Leonhard Euler: Life, Work and Legacy (Studies in the History and Philosophy of Mathematics, Volume 5)

The year 2007 marks the 300th anniversary of the birth of one of the Enlightenment’s most important mathematicians and scientists, Leonhard Euler. This volume is a collection of 24 essays by some of the world’s best Eulerian scholars from seven different countries about Euler, his life and his work. Some of the essays are historical, including much previously unknown information about Euler’s life, his activities in the St. Petersburg Academy, the influence of the Russian Princess Dashkova, and Euler’s philosophy. Others describe his influence on the subsequent growth of European mathematics and physics in the 19th century. Still others give technical details of Euler’s innovations in probability, number theory, geometry, analysis, astronomy, mechanics and other fields of mathematics and science. - Over 20 essays by some of the best historians of mathematics and science, including Ronald Calinger, Peter Hoffmann, Curtis Wilson, Kim Plofker, Victor Katz, Ruediger Thiele, David Richeson, Robin Wilson, Ivor Grattan-Guinness and Karin Reich - New details of Euler's life in two essays, one by Ronald Calinger and one he co-authored with Elena Polyakhova - New information on Euler's work in differential geometry, series, mechanics, and other important topics including his influence in the early 19th century Leonhard Euler: Life, Work and Legacy 4 Copyright Page 5 Table of Contents 8 Foreword 6 Chapter 1 Introduction 10 Chapter 2 Leonhard Euler: Life and Thought 14 1. Lineage, Youth, and Formal Education 16 2. Into the Colossus of the North: The Groundwork of Euler's Research 19 3. In Frederician Berlin: At the Apex of His Career 27 4. During the Reign of Catherine the Great: The Second St. Petersburg Years 59 Acknowledgments 66 References 66 Chapter 3 Leonhard Euler and Russia 70 1. Introduction 70 2. The First Petersburg Period 71 3. The Berlin Period 72 4. The Second Petersburg Period 78 5. Euler's Legacy 79 6. Conclusion 82 Chapter 4 Princess Dashkova, Euler, and the Russian Academy of Sciences 84 1. Princess Dashkova: Life in Brief to 1783 85 2. Academic Governance: Euler, Orlov, and Domashnev 89 3. Princess Dashkova as Imperial Academy Director 93 Acknowledgements 104 Chapter 5 Leonhard Euler and Philosophy 106 Chapter 6 Images of Euler 118 1. Introduction 118 2. Maria Sibylla Merian 119 3. Sokolov's mezzotint of Euler 121 4. Handmann's Pastel Painting of 1753 122 5. Handmann's oil painting of 1756 124 6. Handmann's large oil painting of 1756 (?) 125 7. Darbes' Painting of 1778 126 8. Further Study 129 Chapter 7 Euler and Applications of Analytical Mathematics to Astronomy 130 Introduction 130 1. Euler's first lunar tables, 1746 132 2. Mutual perturbations of Jupiter and Saturn, 1748 133 3. The precession of the Equinoxes and the mechanics of rigid bodies, 1751-1765 143 4. The inverse-square law and the motion of the lunar apse 146 5. Euler's later thoughts on celestial mechanics; his Third Lunar Theory 149 Chapter 8 Euler and Indian Astronomy 156 1. Introduction: Indian astronomy in the Enlightenment 156 2. T. S. Bayer and his work 157 3. Euler and Bayer 159 4. Indian calendrical methods and their representation in the appendices to Bayer's Historia 160 5. Euler's interpretations in the "De Indorum anno" 167 6. The impact of Euler's work 172 References 174 Chapter 9 Euler and Kinematics 176 1. Introduction 176 2. Euler and instantaneous planar kinematics 180 3. Euler, acceleration, spherical and spatial kinematics 184 4. Final remarks 201 Chapter 10 Euler on Rigid Bodies 204 1. Introduction 204 2. Cancellation of forces 205 3. An Error of Euler on Rigid Bodies 215 References 219 Chapter 11 Euler's Analysis Textbooks 222 1. Introduction to Analysis of the Infinite 223 2. Basic Principles of the Differential Calculus 232 3. Basic Principles of the Integral Calculus 237 4. Conclusions 240 References 241 Chapter 12 Euler and the Calculus of Variations 244 1. Expository remarks 244 2. Prehistory 249 3. General remarks on Euler's work 250 4. First period 251 