LESLEY B. CORMACK

THE SIXTEENTH AND SEVENTEENTH CENTURIES have long been seen as fundamentally important to an understanding of the changing study of nature. The changes in this period have been variously categorized by historians as philosophical, methodological, or mathematical, among other explanations. One of the profound transformations that took place was the introduction of mathematics into the language of description and explanation of nature. Many historians and philosophers have investigated this translation, which allowed the description of the occult and unseen forces of nature, introduced a new logical structure and language, and made natural knowledge increasingly useful for technological changes. Once adopted, the advantages of a mathematical lexicon were clear. But how and why did that adoption take place? In the sixteenth century, mathematics was a study separate from (and inferior to) natural philosophy; thus, the story of the mathematization of the worldview is also the story of how mathematics and mathematicians came to have a status previously afforded to philosophy alone. In order to understand how and why those who studied nature came to adopt the language of mathematics, we need to look at the men who were using mathematics in their everyday work—mathematical practitioners. Mathematics and mathematical practitioners played an essential role in the transformation of science in this early modern period, as we can see by examining the role of mathematics and mathematical practice, utility, commerce, and trade on the changing ideology and methodology of science.

The great early and mid-twentieth-century historians of scientific ideas, scholars such as Edwin Burtt, Herbert Butterfield, and especially Alexandre Koyré and E. J. Dijksterhuis, argued that the move to mathematical language, and with this a move toward a mechanization of nature, was the defining characteristic of the age they called the scientific revolution (Burtt 1924; Butterfield 1949; Koyré 1957; Dijksterhuis 1961).1 Indeed, the discipline of history of science really began in the twentieth century by focusing on the problem of the origin of modern science. The work of some of its great founders concentrated on what this important transformation was and how it took place. In the 145 years between Copernicus and Newton, people interested in the Book of Nature developed new methodologies including experimentation; new attitudes toward knowledge, God, and nature; a new ideology of utility and progress; and new institutional spaces and practices.2 They began to view the world as quantifiable, investigable, and controllable. By the end of the period, the investigation of nature, still tied to theological concerns, but increasingly to practical ones as well, was carried out in completely new places, for different ends, and with quite different results.

Why was mathematics so powerful for these historians? First, and perhaps most obviously, modern science was heavily mathematical and so when they sought the origins of modern science, mathematics was a necessary prerequisite. Mathematics, especially Euclidean geometry, introduced a new logical rigor into argumentation. While there had been important disagreements about the status of mathematical truth, with Peter Ramus for example arguing that mathematics was a natural attribute of humans made more abstract and obscure by mathematicians such as Euclid, and Henry Saville insisting that abstraction demonstrated the perfection of mathematical knowledge (Goulding 2010), mathematics seemed to offer a clear language and to demonstrate underlying truths about nature. Those who argued for its importance, like John Dee or Isaac Newton, quoted the familiar biblical passage, “thou hast ordered all things in measure and number and weight.”3

A problem with this grand historical narrative is that in the medieval and early modern periods mathematics was not part of natural philosophy. Mathematics was a separate area of investigation from natural philosophy and those interested in mathematical issues had usually tied such studies to practical applications, such as artillery, fortification, navigation, and surveying.4 The mathematical quadrivium and natural philosophy were studied in different parts of the university curriculum and the status of mathematicians was substantially lower than that of philosophers or theologians (Feingold 1984; Biagioli 1989, 2007). Given this split, it is not self-evident that a natural philosopher would see the benefits of adding mathematics to his explanations of the world.

And yet, it is clear that many who studied and developed explanations of nature did employ mathematics. Studies of the work of Galileo, Huygens, Newton, and Leibniz, among others, point to the profound importance of mathematics to their explanations and worldviews. Such scholars asked different questions of nature—questions about measurement and prediction—because they had access to mathematics. They devised new ways of doing mathematics. So what were the origins of this interdisciplinary moment?

The key to understanding this development must be sought in the socioeconomic transformation of Europe, not simply in a metaphysical gestalt switch. A sociological change in who, where, and why the world was investigated was taking place.5 A crucial category of scientifically inclined men downplayed by most historians of the period, the mathematical practitioners, was crucial to this transformation.6 These mathematical practitioners became more important in the early modern period and provided a necessary ingredient in the transformation of nature studies to include measurement, experiment, and utility (Bennett 1991).7 Their growing importance was a result of changing economic structures, developing technologies, and new politicized intellectual spaces such as courts and merchants’ shops, and thus relates changes in ‘science’ to the development of mercantilism and the nation-state.

