UTA C. MERZBACH is Curator Emeritus of Mathematics at the Smithsonian Institution and Director of the LHM Institute. The late CARL B. BOYER was a professor of mathematics at Brooklyn College and the author of several classic works on the history of mathematics.

"... the book is an essential reference that will help you explore the incredible history of mathematics and the men and women who created it." (Zentralblatt MATH, 2016)"... an 'engaging' read for the mathematically minded." (Inside OR, June 2011)

• Foreword by Isaac Asimov xiPreface to the Third Edition xiiiPreface to the Second Edition xvPreface to the First Edition xvii1 Traces 1Concepts and Relationships 1Early Number Bases 3Number Language and Counting 5Spatial Relationships 62 Ancient Egypt 8The Era and the Sources 8Numbers and Fractions 10Arithmetic Operations 12“Heap” Problems 13Geometric Problems 14Slope Problems 18Arithmetic Pragmatism 193 Mesopotamia 21The Era and the Sources 21Cuneiform Writing 22Numbers and Fractions: Sexagesimals 23Positional Numeration 23Sexagesimal Fractions 25Approximations 25Tables 26Equations 28Measurements: Pythagorean Triads 31Polygonal Areas 35Geometry as Applied Arithmetic 364 Hellenic Traditions 40The Era and the Sources 40Thales and Pythagoras 42Numeration 52Arithmetic and Logistic 55Fifth-Century Athens 56Three Classical Problems 57Quadrature of Lunes 58Hippias of Elis 61Philolaus and Archytas of Tarentum 63Incommensurability 65Paradoxes of Zeno 67Deductive Reasoning 70Democritus of Abdera 72Mathematics and the Liberal Arts 74The Academy 74Aristotle 885 Euclid of Alexandria 90Alexandria 90Lost Works 91Extant Works 91The Elements 936 Archimedes of Syracuse 109The Siege of Syracuse 109On the Equilibriums of Planes 110On Floating Bodies 111The Sand-Reckoner 112Measurement of the Circle 113On Spirals 113Quadrature of the Parabola 115On Conoids and Spheroids 116On the Sphere and Cylinder 118Book of Lemmas 120Semiregular Solids and Trigonometry 121The Method 1227 Apollonius of Perge 127Works and Tradition 127Lost Works 128Cycles and Epicycles 129The Conics 1308 Crosscurrents 142Changing Trends 142Eratosthenes 143Angles and Chords 144Ptolemy’s Almagest 149Heron of Alexandria 156The Decline of Greek Mathematics 159Nicomachus of Gerasa 159Diophantus of Alexandria 160Pappus of Alexandria 164The End of Alexandrian Dominance 170Proclus of Alexandria 171Boethius 171Athenian Fragments 172Byzantine Mathematicians 1739 Ancient and Medieval China 175The Oldest Known Texts 175The Nine Chapters 176Rod Numerals 177The Abacus and Decimal Fractions 178Values of Pi 180Thirteenth-Century Mathematics 18210 Ancient and Medieval India 186Early Mathematics in India 186The Sulbasutras 187The Siddhantas 188Aryabhata 189Numerals 191Trigonometry 193Multiplication 194Long Division 195Brahmagupta 197Indeterminate Equations 199Bhaskara 200Madhava and the Keralese School 20211 The Islamic Hegemony 203Arabic Conquests 203The House of Wisdom 205Al-Khwarizmi 206‘Abd Al-Hamid ibn-Turk 212Thabit ibn-Qurra 213Numerals 214Trigonometry 216Tenth- and Eleventh-Century Highlights 216Omar Khayyam 218The Parallel Postulate 220Nasir al-Din al-Tusi 220Al-Kashi 22112 The Latin West 223Introduction 223Compendia of the Dark Ages 224Gerbert 224The Century of Translation 226Abacists and Algorists 227Fibonacci 229Jordanus Nemorarius 232Campanus of Novara 233Learning in the Thirteenth Century 235Archimedes Revived 235Medieval Kinematics 236Thomas Bradwardine 236Nicole Oresme 238The Latitude of Forms 239Infinite Series 241Levi ben Gerson 242Nicholas of Cusa 243The Decline of Medieval Learning 24313 The European Renaissance 245Overview 245Regiomontanus 246Nicolas Chuquet’s Triparty 249Luca Pacioli’s Summa 251German Algebras and Arithmetics 253Cardan’s Ars Magna 255Rafael Bombelli 260Robert Recorde 262Trigonometry 