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Leonhard Euler: Mathematical Genius in the Enlightenment

Ronald S. Calinger
Princeton University Press
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Frank Swetz
, on

Leonhard Euler (1707–1783) is a name familiar to almost any person who has pursued a study of mathematics beyond elementary analysis. Historians of mathematics remember him as a prolific author and versatile mathematician who had some association with the Russian Academy of Science. Astute students of calculus might recognize his name associated with the notation “\(y = f(x)\)”, the exponential constant \(e\), or a dozen other mathematical concepts attributed to him. He was certainly a busy man who left his mark on calculus and the study of differential equations.

The tercentenary of Euler’s birth in 2007 was celebrated around the world by a series of association meetings and presentations reviewing his mathematical accomplishments; in particular, the Mathematical Association of America published a tribute series of five volumes examining his life and mathematical contributions. But despite this flood of Euler admiration and information, something more was needed to better appreciate Euler, the man, the mathematician and his special form of genius: a comprehensive biography. Ronald Calinger has solved that problem with his Leonard Euler: Mathematical Genius in the Enlightenment.

Calinger, a recognized historian of mathematics and an Euler scholar and researcher, is the only historian of mathematics I know who was actually trained as an historian. This background is obvious in his writing. Euler’s life and work at the Imperial Academy of Sciences in St. Petersburg and the Royal Prussian Academy of Sciences in Berlin is meticulously traced out. His major works and contributions to mathematics — establishing a foundation for differential equations and the calculus of variation; pioneering developments in number theory and topology and strengthening the studies of terrestrial and celestial mechanics through the applications of mathematical analysis — are isolated and examined in detail.

Euler can be considered the major force in the field of modern applied mathematics. In his participation in the work of his resident Scientific Academies and his remote influence on the Académie Royale des Sciences de Paris, especially through the suggestion of “Prize Questions”, he shaped the direction of all of eighteenth century mathematics. He further promoted this effort through a network of active correspondents. Outside of the academic world, Euler’s Letters to a German Princess, 1768–1772, when released to the general public, caused an intellectual stir and advanced the mood of the Enlightenment.

The reader learns about not only Euler’s accomplishments but also the environment in which he grows and labors. This is the beauty of Calinger’s narrative, the daily detail of the times. Life as an academician was not all glamorous. It also had its frustrations and hardships: in Russia, the Orthodox Church and the aristocrats opposed the creation and work of their Scientific Academy; in Prussia [Germany], the Academy’s patron Frederick II was often away fighting wars and did not provide the support needed by his institution.

I learned that Euler enjoyed smoking Virginia tobacco, playing chess and performing on the clavier; also that his wife feared fires in St. Petersburg and that colleagues didn’t want to live in Berlin because it was too cold. These are minor facts, perhaps, but they breathe life into the times and characters. They make Euler, the mathematical genius and innovator, human like us.

Ronald Calinger’s Leonard Euler is an excellent book, a wonderful resource, mathematically informative, well researched and thoughtfully written. It will serve as a definitive reference on Leonhard Euler, his mathematics, his life and his times for many years to come.

Frank Swetz, Professor of Mathematics and Education, Emeritus, The Pennsylvania State University, is the author of several books on the history of mathematics. His research interests focus on societal impact on the development, and the teaching and learning, of mathematics.

Preface ix
Acknowledgments xv
Author's Notes xvii
Introduction 1

1. The Swiss Years: 1707 to April 1727 4
"Das alte ehrwürdige Basel" (Worthy Old Basel) 4
Lineage and Early Childhood 8
Formal Education in Basel 14
Initial Publications and the Search for a Position 27

2. "Into the Paradise of Scholars": April 1727 to 1730 38
Founding Saint Petersburg and the Imperial Academy of Sciences 40
A Fledgling Camp Divided 53
The Entrance of Euler 65

3. Departures, and Euler in Love: 1730 to 1734 82
Courtship and Marriage 87
Groundwork Research and Massive Computations 90

4. Reaching the "Inmost Heart of Mathematics": 1734 to 1740 113
The Basel Problem and the Mechanica 118
The Königsberg Bridges and More Foundational Work in Mathematics 130
Scientia navalis, Polemics, and the Prix de Paris 140
Pedagogy and Music Theory 150
Daniel Bernoulli and Family 160

