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A History of Mathematics

Uta C. Merzbach and Carl B. Boyer
John Wiley
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The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by
Jason M. Graham
, on

Here is the bottom line. The third edition of A History of Mathematics by Merzbach and Boyer (previously by Boyer and Merzbach) is a superlative book. I personally found it to be an extremely pleasurable read and I would recommend it to anyone with even the slightest interest in mathematics.

Now for some perspective. I am not a historian of mathematics, professional or otherwise. However, I have for a long time enjoyed reading and learning about major points or themes in the history of mathematics. I also enjoy biographies of mathematicians. One finds both in A History of Mathematics. Furthermore, the choice of topics is as masterful as their presentation.

I can only imagine how daunting a task it must be to try to compile and arrange in an accessible and interesting manner several millennia of history of a field such as mathematics. Here is where A History of Mathematics shines. The authors present mathematics from the time of the ancient Egyptians up to the twentieth century without a single lull in the excitement. Merzbach and Boyer seamlessly interweave the telling of the major time periods, the most important results, and the essential biographies in mathematics. This book serves as a reminder that mathematics is as much a part of the heritage of human culture as is art, architecture, poetry, etc.

One may worry about the amount of technical detail used in presenting the mathematics in a book about the history of the subject. There is certainly actual mathematics in A History of Mathematics, but not too much to render the book inaccessible to a large audience. Obviously, as one approaches the twentieth century the amount of notation used and the number specific results presented in detail becomes very small. If, while reading it, you find yourself wanting to see greater detail or more notation, you can easily supplement Merzbach and Boyer with more recent books such as Victor Katz’s A History of Mathematics or Grattan-Guiness’s The Rainbow of Mathematics (aka The Norton/Fontana History of the Mathematical Sciences). On some days I might also add to this the historical notes found in the Bourbaki books.

At least two important questions remain to be addressed. First, what is different about the third edition compared with the previous two? I have only compared the third edition with the first. The two biggest differences that I noticed are that the first edition contains exercises at the end of chapters while the third edition does not, and the way in which the material is arranged is slightly different. As far as I can tell, nothing has been removed from previous editions in terms of the basic content in the third edition. Changes seem to have more to do with the way that the topics discussed are emphasized.

A second important question is, could Merzbach and Boyer be used as a text for a course in the history of mathematics? I can’t see any reason why an instructor for a course in the history of mathematics would want students to not read it. As previously noted, there are no exercises, so depending on the nature of and goals for a history of mathematics course A History of Mathematics may not be sufficient as the sole course text. That being said, I think that Merzbach and Boyer is one of the best motivations for students, and perhaps even members of an even broader readership to want to explore more and see greater detail in topics from the history of mathematics.

Jason M. Graham is an assistant professor in the department of mathematics at the University of Scranton, Scranton, Pennsylvania. His current professional interests are in teaching applied mathematics and mathematical biology, and collaborating with biologists specializing in the collective behavior of groups of organisms.

