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.
In Elizabethan London, one of the more surprising things a wealthy owner of a beautifully illustrated folio volume could do was to take a sharp knife and cut it to pieces. John Blagrave’s 1585 Mathematical Jewel, in fact, demands nothing less.1 This work, which introduced an elaborate instrument of Blagrave’s design for performing astronomical calculations, included woodcuts that were specifically provided in order to be cut out and used as surrogates for the brass original:
get very fine pastboord made of purpose, and then spred your paste very fine thereon, & quickly laying on this picture & clappe it streight into a presse before it bee thorowe wette with the paste (fol. ¶6v)
Being a human activity, calculation has a history, even if its operations yield “facts” apparently true in any context. One plus one might always be two, but the methods to arrive at such results, not to mention what they might mean, are another matter. Recent histories involving calculation on this blog include Staffan Müller-Wille and Giuditta Parolini, “Punnett Squares and Hybrid Crosses: How Mendelians Learned Their Trade by the Book”; D. Senthil Babu, “Handbooks of the Mind into Ready Reckoners in Print: The Story of the ‘Encuvati’ in the Nineteenth Century”; and Karine Chemla, “Reading and (Re-)Classifying Canonical Instructions of the Past: Commentaries on ‘The Nine Chapters on Mathematical Procedures’…”
The earliest extant Chinese mathematical writings include two types of components of particular interest for our discussion on manuals and handbooks. On the one hand, there are mathematical problems that often evoke tasks carried out by officials working in the imperial bureaucracy. On the other hand, there are mathematical “procedures,” or “algorithms” in today’s parlance, to solve such problems. This description fits most of the mathematical books composed in China until the seventh century.
The Encuvati was the quintessential Tamil multiplication table book, used in precolonial South Indian schools. Nowadays they are available as palm leaf manuscripts, collected from different geographical locations of the Tamil-speaking region of South India and stored in various manuscript libraries there as well as in other collections inside and outside the country. But why is something as innocuous as a multiplication table book important for us? Their contents look nothing like modern tables, with lines and columns, but the numbers are arranged the same way. Their importance emerges from the simple fact that countless children participated in their making from about the seventeenth century, if not earlier.
In 1901, Erich von Tschermak (1871–1962) produced a critical edition of Gregor Mendel’s (1822–1884) paper on “Versuche über Pflanzenhybriden”; and in the same year, William Bateson (1861–1926) submitted an English translation entitled “Experiments in Plant Hybridization” to the readers of the Journal of the Royal Horticultural Society. Tschermak’s edition appeared as volume 121 of the renowned series Ostwalds Klassiker der exakten Naturwissenschaften (Ostwald’s Classic Texts in the Exact Sciences). Historians have rarely noted the paradox that lies in the fact that a paper, which scientists like von Tschermak and Bateson had lifted from obscurity just a year earlier, was almost instantaneously included in the Pantheon of classical contributions to the “exact” sciences. The discipline that Mendel supposedly founded, namely genetics, did not yet exist in 1901, and his alleged “discovery” of laws of inheritance would remain highly contested for at least another decade, even involving accusations of scientific misconduct.