Gottfried Wilhelm Leibniz: The Polymath Who Brought Us Calculus

Gottfried Wilhelm Leibniz

© 2012 by Taylor & Francis Group, LLC

Gottfried Wilhelm Leibniz The Polymath Who Brought Us Calculus

M. B. W. Tent

© 2012 by Taylor & Francis Group, LLC

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.

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2012 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 2011906 International Standard Book Number-13: 978-1-4398-9224-4 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

© 2012 by Taylor & Francis Group, LLC

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!

© 2012 by Taylor & Francis Group, LLC

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

Index 229

vii © 2012 by Taylor & Francis Group, LLC

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 © 2012 by Taylor & Francis Group, LLC

x Preface

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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Preface xi

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.

© 2012 by Taylor & Francis Group, LLC

xii Preface

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.

© 2012 by Taylor & Francis Group, LLC

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 © 2012 by Taylor & Francis Group, LLC

xiv Acknowledgments

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-

© 2012 by Taylor & Francis Group, LLC

Acknowledgments xv

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.

© 2012 by Taylor & Francis Group, LLC

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

© 2012 by Taylor & Francis Group, LLC

xviii

Figure Credits

The following photographs are courtesy of Gottfried Wilhelm Leibniz Bibibliothek, Niedersäschische Bibliothek, Hannover, Germany: 81 Leibniz’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.

© 2012 by Taylor & Francis Group, LLC

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 Löffler

Friedrich Simon Löffler

xix © 2012 by Taylor & Francis Group, LLC

Johann Friedrich

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

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

© 2012 by Taylor & Francis Group, LLC

Amalia Wilhelmine 1673–1742

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.

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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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Timeline of Events

Year

Age Event

1701

54

Sophie Charlotte becomes Queen of Prussia.

1705

59

Sophie Charlotte dies.

1707

61

Leibniz completes the first volume of the Guelf history.

1711

65

Leibniz meets Czar Peter the Great.

1714

68

Dowager Electress Sophie dies. Queen Anne of England dies. Georg August is crowned George I, King of England. Leibniz is banned from traveling to England.

1716

70

Leibniz dies.

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1646–1661

A Brilliant Child

Gottfried Wilhelm Leibniz was born on July 1, 1646,

in Leipzig, Germany, the heart of the Lutheran reformation. Three days later, his father Friedrich reported that the infant miraculously raised his head at his baptism and gazed at the pastor, seeming to acknowledge the holy water as it fell on his pristine head. “Surely this is a sign,” the baby’s father proclaimed. “I predict that my son will be a good and noble man. In his life, he will always be mindful of God in heaven and of his fellow man on earth! I take this as a sign of his faith—that he will serve God with love and piety and produce many wondrous works.” The Thirty-Years War had been raging throughout Central Europe for 28 years. However, the worst of the fighting in Saxony had ended at least ten years earlier, and the citizens of Leipzig were justified in anticipating peace. The Leibniz home and the university at Leipzig, where Gottfried Wilhelm’s father Friedrich served as professor, were situated safely inside the city walls and 1 © 2012 by Taylor & Francis Group, LLC

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had escaped the worst of the hostilities. Regardless, all were aware of the mayhem that still continued in the western German states, and there was always the fear that it might return to Saxony. Friedrich hoped that his son would grow up in a land that was finally free of war. With the signing of the Peace of Westphalia in 1648 when Gottfried Wilhelm was two years old, the Thirty-Years War came to an end, fulfilling Friedrich’s wish for his child. One Sunday morning of that year, Friedrich lay sick in bed, and once again his child astonished him. This time, Gottfried Wilhelm, who was playing on a tabletop near his father’s bed, somehow fell to the floor. Oh, dear! Is the child hurt? No, not at all! While his father and his aunt cried out in horror, the charmed little boy simply climbed back up onto the table and turned to them, calmly smiling. The father saw this as further evidence that his son was blessed with special favors from God. He immediately sent word to the church, where his wife (the child’s mother) was attending the service, asking that thanks be given to God for the salvation of his precious son. The following week, Friedrich greeted a neighbor in the market square: “I wish you good morning, Herr [Mr.] Schwarz! Have you heard the predictions for my remarkable son?” “Please tell me, Professor Leibniz. What great things do you expect of him?” Herr Schwarz politely asked. “He will set the world on fire with his brilliance,” his father vowed. “He will be renowned for his magnificent achievements and his modest, God-fearing