5. Life Becomes Rather Dangerous: 1740 to August 1741 165
Another Paris Prize, a Textbook, and Book Sales 165
Health, Interregnum Dangers, and Prussian Negotiations 169

6. A Call to Berlin: August 1741 to 1744 176
"Ex Oriente Lux": Toward a Frederician Era for the Sciences 176
The Arrival of the Grand Algebraist 185
The New Royal Prussian Academy of Sciences 189
Europe's Mathematician, Whom Others Wished to Emulate 200
Relations with the Petersburg Academy of Sciences 211

7. "The Happiest Man in the World": 1744 to 1746 215
Renovation, Prizes, and Leadership 215
Investigating the Fabric of the Universe 224
Contacts with the Petersburg Academy of Sciences 234
Home, Chess, and the King 237

8. The Apogee Years, I: 1746 to 1748 239
The Start of the New Royal Academy 241
The Monadic Dispute, Court Relations, and Accolades 247
Exceeding the Pillars of Hercules in the MathematicalSciences 255
Academic Clashes in Berlin, and Euler's Correspondence with the Petersburg Academy 279
The Euler Family 282

9. The Apogee Years, II: 1748 to 1750 285
The Introductio and Another Paris Prize 287
Competitions and Disputes 292
Decrial, Tasks, and Printing Scientia navalis 298
A Sensational Retraction and Discord 303
State Projects and the "Vanity of Mathematics" 308
The König Visit and Daily Correspondence 313
Family Affairs 316

10. The Apogee Years, III: 1750 to 1753 318
Competitions in Saint Petersburg, Paris, and Berlin 320
Maupertuis's Cosmologie and Selected Research 325
Academic Administration 329
Family Life and Philidor 333
Rivalries: Euler, d'Alembert, and Clairaut 335
The Maupertuis-König Affair: The Early Second Phase 337
Two Camps, Problems, and Inventions 344
Botany and Maps 348
The Maupertuis-König Affair: The Late Second and Early Third Phases 350
Planetary Perturbations and Mechanics 359
Music, Rameau, and Basel 360
Strife with Voltaire and the Academy Presidency 363

11. Increasing Precision and Generalization in the Mathematical Sciences: 1753 to 1756 368
The Dispute over the Principle of Least Action: The Third Phase 369
Administration and Research at the Berlin Academy 374
The Charlottenburg Estate 384
Wolff, Segner, and Mayer 385
A New Correspondent and Lessons for Students 391
Institutiones calculi differentialis and Fluid Mechanics 395
A New Telescope, the Longitude Prize, Haller, and Lagrange 399
Anleitung zur Nauturlehre and Electricity and Optimism Prizes 401

12. War and Estrangement, 1756 to July 1766 404
The Antebellum Period 404
Into the Great War and Beyond 409
Losses, Lessons, and Leadership 415
Rigid-Body Disks, Lambert, and Better Optical Instruments 427
The Presidency of the Berlin Academy 430
What Soon Happened, and Denouement 432

13. Return to Saint Petersburg: Academy Reform and Great Productivity, July 1766 to 1773 451
Restoring the Academy: First Efforts 452
The Grand Geometer: A More Splendid Oeuvre 456
A Further Research Corpus: Relentless Ingenuity 471
The Kulibin Bridge, the Great Fire, and One Fewer Distraction 485
Persistent Objectives: To Perfect, to Create, and to Order 488

14. Vigorous Autumnal Years: 1773 to 1782 495
The Euler Circle 496
Elements of Number Theory and Second Ship Theory 497
The Diderot Story and Katharina's Death 499
The Imperial Academy: Projects and Library 502
The Russian Navy, Turgot's Request, and a Successor 504
At the Academy: Technical Matters and a New Director 506
A Second Marriage and Rapprochement with Frederick II 509
End of Correspondence and Exit from the Academy 515
Mapmaking and Prime Numbers 517
A Notable Visit and Portrait 518
Magic Squares and Another Honor 520

15. Toward "a More Perfect State of Dreaming": 1782 to October 1783 526
The Inauguration of Princess Dashkova 526
1783 Articles 529
Final Days 530
Major Eulogies and an Epilogue 532

Notes 537
General Bibliography of Works Consulted 571
Register of Principal Names 625
General Index 657

Comments's picture

See also the review that appeared in The Economist.