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Gottfried Wilhelm Leibniz
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Gottfried Wilhelm Leibniz
The Polymath Who Brought
Us Calculus
M. B. W. Tent
© 2012 by Taylor & Francis Group, LLC
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Cover images: Gottfried Wilhelm Leibniz, courtesy of Gottfried Wilhelm Leibniz Bibliothek, Niedersäschische Landesbibliothek, Hannover; Leibniz’s Staffelwalze (his mechanical calculator), courtesy of Gottfried Wilhelm Leibniz
Bibliothek, Niedersäschische Landesbibliothek, Hannover; medal celebrating Leibniz’s invention of binary arithmetic,
photograph by the author; photograph of the author by Mary Gray Hunter.
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To Joanna Cragin Tent,
Who came to life at the same time as this book.
May Joanna come to enjoy mathematics as she grows up!
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Table of Contents
Preface
Acknowledgments
Figure Credits
Family Trees
Timeline of Events
1
2
3
4
5
6
7
8
9
ix
xiii
xvii
xix
xxiii
A Brilliant Child
A Student at the Universities of Leipzig and Jena
Dr. Leibniz Begins His Career
Paris, London, and Mathematics
Librarian and Councilor to Duke Johann
Friedrich of Hannover
Councilor and Librarian to Duke Ernst August
Writing and Not Writing the History
Court Historian to Elector Georg Ludwig
Alone in Hannover
1
25
45
71
111
131
169
191
219
Index229
vii
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Preface
In constructing this story of Gottfried Wilhelm Leibniz’s life, I was able to take advantage of his written
works, but I found relatively little specific information
on Leibniz himself, with all his foibles and charms.
Consequently this story is partially fabrication based
on the information available. Some of the letters that
I quote are rough translations, while others are simply constructions based on the facts available. The
dialogues are all fabrication, but again they are based
on the historical record on Leibniz and his life experiences. My goal was to present Leibniz as a real person
so that the reader can gain an appreciation of his phenomenal genius.
Gottfried Wilhelm Leibniz, a German who lived
from 1646 to 1716, discovered the calculus by 1675.
Isaac Newton, an Englishman who lived from 1643 to
1727, had already discovered his method of fluxions
and fluents (analysis that is similar to the calculus) a
decade earlier during his annus mirabilis (1664–1665).
ix
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xPreface
Although Leibniz discovered his differential calculus
ten years after Newton’s discovery, he waited only ten
years before he published it in 1684 and 1686—20
years before Newton published his own method in
1704. Both men saw the inverse relation between the
differential and the integral calculus—a critical connection. Although Newton discovered it first, Leibniz
published it first. Because Leibniz’s notation is better
and because it was widely adopted first, Leibniz’s calculus is the standard analysis used today. In a quirk
of history, in the English-speaking world Newton is
often remembered as the founder of the calculus, although it is Leibniz’s calculus—not Newton’s fluxions
and fluents—that scientists, engineers, and economists
use today.
When Jacob Bernoulli (the first mathematician of
the famous Bernoulli family) first read Leibniz’s 1684
article presenting the differential calculus to the world,
he was baffled, describing it as an enigma rather than
an explanation. After diligent study, however, he and
his brother Johann were able to understand it and see
its importance. Together, Leibniz and the two Bernoullis soon made Leibniz’s differential and integral calculus
accessible to all scientists (see M.B.W. Tent’s 2009 book
Leonhard Euler and the Bernoullis). By the time Newton published his fluxions and fluents 20 years later in
1704, Leibniz’s calculus was already in general use on
the continent. In Great Britain, however, Newton continued to be hailed for his fluxions. Because Leibniz’s
calculus was spurned in England for many years, the
British fell behind continental mathematicians in the
discovery of new mathematics. When the British belat-
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Prefacexi
edly adopted Leibniz’s calculus, Newton’s method of
analysis was lost. Regardless, in many people’s minds
Newton’s name is still attached to the calculus.
Leibniz and Newton were contemporaries, and
both were remarkable geniuses who did important
work. Newton’s monumental discoveries in physics
set the stage for Einstein’s theory of relativity, and
Leibniz’s explorations in philosophy were critical in
the evolution of rationalism. In mathematics, both
men constructed algorithms that allow the calculation of areas and volumes of irregular figures. Both
men’s analysis allows a scientist or engineer or economist to figure instantaneous rates of change. Although it seems clear now that both men made their
discoveries independently, in the eighteenth century,
accusations were repeatedly thrown back and forth
across the English Channel, as each side accused the
other of plagiarism. Leibniz and Newton and their
disciples spent many years fighting over who deserved
the credit for the discovery.
Both men had their failings. Newton was generally cantankerous, and Leibniz always committed himself to more than he could accomplish within any given
time. Leibniz was a sociable man who spent much of
his energy ingratiating himself with his noble sponsors
on whom he depended for his livelihood. He enjoyed
publishing his many accomplishments and listening to
the praise they brought. He was the only great German
scientist of the seventeenth century. By contrast, Newton was a loner in a culture with many notable scientists. He jealously guarded his discoveries and was
determined to share nothing with the greater world.
