Why Should You Trust Geometry? The Mathematics of Mining in Early Modern Germany

My book Underground Mathematics tells the story of a discipline that has been forgotten today, subterranean geometry. Known as “the art of setting limits” (Markscheidekunst) in German and geometria subterranea in Latin, it developed in the silver mines of early modern Europe. From the Ore Mountains of Saxony to the Harz, in many mining cities in the Holy Roman Empire and beyond, an original culture of accuracy and measurement developed, paving the way for many technical and scientific innovations.

Mining and metallurgy were of great significance in early modern Europe: if one immediately thinks of gold and silver coins, metals and alloys were further used in countless arts and crafts but also in warfare and medicine. The underground world thus captured early modern imaginations curious about the generation of metals or the existence of mythical creatures such as kobolds.1 More prosaically, the fabulous wealth and dynamism of the great mining cities Freiberg, Annaberg, and Sankt Joachimsthal attracted scholars, preachers, and practitioners.

Georgius Agricola (1494–1555), a humanist physician who, after visiting Italy, spent decades in the Ore Mountains of Saxony, might be the most famous example. In the first chapter of my book, I study in detail the geometry of mines as presented in his famous De re metallica (1556), for centuries the key learned reference on mining.2 Comparing his descriptions and the beautiful engravings in the book with administrative sources and working documents of actual mine surveyors, we see that Agricola’s text is a rich and fascinating account of early modern mining, but that his geometry blends scholarly reflections and practical considerations. When the author wrote—in Latin—that “each method of surveying depends on the measuring of triangles. A small triangle should be laid out, and from it calculations must be made regarding a larger one,”3 he conveyed a sound mathematical principle. I was able to show, however, that its practicability was very limited for contemporaries, and that his text was likely not meant as a set of instructions for prospective miners but as a scholarly interpretation of Agricola’s observations.4

The instruments, methods, and actors of subterranean geometry were very different from the scholarly discipline that was at the time blossoming in universities and Latin schools (see Fig. 1). Subterranean geometry was not based on theory and had little use for proofs since “the laymen should stay unaware of Euclid,” as mining pastor Johannes Mathesius (1504–1565) preached in his Mining Homilies in 1562.5 Still, mining masters, surveyors, and miners faced formidable challenges in trying to keep the mines running: draining tunnels had to be correctly dug, shafts connected to one another, maps drawn, and concessions set according to very specific mining laws and traditions.

Fig. 1. The instruments of subterranean geometry as presented in surveyor Nicolaus Voigtel’s Geometria subterranea (1686). Credit: ETH-Bibliothek Rar 8882.

In the early modern period, in which numeracy was far from being ubiquitous, the social status of mathematics was very different from today. A central argument of the book is that the culture of accuracy that developed in the mining pits of early modern Europe was actively fostered and publicly displayed. This culture of accuracy was not only the privilege of a few but, as it was publicly displayed and discussed, came to be widely shared with the so-called gemeiner Mann (common man) of Reformation Germany.6

Based on extensive archive material, I reconstruct how surveyors supervised the attribution of rectangular concessions, setting limit stones, drawing the boundaries on their maps, and ensuring that they were respected underground. They did all of this with only the aid of their mining compasses, water levels, and surveying chains, working in poor light. Once prospectors had found a promising spot and furnished it “with workers, mining buckets, and rope,” a preliminary survey would be made with a simple cord.7 If the assays of sample ore revealed a rich silver vein, a formal survey (Erbbereiten) would be announced (“three Saturdays in a row”), publicly performed, and recorded in a specific book.8

A central question of the book is actually very simple: why did people trust mathematics when—as a popular saying went—”no one can see through stones”? There was no lack of alternatives to geometry, from anthropomorphic rituals to popular superstition. Divining rods, used by experienced miners relying on the color of stones and other natural signs, could in many cases be very efficient. Conversely, the geometry of surveyors was theoretically sound but was sometimes hampered by geological accidents and real-world conditions. For instance, two outcropping veins located in different concessions might “nevertheless merge together in the depth,” as a jurisprudence book noticed, raising difficult questions about who owned the ore and how to dig it out.9

