Leonhard Euler (eBook)

Life, Work and Legacy

Robert E. Bradley, Ed Sandifer (Herausgeber)

eBook Download: EPUB

2007 | 1. Auflage
542 Seiten
Elsevier Science (Verlag)
978-0-08-047129-7 (ISBN)

Infos

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

Inhalt

Front Cover 1
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
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 Konigsberg 418
1. What Euler didn’t do 418
2. The Konigsberg bridges problem 420
3. Euler’s Konigsberg 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 Schrodinger’s wave function 516
4. Energy, paths and configurations 520
5. Euler’s method of maxima and minima, generalized 525
6. Derivation of the Schrodinger equation 527
7. Summary 532
References 533
Index 536

Textauszug

Leonhard Euler: Life and Thought

Ronald S. Calinger    Department of History, Catholic University of America, Washington, DC, 20064, USA

As the European Enlightenment began in the 1720s, few new accomplishments in mathematics were expected. Although mathematics had not yet become a profession in the previous century, when most of its practitioners came from the aristocracy or positions in medicine or law, that period culminating in the inventions of differential calculus by Isaac Newton and Gottfried Leibniz was considered a great age in mathematics, leaving little to be developed. But some scholars anticipated a fecund era for the field.1 Above all, the research of Leonhard Euler would prove them right. The Swiss-born Euler was to be one of the four preeminent mathematical scientists in history, the other three being Archimedes, Newton, and Carl Friedrich Gauss. Only for Newton and Euler did Gauss reserve the term summus.2

Driven by a passion for mathematics and natural science, a commitment to build a strong institutional base for them, and an insistent defense of reform Christianity, Euler made seminal contributions across the mathematical sciences and was arguably the most prolific mathematician in history. At the core of his research were infinitary analysis, or differential calculus, and rational mechanics. Along with celestial mechanics, he made them the sciences par excellence of the eighteenth century. He was the principal creator of the calculus of variations and differential equations, and he pioneered the differential geometry of surfaces. In mechanics Euler, not Newton, formulated most of the fundamental differential equations before William Rowan Hamilton. Operating within Enlightenment rivalries, in his case with Jean d’Alembert, Alexis Clairaut, Daniel Bernoulli, and Colin Maclaurin, he led in transforming mechanics and astronomy into modern exact sciences based on calculus. Euler founded continuum mechanics and advanced the study of ballistics, cartography, dioptrics, the theory of elasticity, hydraulics, hydrodynamics, music theory, number theory, optics, and ship theory. Massive and fearless computations, an extraordinary application of analysis and analogies, an appeal to his near unerring instinct, and clarity in writing characterize his work. Not since Claudius Ptolemy had a single geometer so dominated all branches of the mathematical sciences. During the eighteenth century four royal science institutions, in Paris, London, St. Petersburg, and Berlin, eclipsed universities in scientific research. It was largely Euler’s efforts that made the St. Petersburg and Berlin Academies of Science prominent European centers. The more than 810 of his articles and books, which fill seventy-four large volumes in the first three series of his Opera omnia, include approximately one-third of the entire corpus of research in mathematics, theoretical physics, and engineering mechanics published from 1726 to 1800, while the equivalent of research articles fill his extensive correspondence.3

1 Lineage, Youth, and Formal Education

Leonhard Euler was born on Friday April 15, 1707 (n. s.), in Basel, Switzerland. While most of Protestant and Orthodox Europe followed the Julian calendar or old style, the city had adopted the current Gregorian style in 1701. Euler’s birth house was probably located in the neighborhood around St. Martin’s Church near the center of the city close to the market quarter and ship landing on the Rhine River. He was the first child of Paul Euler, an Evangelical-Reformed minister, and Margaretha née Brucker. While “reformed” generally refers to the Protestantism of the Calvinists and Lutherans, Basel’s variety was of a pietism stressing love and the inner life. Leonhard’s mother Margaretha, the daughter of a hospital minister, was from a distinguished line of artists and humanistic scholars. Their son was baptized two days after his birth in the same St. Martin’s church as his father had been.