`THE STUDY OF MATHEMATICS`

In the early sixteenth century, few European scholars were interested in questions of mathematics. While Merton College in Oxford, for example, had been famous throughout Europe in the thirteenth and fourteenth centuries for its school of kinetics, this fame had dwindled by the sixteenth century and scientific study had largely been superseded by more humanistic pursuits. Most scholars and educational reformers in the first half of the sixteenth century were more concerned with the introduction of classical languages and literatures, and especially with the religious controversies swirling around them than with the structure of the natural world. This began to change in about the middle of the century and by 1600, natural philosophers and mathematicians were active and innovative. Why then did scholars, educational reformers, and practitioners change their minds?

One of the answers is that the goals of education altered dramatically during the century. Where few courtiers, politicians, or civil servants were university-trained at the beginning of the century, by 1600 a university education was practically a requirement (Stone 1964; see also Cormack 1997; McConica 1986). This changed the content of education as well as its delivery. The direction of education was also linked to patronage and as patronage patterns changed, natural philosophers and mathematicians were increasingly sought (Moran 1991; Biagioli 1993; Smith and Findlen 2002). These patrons, both merchants and courtiers, were responding to an evolving economic imperative, brought about by the rapid expansion of mercantile and trading opportunities. These activities, particularly by the trading companies, were in turn partly made possible by the increasing sophistication of the natural philosophers and mathematical practitioners.

The impetus for increased mathematical education and investigation came from outside the traditional scholarly community. Merchants and trading companies in northern Europe particularly were interested in applied mathematical practice, especially navigation, surveying, and accounting. This need became more pronounced as these merchants changed their focus to the Atlantic trade, putting them in competition with the much better informed Spanish and Portuguese. By the mid-sixteenth century, merchant companies believed that they needed a more theoretical grounding in mathematics, especially because they needed to navigate largely uncharted northern waters, and they began to patronize mathematical practitioners. Eventually, this led to important connections among these practitioners, skilled artisans, and natural philosophers.

The English Muscovy Company provides an interesting example of the new mercantile emphasis on mathematics. The merchants in this trading company recognized their need for mathematical knowledge in order to undertake significant ocean voyages and soon commissioned Robert Recorde to write mathematical books for the use of their navigators. In 1551 Recorde published Pathway to Knowledge, an explication of geometry through the first four books of Euclid’s Elements, and in 1556 (reissued 1596), The Castle of Knowledge, containing the explication of the Sphere, dealing with spherical geometry, astronomy, and navigation (Recorde 1551; 1556). The latter was written and printed for the use of the Muscovy Company, and mentioned the Portuguese discoveries in order to illustrate the positions of the earth with respect to the sun. It was based on Ptolemy’s astronomy and incorporated more recent astronomical work, including a brief, favorable mention of Copernican theory. Recorde’s Whetstone of Witte (1557), his explication of algebra, was dedicated to the governors of the Muscovy Company and written, so Recorde claimed, to encourage the great exploration and trading enterprise on which they were embarked. He even promised to produce a future book (never written), in which “I also will shewe certain meanes how without great difficultie you mai saile to the North-Easte Indies. And so to Camul, Chinchital, and Balor” (Recorde 1557, fol. a3b). Whetstone was never reprinted, perhaps because it dealt with difficult mathematical concepts. It was based on German algebraic texts, including the treatment of the quadratic. Taken as a whole, Recorde’s mathematical books contained a full course of mathematical study and many Elizabethan natural philosophers and mathematicians began their education with Recorde’s books.8 In this way, he was hugely influential in developing the English scientific endeavor, which therefore owed much to mercantile patronage.

The teaching and learning of mathematical knowledge in early modern Europe thus involved a complex interaction among scholars, practitioners, merchants, and gentry. Humanists and scholars at the university saw the value of mathematical knowledge for its intrinsic natural philosophical benefit as a way of understanding God’s handiwork (see, for example, Elyot 1531, sig. 37a; Pace 1517, 109). Mathematics became an important part of both the formal and informal curricula at early modern universities (Feingold 1984). Equally, practitioners—instrument makers, entrepreneurial teachers, mathematical practitioners—were interested in mathematical knowledge for its application, its rhetorical power, and its value to potential patrons and employers (Recorde 1543, sig. A2a). Merchants and gentry needed secure knowledge for navigation, warfare, and investment.

Thus, the teaching and learning of mathematics took place at a variety of venues, some formal and some informal. The formal educational system was itself in a period of expansion and change, moving from an earlier church-based and -oriented institution, to one catering to a wider social demographic and to more political and mercantile career paths. At the same time, many young men (and some young women) had increasing access to printed texts, allowing them to teach themselves, and a new group of entrepreneurial teachers sprang up who could supply alternative instruction, either through personal tutoring, group lessons, or through the writing of self-help texts designed for the autodidact.

