« Les hommes sont dans un État ce que des instrumens de musique sont dans un orchestre: ils rendent des sons plus ou moins agréables, suivant qu’ils sont bien ou mal touchés. »
De La Beaumelle Laurent Angliviel, Mes pensées ou le qu’en dira-t-on, Paris, Rollin Fils, 1753, p. 73.

Introduction

1Laurent Angliviel de La Beaumelle’s metaphor above hints of a dialectic relationship between men and the state administrations of early modern Europe. In these hierarchic states, subjects were expected to be handled so that they produced results agreeable to the public. The cameralists of the developing state apparata, the supposed hands and eyes of the sovereign, were expected to be the handlers of said subjects. Historians have generally understood eighteenth-century “science” and “technology” in relation to a range of processes by which modern phenomena came to be: for example scientific expertise, the rising nation state, and rational industrial production2. However, as recently shown by a number of scholars, long narratives are problematic if we wish to understand how historical actors related to the anticipations of their contemporaries. Their studies have shown how, rather than acting as agents of change at odds with their time, allegedly forward-looking projectors performed to appease contemporary patrons and audiences3. That is, the supposed handlers themselves struggled to be instruments, who produced sounds agreeable to their lords.

2This chapter focuses on mathematics, in an early modern broad sense, encompassing not only pure mathematics such as algebra and geometry but also arts that in various ways were seen as dependent on mathematical principles such as for example mechanics, fortification and optics. It examines such mathematics as a technique for men to become, and be, a part of the Swedish state of the first half of the eighteenth century. More specifically, it studies the becoming of a mathematically based community of men in the Bureau of Mines [Bergskollegium]4. The Bureau’s archive, today stored in the Swedish National Archives (RA) in Stockholm, constitutes a valuable and little-used source for studying the role of mathematics in an early modern state. There, one finds not only protocols of the Bureau’s board, but also applications of young men who wished to auscultate in the Bureau. That is, they wished to become mining officials by participating in the work of the officials and, therefore, they presented themselves in relation to the expectations of the superiors. By relating these protocols and applications to the officials’ publications on the role of mathematics in their community, it is possible to draw a broad picture of the changing roles of such knowledge in an early modern European mining administration.

3The officials of the Bureau were responsible for overseeing two of the most important activities of eighteenth-century Sweden (next to agriculture): the production of, and trade with, metals. At that time, metal production saw a marked increase, and it has been argued that this development can be attributed to the new rational methods put in place by cameralists, chemists and mechanical practitioners. I am less concerned with these changes. Instead, my focus is the norms and skills of these civil servants. During the first half of the eighteenth century, mathematics – alongside natural history, jurisprudence and chemistry – gained an important symbolical role in the Bureau. From having been considered the knowledge of machine builders and makers of subterranean maps, the mathematical sciences formed a common epistemology that united a wider group of men around a common view of not only state and nature, but also of themselves. This group of civil servants, whose objective was to observe nature, had close connections with the universities of the realm and especially with natural philosophers in Uppsala. By studying the Bureau, it is thus possible to understand not only the role of knowledge in an early modern state apparatus, but also aspects of the development and teaching of mathematical sciences in eighteenth-century Sweden.

The mining sciences

4Because of its importance for the economy, the early modern Swedish state took an increasing interest in metal production. From the 1630s, the government took measures to improve the supervision of the mines. In 1637, Queen Christina founded a general mining board [generalbergsamt] that inspected mining and smelting. Its officials were expected to police the mines of the crown and the board was supposed to recruit competent men who could replace dismissed or deceased officials5. The general mining board soon consolidated its place in the state. As early as 1649, it attained the status of an administrative bureau. From this point on, the council was not merely a board of experienced men, but a Bureau of Mines6.

5By the early eighteenth century, the Bureau had become a well-established institution. There, administrative sciences, such as jurisprudence and œconomy, coexisted with other knowledge geared towards metal production, which the officials called mining sciences [bergssciencer or bergswettenskaper]7. These sciences were a way to categorise the activities of the Bureau’s officials as a respectable form of knowledge making, and thus formed a means of status. In controversies concerning their rank, the men of the Bureau repeatedly used their double competence – of juridical matters as well as of the so-called mining sciences – as an argument for why they could claim a higher status, compared with other servants of the state8. These sciences thus constituted a symbolical resource in the Bureau’s interaction with other parts of early modern society.