263Geometry 264Renaissance Trends 271François Viète 27314 Early Modern Problem Solvers 282Accessibility of Computation 282Decimal Fractions 283Notation 285Logarithms 286Mathematical Instruments 290Infinitesimal Methods: Stevin 296Johannes Kepler 29615 Analysis, Synthesis, the Infinite, and Numbers 300Galileo’s Two New Sciences 300Bonaventura Cavalieri 303Evangelista Torricelli 306Mersenne’s Communicants 308René Descartes 309Fermat’s Loci 320Gregory of St. Vincent 325The Theory of Numbers 326Gilles Persone de Roberval 329Girard Desargues and Projective Geometry 330Blaise Pascal 332Philippe de Lahire 337Georg Mohr 338Pietro Mengoli 338Frans van Schooten 339Jan de Witt 340Johann Hudde 341René François de Sluse 342Christiaan Huygens 34216 British Techniques and Continental Methods 348John Wallis 348James Gregory 353Nicolaus Mercator and William Brouncker 355Barrow’s Method of Tangents 356Newton 358Abraham De Moivre 372Roger Cotes 375James Stirling 376Colin Maclaurin 376Textbooks 380Rigor and Progress 381Leibniz 382The Bernoulli Family 390Tschirnhaus Transformations 398Solid Analytic Geometry 399Michel Rolle and Pierre Varignon 400The Clairauts 401Mathematics in Italy 402The Parallel Postulate 403Divergent Series 40417 Euler 406The Life of Euler 406Notation 408Foundation of Analysis 409Logarithms and the Euler Identities 413Differential Equations 414Probability 416The Theory of Numbers 417Textbooks 418Analytic Geometry 419The Parallel Postulate: Lambert 42018 Pre- to Postrevolutionary France 423Men and Institutions 423The Committee on Weights and Measures 424D’Alembert 425Bézout 427Condorcet 429Lagrange 430Monge 433Carnot 438Laplace 443Legendre 446Aspects of Abstraction 449Paris in the 1820s 449Fourier 450Cauchy 452Diffusion 46019 Gauss 464Nineteenth-Century Overview 464Gauss: Early Work 465Number Theory 466Reception of the Disquisitiones Arithmeticae 469Astronomy 470Gauss’s Middle Years 471Differential Geometry 472Gauss’s Later Work 473Gauss’s Influence 47420 Geometry 483The School of Monge 483Projective Geometry: Poncelet and Chasles 485Synthetic Metric Geometry: Steiner 487Synthetic Nonmetric Geometry: von Staudt 489Analytic Geometry 489Non-Euclidean Geometry 494Riemannian Geometry 496Spaces of Higher Dimensions 498Felix Klein 499Post-Riemannian Algebraic Geometry 50121 Algebra 504Introduction 504British Algebra and the Operational Calculus of Functions 505Boole and the Algebra of Logic 506Augustus De Morgan 509William Rowan Hamilton 510Grassmann and Ausdehnungslehre 512Cayley and Sylvester 515Linear Associative Algebras 519Algebraic Geometry 520Algebraic and Arithmetic Integers 520Axioms of Arithmetic 52222 Analysis 526Berlin and Göttingen at Midcentury 526Riemann in Göttingen 527Mathematical Physics in Germany 528Mathematical Physics in English-Speaking Countries 529Weierstrass and Students 531The Arithmetization of Analysis 533Dedekind 536Cantor and Kronecker 538Analysis in France 54323 Twentieth-Century Legacies 548Overview 548Henri Poincaré 549David Hilbert 555Integration and Measure 564Functional Analysis and General Topology 568Algebra 570Differential Geometry and Tensor Analysis 572Probability 573Bounds and Approximations 575The 1930s and World War II 577Nicolas Bourbaki 578Homological Algebra and Category Theory 580Algebraic Geometry 581Logic and Computing 582The Fields Medals 58424 Recent Trends 586Overview 586The Four-Color Conjecture 587Classification of Finite Simple Groups 591Fermat’s Last Theorem 593Poincaré’s Query 596Future Outlook 599References 601General Bibliography 633Index 647

UTA C. MERZBACH is Curator Emeritus of Mathematics at the Smithsonian Institution and Director of the LHM Institute. The late CARL B. BOYER was a professor of mathematics at Brooklyn College and the author of several classic works on the history of mathematics.

