe, sphere, globe “or any such likeheretofore deuise”, and that it was a direct pathway to knowledge in “the wholeArtes of Astronomy, Cosmography, Geography, Topography, Navigation, Longitudesof Regions, Dyalling, ... with great and incredible speede, plainenesse, facillitie,and pleasure”.94 In a way, these instruments can be viewed as embodied forms ofpersuasion. They were physical items, which could be placed in a client’s handsand whose practical uses could be easily depicted. In some cases, however, such aswith Blagrave’s “iewel”, it is unclear how easy to use and how practical such acomplicated instrument could actually be. Additionally, instruments could be usedas persuasive devices in that they can also be clearly meant as a display of wealthand power. For example, part of Richard Delamain’s success in obtaining the pa-tronage of Charles I seems to have been related to his willingness to devise large,silver instruments that could be used as a form of princely display.95 Although theyserved many other purposes, instruments also provided a method for allowing thepractitioners’ clients and patrons to embrace the outward form of mathematicalpractice without the need to delve into theoretical considerations. As StephenJohnston has shown, instruments were, in many ways, as important as texts in shaping


an image of mathematical practice as practical, profitable and pleasurable.96 Thedanger of utilizing instruments in this fashion is, as Zetterburg has pointed out, thatthe descriptions of the instruments’ usefulness might be so exaggerated as to makethe instrument in question seem magical.

An example of an instrument in action, from John Babington’s A short treatise of geometrie(London, 1635), 54.

FIG. 2.


BUILDING ALLIANCES

Another tactic used to persuade people that mathematics was useful, and to helpdisassociate it from occult practices, was to emphasize its links with a particularlypopular and profitable activity. Robert Recorde included references to explorationand navigation in his Castle of knowledge (1556), a work on the sphere in dialogueform that discussed astronomical instruments in general, introducing into his dia-logues “Calicutt, Peru, and the Cape of Good Hope”, along with other newly-foundlands, to illustrate the various positions of the Earth with respect to the Sun.97 Fur-thermore, in his 1557 Whetstone of witte, the first vernacular English algebra text,which was dedicated to the Muscovy Company, he pledged to write a book thatwould concentrate on the problems of northerly navigation. Indeed, he even prom-ised to show them a way to “Northe Easte Indies”, a region more accessible than“southernly Cathay”.98 Thomas Digges claimed to find by “Demonstrations Math-ematical ... the great imperfections in the Arte of Navigation, & grosse Errourspracticised by the masters and Mariners of this our age”.99 He attempted to createan expert role, and to supply advice for the reformation of mariners’ charts, instru-ments, and rules.100 Likewise, John Tapp (fl. 1596–1631), whose mathematical prob-lems were described above, transformed himself from a member of the Draper’sCompany into a teacher of navigation, joined the Stationer’s Company, and beganto publish nautical books and mathematical works that would appeal to marinersand other practical men.101 Tapp began by dedicating his Path-way to knowledge(1613) to “Sir Thomas Smith, Knight, Gouernor of the Company of Marchants ofLondon Trading the East-Indies and the Moscouie Company, as also the Companyof Discouerers for the North-West passage, and treasure for the plantation in Vir-ginia”, making it clear that he was also trying to attract an audience that was inter-ested in navigation. Smith was one of the greatest promoters of overseas enterprises,and he was one of the prime undertakers for the discovery of the North-West pas-sage.102 Tapp appealed to this audience to learn mathematics because it “is the chiefand most effectuall branch” for the “nourishment” of navigation. Additionally, hepraised his patron for providing ships, mariners, settled trade and new discoveries“to the great benefit of many thousands imployed therein, a continvall profitte tothe common-wealth, and a lasting glory to our Nation”.103 The persuasive tech-nique is to emphasize mathematics’ close association with navigation, an activitythat was associated with a similar rhetoric of practicality, profitability, and serviceto the commonwealth.

Indeed, exploration and navigation are both areas that in this period saw a greatlyexpanded amount of mathematical endeavour. As mentioned earlier, David Waters’simportant book, The art of navigation in England in Elizabethan and early Stuarttimes, provides a detailed account of the scope of these mathematical pursuits.104

E. G. R. Taylor, in an equally foundational work, also explores the increasing in-volvement of mathematicians in these spheres of activity.105 These are both seminalworks and much of their material still stands securely, but they share an implicitjudgement that the triumph of mathematical navigation was inevitable. In this period,


it was not yet completely clear what mixture of techniques would be the most effec-tive, or who would win the authority to provide expert advice. It was not that anyoneseriously argued that navigation could completely do without mathematics; it wasmore a question of the usefulness of mathematics when it was not coupled to practi-cal sailing skills, and a struggle over who had the proper qualifications to provideexpert advice. The rhetoric of utility was only one method used to help convince themathematical practitioners’ audience that their services were necessary.