`MATHEMATICAL PRACTITIONERS AND MATHEMATICAL LECTURES`

London was a busy metropolis in the last decades of sixteenth century, both for the numerous and hard-working merchants and for those more interested in mathematical and natural philosophical pursuits (Harkness 2007).9 The Inns of Court, Parliament, and the Royal Court all provided reasons for many young men and women to find their way to the city. Combined with a growing interest in trade, investment, and exploration, London was an increasingly attractive destination for young men from the country, fresh from university or their estates, eager to make their way in the world and to find communities of like-minded individuals. These new inhabitants of London, combined with skilled émigrés fleeing the religious troubles of the continent, ensured that there was both the expertise in mathematics and a ready market for this expertise. In the second half of the sixteenth century, a number of university-trained or self-taught men set themselves up as mathematics teachers and practitioners. These men, who we might call mathematical practitioners, sold their expertise as teachers through publishing textbooks, making instruments, and offering individual and small group tutoring. In the process, they argued for the necessity of practical knowledge of measurement, winds, surveying, and mapping, among others, rather than for a more philosophical and all-encompassing knowledge of the earth.

Most mathematical practitioners were university-trained, showing that the separation of academic and entrepreneurial teaching was one of venue and emphasis, rather than background. Mathematical practitioners claimed the utility of their knowledge, a rhetorical move that encouraged those seeking such information to regard it as useful.10 It is impossible to know the complete audience for such expertise, but English mathematical practitioners seem to have aimed their books and lectures at an audience of London gentry, merchants, and occasionally artisans.11 It is probably this choice of audience that most influenced their emphasis on utility, since London gentry and merchants were looking for practicality and means to improve themselves and their businesses.

Mathematical practitioners professed their expertise in a variety of areas, especially such mathematical applications as navigation, surveying, ballistics, and fortification. For example, Galileo’s early works on projectile motion and his innovative work with the telescope were successful attempts to gain patronage in the mathematical realm.12 Descartes advertised his abilities to teach mathematics and physics. Simon Stevin claimed the status of a mathematical practitioner, including an expertise in navigation and surveying.13 William Gilbert argued that his larger philosophical arguments about the magnetic composition of the earth had practical applications for navigation.

In England, an early example of a mathematician using his expertise to improve the mathematical underpinning of these useful arts was Robert Recorde, employed by the Muscovy Company to give lectures and write a textbook in elementary mathematics in the 1550s (Cormack 2003; Johnston 2004). Recorde’s early foray was to be repeated, especially in London, by mathematical practitioners, many of whom, such as Thomas Hood and Edward Wright, demonstrated an interest in mapping and navigation explicitly.

These mathematical practitioners offered lectures, individual tutelage, and the instruments to explicate the mathematical structure of the world. Sometimes this was done on a completely entrepreneurial model, that is, where the practitioner hung out his shingle and attracted clients through publishing and publicity. At other times, mathematics lectures were founded and supported by a small group of interested men, such as was the case with Thomas Hood.

`THOMAS HOOD AS THE FIRST LONDON MATHEMATICAL LECTURER`

Thomas Hood (1556?–1620) was the first mathematics lecturer paid by the city of London and thus fits a patronage model of mathematics lecturers. However, he also published and encouraged private pupils, and therefore was equally an entrepreneurial mathematics teacher. Hood attended Trinity College, Cambridge, where he received his bachelor’s degree in 1578 and his master’s in 1581.14 In 1588, Hood petitioned William Cecil, Lord Burghley, to support a mathematics lectureship in London, to educate the “Capitanes of the trained bandes in the Citie of London.”15 This was a complex proposition because the Aldermen and Lord Mayor of London would be the ones paying the bills, but the Privy Council had to give its approval in order to allow the lectures to proceed.

Hood received the following positive response from the Privy Council: “The readinge of the Mathematicall Science and other necessarie matters for warlike service bothe by sea and lande, as allso the above saide traninge shalbe continued for the space of 2 yeares frome Michaelmas next to come and so muche longer as the L. Maior and the Citie will give the same alowance or more then at this present is graunted.”16 Hood’s lectureship therefore went forward, held in the home of Sir Thomas Smith, merchant and later governor of the East India Company. The makeup of the audience is now unknown, although from the tone of his introductory remarks, published under the title A Copie of the Speache made by the Mathematicall Lecturer, unto the Worshipfull Companye present … in Gracious Street: the 4 of November 1588, Hood seemed to be talking to his mathematical colleagues and mercantile patrons, rather than to the mariners he insisted needed training (Hood n.d., 1588, sig. A2aff). The contents of Hood’s lectures are also unknown, but the treatises bound with the British Library copy indicate that he stressed navigational techniques, instruments, astronomy, and geometry (Hood 1596, 1598).17

