Leonhard Euler: Mathematical Genius in the Enlightenment

690 Pages·2015·5.46 MB·English

by  Ronald, Calinger, Euler, Leonhard

Leonhard Euler

Leonhard Euler

Mathematical Genius in the Enlightenment

Ronald S. Calinger

Princeton University Press
Princeton and Oxford

Copyright © 2016 by Princeton University Press
Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock,
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Jacket art: Detail of 19th century engraving of Leonhard Euler, private collection. Image
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Library of Congress Cataloging-in-Publication Data

Calinger, Ronald.
Leonhard Euler : mathematical genius in the Enlightenment / Ronald S. Calinger.
pages cm
Includes bibliographical references and index.
ISBN 978-0-691-11927-4 (hardcover : alk. paper) 1. Euler, Leonhard, 1707–1783.
2. Mathematicians—Germany—Biography. 3. Mathematicians—Russia (Federation)
—Biography. 4. Mathematicians—Switzerland—Biography. 5. Physicists—Germany—
Biography. 6. Physicists—Russia (Federation)—Biography. 7. Physicists—
Switzerland—Biography. 8. Mathematics—History—18th century. I. Title.
QA29.E8C35 2015
510.92--dc23
[B]
2014045172

British Library Cataloging- in- Publication Data is available

This book has been composed in Baskerville 10 Pro.

Printed on acid- free paper. ∞

Printed in the United States of America

1 3 5 7 9 10 8 6 4 2

v

Contents

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 Mathematical
Sciences 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

vi | Contents

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

Contents | vii

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

viii | Contents

ix

Preface

F
or his profound and extensive contributions across pure and applied
mathematics, Leonhard Euler (1707–83) ranks among the four great-
est mathematicians of all time, the other three being Archimedes, Isaac
Newton, and Carl Friedrich Gauss. Euler is a principal figure in applying
calculus to create the modern mathematical sciences of celestial mechan-
ics, analytical mechanics, elasticity, and optics. In the twentieth century,
six concise biographies of him were produced, beginning in 1927 with
Louis du Pasquier’s Léonard Euler et ses amis (Leonhard Euler and his
friends) and two years later Otto Spiess’s Leonhard Euler. In 1961 Vadim V.
Kotek published a short biography in Russian, while in 1982 Adolf P.
Yush kevich brought out another in English and Rüdiger Thiele a narra-
tive in German. In 1995 Emil A. Fellmann supplied a text in German of
fewer than two hundred pages. In 2007, the tercentenary of Euler’s birth,
Fellmann’s text was translated into English by Erika and Walter Gautschi,
and Philippe Henry published Leonhard Euler: “incomparable géomètre.” Yet
in no language has a treatment of his life and work appeared that is full
in length and scope.
This book attempts to offer the first detailed and comprehensive
account in the context of Euler’s life, research, computations, and pro-
fessional interactions that centers on his achievements in calculus and
analytical mechanics. The growing body in print of primary sources by
Euler, many long inaccessible, together with the secondary literature on
him and his research, are making this possible. Central to the first effort
is the near completion of the more than eighty large volumes of Euler’s
Opera omnia (Collected works). Series 1, on mathematics, comprises
twenty- nine volumes; series 2, on mechanics and astronomy, comprises
thirty- one volumes; and series 3 comprises twelve volumes on physics
and varia. Series 4A will have eight volumes with annotated versions of
his massive correspondence, and a planned Internet database provision-
ally called Euler Heritage will make accessible his remaining manuscript
catalog, including his twelve notebooks totaling four thousand pages. It
has taken more than a century to near the finish of the Opera omnia. The
catalog of Euler’s complex writings by Gustaf Eneström appears in Die
Schriften Eulers chronologische nach den Jahren geordnet, in denen sie verfasst