Practitioners were called upon to persuade not only the mariners, but also theirpatrons, of the viability of their techniques. Yet even if the promoters of a voyagewere convinced, it did not necessarily mean that the practitioner’s advice would beheeded by the mariners or that their techniques, when they were adopted, wouldwork. In the north-west passage voyages undertaken by Captains Luke Foxe andThomas James in 1631, although James heeded the practitioners’ advice, Foxe didnot. Instead, Foxe argued that mathematical theory is useless without experience atsea.106 Although Foxe’s backers were willing to supply him with the latest newbooks and instruments, he refused them. He felt it would be too late, if there was anemergency at sea, for him to read about the proper course of action. It does notseem, however, that Foxe was unwilling to take advice; he studied with John Tappand Thomas Sterne, as well as learning from Henry Briggs. Nevertheless, he claimedhe would be too busy sailing to bother with books. Yet his voyage was a greatsuccess. James, on the other hand, although he had carefully taken expert advice,seeking out journals, maps, discourses, and setting skilled workmen to making himinstruments, had in many respects a disastrous voyage.107 His ship took on water,and there was a series of accidents. In the end, only James, the Master and theSurgeon were in sound health after wintering in what we would now refer to asJames’ Bay. Besides the Gunner, the Master Warden, the Carpenter, and the Quarter-Master all died. To add to his difficulties, some of his instruments, such as hisclocks, turned out not to work in the harsh climate of the north.108 As James put it,“As for our clock and watch, notwithstanding we still kept them by the fireside, ina chest, wrapped in cloths, yet they were so frozen they could not go”. James’svoyage was a terrorizing one, full of ice, fogs, and storms. Sadly, for all of histroubles, James’s voyage, unlike Foxe’s, made no great discoveries.109

As these north-west passage voyages show, accepting practitioners’ advice didnot necessarily lead to a successful journey. In this period, it was not completelyclear what techniques would be the most effective. Some master mariners, likeWilliam Borough, were advocating creating their own experts who combinedpractical sailing skills with mathematical theory.110 Even what seem to us to begreat successes, such as Edward Wright’s Certaine errors in navigation (1599),display great concern over whether their work will be accepted by mariners. Wrightworried that seafaring men were satisfied with their current methods or that theywould “condemne Vniversities and all in comparison of their manifold experi-ments”.111 Claiming his work would have found favour if it had been written by a“master of the sea”, he believed it would be less popular because a mathematician


wrote it.112 Moreover, as we have seen, captains such as Foxe were still successfullycompleting their voyages without, like James, bringing with them books contain-ing expert advice and all the latest instruments.113 It seems reasonably clear that theseamen were not meekly guided towards improved navigation by mathematicalpractitioners.114 Persuasive techniques, such as the rhetoric of utility, were an im-portant aspect of overcoming resistance to the reform of navigational practice, butit was a negotiated process whose outcome was uncertain.115 It was not until 1673,when The Royal Mathematical School was founded, that mathematical navigationbecame established officially as a necessary component of some mariners’ educa-tion.

CONCLUSION

As we have seen, from at least 1550–1650 mathematics was linked to magic in thepublic’s mind, and it had a dubious reputation. Gradually, throughout this period,the rhetoric of utility helped to reshape mathematics’ associations to profitabilityand pleasure. It is doubtful, however, that the rhetoric of utility alone, or even theactual usefulness of mathematics itself, could have been so successful at thisrefashioning if opinions concerning natural magic had not also been undergoing re-alignment. Recent research in the history of magic has emphasized the differencebetween so-called natural magic on the one hand and symbolic magic and demon-ology on the other.116 While natural magic, which was based on the assumption thatall bodies had various natural powers enabling them to affect various other bodies(namely those that “corresponded” to them in the “great Chain of Being”), waslargely combined with traditional natural philosophy to give rise to somethingrecognizably closer to modern science, symbolic magic and demonology wereincreasingly rejected as beyond the pale. During this transformation of natural magicnot all parts of the tradition proved equally useful to the new philosophers. Themore pragmatic parts of alchemy were preserved at the expense of the more mysti-cal side, giving rise to something we might accept as chemistry, while astrologywas increasingly perceived to be implausible and was rejected.

