Work

The tautochrone of Huygens and Abel: from constructive geometry to fractional calculus

Year: 2026

Type: article

Abstract: In this paper, we explore the connections between Christiaan Huygens and Niels Henrik Abel through the tautochrone problem. The problem—determining the curve along which a particle descends under gravity in the same time, regardless of its starting point—has been a central topic at the intersection of physics, geometry, and analysis. Though these two major figures are separated by nearly two centuries, they approached the problem in radically different ways. While Huygens proposed a physical solution based on geometric construction, Abel approached the problem within the analytic framework of integral equations, employing a procedure that can be seen as anticipating and paving the way for the development of differential calculus of arbitrary order. This contrast highlights a broader historical narrative: the transformation of mathematical thinking from constructive geometry to abstract analysis.

Source: The European Physical Journal Plus

DOI: 10.1140/epjp/s13360-026-07786-3

Authors: Luiz Roberto Evangelista, Francesco Mainardi

Institutions: Universidade Estadual de Maringá, Ca' Foscari University of Venice, Institute for Complex Systems, National Research Council, University of Bologna

Language: English

Topic: History and Theory of Mathematics

Subfield: Theoretical Computer Science

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,890

Funders: Istituto Nazionale di Alta Matematica "Francesco Severi", Universidade Estadual de Maringá, Gruppo Nazionale per la Fisica Matematica