Monographs on Topics of Modern Mathematics

Overview


“Of high merit”—Scientific American
This stimulating approach to several branches of modern mathematics is geared to those with no background beyond elementary algebra and geometry. Its nine essays by leading mathematicians—including Oswald Veblen, Gilbert Ames Bliss, L. E. Dickson, and David Eugene Smith—cover the foundations of geometry, modern pure geometry and non-Euclidean geometry, fundamental propositions of algebra, algebraic equations, functions, fundamentals of ...
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Overview


“Of high merit”—Scientific American
This stimulating approach to several branches of modern mathematics is geared to those with no background beyond elementary algebra and geometry. Its nine essays by leading mathematicians—including Oswald Veblen, Gilbert Ames Bliss, L. E. Dickson, and David Eugene Smith—cover the foundations of geometry, modern pure geometry and non-Euclidean geometry, fundamental propositions of algebra, algebraic equations, functions, fundamentals of calculus, and number theory. Each essay provides wide coverage, with proofs of important results and descriptions of leading methods. 1911 ed.
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Product Details

  • ISBN-13: 9780486438160
  • Publisher: Dover Publications
  • Publication date: 8/25/2004
  • Series: Dover Phoenix Editions Series
  • Pages: 432
  • Product dimensions: 5.80 (w) x 8.88 (h) x 1.06 (d)

Table of Contents

I. The Foundations of Geometry 3
Introduction
The Assumption of Order
Order on a Line
The Triangle and the Plane
Regions in a Plane
Congruence of Point Pairs
Congruence of Angles
Intersections of Circles
Parallel Lines
Mensuration
Three-Dimensional Space
Conclusion
II. Modern Pure Geometry 55
Introduction
Simple Elements in Geometry
The Principle of Duality
Principle of Continuity
Points at Infinity
Fundamental Theorem
Metric Properties
Anharmonic Ratios
Elementary Geometric Forms
Correlation of Elementary Forms
Curves and Sheaves of Rays of the Second Order
Pascal's and Brianchon's Theorems
Pole and Polar Theory
Conclusion
III. Non-Euclidean Geometry 93
Introduction
Parallel Lines
The Euclidean Assumption
The Lobachevskian Assumption
The Riemannian Assumption
The Sum of the Angles of a Triangle
Areas
Non-Euclidean Trigonometry
Non-Euclidean Analytic Geometry
Representation of the Lobachevskian Geometry on a Euclidean Plane
Relation between Projective and Non-Euclidean Geometry
The Element of Arc
IV. The Fundamental Propositions of Algebra 151
Introduction
The Addition of Angles and the Multiplication of Distances
The Abstract Theory of these Operations
Geometric Example of the Algebra of Complex Quantities: The System of Points in the Plane
The Abstract Theory of the Algebra of Complex Quantities
Appendix Other Examples of the Algebra of Complex Quantities
Geometric Proof that Every Algebraic Equation has a Root
V. The Algebraic Equation 211
General Introduction
Historical Sketch and Definitions
Equations with One Unknown and with Literal Coefficients
Equations with One Unknown and with Numerical Coefficients
Simultaneous Equations
A Few References
VI. The Function Concept and the Fundamental Notions of the Calculus 263
Introduction
Variables and Functions
The Fundamental Notions of the Calculus
VII. The Theory of Numbers 307
Introduction
Factors
Diophantine Equations
Congruences
Binomial Congruences
Quadratic Congruences
Bibliography
VIII. Constructions with Ruler and Compasses; Regular Polygons 353
Introduction
Analytic Criterion for Constructibility
Graphical Solution of a Quadratic Equation
Domain of Rationality
Functions Involving no Irrationalities other than Square Root
Reducible and Irreducible Functions
Fundamental Theorem; Duplication of the Cube; Trisection of an Angle; Quadrature of the Circle
Connection between Regular Polygons and Roots of Unity
De Moivre's Theorem
Regular Pentagon and Decagon
Regular Polygon of 17 Sides
Construction of the Regular Polygon of 17 Sides
Gauss's Theory of Regular Polygons
Primitive Roots of Unity
Gauss's Lemma
Irreducibility of the Cyclotomic Equation
Proofs of Theorems Cited Earlier
References
IX. The History and Transcendence of [pi] 389
The Nature of the Problem
The History of the Problem
The Transcendence of e
The Transcendence of [pi]
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