By 1590, Hood had been giving these mathematics lectures for almost two years, as he reported in his 1590 translation of Ramus: “so that the time limited unto me at the first is all most expired.… In this time I have binne diligent to profite, not onlie those yong Gentlemen, whom comonlie we call the captaines of this citie, for whose instruction the Lecture was first under taken, but allso all other whome it pleased to resorte unto the same” (Ramus 1590, sig. 2a). Hood identified himself on the title pages of all his books until 1596 as “mathematical lecturer to the city of London,” sometimes advising interested readers to come to his house in Abchurch Lane for further instruction, or to buy his instruments.18 His books explain the use of mathematical instruments such as globes, the cross-staffe, and the sector, suggesting that his lectures and personal instruction would have emphasized this sort of instrumental mathematical knowledge and understanding. While some historians have questioned what happened at Hood’s lectures (or if indeed they did happen), this larger evidence indicates both that there were such lectures, and that a number of leaders of the community, as well as mathematical practitioners like Hood, thought they were important in creating mathematical literacy and conversation in the city of London.19 This was the beginning of a recognition of the power of mathematics for understanding the answers to practical problems and with it a sense that mathematical answers were as legitimate as philosophical ones.

`MATHEMATICS CHANGES THE GEOGRAPHICAL CONVERSATION`

Given that the connection between mathematics and natural philosophy was a new interdisciplinary interaction, the best place to find such an interconnection would be in a study that blended measurement and larger philosophical theories. One such area of interest was to be found in the mathematical study of geography.

In sixteenth-century England, geography was a flourishing area of investigation. It was studied as part of the arts curriculum at both Oxford and Cambridge and therefore made up part of the worldview of most educated gentlemen and merchants.20 The study of geography included a mathematical model of the earth, descriptions of its distant lands and inhabitants, and the local history of more immediate surroundings, what I have elsewhere labeled mathematical geography, descriptive geography, and chorography.21 Because it relied on geographers of antiquity, such as Ptolemy and Strabo, to provide a backbone for modern investigation, geography was a discipline that used the methods of the humanists and the tradition of university scholars. Equally, geography was a study inspired by and reliant on new discoveries, voyages, and travels and so was integrally connected to the testimony and experience of practical men. Thus, geography existed as a point of contact for theoretical university scholars and practical men of affairs. Equally, it provides an excellent example of how mathematics could change the natural philosophical conversation, as well as the people conversing.

Geography embodied that dynamic tension between the world of the scholar, since geography was an academic subject legitimated by its classical, theoretical, and mathematical roots, and the world of the artisan, since it was inexorably linked with economic, nationalistic, and practical endeavors. It provided a synthesis that enabled its practitioners to move beyond the confines of natural philosophy to embrace a new ideal of science as a powerful tool for understanding and controlling nature. The usefulness of geographical study was of paramount importance to the new men attending the universities in ever greater numbers and it was this concept of utility to the state and to the individual that drove these new university men to investigate and appreciate geography.22 The geographical community, then, was a wide-ranging group, with many different concerns and goals, but with a desire to be useful to the nation and to their own self-interest and a vision of England as an increasingly illustrious player on the world stage.

The English geographical community was complex, due in large part to its necessarily close connection between handwork and brainwork. Even the most theoretical geographer required the information and insight of navigators, instrument makers, cartographers, and surveyors in order to understand the terraequeous globe. This can be seen in the work of Richard Hakluyt, who used sailors’ tales to construct a description of the world and England’s role in its discovery, and who in Principal Navigations created a predominantly practical document with important theoretical insights. Edward Wright, a serious mathematical geographer whose first-hand experience on voyages of discovery deeply affected his research program, also provides an important example of someone who mediated between theory and practice. Equally, the collaboration between John Dee, a university-trained mathematician and geographer, and Henry Billingsley, a London merchant, in the 1570 translation of Euclid indicates the fruitful exchange between the life of the mind and that of the marketplace (Dee 1570). Dee’s career provides a particularly telling example of the importance of the theoretical-practical spectrum, and with it an interest in both mathematics and natural philosophy. Dee would probably have identified himself as a natural philosopher and certainly worked throughout his life to create a new theoretical worldview as well as to achieve a higher social status. Yet he was engaged much of the time in more practical mathematical pursuits, especially astronomical and geographical ones.23 He advised most navigators setting out on northwest or northeast voyages, devised map projections and navigational instruments, and wrote position papers for the Privy Council on the political ramifications of English geographical emplacement.24 Thus Dee, like many other mathematical practitioners, developed multiple and overlapping roles as scholar, craftsman, and statesman. This complex social world encouraged such men to integrate mathematics into their larger natural philosophical investigations and explanations.