"... the book is an essential reference that will help you explore the incredible history of mathematics and the men and women who created it." (Zentralblatt MATH, 2016)"... an 'engaging' read for the mathematically minded." (Inside OR, June 2011)

• Foreword by Isaac Asimov xiPreface to the Third Edition xiiiPreface to the Second Edition xvPreface to the First Edition xvii1 Traces 1Concepts and Relationships 1Early Number Bases 3Number Language and Counting 5Spatial Relationships 62 Ancient Egypt 8The Era and the Sources 8Numbers and Fractions 10Arithmetic Operations 12“Heap” Problems 13Geometric Problems 14Slope Problems 18Arithmetic Pragmatism 193 Mesopotamia 21The Era and the Sources 21Cuneiform Writing 22Numbers and Fractions: Sexagesimals 23Positional Numeration 23Sexagesimal Fractions 25Approximations 25Tables 26Equations 28Measurements: Pythagorean Triads 31Polygonal Areas 35Geometry as Applied Arithmetic 364 Hellenic Traditions 40The Era and the Sources 40Thales and Pythagoras 42Numeration 52Arithmetic and Logistic 55Fifth-Century Athens 56Three Classical Problems 57Quadrature of Lunes 58Hippias of Elis 61Philolaus and Archytas of Tarentum 63Incommensurability 65Paradoxes of Zeno 67Deductive Reasoning 70Democritus of Abdera 72Mathematics and the Liberal Arts 74The Academy 74Aristotle 885 Euclid of Alexandria 90Alexandria 90Lost Works 91Extant Works 91The Elements 936 Archimedes of Syracuse 109The Siege of Syracuse 109On the Equilibriums of Planes 110On Floating Bodies 111The Sand-Reckoner 112Measurement of the Circle 113On Spirals 113Quadrature of the Parabola 115On Conoids and Spheroids 116On the Sphere and Cylinder 118Book of Lemmas 120Semiregular Solids and Trigonometry 121The Method 1227 Apollonius of Perge 127Works and Tradition 127Lost Works 128Cycles and Epicycles 129The Conics 1308 Crosscurrents 142Changing Trends 142Eratosthenes 143Angles and Chords 144Ptolemy’s Almagest 149Heron of Alexandria 156The Decline of Greek Mathematics 159Nicomachus of Gerasa 159Diophantus of Alexandria 160Pappus of Alexandria 164The End of Alexandrian Dominance 170Proclus of Alexandria 171Boethius 171Athenian Fragments 172Byzantine Mathematicians 1739 Ancient and Medieval China 175The Oldest Known Texts 175The Nine Chapters 176Rod Numerals 177The Abacus and Decimal Fractions 178Values of Pi 180Thirteenth-Century Mathematics 18210 Ancient and Medieval India 186Early Mathematics in India 186The Sulbasutras 187The Siddhantas 188Aryabhata 189Numerals 191Trigonometry 193Multiplication 194Long Division 195Brahmagupta 197Indeterminate Equations 199Bhaskara 200Madhava and the Keralese School 20211 The Islamic Hegemony 203Arabic Conquests 203The House of Wisdom 205Al-Khwarizmi 206‘Abd Al-Hamid ibn-Turk 212Thabit ibn-Qurra 213Numerals 214Trigonometry 216Tenth- and Eleventh-Century Highlights 216Omar Khayyam 218The Parallel Postulate 220Nasir al-Din al-Tusi 220Al-Kashi 22112 The Latin West 223Introduction 223Compendia of the Dark Ages 224Gerbert 224The Century of Translation 226Abacists and Algorists 227Fibonacci 229Jordanus Nemorarius 232Campanus of Novara 233Learning in the Thirteenth Century 235Archimedes Revived 235Medieval Kinematics 236Thomas Bradwardine 236Nicole Oresme 238The Latitude of Forms 239Infinite Series 241Levi ben Gerson 242Nicholas of Cusa 243The Decline of Medieval Learning 24313 The European Renaissance 245Overview 245Regiomontanus 246Nicolas Chuquet’s Triparty 249Luca Pacioli’s Summa 251German Algebras and Arithmetics 253Cardan’s Ars Magna 255Rafael Bombelli 260Robert Recorde 262Trigonometry 263Geometry 264Renaissance Trends 271François Viète 27314 Early Modern Problem Solvers 282Accessibility of Computation 282Decimal