x | Preface

worden sind (The Writings of Euler chronologically arranged, according
to the year they were published), which was published in the Jahresber­
icht der Deutschen Mathematiker­ Vereinigung (Annual Report of the Ger-
man Mathematical Society) from 1910 to 1913. Eneström lists 866 of
Euler’s publications, which include eighteen books. Today each has a
number indicating its ordered place from the time of appearance annu-
ally in the Eneström index.1 Thus, E88 stands for “Nova theoria lucis &
colorum,” which was printed in 1746 coming after his first 87 publica-
tions. The Euler Archive, directed by Dominic Klyve, Lee Stemkoski, and
Erik Tou has made the originals of almost all of these writings available
on the Internet.2 One result of the scope and depth of Euler’s research is
that most scholarship on him is fragmentary, centering on one or another
particular subject. The thorough investigation of Euler’s correspondence
and the subjects and reliability of his notebooks is at an early stage.
Euler publications and letters span five languages. Most are in Latin
and French, some are in German and Russian, and Euler himself trans-
lated one book from English into German. Recent Euler scholarship,
especially that published by the Mathematical Association of America,
includes many articles and books in English. Research on Euler requires
expertise ideally in at least the first three of these five languages and col-
laboration among scholars who together address all of them. The bibliog-
raphy in the present work indicates that competence in Italian, Spanish,
Chinese, and Japanese also helps.
The range, depth, and volume of Euler’s work make it highly un-
likely that any one scholar can master all the fields that he pursued. This
comprehensive biography examines the known principal areas of Euler’s
work. In describing, explaining, and summarizing what Euler achieved,
the present book is a scientific biography, but it is not a scientific trea-
tise exploring his central concepts at length. An exhaustive treatment
of Euler’s accomplishments is not yet possible, for there is still much to
learn about them. Instead, this book presents a synoptic study of the full
scope of his research. It stresses the discovery of new information about
contributions, the character of his colleagues and rivals, and the sources
of problems. A future book might have a team of specialists with exper-
tise from different fields write sections of essentially an anthology and be
overseen by an editor to strengthen a coherent perspective. This initial
comprehensive biography does not offer a social, cultural, political, or
economic narrative.
By paying greater attention to Euler’s correspondence and academic
records than did earlier concise biographies, this volume hopes to begin
to bring out Euler’s distinct personality, to remove myths (about, for ex-
ample, his dealings with Denis Diderot), and to note his relations with

Preface | xi

other great Enlightenment figures, particularly Jean- Baptiste le Rond
d’Alembert and Frederick II. Each of these topics remains underdocu-
mented and underexamined.
Two other vital sources exist: the minutes of the Petersburg and Ber-
lin Academies of Sciences, along with the anniversary or jubilee volumes
for dates marking multiples of a full or half century since Euler’s birth
or death—that is, 1907, 1957, 1983, and 2007. From 1897 to 1911, the
minutes of the earlier Petersburg Academy of Sciences, mostly in Latin
and French, were reprinted in four volumes; they were later annotated in
the Chronicles of the Russian Academy of Sciences in another four volumes
(2000–2004). The Trudy (Works), volume 17 (1962), describes all Euler
documents in the Archives of the Russian Academy of Sciences. The
present- day Berlin- Brandenburg Academy of Sciences and Humanities
lacks minutes from the eighteenth century but in 1957 issued extracts, its
Registres. These too merit more examination. The Euler jubilee anniver-
sary volumes published in Basel, Berlin, and Saint Petersburg, along with
recent critical studies published in Washington, DC, have added crucial
information about Euler’s life and work. An excellent account of all these
primary and secondary texts is Gleb K. Mikhaĭlov’s “Euleriana: A Short
Bibliographical Note.”3

