Piotr Błaszczyk: The Pythagorean Theorem Throughout the Ages (with a special focus on the role of Descartes)

Piotr Błaszczyk (Institute of Mathematics, Pedagogical University, Krakow)

Abstract

History. We discuss ancient proofs of the Pythagorean Theorem—Chinese, Arabic, and Hindu—and compare them with Euclid’s demonstrations in Elements I.47 and VI.31. We show how Descartes transformed the meaning of the theorem, shifting it from its ancient geometric interpretation to a modern analytic form. We then highlight the role of the Pythagorean Theorem in the rise of calculus by examining Newton’s derivation of curvature and Euler’s contributions to the development of trigonometry. Finally, we explore the relevance of the theorem in modern Euclidean and unitary spaces.

Epistemology. We discuss various epistemic statuses of the Pythagorean Theorem: as a claim in need of proof, as an intuitively obvious premise, and even as a definition.
Foundations. We highlight the role of the Pythagorean Theorem in axiomatic systems of Euclidean geometry, especially those that depend on the Archimedean axiom.