cover -- copyright page -- title page -- Contents -- Preface -- The Ancient Greeks and the Foundations of Mathematics -- Pythagoras -- Euclid -- Archimedes -- Exercises -- Zeno's Paradox and the Concept of Limit -- The Context of the Paradox -- The Life of Zeno of Elea -- Consideration of the Paradoxes -- Decimal Notation and Limits -- Infinite Sums and Limits -- Finite Geometric Series -- Some Useful Notation -- Concluding Remarks -- Exercises -- The Mystical Mathematics of Hypatia -- Introduction to Hypatia -- What is a Conic Section? -- Exercises -- The Islamic World and the Development of Algebra -- Introductory Remarks -- The Development of Algebra -- The Geometry of the Arabs -- A Little Arab Number Theory -- Exercises -- Cardano, Abel, Galois, and the Solving of Equations -- Introduction -- The Story of Cardano -- First-Order Equations -- Rudiments of Second-Order Equations -- Completing the Square -- The Solution of a Quadratic Equation -- The Cubic Equation -- Fourth-Degree Equations and Beyond -- The Work of Abel and Galois in Context -- Exercises -- René Descartes and the Idea of Coordinates -- Introductory Remarks -- The Life of René Descartes -- The Real Number Line -- The Cartesian Plane -- The Use of Cartesian Coordinates to Study Euclidean Geometry -- Coordinates in Three-Dimensional Space -- Exercises -- Pierre de Fermat and the Invention of Differential Calculus -- The Life of Fermat -- Fermat's Method -- More Advanced Ideas of Calculus: The Derivative and the Tangent Line -- Fermat's Lemma and Maximum/Minimum Problems -- Exercises -- The Great Isaac Newton -- Introduction to Newton -- The Idea of the Integral -- Calculation of the Integral -- The Fundamental Theorem of Calculus -- Some Preliminary Calculations -- Some Examples -- Exercises -- The Complex Numbers and the Fundamental Theorem of Algebra -- A New Number System.

Progenitors of the Complex Number System -- Complex Number Basics -- The Fundamental Theorem of Algebra -- Finding the Roots of a Polynomial -- Exercises -- Carl Friedrich Gauss: The Prince of Mathematics -- Gauss the Man -- The Binomial Theorem -- The Chinese Remainder Theorem -- A Constructive Means for Finding the Solution x -- Quadratic Reciprocity and the Gaussian Integers -- The Gaussian Integers -- Exercises -- Sophie Germain and the Attack on Fermat's Last Problem -- Birth of an Inspired and Unlikely Child -- Sophie Germain's Work on Fermat's Problem -- Exercises -- Cauchy and the Foundations of Analysis -- Introduction -- Why Do We Need the Real Numbers? -- How to Construct the Real Numbers -- Properties of the Real Number System -- Exercises -- The Prime Numbers -- The Sieve of Eratosthenes -- The Infinitude of the Primes -- More Prime Thoughts -- The Concept of Relatively Prime -- Exercises -- Dirichlet and How to Count -- The Life of Dirichlet -- The Pigeonhole Principle -- Ramsey Theory -- Exercises -- Bernhard Riemann and the Geometry of Surfaces -- Introduction -- How to Measure the Length of a Curve -- Riemann's Method for Measuring Arc Length -- The Hyperbolic Disc -- The Use of the Integral -- Exercises -- Georg Cantor and the Orders of Infinity -- Introductory Remarks -- What is a Number? -- The Existence of Transcendental Numbers -- Exercises -- The Number Systems -- The Natural Numbers -- The Integers -- The Rational Numbers -- The Real Numbers -- The Complex Numbers -- Exercises -- Henri Poincaré, Child Phenomenon -- Introductory Remarks -- Rubber Sheet Geometry -- The Idea of Homotopy -- The Brouwer Fixed Point Theorem -- The Generalized Ham Sandwich Theorem -- Exercises -- Sonya Kovalevskaya and the Mathematics of Mechanics -- The Life of Sonya Kovalevskaya -- The Scientific Work of Sonya Kovalevskaya.

Afterward on Sonya Kovalevskaya -- Exercises -- Emmy Noether and Algebra -- The Life of Emmy Noether -- Emmy Noether and Abstract Algebra: Groups -- Emmy Noether and Abstract Algebra: Rings -- Exercises -- Methods of Proof -- Axiomatics -- Proof by Induction -- Proof by Contradiction -- Direct Proof -- Other Methods of Proof -- Exercises -- Alan Turing and Cryptography -- Background on Alan Turing -- The Turing Machine -- More on the Life of Alan Turing -- What is Cryptography? -- Encryption by Way of Affine Transformations -- Digraph Transformations -- Exercises -- Bibliography -- Index -- About the Author.