Foreword by Isaac Asimov xi

Preface to the Third Edition xiii

Preface to the Second Edition xv

Preface to the First Edition xvii

1 Traces 1

Concepts and Relationships 1

Early Number Bases 3

Number Language and Counting 5

Spatial Relationships 6

2 Ancient Egypt 8

The Era and the Sources 8

Numbers and Fractions 10

Arithmetic Operations 12

“Heap” Problems 13 Geometric Problems 14

Slope Problems 18

Arithmetic Pragmatism 19

3 Mesopotamia 21

The Era and the Sources 21

Cuneiform Writing 22

Numbers and Fractions: Sexagesimals 23

Positional Numeration 23

Sexagesimal Fractions 25

Approximations 25

Tables 26

Equations 28 

Measurements: Pythagorean Triads 31

Polygonal Areas 35

Geometry as Applied Arithmetic 36

4 Hellenic Traditions 40

The Era and the Sources 40

Thales and Pythagoras 42

Numeration 52

Arithmetic and Logistic 55

Fifth-Century Athens 56

Three Classical Problems 57

Quadrature of Lunes 58

Hippias of Elis 61

Philolaus and Archytas of Tarentum 63

Incommensurability 65

Paradoxes of Zeno 67

Deductive Reasoning 70

Democritus of Abdera 72

Mathematics and the Liberal Arts 74

The Academy 74 

Aristotle 88

5 Euclid of Alexandria 90

Alexandria 90

Lost Works 91

Extant Works 91

The Elements 93

6 Archimedes of Syracuse 109

The Siege of Syracuse 109

On the Equilibriums of Planes 110

On Floating Bodies 111

The Sand-Reckoner 112

Measurement of the Circle 113

On Spirals 113

Quadrature of the Parabola 115

On Conoids and Spheroids 116

On the Sphere and Cylinder 118

Book of Lemmas 120

Semiregular Solids and Trigonometry 121

The Method 122

7 Apollonius of Perge 127

Works and Tradition 127

Lost Works 128

Cycles and Epicycles 129

The Conics 130

8 Crosscurrents 142

Changing Trends 142

Eratosthenes 143

Angles and Chords 144

Ptolemy’s Almagest 149

Heron of Alexandria 156

The Decline of Greek Mathematics 159

Nicomachus of Gerasa 159

Diophantus of Alexandria 160

Pappus of Alexandria 164

The End of Alexandrian Dominance 170

Proclus of Alexandria 171 

Boethius 171

Athenian Fragments 172

Byzantine Mathematicians 173

9 Ancient and Medieval China 175

The Oldest Known Texts 175

The Nine Chapters 176

Rod Numerals 177

The Abacus and Decimal Fractions 178

Values of Pi 180

Thirteenth-Century Mathematics 182

10 Ancient and Medieval India 186

Early Mathematics in India 186

The Sulbasutras 187

The Siddhantas 188

Aryabhata 189

Numerals 191

Trigonometry 193

Multiplication 194

Long Division 195

Brahmagupta 197

Indeterminate Equations 199

Bhaskara 200

Madhava and the Keralese School 202

11 The Islamic Hegemony 203

Arabic Conquests 203

The House of Wisdom 205

Al-Khwarizmi 206

‘Abd Al-Hamid ibn-Turk 212

Thabit ibn-Qurra 213

Numerals 214

Trigonometry 216

Tenth- and Eleventh-Century Highlights 216

Omar Khayyam 218

The Parallel Postulate 220

Nasir al-Din al-Tusi 220

Al-Kashi 221

12 The Latin West 223

Introduction 223

Compendia of the Dark Ages 224

Gerbert 224

The Century of Translation 226

Abacists and Algorists 227

Fibonacci 229

Jordanus Nemorarius 232

Campanus of Novara 233

Learning in the Thirteenth Century 235

Archimedes Revived 235

Medieval Kinematics 236

Thomas Bradwardine 236

Nicole Oresme 238

The Latitude of Forms 239

Infinite Series 241

Levi ben Gerson 242

Nicholas of Cusa 243

The Decline of Medieval Learning 243

13 The European Renaissance 245

Overview 245

Regiomontanus 246

Nicolas Chuquet’s Triparty 249

Luca Pacioli’s Summa 251

German Algebras and Arithmetics 253

Cardan’s Ars Magna 255

Rafael Bombelli 260

Robert Recorde 262

Trigonometry 263

Geometry 264

Renaissance Trends 271

François Viète 273

14 Early Modern Problem Solvers 282

Accessibility of Computation 282

