THE HISTORY OF ARITHMETIC
CONTENTS THE THEOREM OF PYTHAGORAS ARITHMETIC AND GEOMETRY PYTHAGOREAN TRIPLES RATIONAL POINTS ON THE CIRCLE RIGHT-ANGLED TRIANGLES IRRATIONAL NUMBERS THE DE?NITION OF DISTANCE BIOGRAPHICAL NOTES: PYTHAGORAS GREEK GEOMETRY THE DEDUCTIVE METHOD THE REGULAR POLYHEDRA RULER AND COMPASS CONSTRUCTIONS CONIC SECTIONS HIGHER-DEGREE CURVES BIOGRAPHICAL NOTES: EUCLID GREEK NUMBER THEORY THE ROLE OF NUMBER THEORY POLYGONAL, PRIME, AND PERFECT NUMBERS THE EUCLIDEAN ALGORITHM PELL’S EQUATION THE CHORD AND TANGENT METHODS BIOGRAPHICAL NOTES: DIOPHANTUS IN?NITY IN GREEK MATHEMATICS FEAR OF IN?NITY EUDOXUS’S THEORY OF PROPORTIONS THE METHOD OF EXHAUSTION THE AREA OF A PARABOLIC SEGMENT BIOGRAPHICAL NOTES: ARCHIMEDES NUMBER THEORY IN ASIA THE CHINESE REMAINDER THEOREM LINEAR DIOPHANTINE EQUATIONS PELL’S EQUATION IN BRAHMAGUPTA PELL’S EQUATION IN BHˆASKARA II RATIONAL TRIANGLES BIOGRAPHICAL NOTES: BRAHMAGUPTA AND BHˆASKARA POLYNOMIAL EQUATIONS ALGEBRA LINEAR EQUATIONS AND ELIMINATION QUADRATIC EQUATIONS QUADRATIC IRRATIONALS THE SOLUTION OF THE CUBIC ANGLE DIVISION HIGHER-DEGREE EQUATIONS BIOGRAPHICAL NOTES: TARTAGLIA, CARDANO, AND VI`ETE ANALYTIC GEOMETRY STEPS TOWARD ANALYTIC GEOMETRY FERMAT AND DESCARTES ALGEBRAIC CURVES NEWTON’S CLASSI?CATION OF CUBICS CONSTRUCTION OF EQUATIONS, B´EZOUT’S THEOREM THE ARITHMETIZATION OF GEOMETRY BIOGRAPHICAL NOTES: DESCARTES PROJECTIVE GEOMETRY PERSPECTIVE ANAMORPHOSIS DESARGUES’S PROJECTIVE GEOMETRY THE PROJECTIVE VIEW OF CURVES THE PROJECTIVE PLANE THE PROJECTIVE LINE HOMOGENEOUS COORDINATES PASCAL’S THEOREM BIOGRAPHICAL NOTES: DESARGUES AND PASCAL CALCULUS WHAT IS CALCULUS? EARLY RESULTS ON AREAS AND VOLUMES MAXIMA, MINIMA, AND TANGENTS THE ARITHMETICA IN?NITORUM OF WALLIS NEWTON’S CALCULUS OF SERIES THE CALCULUS OF LEIBNIZ BIOGRAPHICAL NOTES: WALLIS, NEWTON, AND LEIBNIZ 10 IN?NITE SERIES 10 1 EARLY RESULTS 10 2 POWER SERIES 10 3 AN INTERPOLATION ON INTERPOLATION 10 4 SUMMATION OF SERIES 10 5 FRACTIONAL POWER SERIES 10 6 GENERATING FUNCTIONS 10 7 THE ZETA FUNCTION 10 8 BIOGRAPHICAL NOTES: GREGORY AND EULER 11 THE NUMBER THEORY REVIVAL 11 1 BETWEEN DIOPHANTUS AND FERMAT 11 2 FERMAT’S LITTLE THEOREM 11 3 FERMAT’S LAST THEOREM 11 4 RATIONAL RIGHT-ANGLED TRIANGLES 11 5 RATIONAL POINTS ON CUBICS OF GENUS 11 6 RATIONAL POINTS ON CUBICS OF GENUS 11 7 BIOGRAPHICAL NOTES: FERMAT 12 ELLIPTIC FUNCTIONS 12 1 ELLIPTIC AND CIRCULAR FUNCTIONS 12 2 PARAMETERIZATION OF CUBIC CURVES 12 3 ELLIPTIC INTEGRALS 12 4 DOUBLING THE ARC OF THE LEMNISCATE 12 5 GENERAL ADDITION THEOREMS 12 6 ELLIPTIC FUNCTIONS 12 7 A POSTSCRIPT ON THE LEMNISCATE 12 8 BIOGRAPHICAL NOTES: ABEL AND JACOBI 13 MECHANICS 13 1 MECHANICS BEFORE CALCULUS 13 2 THE FUNDAMENTAL THEOREM OF MOTION 13 3 KEPLER’S LAWS AND THE INVERSE SQUARE LAW 13 4 CELESTIAL MECHANICS 13 5 MECHANICAL CURVES 13 6 THE VIBRATING STRING 13 7 HYDRODYNAMICS 13 8 BIOGRAPHICAL NOTES: THE BERNOULLIS 14 COMPLEX NUMBERS IN ALGEBRA 14 1 IMPOSSIBLE NUMBERS 14 2 QUADRATIC EQUATIONS 14 3 CUBIC