Underground Mathematics tells the story of subterranean geometry, a forgotten discipline that developed in the silver mines of early modern Europe. Seven case studies describe how an original culture of accuracy and measurement paved the way for technical and scientific innovations. Based on a variety of original manuscripts, maps and archive material, it recounts how knowledge was crafted and circulated among practitioners in the Holy Roman Empire and beyond. Specific chapters deal with the material culture of surveying, map-making, expertise and the political uses of quantification. By carefully reconstructing the religious, economic and cultural context of mining cities, Underground Mathematics argues that practical mathematics fruitfully interacted with the world of humanists, scholars and courts. In doing so, it contextualizes the rise of a culture of accuracy and quantification from 1500 to 1800. Subterranean geometry thus proves relevant to broader discussions in the history of science, technology, and knowledge.

Chapter 7, ‘One of Geometry’s Nicest Applications’, relates the digging of the Deep-George draining tunnel (1771–1800), named after George III, King of Great Britain and Hanover. During the planning phase, surveyors designed this engineering project with a previously unknown level of detail. Jean-André Deluc, Fellow of the Royal Society and reader to Queen Charlotte, visited the Harz mines three times as the operations were under way. Deluc’s geological and meteorological inquiries led him to perform barometric experiments in these mines. He relied on practitioners’ data to test and calibrate his instruments, marvelling about their precision in the Philosophical Transactions and the Journal des Sçavans. Scholars and amateurs – from Goethe to Watt – but also merchants and their wives rushed to visit the project. Year after year, journals reported to the public how the various sections of the tunnel connected seamlessly. In the late eighteenth century, the two worlds of natural scientists and mine engineers were meeting one last time, this time around common issues of precision, data gathering, and instrumentation.

Chapter 2, ‘A Mathematical Culture: The Art of Setting Limits’ brings the reader directly into early modern metal mines. The birth of a vernacular culture of geometry is described, detailing the daily work of craftsmen and insisting on the materiality of measuring practices. Surveys, carried out in public during solemn ceremonies, were a keystone of mining laws. The chapter exposes a central hypothesis of this book: At the time, mathematical accuracy acquired a dual meaning. Measurements had to be precise enough to solve intricate technical problems, while at the same time respecting procedures codified in mining customs and laws. Far from being a mere tool, geometry was meant to ensure trust; it was ubiquitous and pervaded many aspects of a miner’s life. In the early years of the Protestant Reformation, Lutheran pastors actively fostered the rise of practical mathematics. Mathematical and religious rationality were equated, making subterranean geometry accurate in a third way, this time as an expression of divine will. The omnipresence of measurements, combined with their legal and religious recognition, ultimately conferred a higher status to the discipline.

Chapter 4, ‘Writing It Down: Innovation, Secrecy, and Print’ explains how mining, and subterranean geometry, evolved during the troubled time of the Thirty Years War (1618–1648). It brings together issues related to book history as well as the history of training and teaching practices. Balthasar Rösler (1605–1673) introduced numerous innovations, and his teaching was disseminated by his students among mining regions, in a series of beautifully illustrated and hitherto unstudied manuscripts. The birth of this technical genre is presented in detail, with its evolution and uses within the training system of mining regions. In 1686, Nicolaus Voigtel then published the first practical textbook on the topic. Surprisingly, the craftsmen’s manuscripts weathered the rise of the printed press. I argue that authoring and publishing books failed to supersede the authority of practitioners precisely because their know-how was embedded in a specific technical and cultural setting. Subterranean geometry would stay an underground knowledge for another century, as most innovations arose within this handwritten tradition.

Chapter 3, ‘The Mines and the Court’ is set both in the Ore Mountains and at the court of Dresden under August of Saxony (1553–1586). It offers a broader picture of the vibrant intellectual life in mining cities, illustrated here by the example of Annaberg. Local officials and technicians developed remarkable skills in arithmetic and geometry, on which rulers came to rely to map their realms and tame their capricious landscapes. I focus on the careers of two dynasties of practitioners, respectively subterranean surveyors (the Öders) and reckoning masters (the Rieses). After both patriarchs contributed to the economic rise of the city, their descendants became versatile engineers and courtiers of the Saxon Electors. They collaborated with university professors and instrument-makers, using their skills all over the Electorate and beyond, temporarily turning the court of Dresden into a centre of practical mathematics.

Chapter 6, ‘How to Teach It: Finding the Right Direction’, offers a reappraisal of the foundation of mining academies. Subterranean geometry merges here with broader questions about technical education in the eighteenth century. Early attempts to replace the guild-like training and to establish brick-and-mortar institutions prompt a familiar debate between professors and practitioners. Who could best formalize and improve a century-old corpus? Moreover, what was the right way to teach it? Major mining centres, I argue, offered varied solutions to improve theoretical teaching, of which mining academies were but the ultimate step. I focus here on the biography of Johann Andreas Scheidhauer (1718–1784), mining master and autodidact mathematician. His vast geometrical production – unpublished and long forgotten – looms large in the early projects of mining academies, not least through the influence of his student Johann Friedrich Lempe (1757–1801), emblematic professor of the Bergakademie Freiberg.

Chapter 1, ‘Of Scholars and Miners’, introduces the discipline of subterranean geometry from the point of view of Renaissance scholars. Early modern humanists were fascinated by the underground world of metal mines. The richness of the geometrical thinking contained in Georgius Agricola’s De re metallica (1556) or Erasmus Reinhold’s On Surveying (1574) is presented. By comparing them with actual productions of contemporary mine surveyors, I further show that these books, despite their lifelike descriptions and illustrations, did not limit themselves to straightforward, faithful depictions of actual practices. Early modern readers were presented with rational reconstructions and pseudo-technical procedures. In spite of a thorough knowledge and a genuine interest for the underground world, scholars mainly used their writings on mines in a patronage context, or to display their interpretation of Euclidean geometry.

Underground Mathematics tells the story of subterranean geometry, a forgotten discipline that developed in the silver mines of early modern Europe. Seven case studies describe how an original culture of accuracy and measurement paved the way for technical and scientific innovations. Based on a variety of original manuscripts, maps and archive material, it recounts how knowledge was crafted and circulated among practitioners in the Holy Roman Empire and beyond. Specific chapters deal with the material culture of surveying, map-making, expertise and the political uses of quantification. By carefully reconstructing the religious, economic and cultural context of mining cities, Underground Mathematics argues that practical mathematics fruitfully interacted with the world of humanists, scholars and courts. In doing so, it contextualizes the rise of a culture of accuracy and quantification from 1500 to 1800. Subterranean geometry thus proves relevant to broader discussions in the history of science, technology, and knowledge.

Chapter 5, ‘“So Fair a Subterraneous City”: Mapping the Underground’, focuses on map-making and the visualization of the underground. It argues that these developments were deeply linked to broad changes in the political structure of mining regions. Drawing mining maps and working on them became widespread in the second half of the seventeenth century, gradually replacing alternative tools such as written reports of visitations, wood models, or annotated sketches. In Saxony, Captain-general Abraham von Schönberg (1640–1711) put his weight and reputation behind the new cartographic technology, hoping that its acceptance would in turn help him advance his reform agenda. At-scale representations were instrumental in justifying new investments, while offering technical road maps to implement them. Johann Berger (1649–1695) spent years producing a monumental cartographic enterprise, the Freiberga subterranea (1693) to support his patron’s ambitions. As surveyors finally realized the old dream of ‘seeing through stones’, the administrations rapidly seized their skills to reform and police their subterraneous cities.