6What constituted the mining sciences was historically contingent. Over time, they came to consist of, for example, physics (in the broad early modern meaning of physica, denoting knowledge of nature), chemistry and mathematics (also in a broad sense)9. Previous studies have primarily discussed the Bureau of Mines as a place of chemistry10. As shown by Hjalmar Fors, in the late seventeenth century, Urban Hiärne introduced chemistry into the Bureau’s work in a Laboratorium Chymicum11. Still, in the applications from the first half of the eighteenth century, few aspiring auscultators discussed chemistry explicitly. Instead, the craft of assaying was a reoccurring topic. The lack of applicants discussing chemistry in their applications thus supports the claim that chemistry developed as an “art” from the 1720s to 1740s and a “systematic science” from the mid-1700s to the end of the century12. In comparison, in the applications of the early 1700s, mathematics was repeatedly used as a unifying concept encompassing geometry, arithmetic but also so-called “practical mathematics” such as mechanics. Although uncommon during the first decade, from 1700 to 1750, on average 18 % of the applicants mentioned mathematics, roughly comparable to the 16 % who mentioned assaying.

7But such statistical figures do not show how, as the number of vernacular publications increased, the roles of mathematics changed. In an extractum protocolli of 28 November 1715, issued to all auscultators of the Bureau, the board argued that an auscultator was required to show that he had studied mining and smelting [bergsväsendet] and that he knew how to apply his acquired knowledge. The board stressed that “mathesis” was an important basis for a future official13. Correspondingly, between 1710 and 1750, it was relatively common to mention mathematics in one’s application to the Bureau. But what was meant by mathesis changed from the early to mid-eighteenth century. By following the uses of mathematics in the Bureau, it is possible to delineate how it, in the words of Liliane Hilaire-Pérez, became “embedded within specific communities because of their needs and constraints, their habits and symbols, and their territories14”. Interestingly, the roles of mathematics in the Bureau changed in ways hard to reconcile with simple long narratives of the rise of applied useful knowledge or expertise.

Mathematics as mechanics (1700-1720)

8From the 1690s, practical mathematics found a place in the Bureau in the form of mechanics. In 1697, the mechanicus Christopher Polhem managed to convince the king, Karl XI, to establish a Laboratorium Mechanicum in the Bureau under his command, formed on the model of Hiärne’s chemical counterpart15. In Polhem’s Laboratorium, young men were to be educated in mechanics and mathematics. To encourage such apt young men, on 22 July 1699, two stipends in mechanics were established in the Laboratorium16. The establishment of the Laboratorium mechanicum can be seen as a manifestation of a shift in the Bureau from curious chemical knowledge to useful knowledge based on mechanics, when mechanics replaced chemistry as the science favoured by the state17.

9In the first two decades of the century, mathematics and mechanics were mostly discussed by applicants who aimed for the stipends in mechanics. In the applications of the early 1700s, mathematics was thus almost exclusively associated with mining mechanics, and the discussions of mathematics can be found among a clearly delineated subgroup of auscultators with personal links to Polhem18. These applicant’s relationships with Polhem are unsurprising, given his role at that time as the final authority on identifying mechanical competence in the Bureau, and it was difficult for young men to assert their mechanical and mathematical skill without his support.

10In 1715, when Anders Gabriel Duhre and Georg Rudin applied for the stipendium mechanicum, Polhem instead recommended his younger son Gabriel. He considered Duhre to be competent in “mathesi pura” but argued that he was too old to learn the “mechanical praxis”. Instead, Polhem argued, Duhre would “provide his Majesty a much greater service, if he were used as a teacher for the young [at the naval base] in Karlskrona in the things that belong to navigation”. The board agreed that one of the stipends should be given to Gabriel Polhem, but sidestepped the opinions of Polhem Senior in the case of Duhre. Although Polhem Senior gave a negative review of Duhre’s mechanical skill, the board responded as if he had been in favour of him. They wished to give Duhre the other mechanical stipend because of “the good recommendation from [Polhem] […] that he has put down an uncommon diligence in mathesis”. The Bureau proposed that Polhem should try to find Duhre a position in Karlskrona himself, and decided that they would give Duhre the stipend should he fail19.