Decimal Fractions 283

Notation 285

Logarithms 286

Mathematical Instruments 290

Infinitesimal Methods: Stevin 296

Johannes Kepler 296

15 Analysis Synthesis the Infinite and Numbers 300

Galileo’s Two New Sciences 300

Bonaventura Cavalieri 303

Evangelista Torricelli 306

Mersenne’s Communicants 308

René Descartes 309

Fermat’s Loci 320 Gregory of St. Vincent 325

The Theory of Numbers 326

Gilles Persone de Roberval 329

Girard Desargues and Projective Geometry 330

Blaise Pascal 332

Philippe de Lahire 337

Georg Mohr 338 

Pietro Mengoli 338

Frans van Schooten 339

Jan de Witt 340 

Johann Hudde 341

René François de Sluse 342

Christiaan Huygens 342

16 British Techniques and Continental Methods 348

John Wallis 348

James Gregory 353

Nicolaus Mercator and William Brouncker 355

Barrow’s Method of Tangents 356

Newton 358

Abraham De Moivre 372

Roger Cotes 375

James Stirling 376

Colin Maclaurin 376

Textbooks 380

Rigor and Progress 381

Leibniz 382

The Bernoulli Family 390

Tschirnhaus Transformations 398

Solid Analytic Geometry 399

Michel Rolle and Pierre Varignon 400

The Clairauts 401

Mathematics in Italy 402

The Parallel Postulate 403

Divergent Series 404

17 Euler 406

The Life of Euler 406

Notation 408

Foundation of Analysis 409

Logarithms and the Euler Identities 413

Differential Equations 414

Probability 416

The Theory of Numbers 417

Textbooks 418 

Analytic Geometry 419

The Parallel Postulate: Lambert 420

18 Pre- to Postrevolutionary France 423

Men and Institutions 423

The Committee on Weights and Measures 424

D’Alembert 425

Bézout 427

Condorcet 429

Lagrange 430

Monge 433

Carnot 438

Laplace 443

Legendre 446

Aspects of Abstraction 449

Paris in the 1820s 449

Fourier 450

Cauchy 452

Diffusion 460

19 Gauss 464

Nineteenth-Century Overview 464

Gauss: Early Work 465

Number Theory 466

Reception of the Disquisitiones Arithmeticae 469

Astronomy 470

Gauss’s Middle Years 471

Differential Geometry 472

Gauss’s Later Work 473

Gauss’s Influence 474

20 Geometry 483

The School of Monge 483

Projective Geometry: Poncelet and Chasles 485

Synthetic Metric Geometry: Steiner 487

Synthetic Nonmetric Geometry: von Staudt 489

Analytic Geometry 489

Non-Euclidean Geometry 494

Riemannian Geometry 496

Spaces of Higher Dimensions 498

Felix Klein 499

Post-Riemannian Algebraic Geometry 501

21 Algebra 504

Introduction 504

British Algebra and the Operational Calculus of Functions 505

Boole and the Algebra of Logic 506

Augustus De Morgan 509

William Rowan Hamilton 510

Grassmann and Ausdehnungslehre 512

Cayley and Sylvester 515

Linear Associative Algebras 519

Algebraic Geometry 520

Algebraic and Arithmetic Integers 520

Axioms of Arithmetic 522

22 Analysis 526

Berlin and Göttingen at Midcentury 526

Riemann in Göttingen 527

Mathematical Physics in Germany 528

Mathematical Physics in English-Speaking Countries 529

Weierstrass and Students 531

The Arithmetization of Analysis 533

Dedekind 536

Cantor and Kronecker 538

Analysis in France 543

23 Twentieth-Century Legacies 548

Overview 548 

Henri Poincaré 549

David Hilbert 555

Integration and Measure 564

Functional Analysis and General Topology 568

Algebra 570

Differential Geometry and Tensor Analysis 572

Probability 573

Bounds and Approximations 575

The 1930s and World War II 577

Nicolas Bourbaki 578

Homological Algebra and Category Theory 580

Algebraic Geometry 581

Logic and Computing 582

The Fields Medals 584

24 Recent Trends 586

Overview 586

The Four-Color Conjecture 587

Classification of Finite Simple Groups 591

Fermat’s Last Theorem 593

Poincaré’s Query 596

Future Outlook 599

References 601

General Bibliography 633

Index 647