Unfortunately, the exact reasons why some portions of the magical tradition wereabsorbed into natural philosophy while others were rejected have yet to be totallyascertained. As we have already noted, magic had a dubious public image, and, asJohn Henry points out, it “made sense for reforming natural philosophers to addtheir own voices to the denunciation of magic, while they extracted what they rec-ognized to be useful out of the tradition”.117 We should keep in mind, however, thatthis was an uneven, complex process. In the second half of the seventeenth centurynatural philosophers such as Boyle and Newton were still drawing upon the oldernatural magic traditions. It is perhaps worth adding that the rhetoric of utility itselfowed something to the older natural magical tradition. Magic was always used forpractical benefit and the rhetorical emphasis on utility by propagandists for thereform of natural philosophy can be seen simply as yet another appropriation from


magic.118 It is hardly surprising, therefore, that mathematicians, previously seen asworking within the occultist tradition, should have at their fingertips a repertoire ofarguments for the usefulness of their art.

Thus practitioners were aided in their quest to associate mathematics with usefulactivities instead of occult practices by the overall shift in perception concerningwhat was meant by magic. Many of the negative connotations could be pushedentirely over to astrology, which was eventually rejected by both mathematics andthe new philosophy. The refashioning of the image of mathematics, and its disso-ciation from astrology, however, was also a complicated, long-drawn-out process.A 1657 list of “vendible”, or best-selling, books includes astrological and magicaltexts in with the “Books Mathematicall”.119 The push, on the other hand, in the1640s and 1650s, to add practical mathematics to various curricula by certain fac-tions in the education reform movements, shows that mathematics was accepted bymany as a profitable, pleasurable activity.120 Considering how few English math-ematical practitioners were active in 1550, and the doubtful reputation of math-ematics in the second half of the sixteenth and the beginning of the seventeenthcentury, versus both the greater number of practitioners and the improved image ofmathematical practice by 1650, it seems that the frequent emphasis on utility wasindeed a persuasive defence against charges of dangerous occultism.

ACKNOWLEDGEMENTS

I am very grateful for comments on earlier drafts of this paper by John Henry,Frances Willmoth, and the anonymous referees. I would particularly like to thankJohn Henry; his encouragement and advice were vital to the rewriting of my paper.Any faults that remain are, of course, my own.

REFERENCES

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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

18. The famous story alleges that when Dee left Mortlake privately in 1583 to embark for Holland, amob, believing him to be a magician, broke into his house and destroyed a great part of hisfurniture 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 Julian Roberts and AndrewG. Watson (eds), John Dee’s library catalogue (London, 1990), for a refutation.

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

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

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

22. Edward Worsop, A discoverie of sundrie errours and faults daily committed by landemeaters(London, 1582), sig. Fr, sig. E3l.

23. Worsop, op. cit. (ref. 22), sig. F2r.

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

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

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

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

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

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

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

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

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

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

34. Thomas, op. cit. (ref. 15), 358–68.

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

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

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

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

39. Roger Ascham, The scholemaster, ed. by John E. B. Mayor (London, 1863; rpr. New York, 1967),14–15. Ascham’s text was printed in 1570, with two more editions in 1571, one edition in1573, and one edition in 1589.

40. For Ascham’s career as a mathematical lecturer, see P. L. Rose, “Erasmians and mathematics atCambridge 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 letterto the Earl of Leicester, stating “I think you did yourself injury in changing Tully’s wisdomwith Euclid’s pricks and lines” (in J. A. Giles (ed.), The whole works of Roger Ascham (3 vols,London, 1864–65), ii, 103).

41. Stephen Shapin, A social history of truth: Civility and science in seventeenth-century England


(Chicago, 1994); Peter Dear, “Totius in verba: Rhetoric and authority in the early RoyalSociety”, Isis, lxxvi (1985), 145–61. See also Marello Pera and William Shea (eds), Persuadingscience (Canton, Mass., 1991) and Peter Dear (ed.) The literary structure of scientific argument:Historical studies (Philadelphia, 1991).

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

43. 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.