Mathematics as the basis of a community (1720-1750)

11The admittance of Duhre, against Polhem’s recommendation, could be seen as indicating that the authority of the old mechanicus was diminishing by the late 1710s. By 1715, when Duhre applied for the stipend, the Laboratorium Mechanicum had became relatively inactive. Svante Lindqvist has argued that between 1715 and 1725 “mechanical engineering was […] in poor shape” in the Bureau20. Interestingly, whereas the activity in the Laboratorium dwindled, the number of applicants who mentioned mathematics grew. Tis is indicative of a development where the term “mathematics” gained a new meaning for the Bureau’s officials. It is this new role, related to much more than mechanics, which interests me here.

12In contrast to the applications of the first decades, none of these latter ones mentioned Christopher Polhem. Lindqvist points out that, by the 1720s, men such as Urban Hiärne and Christopher Polhem became representatives of a past era, when men had been able to rise in the Bureau through royal command instead of through the “normal channels”. In the 1720s, however, having been a client of the former absolute monarch hindered rather than facilitated advancement21. Furthermore, few young men of the 1720s wished to present a relationship with a client of the old king, such as Polhem. Thus, it was not in order to display their allegiance to the old mechanicus that young applicants presented their mathematical competence to the board.

13Moreover, by the 1730s the strong correlation between mentions of mechanics and mathematics, found in previous applications, disappeared. In this decade, 19 of the 76 applicants (25 %) mentioned having studied mathematics and these studies’ importance to mining work. Only six mentioned mechanics, and only one, Samuel Sohlberg, received the stipendium mechanicum. At this time, there was therefore a large group of young men who presented mathematical skill to the Bureau, but who did not necessarily associate this skill with mining mechanics. In the 1740s, in total 18 out of the 88 applicants mentioned mathematics in their application, but only three mentioned mechanics. These applicants of the 1730s and 1740s did not necessarily just see mathematical exercise as a way to become a mining mechanicus. Instead, they discussed mathematics as relevant to the whole business of the Bureau. The applicants described how they had studied things “that facilitate mining, and the adherent mathematical and mechanical sciences.” Or how, because of their interest in and inclination for mining, they “especially had endeavoured to learn Mathesis and Physics”22. Tis function of mathematics post-Polhem could be seen as a framework based on a common way of seeing and thinking about the world and of categorising knowledge. This new role of mathematics is very similar to that given by contemporary eclectic philosophers in Helle23.

14This new role of mathematics is evident in the prefaces to two mathematical textbooks, written by auscultators of the Bureau in the late 1710s. In A Fundamental Guide to Mathesis Universalem and Algebra (1718), Georg Brandt, accepted as an auscultator in 1715, argued to his superiors that mathematics was the fundamental structuring principle of the Bureau’s work. Brandt which based his book on the lectures of his fellow auscultator Anders Gabriel Duhre, can be seen as an early forceful argument for the relevance of mathematics beyond mining mechanics. As argued by Fors, Brandt would later become the “grand old man” of the group of chemists who saw mechanical philosophy as a basis for their art24. But, in Brandt’s preface, it is possible to discern an even wider symbolical role of mathematics as the basis for all mining work25.

15Brandt’s publication was dedicated to the Bureau’s vice president, the mine councillors [Bergsråd] and the Bureau’s assessors. In his preface, Brandt firmly placed himself and his publication within the hierarchies of the Bureau: it was meant for Brandt’s fellow civil servants, rather than for a European scholarly audience. From the start, Brandt conformed to the style of writing expected of a Stockholm cameralist. He humbly pointed out that his book should not be seen as a work “of any perfection”. His only intention was to give his “gracious superior” a “simple specimen of the diligence” that he had put into the mathematical sciences while being an auscultator. The mathematical sciences, he continued, were the foundation of “as many parts of the correct practice of mining, as of much other highly useful knowledge26”.

