Bernard Bolzano Gesamtausgabe / Reihe II: Nachlaß. B. Wissenschaftliche Tagebücher. Band 12,2: Miscellanea Mathematica 22

The notes of part 22 of the ‹Miscellanea Mathematica‹ were written between July of 1830 and August of 1832. Bolzano‹s natural philosophy of his later years was based on the assumption that the entire universe had a nuclear particle structure. Unlike the classic atomist, he believed that there was an atom at every point in space. Thus there is no vacuum. This means that no substance is isolated. This made him question the opposite view which he had expressed previously in the ‹Athanasia‹. His refutation in the second edition of the ‹Athanasia‹ is given in detail in the ‹Miscellanea Mathematica‹. At the beginning of the ‹Miscellanea Mathematica 22‹, he dealt with the definition of the concepts of space and time. According to Bolzano, the traditional definitions of the concepts of line, surface and solid were in need of improvement, and this induced him to give a detailed explanation. During the years of his reclusiveness after his dismissal in 1819 from his position as head of the department for religious studies at the University of Prague, Bolzano had been studying the logical foundations of mathematics and the foundations of logic. These studies were reflected in his theory of science. When he returned to pure mathematics after about a decade, he began work on a new encyclopedic work, the ‹Größenlehre‹, (Theory of Quantities). The notes in this volume contain a precisely structured plan of this work from the year 1831. In explaining the first arithmetical terms, Bolzano tries to avoid abstract terms. Some abstract terms are imaginary. Referring to contemporary scientific journals, he gives a report on imaginary and irrational quantities in the theory of numbers. On the basis of Legendre‹s discussion of Galois‹ problem of the solvability of algebraic equations, Bolzano suggests a simple method. He gives a detailed account of Gergonne‹s geometric treatment of Newton‹s method of finding the roots of a numerical equation by approximation and provides a counterexample for Fourier‹s assertion that a characteristic of