The mathematical works of Bernard Bolzano

Steve Russ0198539304, 9781429459952, 9780198539308

Bernard Bolzano (1781-1848, Prague) was an outstanding thinker and reformer, far ahead of his times in many areas, including philosophy, ethics, politics, logic, theology and physics, and mathematics. Aimed at historians of mathematics, philosophy, ethics and logic, this volume contains the first English translations of some of his most significant mathematical writings, which contain the details of many celebrated insights and anticipations: clear topological definitions of various geometric extensions, an effective statement and use of the Cauchy convergence before it appears in Cauchy’s work, remarkable results on measurable numbers (a version of real numbers), on functions (the construction of a continuous, non-differentiable function around 1830) and on infinite collections.

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Table of contents :
Contents……Page 8
List of Illustrations……Page 10
Abbreviations……Page 11
Preface……Page 13
Note on the Texts……Page 21
Note on the Translations……Page 24
Introduction……Page 32
GEOMETRY AND FOUNDATIONS……Page 42
Considerations on Some Objects of Elementary Geometry (BG)……Page 56
Contributions to a Better-Grounded Presentation of Mathematics (BD)……Page 114
EARLY ANALYSIS……Page 170
The Binomial Theorem, and as a Consequence from it the Polynomial Theorem, and the Series which serve for the Calculation of Logarithmic and Exponential Quantities, proved more strictly than before (BL)……Page 186
Purely Analytic Proof of the Theorem, that between any two Values, which give Results of Opposite Sign, there lies at least one real Root of the Equation (RB)……Page 282
The Three Problems of Rectification, Complanation and Cubature, solved without consideration of the infinitely small, without the hypotheses of Archimedes and without any assumption which is not strictly provable. This is also being presented for the scrutiny of all mathematicians as a sample of a complete reorganisation of the science of space (DP)……Page 310
LATER ANALYSIS AND THE INFINITE……Page 376
Pure Theory of Numbers Seventh section: Infinite Quantity Concepts (RZ)……Page 386
Theory of Functions (F)……Page 460
Improvements and Additions to the Theory of Functions (F+)……Page 604
Paradoxes of the Infinite (PU)……Page 622
Selected Works of Bernard Bolzano……Page 710
Bibliography……Page 716
L……Page 722
Y……Page 723
C……Page 724
F……Page 725
L……Page 726
P……Page 727
S……Page 728
Z……Page 729