`WRIGHT AND HARRIOT AS MATHEMATICAL PRACTITIONERS`

There are many examples of these interactive roles and disciplines. Two geographers who combined the life of the natural philosophical scholar with that of the mathematical practitioner were Edward Wright (Apt 2004) and Thomas Harriot (Roche 2004). Both were university-educated men, who had learned the classical foundations of their subject, as well as recent discoveries and theories. But these two were not isolated or traditional scholastics. Both went on prolonged voyages of discovery and learned navigation and its problems from the rude mechanicals and skilled navigators they encountered. They recognized the need to use mathematics to measure and understand the practical problems they encountered. They went beyond this practical knowledge, however, to try and formalize the structure of the globe and the understanding of the new world. Both were connected with important courts and patrons, and both used the cry of utility and imperialism to argue the need for geographic knowledge.

Edward Wright, the most famous English geographer of the period, was educated at Gonville and Caius College, Cambridge, receiving his bachelor’s degree in 1581 and his master’s in 1584. He remained at Cambridge until the end of the century, with a brief sojourn to the Azores with the Earl of Cumberland in 1589.25

In 1599 Edward Wright translated Simon Stevin’s The Haven-finding Arte from the Dutch (Wright 1599a; Taylor 1954, #100). In this work Stevin claimed that magnetic variation could be used as an aid to navigation in lieu of the calculation of longitude (Wright 1599a, 3).26 He set down tables of variation, means of finding harbors with known variations, and methods of determining variations. In his translation Wright called for systematic observations of compass variation to be conducted on a worldwide scale, “that at length we may come to the certaintie that they which take charge of ships may know in their navigations to what latitude and to what variation (which shal serve in stead of the longitude not yet found) they ought to bring themselves” (Wright 1599a, preface, B3a).27

Wright’s work demonstrates a close connection between navigation and the promotion of a “proto-Baconian” tabulation of facts meant both for practical application and scientific advancement. Here appears the foundation of an experimental science, grounded in both practical application and theoretical mathematics, quite separate from any more traditional Aristotelian natural philosophy or Neoplatonic mathematics. Unfortunately, Wright’s scheme was not entirely successful. By 1610, in his second edition of Certaine Errors in Navigation, Wright had constructed a detailed chart of compass variation—but he had also become more hesitant in his claims concerning the use of variation to determine longitude (Wright 1610a, sigs. 2P1a-8a; Waters 1958, 316).

Wright’s greatest achievement was Certaine Errors in Navigation (1599), his appraisal of the problems of modern navigation and the need for a mathematical solution. In this book, Wright explained Mercator’s map projection for the first time, providing an elegant Euclidean proof of the geometry involved. He also published a table of meridian parts for each degree, which enabled cartographers to construct accurate projections of the meridian network, and offered straightforward instructions on map construction (Wright 1599b, sigs. D3a-E4a; Taylor 1954, #99). He also constructed his own map using this method. Wright’s work was the first truly mathematical rendering of Mercator’s projection and placed English mathematicians, for a time, in the vanguard of European mathematical geography. It was equally significant for the close communication it claimed and required of theoretical mathematicians and practical navigators.

At about the turn of the century, Wright moved from Cambridge to London, where he established himself as a teacher of mathematics and geography, following in the footsteps of Robert Hood. At about the same time, he contributed to Gilbert’s work on magnetism, providing a practical perspective to Gilbert’s more natural philosophical outlook (Pumfrey 2002, 175–81). He created a world map using Mercator’s techniques and probably aided in the construction of the Molyneux globes (Wallis 1952; 1989, 94–104). In the early seventeenth century, he is said to have become a tutor to Henry, Prince of Wales (elder son of James), a claim strengthened by Wright’s dedication of his second edition of Certaine Errors to Henry in 1610 (Wright 1610a, sigs. *3a–8b, X1–4; Birch 1760, 389). Upon becoming tutor, Wright “caused a large sphere to be made for his Highness, by the help of some German workmen; which sphere by means of spring-work not only represented the motion of the whole celestial sphere, but shewed likewise the particular systems of the Sun and Moon, and their circular motions, together with their places, and possibilities of eclipsing each other. In it was a work by wheel and pinion, for a motion of 171000 years, if the sphere could be kept to long in motion” (Birch 1760, 389).28

Henry had a decided interest in such devices and rewarded those who could create them.29 In addition, Wright designed and constructed a number of navigational instruments for the prince and prepared a plan to bring water down from Uxbridge for the use of the royal household (Strong 1986, 218; Wright 1610b, identified by Taylor 1934). In or around 1612, Wright was appointed librarian to Prince Henry, but Henry died before Wright could take up the post (Strong 1986, 212). In 1614, Wright was appointed by Sir Thomas Smith, governor of the East India Company, to lecture to the company on mathematics and navigation, for which he was paid £50 per annum (Waters 1958, 320–21). There is some speculation as to whether or not Wright actually gave these lectures, since he died the following year.