© 2012 by Taylor & Francis Group, LLC

A Brilliant Child

3

soul. I have received two unmistakable signs of this from the Heavenly Father.” “I am so happy for you,” Herr Schwarz said as they parted company. Another neighbor who had witnessed this encounter smiled knowingly. “I see Professor Leibniz has been sharing with you his ambitious plans for his young son,” he said. “Oh, yes,” Herr Schwarz replied. “I would say that his prediction is a tall order for a small child. Quite naturally, I expected great things of each of my own children as well, but you know how it is. One hopes that a child will grow healthy and strong, and live a godly life, but anything beyond that usually turns out to be nothing more than a dream, perhaps to be fulfilled but probably not. My children have done well enough, but I’m glad I didn’t raise such high expectations for them.” “The same is true of my children. You are a wise man, Herr Schwarz,” the other man observed.

Gottfried, this truly brilliant child, easily learned to read at an early age merely by watching and listening to his father, who often read to his son from books on German and European history. When his father told him stories from the history of the German people drawing on his own vast knowledge of the subject, Gottfried William continued to listen, in this way learning to appreciate his culture as well as his father’s attainments. The boy was a wonderful companion to

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his father, while his mother, the third wife of this distinguished citizen and professor of moral philosophy, looked on. It was a joy for her to see the warm relationship that developed between the dear little boy and his father through their shared love of history. Professor Friedrich Leibniz died in 1652 at the age of 55, leaving his six-year-old son to mature without his father—his favorite companion. Predictions of his son’s many gifts, which were often repeated to the boy as he grew up, provided the townsfolk with material for jokes throughout Gottfried Wilhelm’s childhood. What a silly man his father had been, with his unrealistic expectations for the boy! No child could ever live up to the professor’s fantastic dreams. Nevertheless, the boy did almost everything his father had predicted. He was truly a marvel, fulfilling his father’s ambitions time after time.

Gottfried Wilhelm’s mother, Catharina Schmuck Leibniz, was 20 years younger than her husband. The devout young widow, who chose not to remarry after her husband’s death, devoted her life to fostering the development of their son Gottfried Wilhelm and his younger sister Anna Catharina. Never doubting her husband’s predictions for the boy, she did her best to see that her children developed according to their father’s plans. Anna Catharina received her schooling at home because the city provided formal schooling only for boys—a tradition that would change slowly over the coming centuries in Germany. Since Anna Catha-

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rina needed to be prepared to serve as the wife of an appropriately educated gentleman, she would have no use for the Latin language. All she needed to learn was reading and writing in German, rudimentary arithmetic, efficient household management, and effective child-rearing. She also learned to play the piano acceptably and to create beautiful pieces of needlework. Like most mothers in the city, Anna Catharina’s mother was fully qualified to teach her those skills. Bringing up a boy was a different challenge entirely. Following her late husband’s plans, Catharina sent young Gottfried Wilhelm to an elite Latin school (one of two such schools in the city) that was dedicated to preparing the city’s future leaders to be upstanding members of the Lutheran church and to go on to study at the university. Gottfried Wilhelm Leibniz entered the Nicolaischule [Nicolai School] at the age of seven and studied there for almost eight years. The central curriculum was the classical trivium—the three classical liberal arts consisting of Latin grammar (learning to read and write correct, intelligent prose and poetry in Latin), rhetoric (mastering the art of writing and speaking persuasively and impressively in Latin), and logic (learning the art of constructing a valid argument in the classical style in Latin). In fact, the first few years of schooling were devoted almost exclusively to the mastery of Latin, the only language of instruction until later years when it would be supplemented with a modest amount of Greek. Neither the boys nor their teachers were allowed to speak any language other than classical Latin or Greek from the first day of school.

