On the history of nested intervals: from Archimedes to Cantor

TL;DR

This paper traces the historical development of the nested intervals principle and convergent sequences from ancient times through the 19th century, highlighting key contributions from mathematicians like Archimedes, Fermat, and Cantor.

Contribution

It provides a comprehensive historical analysis of the evolution of the nested intervals concept and its role in the foundation of real analysis.

Findings

The nested intervals idea originated in ancient mathematics.

Key developments occurred through the works of mathematicians from Archimedes to Cantor.

The concept became central to the formalization of real numbers in the 19th century.

Abstract

The idea of the principle of nested intervals or the concept of convergent sequences which is equivalent to this idea dates back to the ancient world. Archimedes calculated the unknown in excess and deficiency, approximating with two sets of values: ambient and nested values. Buridan came up with a concept of a point lying within a sequence of nested intervals. Fermat, Gregory, Newton, MacLaurin, Gauss,and Fourier used to search for an unknown value with the help of approximation in excess and deficiency. In the 19-th century, in works of Bolzano, Cauchy, Lejeune Dirichlet, Weierstrass, and Cantor, this logical construction turned into the analysis argumentation method. The concept of a real number was elaborated in the 1870-s in works of Meray, Weierstrass, Heine, Cantor, and Dedekind. Cantor elaboration was based on the notion of a limiting point and principle of nested intervals. What we are going to consider now, is the genesis of this idea which dates back to the ancient world.