Wright thus provides a nice example of a mathematical practitioner who provided both intellectual and social connections between theory and practice. He was university-trained and worked as a teacher at various points in his career. He was interested in theoretical problems, including the mathematically sophisticated construction of map projections, and aided Gilbert in his philosophical enterprise. On the other hand, this was an academic who respected practical experience. He himself experienced the problems of ocean navigation, he built instruments, and he solicited the help and opinion of sailors and navigators. His motivation for this balancing of handwork and brainwork were many, probably including financial gain and social prestige as well as more intellectual concerns. He was certainly concerned with the usefulness of his investigations and, through the patronage support of aristocrats, Prince Henry, and the East India Company (somewhat latterly), was able to argue the utility of geographical knowledge both to imperial and mercantile causes. Mathematics provided a language for both his practical and theoretical pursuits, demonstrating a new integration of these different branches of knowledge.

Another preeminent figure in mathematical geography, also connected with Prince Henry, was Thomas Harriot (Shirley 1983). Harriot attended Oxford at the same time as Wright was at Cambridge. He matriculated from St. Mary’s Hall in 1577 and received his bachelor’s degree in 1580. By 1582 he was in the employ of Sir Walter Ralegh, who sent him to Virginia in 1585. Harriot, like Wright, was an academic and theoretical geographer whose sojourn into the practical realm of travel and exploration helped form his conception of the vast globe and of what innovations were necessary to travel it. Harriot’s description of Virginia, seen in his Brief report of … Virginia (1588),30 was “the first broad assessment of the potential resources of North America as seen by an educated Englishman who had been there” (Quinn 1974, 45).31 Harriot compiled the first word list of any North American Indian language (probably Algonquin) (Shirley 1983, 133), a necessary first step of classifying in order to control, thus illustrating that inductive spirit never far from the heart of even the most mathematical geographer. He saw Virginia’s great potential for English settlement, provided that the natives were treated with respect and that missionary zeal and English greed were kept to a minimum.32 His advice concerning Virginian settlement was to prove important as the Virginia companies of the seventeenth century were established. This was the work of a man very aware of the practical and economic ramifications of the intellectual work of describing the larger world, as well as the imperial imperatives at work.

More important for Harriot were issues of the mathematical structure of the globe. Indeed his mathematics was bound up closely with his imperial attitude generally and the experience of his Virginian contacts in particular.33 He was deeply concerned about astronomical and physical questions, including the imperfection of the moon and the refractive indexes of various materials (Shirley 1983, 381–416). Harriot was inspired by Galileo’s telescopic observations of the moon and produced several fine sketches himself after The Starry Messenger appeared. He also investigated one of the most pressing problems of seventeenth-century mathematical geography—the problem of determining longitude at sea. Harriot worked long and hard on the longitude question and on other navigational problems, relating informally to many mathematical geographers his conviction that compass variation contained the key to unraveling the longitude knot (Harriot 1596).

Harriot was a mathematical tutor to Sir Walter Ralegh for much of the last two decades of the sixteenth century, advising his captains and navigators, as well as pursuing research interesting to Ralegh. As Richard Hakluyt said of Harriot, in a dedication to Ralegh: “By your experience in navigation you saw clearly that our highest glory as an insular kingdom would be built up to its greatest splendor on the firm foundation of the mathematical sciences, and so for a long time you have nourished in your household, with a most liberal salary, a young man well trained in those studies, Thomas Hariot, so that under his guidance you might in spare hours learn those noble sciences.”34

As Ralegh fell from favor, eventually ending up in the Tower, Harriot began to move his patronage expectation to another aristocrat interested in mathematical and geographical pursuits, the ninth Earl of Northumberland (the so-called Wizard Earl). Although Harriot’s relationship with Northumberland is somewhat obscure, he appears to have conducted research within Northumberland’s circle and occasionally his household, as well as acting as a tutor as needed. Finally, Harriot was also connected with Henry, Prince of Wales, as a personal instructor in applied mathematics and geography, just as Wright had been (Shirley 1985, 81). It is likely that Wright and Harriot met at Henry’s court. As two university-trained contemporaries, with very similar interests and experiences, they would have gained much from their association. Given their mutual interests, it would have made sense for them to discuss matters of mutual geographical and mathematical interest while at court together.