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

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

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

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

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

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

50. Edmund Wingate, Arithmetique made easie, in tvvo bookes (London, 1630), sig. Ar.

51. Ibid.

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

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

54. M. Blundevile, M Blundevile His exercises, containing sixe treatises ... (London, 1594), sig. A3r;Worsop, A discoverie of sundrie errours ... (London, 1582), sig. A2r–A2l.

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

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

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

58. Robert Brenner, Merchants and revolution: Commercial change, political conflict, and London’soverseas traders, 1550–1653 (Cambridge, 1993), 5.

59. Brenner, op. cit. (ref. 58), 12–14.

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

61. Waters, op. cit. (ref. 5), 542–3.

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

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

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

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

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

67. Elyot, op. cit. (ref. 66), 45–46.


68. Sir Humphrey Gilbert, “Queene Elizabethes Academy”, in F. J. Funivell (ed.), Queene ElizabethesAcademy (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, Apoem on heraldry, Occleve on Lords’ men &c (London, 1969), 4–5.

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

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

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

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

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

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

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

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

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

78. Robert Recorde, The grounde of artes (London, 1542), sig. Aivr.

79. Recorde, Grounde (ref. 78), sig. Biiir.

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

81. Recorde, Grounde (ref. 78), sig. Biiir.

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

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

84. Thomas Digges, Stratioticos (London, 1579), sig. Aiijr.

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

86. Blundevile, His exercises (ref. 54), sig. A4l.

87. Richard Witt, Arithmeticall qvestions ... (London, 1613), sig. A3r.

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

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

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

91. Tapp, op. cit. (ref. 89), 305–6.

92. Edward Worsop, A discoverie of sundrie errours (London, 1582), sig. A2l–r.

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

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

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

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

97. John Parker, Books to build an empire: Bibliographical history of English overseas interest to1620 (Amsterdam, 1965), 55–56.

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

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


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

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

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

103. Tapp, op. cit. (ref. 89), sig. A2l.

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

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

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

107. Thomas James, The strange and dangerous voyage of Captain Thomas James (London, 1633),sig. A2v.

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

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

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

111. Edward Wright, Certaine errors in navigation; The voyage of ... to the Azores (London, 1599),sig. ||3r.

112. Wright, op. cit. (ref. 111), sig. ||3r–l.

113. James does not mention very precisely which books he brought with him, stating merely “AChest full of the best and choicest Mathematicall bookes that could be got for money in England;as likewise Master Hackkluite and Master Purchas, 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 broughtvery few instruments. He mentions a compass, which he said was unreliable (p. 309), a log-line, and a semi-circle.

114. William Barlow, The navigators svpply (London, 1597) is another example of suggestions offeredcombined with concern over acceptance by sailors. He relates a story about one of Sir FrancisDrake’s voyages to show what happens to those who fail to listen to their experts; Drake endedup sailing in a circle, arriving back where he began after 16 days, because he would not listento his navigator who was aware of the difficulties that ensued due to the variation of the compass(sig. a3l). Moreover, he realized that some would suggest that his lack of experience made himan unfit guide, but he claimed “And in the minde onely, pure and true Arte, refined from thedroße of sensible or experimental knowledge, is to be found” (sig. a5r–br ). In the end, heassumed that the more skilful sailors would accept his suggestions.

115. 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), writtenin response to John Chamber’s A treatise against judicial astrology (London, 1601), where themain argument seemed to be that astronomy was the highest form of human knowledge in thatthe “ravishing beautie, constant order, and powerfull efficacy of the celestiall bodyes ... lead


us to God” (sig. &3r). Additionally, he claimed that astrology and astronomy were in fact thesame thing (sig. A1r), and also maintained that he was only defending ‘proper’ astrologerswho “contain themselves within the bounds of naturall Philosophie”.

116. Stuart Clark, “The rational witchfinder: Conscience, demonological naturalism and popularsuperstitions”, in Pumfrey, Rossi and Slawinski (eds), op. cit. (ref. 2), 222–48; idem, “Thescientific status of demonology”, in Vickers (ed.), op. cit. (ref. 12), 351–74; and John Henry,“‘The touch of cold philosophy? ...’: The fragmentation of Renaissance occultism and theorigin of the Enlightenment”, unpublished manuscript. I thank John Henry for letting me readthis work.

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

118. Paolo Rossi, Francis Bacon: From magic to science (Chicago, 1968); and Henry, opera cit. (refs9 and 117).

119. William London, A catalogue of the most vendible books in England, sig. Dd2r–Dd3l.

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

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