Fractions 283Notation 285Logarithms 286Mathematical Instruments 290Infinitesimal Methods: Stevin 296Johannes Kepler 29615 Analysis, Synthesis, the Infinite, and Numbers 300Galileo’s Two New Sciences 300Bonaventura Cavalieri 303Evangelista Torricelli 306Mersenne’s Communicants 308René Descartes 309Fermat’s Loci 320Gregory of St. Vincent 325The Theory of Numbers 326Gilles Persone de Roberval 329Girard Desargues and Projective Geometry 330Blaise Pascal 332Philippe de Lahire 337Georg Mohr 338Pietro Mengoli 338Frans van Schooten 339Jan de Witt 340Johann Hudde 341René François de Sluse 342Christiaan Huygens 34216 British Techniques and Continental Methods 348John Wallis 348James Gregory 353Nicolaus Mercator and William Brouncker 355Barrow’s Method of Tangents 356Newton 358Abraham De Moivre 372Roger Cotes 375James Stirling 376Colin Maclaurin 376Textbooks 380Rigor and Progress 381Leibniz 382The Bernoulli Family 390Tschirnhaus Transformations 398Solid Analytic Geometry 399Michel Rolle and Pierre Varignon 400The Clairauts 401Mathematics in Italy 402The Parallel Postulate 403Divergent Series 40417 Euler 406The Life of Euler 406Notation 408Foundation of Analysis 409Logarithms and the Euler Identities 413Differential Equations 414Probability 416The Theory of Numbers 417Textbooks 418Analytic Geometry 419The Parallel Postulate: Lambert 42018 Pre- to Postrevolutionary France 423Men and Institutions 423The Committee on Weights and Measures 424D’Alembert 425Bézout 427Condorcet 429Lagrange 430Monge 433Carnot 438Laplace 443Legendre 446Aspects of Abstraction 449Paris in the 1820s 449Fourier 450Cauchy 452Diffusion 46019 Gauss 464Nineteenth-Century Overview 464Gauss: Early Work 465Number Theory 466Reception of the Disquisitiones Arithmeticae 469Astronomy 470Gauss’s Middle Years 471Differential Geometry 472Gauss’s Later Work 473Gauss’s Influence 47420 Geometry 483The School of Monge 483Projective Geometry: Poncelet and Chasles 485Synthetic Metric Geometry: Steiner 487Synthetic Nonmetric Geometry: von Staudt 489Analytic Geometry 489Non-Euclidean Geometry 494Riemannian Geometry 496Spaces of Higher Dimensions 498Felix Klein 499Post-Riemannian Algebraic Geometry 50121 Algebra 504Introduction 504British Algebra and the Operational Calculus of Functions 505Boole and the Algebra of Logic 506Augustus De Morgan 509William Rowan Hamilton 510Grassmann and Ausdehnungslehre 512Cayley and Sylvester 515Linear Associative Algebras 519Algebraic Geometry 520Algebraic and Arithmetic Integers 520Axioms of Arithmetic 52222 Analysis 526Berlin and Göttingen at Midcentury 526Riemann in Göttingen 527Mathematical Physics in Germany 528Mathematical Physics in English-Speaking Countries 529Weierstrass and Students 531The Arithmetization of Analysis 533Dedekind 536Cantor and Kronecker 538Analysis in France 54323 Twentieth-Century Legacies 548Overview 548Henri Poincaré 549David Hilbert 555Integration and Measure 564Functional Analysis and General Topology 568Algebra 570Differential Geometry and Tensor Analysis 572Probability 573Bounds and Approximations 575The 1930s and World War II 577Nicolas Bourbaki 578Homological Algebra and Category Theory 580Algebraic Geometry 581Logic and Computing 582The Fields Medals 58424 Recent Trends 586Overview 586The Four-Color Conjecture 587Classification of Finite Simple Groups 591Fermat’s Last Theorem 593Poincaré’s Query 596Future Outlook 599References 601General Bibliography 633Index 647

UTA C. MERZBACH is Curator Emeritus of Mathematics at the Smithsonian Institution and Director of the LHM Institute. The late CARL B. BOYER was a professor of mathematics at Brooklyn College and the author of several classic works on the history of mathematics.