Harriot’s career displays many of the same characteristics as Wright’s. Harriot too was a man who drifted in and out of academic pursuits, from university, to Virginia, to positions as researcher and tutor for Ralegh and Northumberland. In some ways, he was less connected to practical pursuits than Wright, although his trip to Virginia and his work on longitude indicate his engagement with issues of practical significance. Harriot was also dependent on patronage, especially that of Ralegh and of Northumberland (poor choices as they turned out to be), and used this patronage to help create an intellectual community in which mathematical theory and imperial utility could be considered equally important.

Wright and Harriot, as well as a host of other geographers interested in this interconnection between theoretical and practical issues, combined an interest in the mathematical construction of the globe and a new, more wide-reaching understanding of basic geographical concepts with a desire for political and economic power on the part of princes, nobles, and merchants. This wide-ranging area of investigation encouraged associations to develop between academic geographers, instrument makers, navigators, and investors. The result was a negotiation between theoretical and practical issues, which helped introduce mathematics as a common language and rigorous means of analysis. This fruitful association between theory and practice helped to determine the kinds of questions these men asked, the kinds of answers that were acceptable, and the model of the world that would be developed. It was the work of mathematical practitioners such as these geographers that introduced mathematics to natural philosophical questions and to questions of natural knowledge more broadly. The utility, or at least the perceived utility, of such a language is part of the reason that mathematics became the language of nature in the years to come.

`CONCLUSION`

Hood, Wright, and Harriot provide good examples of the kind of investigators necessary for the introduction of the language of mathematics into the study of nature. These three men, and many other mathematical practitioners, represent the communication between theory and practice, both within their own careers and ideas, and between universities, courts, print shops, the shops of instruments makers, and many other liminal venues. Their lives and careers show that new locales were becoming important for the pursuit of natural knowledge, including urban shops and houses on the one hand, and the courts and stately homes of aristocratic and noble patrons on the other. Wright and Harriot also demonstrate within their scientific worldviews an interesting mixture of theory, inductive fact gathering, and quantification, which provided part of the changing view toward nature and its investigation so important for the changing emphasis on mathematization. They were both concerned with practicality and utility, especially within the rhetoric they employed to argue their cause, and mathematics seemed to them to provide the useful answers necessary to their careers and their ambitions. But they were also convinced that mathematics would help them to understand the natural world more sufficiently. Their connections to mercantilism were important, but do not provide a complete answer to the changing emphasis of the study of nature (as Edgar Zilsel [1942] once suggested).35 This was not science directed by the bottom line of mercantilist expenditure, but rather a more complex interaction among court, national and international intellectual communities, and mercantilist enterprise.

Thus, the mathematical practitioners provided an agent for the changing nature of the scientific enterprise in the early modern period. They created a fundamental step toward the introduction of the language of mathematics into the larger study of nature. They did this by combining theory and practice in a new and interesting way. They did so for reasons that included the economic and bourgeois changes that were directly affecting Europe. These men were also concerned with issues of nationalism, imperialism, cultural credit, and status, issues that do not fit easily into a more Marxist and materialist interpretation.

Did this change the enterprise of natural philosophy? Yes. Because these men were interested in mathematics, measurement and quantification became increasingly more significant. Their social circumstances ensured that the investigation of nature must be seen to be practical, using information from any available source, and science developed a rhetoric of utility and progress, as well as an inductive methodology, in response. Intimately connected to national pride and mercantile profit, the science that developed in this period reflected those concerns. In essence, in large part because of the work of mathematical practitioners like Hood, Wright, and Harriot, the investigation of nature began to take place away from the older university venue (though there remained important connections), with new methodologies, epistemologies, and ideologies of utility and progress. The scientific revolution had begun.

But there was still something missing. Hood, Wright, and Harriot did not make the transition to natural philosophers. Despite their best efforts, they remained mathematical practitioners. And by the end of the seventeenth century, mathematical practitioners had been reduced to technicians, whose presence became less and less visible.36 Meanwhile, natural philosophers such as Robert Boyle and Isaac Newton removed themselves from the company of mathematical practitioners, even as they utilized the fruits of their labor. In other words, the final translation of mathematics as a tool and language of natural philosophy involved another social transformation, which devalued the very group that had made it possible.

`NOTES`

1. See Lindberg (1990, 1–26) for a discussion of early uses of the scientific revolution as a concept, and Lindberg (16) and Cohen (1994, 88–97) for a fuller treatment of Burtt.

2. Steven Shapin (1996), despite his opening caveat, does a good job of laying out some of the changes taking place that made up the scientific revolution, as more recently has John Henry (2001).

3. Wisdom of Solomon 11:20.

4. Kuhn (1977, 31–65) separates these two traditions. See also Cunningham and Williams (1993) and Dear (2001) for discussion of this separation.