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A few hours per week in the last four years were devoted to the four lesser arts—the quadrivium. These included a small amount of arithmetic and geometry, as well as astronomy and physics, which would be taught for four hours each week in only the final year of schooling. The most abstract of the trivium, logic, was also saved for the last year, because it was considered the most difficult subject for a young boy to master. This was by no means a practical education—these children were not being prepared for lives as artisans or tradesmen. Instead, they were being groomed to be members of the upper class. Although very few would actually become scholars, most would know enough to communicate with scholars. The only break from the liberal arts in the Latin school program was time spent learning the Lutheran catechism and reading the Gospels. The German language, which all the children grew up speaking at home, had no role in the school. If a child learned to read German (as young Leibniz had done before he entered school), it was the result of instruction from someone at home, not in school. Leibniz’s later arguments in favor of the use of the German language in law and public life were a direct reaction to his educational background. With what we know of Leibniz’s later work in mathematics and law, we can only wonder that his early education was so narrow. His instruction in mathematics was elementary at best—he could add and subtract, and perhaps even multiply—and this was later supplemented with an introduction to Euclid’s geometry. However, young Gottfried Wilhelm’s broad intelligence always welcomed learning wherever he found © 2012 by Taylor & Francis Group, LLC

A Brilliant Child

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it. Although his education in school was probably no better than other privileged boys in Germany at the time, his thirst for learning allowed him to fill many of the gaps. When seven-year-old Gottfried Wilhelm entered the Nicolaischule in 1654, the exhausted, war-torn German states were beginning to rebuild. Immediately after his father’s death, his mother had locked his father’s library because she had been told that it might corrupt his young mind. Gottfried Wilhelm expected his schooling to fill that vacuum, but he was met with disappointment. “Hmmm,” young Gottfried Wilhelm said to himself when he found two tempting books lying on a hall table one afternoon. “These books must belong to the students who rent rooms in our house. I wonder what they are.” Upon opening the first book, he discovered that it was not written in German, like the books he had read with his father. Instead, it was in Latin, the language that he was just beginning to learn in school. Even this very bright child hadn’t been able to master that language in only the first few weeks of school. As he looked more closely, however, he saw that the book was full of columns of numbers—could they be dates?—as well as narrative, and he began to decipher parts of it. He recognized the names of people, places, and events from the Bible. The fact that the sentences were relatively short helped too. He soon realized that this work concerned the dates of events in the Bible. The book was called Opus Chronologicum [Chronological Work], and the author was Sethus Calvisius. Gottfried Wilhelm was pleased to find the Latin fairly straightforward, and using his knowledge of stories from the Bible © 2012 by Taylor & Francis Group, LLC

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he soon made genuine progress in the text as he stood in the hall studying it. “Eclipses?” he asked himself. “Oh, I remember. The word Eclipsis in Latin means a darkening. Yes, eclipses are described in my schoolbook. An eclipse must be when the moon comes between the earth and the sun, causing a darkening—a shadow—on the earth. Yes, that would be a fearsome event. I guess it would be like night falling in the middle of the day. It is something that I have never seen—my father never talked about it either. After it happens, people must talk about it a lot. People of earlier times must have worried that the sun was going to abandon the earth forever. But in an eclipse, my schoolbook says that the sun doesn’t die—its light is just hidden from us by the moon because the moon is closer to the earth than the sun is as they both orbit the earth.” He would learn a few years later that his schoolbook was actually wrong on this point—the earth orbits the sun rather than the other way around, although that was not common knowledge at that time. Gottfried Wilhelm continued to ponder, “I wonder why this author is writing about eclipses anyway.” Fearing that someone might discover his illicit reading in the hall, he decided to gather up the books in his arms and take them to his room. Clearly these books deserved more than a passing glance. Once he was seated behind his own closed door, he again opened the Calvisius book. “Aha! I see what the author is doing!” he said to himself. “He is establishing the dates of these events in history. Yes! Perhaps the dates given in the Bible are not entirely reliable. I sup-

