Table Of ContentLeonhard Euler and the Bernoullis
Leonhard Euler and the Bernoullis
Mathematicians from Basel
M.B.W. Tent
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Library of Congress Cataloging-in-Publication Data
Tent, M. B. W. (Margaret B. W.), 1944-
Leonhard Euler and the Bernoullis : mathematicians from Basel /
M.B.W. Tent.
p. cm.
Includes index.
ISBN 978-1-56881-464-3 (alk. paper)
1. Mathematicians--Switzerland--Basel--Biography. 2. Euler, Leonhard,
1707-1783. 3. Bernoulli, Jakob, 1654-1705. 4. Bernoulli family. 5.
Mathematics--Switzerland--History--17th century. 6. Mathematics--
Switzerland--History--18th century. I. Title.
QA28.T46 2009
510.92’2494--dc22
[B]
2009010076
Cover Illustrations: From left to right, Daniel Bernoulli (p. 196), Jacob
Bernoulli (p. 66), and Leonhard Euler (p. 259).
Printed in India
13 12 11 10 09 10 9 8 7 6 5 4 3 2 1
To our friends,
Sabine and Christian Koch
Contents
Preface xi
Acknowledgements xv
Figure Credits xix
1 The Bernoullis as Huguenots 1
2 The Bernoulli Family in Frankfurt and Then Basel 7
3 Jacob Makes His First Steps in the Study of
Mathematics 13
4 His Little Brother Johann “Helps” Jacob
with Mathematics 21
5 Having Completed His Studies in Philosophy and
Theology, Jacob Moves On 27
6 Jacob Travels to Geneva and Meets Elizabeth
Waldkirch and Her Family 33
7 Jacob Teaches Elizabeth Waldkirch to
Read and Write Numbers and Words 39
8 Sundials, and Tutoring in France 47
9 Jacob Meets with Mathematicians in Paris 53
10 Jacob Travels to Holland and England 61
11 Jacob Settles into Life in Basel to
Lecture and Learn 65
vii
viii Contents
12 Leibniz’s Calculus vs. Newton’s Fluxions 77
13 Johann Bernoulli Grows Up 87
14 Two Curves Studied by the Bernoullis:
The Isochrone and the Catenary 97
15 More Mathematical Challenges
from the Bernoullis 103
16 Jacob Bernoulli’s Mathematics 109
17 Johann Bernoulli Returns to Basel with His Family 117
18 Johann Bernoulli’s Son Daniel Grows Up 123
19 Daniel Bernoulli, the Paris Prize,
and the Longitude Problem 129
20 Leonhard Euler 133
21 Leonhard Euler’s Early Education 139
22 Leonhard Euler Goes to the Latin School
in Basel and Then on to the University 143
23 Daniel and Nicolaus Bernoulli Receive a Call to
the Academy at St. Petersburg 149
24 The Academy of Sciences at St. Petersburg 157
25 Euler Begins His Career and Moves to
St. Petersburg 161
26 Daniel Bernoulli and Leonhard Euler:
An Active Scientifi c Partnership 169
27 The St. Petersburg Paradox 177
28 Euler’s Early Work in St. Petersburg 181
29 Daniel Returns to Basel, and Leonhard Euler
Becomes Professor of Mathematics
at St. Petersburg 191
30 Daniel Bernoulli: A Famous Scholar 201
Contents ix
31 Leonhard Euler: Admired Professor
at St. Petersburg 207
32 Euler Becomes Blind in His Right Eye 215
33 St. Petersburg Loses Euler to Frederick
the Great of Prussia 219
34 The Eulers Arrive at the Court of Frederick
the Great in Berlin 225
35 Euler’s Scientifi c Work in Berlin 237
36 Euler’s Work in Number Theory 245
37 Magic Squares 251
38 Catherine the Great Invites Euler to
Return to St. Petersburg 255
39 The Basel Clan 263
Index 269
Preface
Th ese mathematicians, who lived between 1650 and 1800, all grew
up in Basel, Switzerland. Th e fi rst two—Jacob and Johann Bernoul-
li—were important Bernoulli mathematicians who made their ca-
reers mainly in Basel. Jacob’s name is sometimes given as James in
English or Jacques in French, and Johann’s name is sometimes given
as John in English or Jean in French. Johann’s son Daniel, the third
great mathematical Bernoulli, spent some years early in his career
fi rst in Venice and then in St. Petersburg but returned to Basel as
soon as he was able to arrange it. All the other Bernoullis made their
careers in Switzerland whenever possible. Only Euler (pronounced
“oiler”), who made his career in St. Petersburg and Berlin, chose not
to return to Basel. Since the Bernoullis were all related and tended to
use the same fi rst names over and over, their names can be confusing,
but I hope the family trees in the text will help the reader keep them
straight. Th ere is only one Euler who made a career as a mathemati-
cian, but he was as important to the development of mathematics as
all the Bernoullis taken together. It is unfortunate that most Ameri-
cans, unless they are crossword puzzle enthusiasts, have never even
heard the name Euler. And for that matter, most Americans have
heard only of Daniel Bernoulli even though his father Johann and
his uncle Jacob were probably equally important.
As I assembled this story, I was disturbed by the minor role
played by the women. If the Bernoullis had what might be called
the “math gene,” surely that was present in the females as well as the
xxii
xii Preface
males. I assume that the mothers were signifi cant in the upbring-
ing of both boys and girls, although there is also little indication of
that infl uence in the historical record. I think it is likely that Daniel
Bernoulli’s older sister Anna Catharina was at least partially involved
in Daniel and Nicolaus’ discussions of mathematics when they were
growing up, but that is conjecture on my part. Th e Bernoulli girls,
like other girls at that time, were probably barred from serious educa-
tion and from later life in the academic world simply because of their
gender and the time that they lived. Th at is regrettable.
Th e information available on Euler and the Bernoullis is spotty,
and in order to make a coherent story I had to fabricate some of the
details of their lives and the dialogues that portray their interactions.
In general I have tried to convey the interactions of the families and
the mathematicians in a way that is compatible with the available
records, but there is certainly an element of fi ction throughout this
work. Th e quotations from letters are only loose translations, but I
have tried to convey both the gist and the mood of the letters. Th ey
were written in German, Latin, and French, and I have not made a
note of the languages except in one letter that Daniel Bernoulli wrote
to Euler, in which he switched repeatedly from one language to an-
other. It is interesting that the correspondents generally preserved
the grammar of the disparate languages as they switched from lan-
guage to language. Th eir formal letters were all written exclusively
in Latin, the language of the scientifi c community of Europe at the
time, while many of their casual letters were written in one or more
languages.
Many of the “brilliant but bickering Bernoullis,” as William
Dunham called them, were indeed cantankerous, particularly when
it came to guiding their sons into their careers. For some reason,
each succeeding generation apparently tried to force sons into busi-
ness, law, or medicine rather than mathematics. I have indicated
that attitude in part by showing the Bernoulli patriarchs often re-
sponding to their sons with a resounding “no!” Leonhard Euler and
his father, by contrast, were apparently always kind and supportive