5. Steven Shapin (1982) made a case for this new interpretation, and then, with Simon Schaffer, provided an extremely influential case study (1985).

6. With some modification, I take the important classification of the more practical men in Taylor (1954). For modern treatment of these crucial figures, see Bennett (1986) and Johnston (1991; 1994). Most recently, Pamela Long (2011) discusses these issues.

7. Kuhn (1977) provides an early attempt to claim a different history for mathematics and natural philosophy.

8. In Cormack (1997, 108, 110) I argue that Recorde’s books were owned by many university students and college libraries in the late sixteenth and early seventeenth centuries.

9. For a more general discussion of early modern London, see Rappaport (1989) and Archer (1991).

10. Neal (1999) discusses some attempts to make mathematics appear useful.

11. Thomas Hood’s lecture, (n.d. 1588) is a good example. See Harkness (2007) for a discussion of the complex interactions among London merchants, artisans, and scholars.

12. Of course, once Galileo successfully gained a patronage position, particularly with the Florentine Medici court, he left his mathematical practitioner roots behind and became a much higher status natural philosopher (Biagioli 1993).

13. Descartes was Jesuit-trained (Dear 1995).

14. Biographical material on Thomas Hood can be found in Taylor (1954, 40–41); Waters (1958, 186–89); Higton (2004).

15. British Library, Lansdowne 101, f. 56.

16. British Library, Lansdowne 101, f. 58.

17. Hood (1596; 1598) are bound together in BL 529, g. 6.

18. Thomas Hood lists himself as a mathematical lecturer on the frontispiece of the following books: Hood (n.d. [1588]; 1590); Ramus (1590); Hood (1592; 1596).

19. Further evidence of Hood’s lectures is the fact that John Stow mentions them (1598, 57).

20. See Cormack (1997, 17–47) for a full treatment of the place of geography in the university curriculum.

21. See Cormack (1991) for a description of the different types of geography studied in sixteenth- and seventeenth-century England.

22. See Stone (1964) for an evaluation of the growing numbers of new men at the universities in this period.

23. For the natural philosophical work, see Clulee (1988). For his practical advising, see Sherman (1995).

24. In many ways, Dee is an English Galileo, providing a crossover from mathematical practitioner to court natural philosopher. It is no surprise, however, that Zilsel did not mention him, since his magical heritage, made famous by Frances Yates (1972), among others, discounted him in Zilsel’s mind as a true scientist (Zilsel 1942; Harkness 1999).

25. As a result of this voyage, Wright wrote (1589), which was later printed by Richard Hakluyt, “written by the excellent Mathematician and Enginier master Edward Wright” (1598–1600). Hall (1962, 204), Waters (1958, 220), and Shirley (1985, 81) all cite this trip to the Azores as the turning point in Wright’s career, his road to Damascus, since it convinced him in graphic terms of the need to revise completely the whole navigational theory and procedure.

26. Bennett (1991, 186) marks the relationship between magnetism and longitude as one of the important sites of the scientific revolution.

27. Waters (1958, 237).

28. “Mr. Sherburne’s Appendix to his translation of Manilius, p. 86” in Birch (1760, 389).

29. Smuts (1987) especially mentions Salomon de Caus’s La perspective avec la raison des ombres et miroirs (London, 1612), dedicated “Au Serenissime Prince Henry,” 157.

30. Harriot, Briefe and True Report, reproduced verbatim in T. de Bry, America. Pars I, published concurrently in English (Frankfurt, 1590) and in Hakluyt (1598–1600, 3:266–80).

31. See Alexander (2002) for an interesting interpretation of Harriot’s mathematics.

32. The manuscript information concerning this expedition is gathered together in Quinn (1955, 36–53). Shirley (1983, 152ff) discusses Harriot’s desire for noninterference. To see White’s illustrations of this expedition, see White (2006).

33. Alexander argued that Harriot’s work on the continuum was influenced by his view of geographical boundaries and the “other.” “The geographical space of the foreign coastline and the geometrical space of the continuum were both structured by the Elizabethan narrative of exploration and discovery” (1995, 591). Alexander (2002) develops this further.

34. Richard Hakluyt, introduction to Peter Martyr, as quoted in Shirley (1985, 80). See Shirley (1983) for a fuller discussion.

35. See Raven and Krohn (2000, xx–xlvi) for an appraisal of the intellectual climate in which Zilsel worked.

36. Brotton (1997, 186) shows that cosmographers had become employees of the joint stock companies by the end of the seventeenth century, while Shapin (1994, 355–408) argues for the increasing invisibility of technicians. Sprat (1667, 392) celebrates the distance between gentlemen who create new knowledge and technicians who can only do as they are told. Jardine (2003) suggests that Hooke remained a technician.

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