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xiiPreface
Nevertheless, he was respected in England almost as a
god, and he accepted that adulation as his due.
Although at the time of their deaths it was not
clear who would be the winner in the “calculus wars,”
it is obvious now that Leibniz’s calculus won—although Leibniz the man did not. It is his calculus that
all university students learn and his calculus that is
used in all scientific work today. Most people have
never heard of Newton’s fluxions, but his name continues to be celebrated in connection with the calculus.
Leibniz’s name, alas, has been forgotten.
Leibniz and his work deserve serious attention.
Certainly he was vain, and certainly he wasted much
of his genius on trivial projects. However, his accomplishments are phenomenal, and even as we use his
calculus—and everyone everywhere benefits from its
fruits—we should remember the man who discovered
and published it.
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Acknowledgments
I received help from many people as I pulled together the materials to tell the story of Gottfreid Wilhelm
Leibniz. I would not have been able to write this book
without their help.
First, I would like to thank my photographer,
Lizanne Gray. She travelled to Germany with me and
photographed all the things that I asked her to, but
she also discovered several noteworthy subjects on her
own. The result is a collection of photos that both of
us are proud of. Thank you, Lizanne.
Once again I had linguistic assistance from the faculty of the Altamont School in Birmingham, Alabama.
Jake Linder helped me with Latin translations and
Jeanne Classé helped me with French. Once again, my
husband Jim helped me with my German. Although I
know all of those languages, I still needed help. I thank
you all.
I also received help from people in Hannover,
Germany, the city where Leibniz spent his last years
xiii
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xivAcknowledgments
and where his archive is located. Jürgen Herbst at the
archive helped me navigate the archive and locate the
material I needed, provided me with material, and
read through an earlier version of the manuscript. Our
friends Günther and Traute Eisenhauer were also very
helpful during our stay in Hannover, giving me a variety of sources that were useful and driving me to important landmarks. I thank these citizens of Hannover
heartily.
My friends in my early morning walking group in
Birmingham (Barbara Morgan, Patsy Straka, and Eve
Graham) helped me many mornings as we walked and
I bounced ideas off them. At the beginning they had
no idea what I was talking about, but they came to
understand the project and they helped me see it more
clearly as I explained it to them. Thank you, girls!
Once again I found help at the libraries of Birmingham. The Avondale branch of the city library and
the Stearn Library at the University of Alabama at Birmingham provided valuable help. Thank you all.
Axel Wittmann, whom I first came to know as I
was working on my first book on Carl Friedrich Gauss,
helped me by arranging a tour of the library at Wolfenbüttel. It was a wonderful experience, and it helped
me see the other environment that Leibniz enjoyed in
Lower Saxony. Thank you, Axel.
Thanks also to our friends Sabine and Christian
Koch, who serve as our base of operations when we
travel in Germany. They are wonderful hosts and I
thank them for our many visits.
Mary Catherine Phinney, a Latin scholar and
friend of my father and stepmother, helped me com-
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Acknowledgmentsxv
pose the Latin conversation of the schoolboy Leibniz.
Thank you, Mary Catherine.
I greatly appreciate Brenda Bredin’s expert editing
on the preface to this book. It was difficult but important for me to write. Thank you, Brenda.
Klaus Peters at A K Peters helped me in many
little ways as I wrote this manuscript. His knowledge
of mathematics often saves me from mathematical embarrassment, and his contacts in the field have helped
in that too. I can write the story, but I am a mathematics teacher—not a mathematician—and the mathematics in my books must be correct.
Finally, I would like to thank my husband, Jim.
He read the entire manuscript even though he claims
to be baffled by mathematics, and he provided the historical background that I needed. He also put up with
my odd, early-morning working habits and accompanied me on our travels in Germany. Thank you, Jim.
There are undoubtedly many others whom I
should include in these acknowledgments. I apologize
for omitting you, but I am still grateful for your help.
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Figure Credits
Unless otherwise noted below, photographs are by
Lizanne Gray and illustrations are by the author.
88
Wooden model of Napier’s bones. In the
possession of the Mathematics Department,
Altamont School, Birmingham, AL. Photograph by the author.
133
Electress Sophie of Hannover. Statue in the
Herrenhauser Gardens in Hannover, Germany.
183
Medal celebrating Leibniz’s invention of binary arithmetic. Photograph of duplicate
medal in author’s possession.
196
Leibniz Statue in the Innenhof at Leipzig University. Photo by James F. Tent.
203
Street sign honoring Sophie Charlotte, Berlin. Photograph by the author.
xvii
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xviii
Figure Credits
The following photographs are courtesy of Gottfried Wilhelm Leibniz Bibibliothek, Niedersäschische
Bibliothek, Hannover, Germany:
81Leibniz’s Staffelwalze [step cylinder], his mechanical calculator.
124
Model of Leibniz’s windmill on display at the
Gottfried Wilhelm Leibniz Bibliothek.
132
Ernst August, Duke of Hannover.
141
King Frederick I of Prussia.
159
Leibniz’s travel chair.