5. Second period 255 6. Third period 258 References 260 Chapter 13 Euler, D'Alembert and the Logarithm Function 264 1. Introduction 264 2. Euler and D'Alembert 265 3. A History of the Logarithm Function 268 4. The Introductio 270 5. The Debate between Euler and d'Alembert 272 6. Euler's First Memoir 278 7. Euler's Second Memoir 281 8. Conclusion: D'Alembert’s Memoir 282 Acknowledgement 284 References 284 Chapter 14 Some Facets of Euler's Work on Series 288 1. Introduction 288 2. Euler and the Bernoulli numbers 290 3. Euler and the lack of subscripts 297 References 308 Chapter 15 The Geometry of Leonhard Euler 312 1. Introduction 312 2. The Tour 313 3. Conclusion 329 Acknowledgements 330 References 330 Chapter 16 Cyclotomy: From Euler through Vandermonde to Gauss 332 1. Euler 334 2. Vandermonde 340 3. Gauss 356 Acknowledgement 366 References 366 Chapter 17 Euler and Number Theory: A Study in Mathematical Invention 372 1. Introduction 372 2. Fermat and Number Theory 372 3. Goldbach and Euler 376 4. Fermat's Little Theorem: First Proof 380 5. Fermat's Little Theorem: Second Proof 382 6. The Sum of Four Squares 384 7. Fermat's Little Theorem: Third Proof 385 8. Fermat's Little Theorem: Fourth Proof 386 9. Results on Quadratic Forms 388 10. Fermat's Last Theorem 390 References 392 Chapter 18 Euler and Lotteries 394 1. Introduction 394 2. Euler's First Analysis 396 3. Euler's Later Analyses 399 References 402 Chapter 19 Euler's Science of Combinations 404 1. Partitions 404 2. Squares 409 3. Other Topics 412 References 416 Chapter 20 The Truth about Königsberg 418 1. What Euler didn't do 418 2. The Königsberg bridges problem 420 3. Euler's Königsberg letters 422 4. Euler's 1736 paper 424 5. The modern solution 427 References 429 Chapter 21 The Polyhedral Formula 430 1. The polyhedral formula 431 2. The flaw and the repair 434 3. Legendre's proof 437 4. The exceptions of Lhuilier, Hessel, and Poinsot 438 5. Cauchy's proof 440 6. Von Staudt's proof 443 7. Prehistory of the polyhedral formula: Descartes' lost notes 444 8. After 1850 445 References 446 Chapter 22 On the Recognition of Euler among the French, 1790-1830 450 1. The rise of Paris to mathematical eminence 450 2. Varieties in the calculus and mechanics 451 3. Euler's place: preliminary remarks 452 4. Euler or Lagrange in the calculus and analysis? 453 5. Euler or Lagrange in mechanics? 455 6. On Laplace and his own place 457 7. Laplace's programme of molecular physics, and the alternatives 458 8. Continuum mechanics, molecular and otherwise 460 9. A new tradition for the calculus: the impact of Cauchy 461 10. Three smaller topics 461 11. Three general surveys 462 12. Concluding remark 464 References 464 Chapter 23 Euler's Influence on the Birth of Vector Mechanics 468 1. Introduction 468 2. On Euler's conception of vectors 469 3. Euler's first memoir: the solution by pure geometry 473 4. Euler's second memoir: the solution by the first principles of statics 476 5. Impact and influence of the work 477 Acknowledgments 481 References 482 Chapter 24 Euler's Contribution to Differential Geometry and its Reception 488 1. Leonhard Euler's various contributions to differential geometry 488 2. Reception 496 3. Final Remarks 507 References 508 Chapter 25 Euler's Mechanics as a Foundation of Quantum Mechanics 512 1. Introduction 512 2. The contribution of Euler to mechanics 513 3. From Euler's mechanics to Schrödinger's wave function 516 4. Energy, paths and configurations 520 5. Euler's method of maxima and minima, generalized 525 6. Derivation of the Schrödinger equation 527 7. Summary 532 References 533 Index 536 Content: Foreword Pages v-vi Robert E. Bradley, C. Edward Sandifer Introduction Original Research Article Pages 1-4 C. Edward Sandifer, Robert E. Bradley Leonhard Euler: Life and thought Original Research Article Pages 5-60 Ronald S. Calinger Leonhard Euler and Russia Original Research Article Pages 61-73 Peter Hoffmann Princess dashkova, euler, and the russian academy of sciences Original Research Article Pages 75-95 Ronald S. Calinger, Elena N. Polyakhova Leonhard Euler and philosophy Original