16Brandt’s preface put forth an epistemic framework in which mathematical knowledge, work and production were intertwined27. An important aim of the preface was to establish mathematics as a foundation for several branches of the Bureau’s work. Through mathematics, mining could “be noticeably improved28”. The preface argued that what “service and usefulness the Mathematical sciences provide for the mines is such a well-known matter, especially what concerns Mechanics, that there is no need to describe it further29”.

17Thus, Brandt’s aim was not to link mathematics to mechanics, nor to contest that mathematics was relevant to this field. The connection between mathematics and mining mechanics had already been done (for example by Polhem) and, according to Brandt, it was self-evident for the Bureau’s officials. For Brandt, it was instead central to show that by applying mathematics to the experience of nature it was possible to establish facts about nature. The prime example of such a practice, according to him, was “the splendid mathematical and philosophical works by the Englishman Newton, which clearly show how much in physics that can be discovered, using the help of mathematic30”.

18Duhre, whose lectures Brandt’s publication was based on, was a mathematician with an interest in mechanical inventions. Like the other officials of the Bureau, and following Polhem, Duhre considered mathematics, and more specifically geometry, to be the foundation of mechanical work. But in 1722, when arguing for an educational institution for the teaching of œconomical sciences using mathematical principles, Duhre, similarly to Brandt, argued for a broader role for mathematics. He pointed out how “chemical praxis and the handling of metals”, would be “improved through the application of geometry31”. Whereas Polhem had been the director of the Laboratorium mechanicum in the Bureau, Duhre proposed that he should be the director of a Laboratorium mathematico-œconomicum, signalling that mathematics was the basis for all useful œconomical knowledge. This institution would be even more short-lived than Polhem’s mechanical counterpart, in 1731 it was discontinued after only having been active for eight years. Nonetheless, it was followed by a Teatrum œconomicum, established at Uppsala University in 1741, which housed the professor in œconomy Anders Berch, a former official of the Bureau of Commerce32.

19A Fundamental Guide might be seen as the joint product of the mathematically inclined men Brandt and Duhre, both inspired by contemporary Newtonianism as well as the eclectic philosophy of Halle in their views on the role of mathematics in material and knowledge production. Duhre and Brandt presented the Bureau as a community that was (or at least should be) formed around a natural philosophy based on mathematical principles. This community should act according to common and predictable methods, knowledge and norms33. As seen from the protocols of the board from 1715, as well as from the applications to auscultate from the 1710s onward, Brandt’s linking of mathematics and mining in 1718 was hardly original. Still, what the unifying term of “mathematics” contained and how its contents were related to mining was not static. We could thus see the statements of Brandt and Duhre as arguments for the specific epistemology that gave mathematics a new and prominent position in the Bureau. What the two argued for in their publications was the reconfiguration of the category of the man of the state into one who based his work on an experimentalist framework underpinned by mathematics and especially geometry. Generally, their aim was to relate a much broader range of activities – such as œconomy, state administration and examination of nature – to mathematical principles in the same way as practical mathematical arts, such as mechanics, had been up until this point.

The mathematical man of metals

20The new role and meaning of mathematics in the Bureau of the mid 1700s can also be found in the Uppsala dissertation The Characteristics of an Ironmaster written in 1750, which describes this mathematical man of metals. The thesis was presented by the auscultator of the Bureau, Isaac Johan Uhr, under the praeses Berch, the previously mentioned professor of œconomy in Uppsala. In his dissertation, Uhr discussed what character, knowledge and skills an owner and director of an ironworks should possess, and argued for the importance of mathematics and physics in the making of such a man. Although the ironmasters were not the same men as the civil servants within the Bureau (however, many auscultators came from families of ironmasters and many auscultators eventually became ironmasters), it is clear that Uhr’s beliefs as to the knowledge and skills an ironmaster should possess reflected the norms of the community that he was becoming part of34.