If mathematics ultimately prevailed as the most efficient solution—and became a pillar of the mining states (the Harz, the Ore Mountains, and other regions had such a specific culture that they were referred to as Bergstaaten)—I argue that it was the result of a social process. Measurements were publicly displayed and could be contested in many ways, ensuring a broad social consensus. At the same time, mining preachers used their sermons to remind their audiences that God had “ordered all things in measure and number and weight” (Wisdom of Salomon) and presented Luther as “a trustworthy geometer on the mountain of our Lord.”10 Geometry was closely integrated in the legal systems, and the closeness of the mining regulations (Bergordnungen) all across central Europe enabled experts to circulate easily between mining regions.

Who were these surveyors, and how did they learn mathematics? Here again, the vast archives of the mining administrations—among the oldest of their kind—show a vernacular geometry that developed slowly by and for practitioners. Reports and working sketches from the sixteenth century can be cross-studied with surveyors’ oaths and the Instruktionen they had to respect. For the subsequent generations, mining maps and the discipline developed through new genre of manuscript, the Geometria subterranea (resp. Fig. 2 and Fig. 3), gradually began to complement these administrative sources.

Fig. 2. Mining map by Balthasar Rösler representing the concession Sun and God’s Gift in Erbisdorf, Saxony in 1656 (SächsStA-F 40040 H4636, photo Thomas Morel)

With the introduction of a new measuring instrument—the suspended compass (see Fig. 1 above)—the precision and complexity of mining mathematics suddenly increased in the mid-seventeenth century. In order to learn how to use trigonometric tables, to draw mining maps, and to solve the ever more complex technical challenges, young boys had to undergo a companionship system supervised by the state and delivered by mine surveyors (Markscheider), such as Balthasar Rösler (1605–1673) and August Beyer (1677–1753).11 In chapter six of the book, I retrace the development of this mathematical education, which sheds new light on the foundation of mining academies in the late eighteenth century.

Fig. 3. Teaching manuscript used by Samuel Gottlieb Rausch, surveyor in Clausthal in the middle of the eighteenth century (TU BAF – UB XVII 13, photo Thomas Morel).

A last central theme of the book is the relationships between scholars and practitioners, although these labels should not be taken too rigidly. Even as some professors openly despised the mine surveyors, there were preachers and natural scientists who were curious and admired the surveyors’ mindful hands.12 Among practitioners, there was also a considerable range of skill, varying from a handful of superior craftsmen knowledgable in university mathematics to more modest surveyors who relied primarily on their experience.

In some rare cases, the interactions among surveyors are documented, as when Jean-André Deluc (1727–1817) visited the Harz mines thrice in the 1770s, extensively describing the work of subterranean geometers. Deluc, a fellow of the Royal Society and one of Queen Charlotte’s university lecturers, was a prime naturalist experimenting with his barometer to assess the height of mountains and the depth of mines. In Clausthal, he could observe the digging of the longest gallery ever planned. When local surveyors assisted him with his observations and provided him with extremely accurate data taken from their registers, Deluc documented their methods and praised their ability in the Philosophical Transactions of the Royal Society of London.13

Underground Mathematics ultimately argues that practical mathematics fruitfully brought together the worlds of humanists, scholars, and courts. In doing so, it contextualizes the rise of a culture of accuracy and quantification. In the early modern period, mathematics was not only trusted for its inner consistency or for its demonstrative character—beautiful things that went far beyond the appreciation of most contemporaries. As basic numeracy grew slowly but steadily, most people saw its efficiency at work in the mines and heard about it in church. Subterranean geometry, thus, proves to be relevant to broader discussions in the history of science, technology, and knowledge.

Thomas Morel, who is the new professor in the history of mathematics and its instruction in the Faculty of Mathematics at the University of Wuppertal, specializes in the history of science and, specifically, the history of mathematics from the sixteenth to the nineteenth century. Underground Mathematics (Cambridge UP, 2022) is his latest book among his many publications.

Featured image: Excerpt from the cover of Thomas Morel’s new publication, Underground Mathematics (Cambridge: Cambridge UP, 2022).

