What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)
CHAPTER I. THE NATURAL NUMBERS
§2. THE INFINITUDE OF THE NUMBER SYSTEM. MATHEMATICAL INDUCTION
SUPPLEMENT TO CHAPTER I. THE THEORY OF NUMBERS
§3. PYTHAGOREAN NUMBERS AND FERMAT’S LAST THEOREM
CHAPTER II. THE NUMBER SYSTEM OF MATHEMATICS
§2. INCOMMENSURABLE SEGMENTS, IRRATIONAL NUMBERS, AND THE CONCEPT OF LIMIT
§3. REMARKS ON ANALYTIC GEOMETRY
§4. THE MATHEMATICAL ANALYSIS OF INFINITY
§6. ALGEBRAIC AND TRANSCENDENTAL NUMBERS
SUPPLEMENT TO CHAPTER II. THE ALGEBRA OF SETS
CHAPTER III. GEOMETRICAL CONSTRUCTIONS. THE ALGEBRA OF NUMBER FIELDS
PART I. IMPOSSIBILITY PROOFS AND ALGEBRA
§1. FUNDAMENTAL GEOMETRICAL CONSTRUCTIONS
§2. CONSTRUCTIBLE NUMBERS AND NUMBER FIELDS
§3. THE UNSOLVABILITY OF THE THREE GREEK PROBLEMS
PART II. VARIOUS METHODS FOR PERFORMING CONSTRUCTIONS
§4. GEOMETRICAL TRANSFORMATIONS. INVERSION
§5. CONSTRUCTIONS WITH OTHER TOOLS. MASCHERONI CONSTRUCTIONS WITH COMPASS ALONE
§6. MORE ABOUT INVERSION AND ITS APPLICATIONS
CHAPTER IV. PROJECTIVE GEOMETRY. AXIOMATICS. NON-EUCLIDEAN GEOMETRIES
§7. PROBLEMS ON CONSTRUCTIONS WITH THE STRAIGHT-EDGE ALONE
§8. CONICS AND QUADRIC SURFACES
§9. AXIOMATICS AND NON-EUCLIDEAN GEOMETRY
APPENDIX. GEOMETRY IN MORE THAN THREE DIMENSIONS
CHAPTER V. TOPOLOGY
§1. EULER’S FORMULA FOR POLYHEDRA
§2. TOPOLOGICAL PROPERTIES OF FIGURES
§3. OTHER EXAMPLES OF TOPOLOGICAL THEOREMS
§4. THE TOPOLOGICAL CLASSIFICATION OF SURFACES
CHAPTER VI. FUNCTIONS AND LIMITS
§3. LIMITS BY CONTINUOUS APPROACH
§4. PRECISE DEFINITION OF CONTINUITY
§5. TWO FUNDAMENTAL THEOREMS ON CONTINUOUS FUNCTIONS
§6. SOME APPLICATIONS OF BOLZANO’S THEOREM
SUPPLEMENT TO CHAPTER VI. MORE EXAMPLES ON LIMITS AND CONTINUITY
CHAPTER VII. MAXIMA AND MINIMA
§1. PROBLEMS IN ELEMENTARY GEOMETRY
§2. A GENERAL PRINCIPLE UNDERLYING EXTREME VALUE PROBLEMS
§3. STATIONARY POINTS AND THE DIFFERENTIAL CALCULUS
§4. SCHWARZ’S TRIANGLE PROBLEM
§7. THE EXISTENCE OF AN EXTREMUM. DIRICHLET’S PRINCIPLE
§10. THE CALCULUS OF VARIATIONS
§11. EXPERIMENTAL SOLUTIONS OF MINIMUM PROBLEMS. SOAP FILM EXPERIMENTS
CHAPTER VIII. THE CALCULUS
§3. THE TECHNIQUE OF DIFFERENTIATION
§4. LEIBNIZ’ NOTATION AND THE “INFINITELY SMALL”
§5. THE FUNDAMENTAL THEOREM OF THE CALCULUS
§6. THE EXPONENTIAL FUNCTION AND THE LOGARITHM
SUPPLEMENT TO CHAPTER VIII
§3. INFINITE SERIES AND PRODUCTS
§4. THE PRIME NUMBER THEOREM OBTAINED BY STATISTICAL METHODS
CHAPTER IX. RECENT DEVELOPMENTS
§2. THE GOLDBACH CONJECTURE AND TWIN PRIMES
§7. HAUSDORFF DIMENSION AND FRACTALS
§11. SOAP FILMS AND MINIMAL SURFACES
APPENDIX. SUPPLEMENTARY REMARKS, PROBLEMS, AND EXERCISES