Research Article Pages 97-108 Wolfgang Breidert Images of Euler Original Research Article Pages 109-120 Florence Fasanelli Euler and applications of analytical mathematics to astronomy Original Research Article Pages 121-145 Curtis Wilson Euler and indian astronomy Original Research Article Pages 147-166 Kim Plofker Euler and kinematics Original Research Article Pages 167-194 Teun Koetsier Euler on rigid bodies Original Research Article Pages 195-211 Stacy G. Langton Euler's Analysis Textbooks Original Research Article Pages 213-233 Victor J. Katz Euler and the Calculus of Variations Original Research Article Pages 235-254 Rüdiger Thiele Euler, D'Alembert and the Logarithm Function Original Research Article Pages 255-277 Robert E. Bradley Some Facets of Euler's Work on Series Original Research Article Pages 279-302 C. Edward Sandifer The Geometry of Leonhard Euler Original Research Article Pages 303-321 Homer S. White Cyclotomy: From Euler through Vandermonde to Gauss Original Research Article Pages 323-362 Olaf Neumann Euler and number theory: A study in mathematical invention Original Research Article Pages 363-383 Jeff Suzuki Euler and lotteries Original Research Article Pages 385-394 D.R. Bellhouse Euler's science of combinations Original Research Article Pages 395-408 Brian Hopkins, Robin Wilson The truth about königsberg Original Research Article Pages 409-420 Brian Hopkins, Robin Wilson The polyhedral formula Original Research Article Pages 421-439 David Richeson On the recognition of euler among the french, 1790-1830 Original Research Article Pages 441-458 I. Grattan-Guinness Euler's influence on the birth of vector mechanics Original Research Article Pages 459-477 Sandro Caparrini Euler's contribution to differential geometry and its reception Original Research Article Pages 479-502 Karin Reich Euler's mechanics as a foundation of quantum mechanics Original Research Article Pages 503-525 Dieter Suisky Index Pages 527-534 "The year 2007 marks the 300th anniversary of the birth of one of the Enlightenment's most important mathematicians and scientists, Leonhard Euler. This volume is a collection of 24 essays by some of the world's best Eulerian scholars from seven different countries about Euler, his life and his work. Some of the essays are historical, including much previously unknown information about Euler's life, his activities in the St. Petersburg Academy, the influence of the Russian Princess Dashkova, and Euler's philosophy. Others describe his influence on the subsequent growth of European mathematics and physics in the 19th century. Still others give technical details of Euler's innovations in probability, number theory, geometry, analysis, astronomy, mechanics and other fields of mathematics and science. - Over 20 essays by some of the best historians of mathematics and science, including Ronald Calinger, Peter Hoffmann, Curtis Wilson, Kim Plofker, Victor Katz, Ruediger Thiele, David Richeson, Robin Wilson, Ivor Grattan-Guinness and Karin Reich - New details of Euler's life in two essays, one by Ronald Calinger and one he co-authored with Elena Polyakhova - New information on Euler's work in differential geometry, series, mechanics, and other important topics including his influence in the early 19th century."--Publisher's description The year 2007 marks the 300th anniversary of the birth of one of the Enlightenment's most important mathematicians and scientists, Leonhard Euler. This volume is a collection of 24 essays by some of the world's best Eulerian scholars from seven different countries about Euler: his life, his work and his legacy. Some of the essays are historical, including much previously unknown information about Euler's life, his activities in the St. Petersburg Academy, the influence of the Russian Princess Dashkova, and Euler's philosophy. Others describe his influence on the subsequent growth of European mathematics and physics in the 19th century. Still others give technical details of Euler's innovations in probability, number theory, geometry, analysis, astronomy, mechanics and other fields of mathematics and science