"... the book is an essential reference that will help you explore the incredible history of mathematics and the men and women who created it." (Zentralblatt MATH, 2016)"... an 'engaging' read for the mathematically minded." (Inside OR, June 2011)

• Foreword by Isaac Asimov xiPreface to the Third Edition xiiiPreface to the Second Edition xvPreface to the First Edition xvii1 Traces 1Concepts and Relationships 1Early Number Bases 3Number Language and Counting 5Spatial Relationships 62 Ancient Egypt 8The Era and the Sources 8Numbers and Fractions 10Arithmetic Operations 12“Heap” Problems 13Geometric Problems 14Slope Problems 18Arithmetic Pragmatism 193 Mesopotamia 21The Era and the Sources 21Cuneiform Writing 22Numbers and Fractions: Sexagesimals 23Positional Numeration 23Sexagesimal Fractions 25Approximations 25Tables 26Equations 28Measurements: Pythagorean Triads 31Polygonal Areas 35Geometry as Applied Arithmetic 364 Hellenic Traditions 40The Era and the Sources 40Thales and Pythagoras 42Numeration 52Arithmetic and Logistic 55Fifth-Century Athens 56Three Classical Problems 57Quadrature of Lunes 58Hippias of Elis 61Philolaus and Archytas of Tarentum 63Incommensurability 65Paradoxes of Zeno 67Deductive Reasoning 70Democritus of Abdera 72Mathematics and the Liberal Arts 74The Academy 74Aristotle 885 Euclid of Alexandria 90Alexandria 90Lost Works 91Extant Works 91The Elements 936 Archimedes of Syracuse 109The Siege of Syracuse 109On the Equilibriums of Planes 110On Floating Bodies 111The Sand-Reckoner 112Measurement of the Circle 113On Spirals 113Quadrature of the Parabola 115On Conoids and Spheroids 116On the Sphere and Cylinder 118Book of Lemmas 120Semiregular Solids and Trigonometry 121The Method 1227 Apollonius of Perge 127Works and Tradition 127Lost Works 128Cycles and Epicycles 129The Conics 1308 Crosscurrents 142Changing Trends 142Eratosthenes 143Angles and Chords 144Ptolemy’s Almagest 149Heron of Alexandria 156The Decline of Greek Mathematics 159Nicomachus of Gerasa 159Diophantus of Alexandria 160Pappus of Alexandria 164The End of Alexandrian Dominance 170Proclus of Alexandria 171Boethius 171Athenian Fragments 172Byzantine Mathematicians 1739 Ancient and Medieval China 175The Oldest Known Texts 175The Nine Chapters 176Rod Numerals 177The Abacus and Decimal Fractions 178Values of Pi 180Thirteenth-Century Mathematics 18210 Ancient and Medieval India 186Early Mathematics in India 186The Sulbasutras 187The Siddhantas 188Aryabhata 189Numerals 191Trigonometry 193Multiplication 194Long Division 195Brahmagupta 197Indeterminate Equations 199Bhaskara 200Madhava and the Keralese School 20211 The Islamic Hegemony 203Arabic Conquests 203The House of Wisdom 205Al-Khwarizmi 206‘Abd Al-Hamid ibn-Turk 212Thabit ibn-Qurra 213Numerals 214Trigonometry 216Tenth- and Eleventh-Century Highlights 216Omar Khayyam 218The Parallel Postulate 220Nasir al-Din al-Tusi 220Al-Kashi 22112 The Latin West 223Introduction 223Compendia of the Dark Ages 224Gerbert 224The Century of Translation 226Abacists and Algorists 227Fibonacci 229Jordanus Nemorarius 232Campanus of Novara 233Learning in the Thirteenth Century 235Archimedes Revived 235Medieval Kinematics 236Thomas Bradwardine 236Nicole Oresme 238The Latitude of Forms 239Infinite Series 241Levi ben Gerson 242Nicholas of Cusa 243The Decline of Medieval Learning 24313 The European Renaissance 245Overview 245Regiomontanus 246Nicolas Chuquet’s Triparty 249Luca Pacioli’s Summa 251German Algebras and Arithmetics 253Cardan’s Ars Magna 255Rafael Bombelli 260Robert Recorde 262Trigonometry 