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Congruence of Sets and Other Monographs
W. Sierpiński, F. Klein, C. Runge, and L. E. Dickson

AMS Chelsea Publishing
1967; 441 pp; hardcover
Volume: 209
ISBN-10: 0-8218-7567-1
ISBN-13: 978-0-8218-7567-4
List Price: US$55
Member Price: US$49.50
Order Code: CHEL/209.H
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Four volumes in one: On the Congruence of Sets and Their Equivalence by Finite Decomposition, by W. Sierpinski; The Mathematical Theory of the Top, by F. Klein--well-known lectures on the analytical formulas relating to the motion of the top; Graphical Methods, by C. Runge; Introduction to the Theory of Algebraic Equations, by L. E. Dickson--Dickson's earliest account of the subject, presented with a historical, elementary perspective.

Table of Contents

On the Congruence of Sets and Their Equivalence by Finite Decomposition by Waclaw Sierpinski
  • Congruence of sets; Translation of sets; Equivalence of sets by finite decomposition; Some theorems on the equivalence of sets by decomposition into two parts; Equivalence of a rectangular isosceles triangle to a square by finite decomposition; Equivalence of a segment of the straight line to one of its proper subsets by finite decomposition; Paradoxical decompositions; Decomposition of the interval \([0\leq x < 1]\) into an enumberable infinity of disjoint sets each two of which are equivalent by decomposition into two parts; Some theorems on the equivalence of sets by finite decomposition; The Hausdorff paradox; The paradox of Banach and Tarski; Banach measure. The generalized problem of measure; The absolute measure; The paradox of J. von Neumann; Appendix
The Mathematical Theory of the Top by Felix Klein
  • Lecture I; Lecture II; Lecture III; Lecture IV
Graphical Methods by C. Runge
  • Graphical Calculation: Graphical arithmetic; Integral functions; Linear functions of any number of variables; The graphical handling of complex numbers
  • The Graphical Representation of Functions of One or More Independent Variables: Functions of one independent variable; The principle of the slide rule; Rectangular coördinates with intervals of varying size; Functions of two independent variables; Depiction of one plane on another plane; Other methods of representing relations between three variables; Relations between four variables
  • The Graphical Methods of the Differential and Integral Calculus: Graphical integration; Graphical differentiation; Differential equations of the first order; Differential equations of the second and higher orders
Introduction to the Theory of Algebraic Equations by Leonard Eugene Dickson
  • Solution of the general quadratic, cubic, and quartic equations. Lagrange's theorem on the irrationalities entering the roots; Exercises
  • Substitutions; Rational functions; Exercises
  • Substitution groups; Rational functions; Exercises
  • The general equation from the group standpoint; Exercises
  • Algebraic introduction to Galois' theory
  • The group of an equation; Exercises
  • Solution by means of resolvent equations
  • Regular cyclic equations; Abelian equations
  • Criterion for algebraic solvability
  • Metacyclic equations; Galoisian equations
  • An account of more technical results
  • Appendix; Symmetric functions; On the general equation; Index
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