On the history of nested intervals: from Archimedes to Cantor

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.