EQUATIONS 14 4 WALLIS’S ATTEMPT AT GEOMETRIC REPRESENTATION 14 5 ANGLE DIVISION 14 6 THE FUNDAMENTAL THEOREM OF ALGEBRA 14 7 THE PROOFS OF D’ALEMBERT AND GAUSS 14 8 BIOGRAPHICAL NOTES: D’ALEMBERT 15 COMPLEX NUMBERS AND CURVES 15 1 ROOTS AND INTERSECTIONS 15 2 THE COMPLEX PROJECTIVE LINE 15 3 BRANCH POINTS 15 4 TOPOLOGY OF COMPLEX PROJECTIVE CURVES 15 5 BIOGRAPHICAL NOTES: RIEMANN 16 COMPLEX NUMBERS AND FUNCTIONS 16 1 COMPLEX FUNCTIONS 16 2 CONFORMAL MAPPING 16 3 CAUCHY’S THEOREM 16 4 DOUBLE PERIODICITY OF ELLIPTIC FUNCTIONS 16 5 ELLIPTIC CURVES 16 6 UNIFORMIZATION 16 7 BIOGRAPHICAL NOTES: LAGRANGE AND CAUCHY 17 DI?ERENTIAL GEOMETRY 17 1 TRANSCENDENTAL CURVES 17 2 CURVATURE OF PLANE CURVES 17 3 CURVATURE OF SURFACES 17 4 SURFACES OF CONSTANT CURVATURE 17 5 GEODESICS 17 6 THE GAUSS–BONNET THEOREM 17 7 BIOGRAPHICAL NOTES: HARRIOT AND GAUSS 18 NON-EUCLIDEAN GEOMETRY 18 1 THE PARALLEL AXIOM 18 2 SPHERICAL GEOMETRY 18 3 GEOMETRY OF BOLYAI AND LOBACHEVSKY 18 4 BELTRAMI’S PROJECTIVE MODEL 18 5 BELTRAMI’S CONFORMAL MODELS 18 6 THE COMPLEX INTERPRETATIONS 18 7 BIOGRAPHICAL NOTES: BOLYAI AND LOBACHEVSKY 19 GROUP THEORY 19 1 THE GROUP CONCEPT 19 2 SUBGROUPS AND QUOTIENTS 19 3 PERMUTATIONS AND THEORY OF EQUATIONS 19 4 PERMUTATION GROUPS 19 5 POLYHEDRAL GROUPS 19 6 GROUPS AND GEOMETRIES 19 7 COMBINATORIAL GROUP THEORY 19 8 FINITE SIMPLE GROUPS 19 9 BIOGRAPHICAL NOTES: GALOIS 20 HYPERCOMPLEX NUMBERS 20 1 COMPLEX NUMBERS IN HINDSIGHT 20 2 THE ARITHMETIC OF PAIRS 20 3 PROPERTIES OF + AND × 20 4 ARITHMETIC OF TRIPLES AND QUADRUPLES 20 5 QUATERNIONS, GEOMETRY, AND PHYSICS 20 6 OCTONIONS 20 7 WHY C, H, AND O ARE SPECIAL 20 8 BIOGRAPHICAL NOTES: HAMILTON 21 ALGEBRAIC NUMBER THEORY 21 1 ALGEBRAIC NUMBERS 21 2 GAUSSIAN INTEGERS 21 3 ALGEBRAIC INTEGERS 21 4 IDEALS 21 5 IDEAL FACTORIZATION 21 6 SUMS OF SQUARES REVISITED 21 7 RINGS AND FIELDS 21 8 BIOGRAPHICAL NOTES: DEDEKIND, HILBERT, AND NOETHER 22 TOPOLOGY 22 1 GEOMETRY AND TOPOLOGY 22 2 POLYHEDRON FORMULAS OF DESCARTES AND EULER 22 3 THE CLASSI?CATION OF SURFACES 22 4 DESCARTES AND GAUSS–BONNET 22 5 EULER CHARACTERISTIC AND CURVATURE 22 6 SURFACES AND PLANES 22 7 THE FUNDAMENTAL GROUP 22 8 THE POINCAR´E CONJECTURE 22 9 BIOGRAPHICAL NOTES: POINCAR´E 23 SIMPLE GROUPS 23 1 FINITE SIMPLE GROUPS AND FINITE FIELDS 23 2 THE MATHIEU GROUPS 23 3 CONTINUOUS GROUPS 23 4 SIMPLICITY OF SO(3) 23 5 SIMPLE LIE GROUPS AND LIE ALGEBRAS 23 6 FINITE SIMPLE GROUPS REVISITED 23 7 THE MONSTER 23 8 BIOGRAPHICAL NOTES: LIE, KILLING, AND CARTAN 24 SETS, LOGIC, AND COMPUTATION 24 1 SETS 24 2 ORDINALS 24 3 MEASURE 24 4 AXIOM OF CHOICE AND LARGE CARDINALS 24 5 THE DIAGONAL ARGUMENT 24 6 COMPUTABILITY 24 7 LOGIC AND G¨ODEL’S THEOREM 24 8 PROVABILITY AND TRUTH 24 9 BIOGRAPHICAL NOTES: G¨ODEL 25 COMBINATORICS 25 1 WHAT IS COMBINATORICS? 25 2 THE PIGEONHOLE PRINCIPLE 25 3 ANALYSIS AND COMBINATORICS 25 4 GRAPH THEORY 25 5 NONPLANAR GRAPHS 25 6 THE K?ONIG IN?NITY LEMMA 25 7 RAMSEY THEORY 25 8 HARD THEOREMS OF COMBINATORICS 25 9 BIOGRAPHICAL NOTES: ERD?OS