202
Sophie Charlotte, Queen of Prussia.
226
Gottfried Wilhelm Leibniz.
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Family Trees
first wife
Friedrich Leibniz
1597–1652
Anna Rosina
Heinrich Freiesleben
Friedrich Leibniz
1597–1652
Gottfried Wilhelm
Leibniz
1646–1716
Catharina Schmuck
(third wife)
1621–1663
Anna Catharina
Leibniz
1648–1672
Simon Lưffler
Friedrich Simon
Lưffler
xix
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Johann Friedrich
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xx
Family Trees
King James I of
England
1566–1625
King Frederick V
of Bohemia
1596–1632
Elizabeth,
Princess and
Abbess of
Herford
1618–1680
King George I
of England
1660–1727
Queen Anne of
Denmark
1574–1619
Queen Elizabeth Stuart
of Bohemia
1596–1662
Duke Georg von
Brunswick-Lüneburg
1582–1641
Sophie, Princess and
Electress of
Hannover
1630–1714
Ernst August, Duke
of BrunswickLüneburg
1629–1698
Johann
Friedrich, Duke
of Hannover
1625–1679
Maximilian Wilhelm,
Prince of Hannover
1666–1726
Karl Philipp,
Prince of
Hannover
1669–1690
Ernst August,
Duke of York
1674–1728
Frederick
August
1661–1691
Sophie Charlotte,
Queen of Prussia
1668–1705
Caroline of Ansbach
1683–1737
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Christian Heinrich,
Prince of Hannover
1671–1703
King George II of
England
1683–1760
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Family Trees
xxi
Duke Georg von
Brunswick-Lüneburg
1582–1641
Christian
Ludwig, Duke
of BrunswickLüneburg
1622–1765
Georg Wilhelm,
Duke of BrunswickLüneburg
1624–1705
Anna Eleonore von
Hessen-Darmstadt
1601–1659
Queen Sophie
Amalie of
Denmark
1628–1685
King Friedrich III
of Denmark
1609–1670
Ernst August, Duke of
Brunswick-Lüneburg
1629–1698
Johann
Friedrich, Duke
of Hannover
1625–1679
Anne Sophie
1670–1720
Charlotte Felicitas
1671–1710
Benedicta Henriette
von der Pfalz
1652–1730
Henriette Marie
1672–1757
Holy Roman
Emperor Joseph I
1678–1711
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Amalia Wilhelmine
1673–1742
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Timeline of Events
Year
Age Event
1646
Leibniz is born.
1648
2
The Thirty-Years War ends.
1654
7
Leibniz enters the Nicolaischule.
His father’s library is opened.
1661
14
Leibniz enters the university at Leipzig.
1662
16
Leibniz completes his Bachelor’s degree in philosophy.
He studies at Jena and joins the Societas Quaerentium.
1663
17
Leibniz returns to Leipzig and chooses to study philosophy
and law.
He completes his Bachelor’s degree in law.
His mother, Catharina Leibniz, dies.
1666
20
Leibniz completes his Master’s degree in both philosophy
and law.
He completes his habilitation in philosophy.
1666
21
Leibniz earns his Doctorate in Law at Altdorf.
1667
21
Leibniz serves as secretary to the Society of Alchemists in
Nürnberg.
1667
21
Leibniz travels to Frankfurt and Mainz and begins his
work with Schönborn, Elector of Mainz.
He meets and begins working with Boineburg.
He makes his first contact with Duke Johann Friedrich.
xxiii
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xxiv
Timeline of Events
Year
Age Event
1670
24
Leibniz makes his first contact with Heinrich Oldenburg.
1672
26
Leibniz’s sister, Anna Catharina, dies.
He travels to Paris where he meets Christian Huygens and
shows him the calculating machine.
He begins to tutor Boineburg’s son, Philipp.
Boineburg dies.
1673
27
Elector Schönborn dies.
Leibniz loses his job tutoring Philipp.
He is elected to the Royal Society.
1674
28
Leibniz discovers differential and integral calculus.
1676
30
Newton sends his epistola priori.
Leibniz travels to London and delivers his calculating machine.
He meets Baruch de Spinoza.
1677
31
Newton sends his epistora posterior.
1679
32
The Duke approves Leibniz’s mining project.
Duke Johann Friedrich dies.
Leibniz meets Sophie, who becomes the Electress of Hannover.
1684
38
Leibniz publishes his differential calculus.
Sophie Charlotte marries Frederich III of Prussia.
1685
39
Leibniz begins his work on the Guelf history.
The mining project ends.
1686
40
Leibniz publishes his integral calculus.
1687
41
Leibniz begins his travels to research the Guelf history.
1688
42
Leibniz arrives in Munich and travels by boat to Vienna.
1690
44
Leibniz returns to Hannover and begins his work at Wolfenbüttel.
1692
46
The new electorate of Hannover and Celle is created.
Leibniz is offered a job as librarian to King Louis XIV.
1693
47
The mining project is renewed.
1698
52
Duke Ernst August dies.
1700
53
Leibniz is elected as a member of the Académie in Paris.
The Societät of Sciences is established.
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