<p>In this article, I propose a new approach to analyze the interrelations between mathematics and technology. It has the potential to contribute methodologically to both the fields of history of mathematics as well as the study of computational technologies in the current context. Based on the conception of mathematics as a contingent human practice, I claim that the practical engagement with technology not only subjects new fields, materials, and problems to mathematical scrutiny but might even shape mathematics from within. To illustrate my approach and corroborate my thesis, I present a historical case study on the mathematical works of the Swiss clock- and instrument-maker Jost Bürgi (1552–1632). Besides being a practicing artisan, he left three mathematical treatises. The advancements in fine metal working at his time, exemplified in clockwork mechanisms and measuring instruments, not only motivated and directed Bürgi’s mathematical inquiries. Instead, I argue that the interaction with these technical apparatuses in practice has shaped the internal structure and workings of his mathematics, that is, its entities, justifications, presentations, proofs, and procedures. The close analysis of some aspects of his oeuvre, especially his notion(s) of the sine, his way of explaining the occurrence of multiple solutions in algebra, and his visual depiction of the bridging of ten in his logarithmic computational tool, reveals a potential integration of the experience and practical knowledge of a clockmaker into mathematics. I therefore make the point that his mathematical writings portray a clockmaker’s mathematics.</p>