  1. About early modern mining and its significance in arts and crafts, see Pamela H. Smith, From Lived Experience to the Written Word: Reconstructing Practical Knowledge in the Early Modern World (Chicago: University of Chicago Press, 2022). About early modern imaginations and the mines, see Warren Alexander Dym, Divining Science Treasure Hunting and Earth Science in Early Modern Germany (Leiden & Boston: Brill, 2011). ↩︎
  2. Pamela O. Long, Openness, Secrecy, Authorship: Technical Arts and the Culture of Knowledge from Antiquity to the Renaissance (Baltimore: Johns Hopkins University Press, 2001), contains a beautiful chapter on Georgius Agricola and related mining writings, chapter 6, “Openness and Authorship I: Mining, Metallurgy, and the Military Arts,” 175–91. ↩︎
  3. Georgius Agricola, De re metallica, translated from the first Latin edition of 1556, edited by Lou Henry Hoover (London: Mining Magazine, 1912), 128–29. ↩︎
  4. Thomas Morel, “De Re Geometrica: Writing, Drawing and Preaching Mathematics in Early Modern Mines,” Isis, A Journal of the History of Science Society 111, no. 1 (2020): 22–45, https://doi.org/10.1086/707640. ↩︎
  5. Johannes Mathesius, Sarepta oder Bergpostill: sampt der Jochimßthalischen kurtzen Chroniken (Nuremberg: Berg and Newber, 1562), 204. See also Thomas Morel, “Bringing Euclid into the Mines: Classical Sources and Vernacular Knowledge in the Development of Subterranean Geometry,” in Translating Early Modern Science, by S. Fransen, N. Hodson, and K. Enenkel, Intersections 51, 154–81 (Leiden: Brill, 2017). ↩︎
  6. On the social aspect of geometry and measures in the early modern period, see Witold Kula, Measures and Men (Princeton: Princeton University Press, 1986). ↩︎
  7. Hubert Ermisch, Das sächsische Bergrecht des Mittelalters (Leipzig: Giesecke & Devrient, 1887), 171. ↩︎
  8. Abraham von Schönberg, Ausführliche Berg-Information (Leipzig and Zwickau: Fleischer und Büschel, 1693), 28. ↩︎
  9. Annaberg Mining Law of 1509, §92, as transcribed in Ermisch, Das sächsische Bergrecht des Mittelalters, 195. ↩︎
  10. Book of Wisdom 11:20 (King James Bible); Cyriacus Spangenberg, Die XIX. Predigt von Doctore Martino Luthero, wie er so ein getreuwer Marscheider auff vnsers Herrn Gottes Berge gewesen (Frankfurt am Main: Nicolaus Bassee, 1574), title page. ↩︎
  11. A facsimile and an analysis of Rösler’s book can be found in Heinz Meixner et al., Balthasar Rösler: Persönlichkeit und Wirken für den Bergbau des 17. Jahrhunderts: Kommentarband zum Faksimiledruck “Hell-polierter Berg-Bau-Spiegel” (Leipzig: Deutscher Verlag für Grundstoffindustrie, 1980). ↩︎
  12. Lissa Roberts, Simon Schaffer, and Peter Robert Dear, eds., The Mindful Hand: Inquiry and Invention from the Late Renaissance to Early Industrialisation (Amsterdam: Koninklijke Nederlandse Akademie van Wetenschappen, 2007); Pamela O Long, Artisan/Practitioners and the Rise of the New Sciences, 1400–1600 (Corvallis: Oregon State University Press, 2011). ↩︎
  13. Jean André Deluc, “Barometrical Observations on the Depth of the Mines in the Hartz,” Philosophical Transactions of the Royal Society of London 67 (1777): 401 49, https://doi.org/10.1098/rstl.1777.0023; Jean André Deluc, “A Second Paper Concerning Some Barometrical Measure in the Mines of the Hartz,” Philosophical Transactions of the Royal Society of London 69 (1779): 485 504, https://doi.org/10.1098/rstl.1779.0032. ↩︎
Suggested Citation: Thomas Morel, “Why Should You Trust Geometry? The Mathematics of Mining in Early Modern Germany,” History of Knowledge, August 25, 2023, https://historyofknowledge.net/2023/08/25/mathematics-of-mining/.