Henri Poincar\'e and his "model" of hyperbolic geometry

TL;DR

This paper explores Henri Poincaré's development and use of non-Euclidean geometries, focusing on his creation of a model for hyperbolic geometry and its influence on his work with Fuchsian functions.

Contribution

It provides a detailed historical and mathematical analysis of Poincaré's application of non-Euclidean geometry and his conceptualization of a hyperbolic model, highlighting its novelty in mathematical thought.

Findings

Poincaré's use of NEG in Fuchsian functions

Development of a hyperbolic model by Poincaré

Analogy between elliptic and Fuchsian functions

Abstract

The aim of the talk is to trace how and when Henri Poincar\'e used non-Euclidean geometries (NEG) in his mathematical and philosophical works, with a particular attention to the genesis and the description of his model. We begin by a short presentation of the context of NEG in France around the 1870-80s. Then we expound from several sources the introduction and use of NEG in Poincar\'e's work about Fuchsian functions and we stress on the analogy between elliptic functions and fuchsian functions.