263Geometry 264Renaissance Trends 271François Viète 27314 Early Modern Problem Solvers 282Accessibility of Computation 282Decimal Fractions 283Notation 285Logarithms 286Mathematical Instruments 290Infinitesimal Methods: Stevin 296Johannes Kepler 29615 Analysis, Synthesis, the Infinite, and Numbers 300Galileo’s Two New Sciences 300Bonaventura Cavalieri 303Evangelista Torricelli 306Mersenne’s Communicants 308René Descartes 309Fermat’s Loci 320Gregory of St. Vincent 325The Theory of Numbers 326Gilles Persone de Roberval 329Girard Desargues and Projective Geometry 330Blaise Pascal 332Philippe de Lahire 337Georg Mohr 338Pietro Mengoli 338Frans van Schooten 339Jan de Witt 340Johann Hudde 341René François de Sluse 342Christiaan Huygens 34216 British Techniques and Continental Methods 348John Wallis 348James Gregory 353Nicolaus Mercator and William Brouncker 355Barrow’s Method of Tangents 356Newton 358Abraham De Moivre 372Roger Cotes 375James Stirling 376Colin Maclaurin 376Textbooks 380Rigor and Progress 381Leibniz 382The Bernoulli Family 390Tschirnhaus Transformations 398Solid Analytic Geometry 399Michel Rolle and Pierre Varignon 400The Clairauts 401Mathematics in Italy 402The Parallel Postulate 403Divergent Series 40417 Euler 406The Life of Euler 406Notation 408Foundation of Analysis 409Logarithms and the Euler Identities 413Differential Equations 414Probability 416The Theory of Numbers 417Textbooks 418Analytic Geometry 419The Parallel Postulate: Lambert 42018 Pre- to Postrevolutionary France 423Men and Institutions 423The Committee on Weights and Measures 424D’Alembert 425Bézout 427Condorcet 429Lagrange 430Monge 433Carnot 438Laplace 443Legendre 446Aspects of Abstraction 449Paris in the 1820s 449Fourier 450Cauchy 452Diffusion 46019 Gauss 464Nineteenth-Century Overview 464Gauss: Early Work 465Number Theory 466Reception of the Disquisitiones Arithmeticae 469Astronomy 470Gauss’s Middle Years 471Differential Geometry 472Gauss’s Later Work 473Gauss’s Influence 47420 Geometry 483The School of Monge 483Projective Geometry: Poncelet and Chasles 485Synthetic Metric Geometry: Steiner 487Synthetic Nonmetric Geometry: von Staudt 489Analytic Geometry 489Non-Euclidean Geometry 494Riemannian Geometry 496Spaces of Higher Dimensions 498Felix Klein 499Post-Riemannian Algebraic Geometry 50121 Algebra 504Introduction 504British Algebra and the Operational Calculus of Functions 505Boole and the Algebra of Logic 506Augustus De Morgan 509William Rowan Hamilton 510Grassmann and Ausdehnungslehre 512Cayley and Sylvester 515Linear Associative Algebras 519Algebraic Geometry 520Algebraic and Arithmetic Integers 520Axioms of Arithmetic 52222 Analysis 526Berlin and Göttingen at Midcentury 526Riemann in Göttingen 527Mathematical Physics in Germany 528Mathematical Physics in English-Speaking Countries 529Weierstrass and Students 531The Arithmetization of Analysis 533Dedekind 536Cantor and Kronecker 538Analysis in France 54323 Twentieth-Century Legacies 548Overview 548Henri Poincaré 549David Hilbert 555Integration and Measure 564Functional Analysis and General Topology 568Algebra 570Differential Geometry and Tensor Analysis 572Probability 573Bounds and Approximations 575The 1930s and World War II 577Nicolas Bourbaki 578Homological Algebra and Category Theory 580Algebraic Geometry 581Logic and Computing 582The Fields Medals 58424 Recent Trends 586Overview 586The Four-Color Conjecture 587Classification of Finite Simple Groups 591Fermat’s Last Theorem 593Poincaré’s Query 596Future Outlook 599References 601General Bibliography 633Index 647