21Uhr began his work with the common metaphor of society as a body consisting of diverse parts. As a part of this body politic, the role of the ironwork’s owner, much like that of civil servants, was to act in the interest of the whole of the state. Because the ironmaster was an important member of the social body, it was vital that he was knowledgeable in his trade and that he constantly strived to perfect it. According to Uhr, it was because of the ironmaster’s desire for continuous improvement, in the interest of the state, that metal making and mathematics merged. Such practices of self-betterment not only improved his ironworks: they were also exercises by which he bettered himself. Uhr presented mathematical studies as something that could be understood as a dialectic process of becoming; he explicitly pointed out that natural knowledge, good quality metal and virtuous men were created in parallel:

“Thus, when [the ironmaster] examines ores, and wishes to perfect the knowledge about them, he also examines and perfects himself to quite a useful member of society, and himself becomes both the worker in, and the product of, his workshop35.”

22Such examination of nature, which made an ironmaster into someone other than a commoner, was based on mathematical and physical principles. Uhr argued that common men, who only had the bodily skills required for menial labour, were merely able to carry out “what cultivated minds had conceived36”. However, men whose minds had been nurtured by the mathematical sciences had an obligation “to cultivate their natural senses to advance the public good37. Such men should have as their objective to “perfect their knowledge of the subject [of mining], and to constantly work to carry the arts to greater heights and perfection38”. Uhr specified how “the skill of an ironmaster begins with knowledge in mathematics, physics and chemistry39”. These skills identified an ironmaster as “a father, who gave birth to and nurtured as many useful members [of the state], as the number of young men he made into useful workers40”. Mathematical exercises would thus make a man into the mature patriarch whom the sovereign could trust to handle other men in ways that made them into useful subjects.

23In Uhr’s narrative, mathematical and natural philosophy constituted a boundary between the ironmaster and other men found in metal production. Likewise, by the 1730s, the officials of the Bureau identified themselves as socially and epistemically distinct by the manners in which they related mathematical and physical knowledge to their work. Mathematical proficiency was not only evidence of a set of skills, but also proved that you were a specific kind of man. In applications to auscultate from the 1730s and 1740s, we can see a rising number of young men who wished to adhere to the expectations of a civil servant who handled the realm according to mathematical principles.

Conclusions

24The archive of the Swedish Bureau of Mines and the publications of its members hint at a process, over the first half of the 18^th century, by which mathematics became relevant to a broader group of mining officials. At the same time, what the Bureau’s officials actually meant by mathematics changed too. At the onset of the eighteenth century, the term mathematics primarily denoted practical mathematical arts – such as mechanics and subterranean architecture – practised by artisans and mining mechanici. By mid-century however, the ability to present a foundation for one’s work based on mathematical, and especially geometric, principles became a shibboleth by which men portrayed themselves as useful and loyal hands of the king. Over time, mathematics – together with jurisprudence and natural history – became a means by which young men not only constructed useful machinery or understood natural phenomena, but also formed themselves in relation to the expectations of their time. That is, mathematics became part of intertwined dialectic processes of seeing, acting and of becoming.

25This narrative, situated in the state apparatus of a dwindling Baltic empire, is difficult to reconcile with long narratives of industrialisation and the rise of modern technology, or of how, during the eighteenth century, propositional knowledge was made useful by being integrated with production. In the Bureau, from having been a knowledge primarily connected with mining work, mathematics became a symbolical foundation for a group of men trusted to police the mining in the realm. In this community, mathematics thus carried a different meaning than among the entrepreneurs that economic historians such as Mokyr and Jacob have examined. Its role has more similarities with the role of œconomical knowledge among early modern cameralists. In the Bureau of the eighteenth century, mathematics was not primarily made prevalent by the mathematisation of artisanal work, but through the mathematisation of a large number of its officials. Whereas mathematics in the early 1700s was the main form of knowledge by which a mining mechanicus separated himself from a mere craftsman, in the mid-eighteenth-century Bureau mathematics also became a way for an official to separate himself from a mere commoner of the metal trade.