The Euler (Äuler, Ewler, Öwler) family came from the town of Lindau on Lake Constance (the Bodensee) in the German Swiss Canton. Au is the diminutive of Äule, which refers to a small, wet field or meadow. Au appears in the names of many small German towns, such as Dessau and Nassau. The owner of an Äule was an Äuler (oyler). The Eulers were variously called Euler-Schoelpin, signifying squint-eyed, which suggests that they were susceptible to an eye malady. The first written record of an Euler appeared in 1287, but a documented continuous line did not commence until 1458. Lindau, though on the far side of the Canton from Basel, had many close economic, political, and religious ties with the town. Hans Georg Euler, the great-great grandfather of Leonhard and the grandson of the German-speaking patriarch Hans Euler, moved to Basel in 1594. Hans Georg obtained citizenship in Basel, became a comb- and brush-maker, fathered fifteen children in two marriages, and lived to be ninety.4 Apparently the next three generations were artisans, most of them comb-makers or tradesmen belonging to hospitality guilds. They built the family’s modest financial base. In the fourth generation, four of the fourteen male cousins were able to become Basler Evangelical-Reformed ministers. These included Paul Euler, who matriculated at the University of Basel in 1685 at the age of fifteen. While at the university, Paul resided at the home of Jacob Bernoulli, under whose direction he wrote his senior thesis on ratios and proportions. He shared rooms with the young Johann I Bernoulli. Paul Euler completed his theological studies in 1693.

Leonhard did not spend his early youth in Basel. In June 1708 his father was named pastor-designate to St. Martin’s church in nearby Riehen-Bettingen. In November he was installed and the family moved to Riehen, about five kilometers northeast of Basel. It and Bettingen combined had a population of fourteen hundred. Supported by a sub-Mediterranean climate, these small villages were known for their rich vegetation, especially the white blossoms of the cherry trees in the late spring and the gold and red leaves on the grapes in the vineyards. The Eulers lived in a two-room parsonage until it was enlarged in 1712. One room was a study and the other living quarters. Of Leonhard’s two younger sisters, Anna Marie was born in 1708 and Marie Magdalena in 1711. His paternal grandmother lived with the family to her death in 1712. Johann Heinrich, the fourth child of the Eulers, was not born until 1719, after his brother departed for studies at the Basel Gymnasium. Leonhard was a talented child, apparently cheerful and sociable. The simplicity of rural life together with the model of his parents has its reflection in the forthright nature and even disposition of the adult Euler.

Leonhard’s parents were his first teachers. Familiar with the humanistic tradition, his mother Margaretha introduced him to Greek and Roman classics. The elementary instruction that his father Paul offered included mathematics, conceived as a subject underlying all natural knowledge. Paul began not with a geometry text but with Christoff Rudolff’s Coss, or algebra, the German equivalent of the Italian cosa or unknown, a two-part work that Michael Stifel had expanded from 208 to 484 pages. Paul possibly employed a reprint from 1615 of the first edition published in 1553. After explaining place-value notation and the four basic arithmetical operations, it examines in verbal form first, second, and third degree equations. Euler’s unfinished autobiography notes that he diligently studied the text for several years and made progress in solving its 434 problems, almost all of them first- or second-degree equations.5 He did this before moving to lodge with his maternal grandmother in Basel and enroll in the city’s Gymnasium, probably at age eight. Only an exceptional child of this age could have advanced previously through the difficult Coss.

The Basel Gymnasium, a Latin school, was in a pitiful state. Students were taught the Latin language and selections from ancient classics. Greek was optional. Teachers did not spare the rod, and fistfights broke out in the classroom. Like most parents, Euler’s hired a tutor, in their case a young theologian named Johann Burckhardt, who sided with Johann I Bernoulli in disputes with British geometers and natural philosophers, especially Brook Taylor. Burckhardt taught Euler the humanities and mathematics, a subject earlier struck from the curriculum by a vote of the townspeople.

In 1720 Euler matriculated at the University of Basel into the Philosophical Faculty, essentially the school of arts and sciences. He was thirteen, at the time roughly the normal age for entering a university. The university was in decline. Its enrollment had fallen from over a thousand students a century earlier to just above a hundred. It had only nineteen professors, underpaid and most of them mediocre. The exception was Johann I Bernoulli. The Philosophical Faculty provided the general education preparatory to choosing a specialty for a higher degree. Through industry and a powerful memory, Euler mastered all his subjects. Apparently he skipped the dry introductory mathematics lectures of Bernoulli, as Charles Darwin and Albert Einstein would later avoid sessions of tedious college courses. At fourteen Euler gave a speech titled “Declamatio: De Arithmetica et...

Produkt-Details

• Erscheint lt. Verlag: 20.3.2007
• Sprache: englisch
• Themenwelt: Literatur ► Biografien / Erfahrungsberichte
• Themenwelt: Literatur ► Romane / Erzählungen
• Themenwelt: Sachbuch/Ratgeber ► Natur / Technik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Finanz- / Wirtschaftsmathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geschichte der Mathematik
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik
• ISBN-10: 0-08-047129-3 / 0080471293
• ISBN-13: 978-0-08-047129-7 / 9780080471297