On Bolzano and Greek Concepts of Continuity

In ancient Greek, continuity referred to lines, figures, and solids, or time, place, and motion. All these objects were considered continuous in the same sense: they were divisible into infinitely divisible parts. Bolzano 1817 Rein analytischer Beweis (Bolzano, Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein ent-gegengesetzes Resultat gewäihren, wenigstens eine reelle Wurzel der Gleichung liege. Gottlieb Hasse, Prague. In: [53], 253–277, 1817a) gave rise to two brand new ideas: εδ concept of continuous function and continuity of total order in terms of the least upper bound principle. The first was related to motion, the second to a straight line. The chapter aims to reveal the mathematical motives that lead Bolzano to these ideas. In the first part, we discuss Aristotle’s definition and examples of continuity, Euclid’s concept of magnitude and number, and Pappus’ account of mechanical curves. Then we show how the idea of the punctiform curve emerged in Descartes’ (Descartes, La Géométrie. Jan Maire, Leiden, 1637). The second part includes a detailed analysis of (Bolzano, Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein ent-gegengesetzes Resultat gewäihren, wenigstens eine reelle Wurzel der Gleichung liege. Gottlieb Hasse, Prague. In: [53], 253–277, 1817a).