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Congruence of Sets and Other Monographs
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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

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.

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