In the investigation of the intellectual, social, and personal odyssey
of Leonhard Euler, this biography follows a chronological order, and its
chapters are broken down into sections that follow episodes from his life
and research.4 Of the settings, cultures, and circumstances that influence
his studies,5 the institutional is the foremost, involving royal financing,
programs, and administration at the Berlin and the Petersburg academies.
These offered libraries, opportunities for publication, the safety of the
laboratory, foreign connections, and a measure of freedom.6 The Royal
Academy of Sciences in Paris, with its prestigious annual prizes, set out
important topics and promoted competitions across Europe.
The sections of this biography emphasize a few significant problems,
innovative and daring computations, and proofs. These draw upon basic
concepts; new methods; disciplinary intuition; symbols such as e, п, and i
that Euler invented or made standard; and guesswork. Readers who do not
wish to review the technical steps in computations and proofs may proceed
directly to results. The time divisions for chapters largely follow Euler’s
places of residence, major developments in his career, or times of war. The
first Saint Petersburg period was when he laid the groundwork of his re-
search; during his Berlin years, he attained the summit of his career when
he presided over mathematical research and the transformation of the old
geometric exact sciences into the modern mathematical sciences; his sec-
ond Saint Petersburg stay, according to his grandson Paul Heinrich von

xii | Preface

Fuss, was a time of “prodigious activity” when the degree of Euler’s writing
increased.7 Although the preparation of many of Euler’s books and articles
came years before their actual dates of publication, this biography will date
them according to when they were printed but occasionally also give the
times of composition. The book generally follows the Gregorian calendar.
Biographies in the mathematical sciences have open and protean
boundaries. In the development of mathematics, two principal interpre-
tations exist: the Platonic, in which the subject advances independently
of time and setting by the force of its interior logic; and the external, in
which cultural, intellectual, political, social, and institutional forces shape
and support the subject’s growth or impede it.8 But no such simple dichot-
omy alone can capture the diversity and nuances of a vigorous discipline.
Prominent among other sources are aesthetics or beauty and beginning
studies with the final proofs, computations, and other results of mathema-
ticians.9 Mathematics is the most certain and exact of all the sciences, and
studies of its history benefit greatly from an interpretation that centers
on the close and critical reading of original written sources. The rigor-
ous examination of these materials can elucidate the methods that guided
Euler’s research and can identify persistent problems, the pattern of his
breakthroughs, his orderly arrangement of fields, and the few errors that
escaped his methods and near flawless intuition in mathematics.
Since Ernst Mach’s Die Mechanik in ihrer Entwicklung historisch­ kritisch
dargestellt (The science of mechanics: a critical and historical account of its
development), first published in 1883, it was often believed that Newton’s
Principia mathematica had provided the complete framework for classical
mechanics, with the eighteenth century adding little. But René Dugas,
Craig Fraser, Walter Habicht, Thomas Hankins, Gleb K. Mikhaĭlov, Ist-
ván Szabo, Stephen Timoshenko, Clifford Truesdell, Eduard Winter, and
others have brought out the vigorous, inventive research during the En-
lightenment. Upon Newton’s Principia mathematica Euler and his compet-
itors built a coherent foundation for classical mechanics, proposed novel
procedures for solving problems, and successfully applied differential
calculus.10

Truesdell, a master of six languages, including Greek and Latin, skill-
fully turned to a critical inquiry into Euler’s writings and especially rees-
tablished the significance of Euler’s work in theoretical physics, editing
five volumes of the Opera omnia.11 For these he provided exemplary his-
torical and scientific introductions on rational mechanics, elasticity, and
fluid mechanics. Truesdell has also written Essays in the History of Mechan­
ics, published in 1968, and founded the journal Archive for History of Exact
Sciences.12

Preface | xiii

Although the importance of what was called the progress of the sci-
ences to the Enlightenment is recognized, the history of science scarcely
appears in general histories of the period or biographies of Enlighten-
ment leaders associated with Euler. Thorough studies of his correspon-
dence and much of his Opera omnia could illuminate points in common
with the French Enlightenment and ideas distinctive to Berlin and Saint
Petersburg. Neither Tim Blanning’s impressive The Pursuit of Glory: Eu­
rope 1648–1815 (2007) nor Robert Massie’s Catherine the Great: Portrait of
a Woman (2011) mention Euler, while David Fraser’s Frederick the Great
(2000) has only one sentence on him.13 Thus, the present study of the life
of Euler and the importance of his research seeks to remove a major gap
in the history of science and of the Enlightenment.