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pose that would make sense. If those events happened before the birth of Christ, how could their dates fit in with the Christian calendar? The writers in the Old Testament couldn’t possibly have known when the Messiah would come, so they certainly couldn’t have given dates based on his birth.” As the child looked further into the book, he found listings of many events in ancient times paired up with their dates, and later in the book he saw events that were more recent. All these happenings seemed to fit into a big listing of events over time. “I love it!” he said to himself. “I will enjoy studying this some more.” Then he turned to the other book on the table. On the cover of the second book he saw the title, Ab Urbe condita. “I wonder what that means,” he asked himself. The first page began, “Iam premium omnium satis constat Troia capta in ceteros….” He studied the words in dismay. “No, this is beyond me,” he admitted to himself. “I don’t recognize any of those words. This is much harder than the other book.” Then he looked at the next few pages in the book. “Ah, but there are pictures,” he observed with delight. “In fact, there are lots of pictures—they’re lovely woodcuts. I’ll bet Father would have liked them. This must be history—at least I hope it’s history! I wonder if I can figure it out.” Over the next several days, through studying the pictures carefully while looking at the mysterious words, he finally managed to make out some of the text in the second book. He saw a drawing of two babies and a wolf, showing the wolf apparently taking care of the babies. “What an odd thought,” Gottfried Wilhelm said to himself. “I wonder if a wild animal

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could actually care for the babies of people.” The wolf was allowing the babies to drink her milk, and she was licking them gently. “This looks like a wonderful story,” he reflected happily. During the next few weeks he figured out more and more, repeatedly working from picture to text and back to picture again. Although he had no dictionary to help him, he had a fertile imagination and a compulsion to find out what the story might mean. “Hmmm, this picture shows two boys—the babies must have grown bigger by now—in a hut, being cared for by a man and a woman. Oh, I see! Romulus and Remus must be the boys’ names. I wonder if the city of Rome is named for Romulus. I’ll bet it is.” The book was Livy’s History of Rome, which was full of amazing stories of life in ancient times. Gottfried Wilhelm already knew the great rewards of reading, and now at last, he could revisit those delights. He had felt his father’s absence keenly since his death—a death that had taken not only his father, but also his father’s library, away from him. In school, his teacher taught from the revolutionary textbook for schoolboys of the day—Comenius’ Janua linguarum [The Door to Languages]. Comenius (1592–1670) was an educator who grew up in central Europe. He disagreed strongly with traditional approaches to education, with their emphasis on mindless memorization to the exclusion of almost everything else. He came to be known as the educator of Europe as he moved from country to country, seeing town after town adopt his books for its school. The Janua presented general science and history—fortunately in-

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cluding an explanation of eclipses, which had allowed young Gottfried Wilhelm to begin reading the Opus Chronologicum with that key word. The pages of the Janua presented the Latin language version side by side with a translation in German, and it presented vocabulary in sets. On the title page, Comenius encouraged his readers to work seriously in order to learn—Leibniz had already decided to do that!—before presenting an introduction to the origin of the world in the first chapter. Then Comenius proceeded to describe the basic elements before moving on to the heavens and much, much more. This book was written to challenge a bright child like Leibniz. Another book by Comenius—this time for younger children—was Orbis Pictus [Picture of the World], with text also in both German and Latin. It was more of a picture book, illustrated with line drawings that appeared throughout the text, presenting basic facts about people and animals. Although most of the boys in Leibniz’s class would probably have preferred the more elementary Orbis, the textbook Janua was just right for Leibniz. Gottfried Wilhelm’s teacher tried to make headway in the Janua as he taught the class. Unfortunately, progress was slow with all those squirming boys, most of whom paid little attention. Every day the teacher had to repeat himself, explaining the previous days’ lessons again and again before tentatively moving on to at least one new line of the text. Perhaps a few of the boys were listening and might even memorize the material he was spoon-feeding them. Although the author Comenius would have recommended instead involving the boys in

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a discussion through which they could use the new vocabulary, this teacher had more limited goals: “Memorize, boys. Memorize!” By this time, however, young Leibniz—frustrated by the tedium of his teacher’s sing-song repetitions of the lessons—had already raced far ahead of the class. Since books were his favorite play things, he furtively moved on in the textbook whenever his teacher’s gaze was not directed at him. The combination of his reading of Calvisius’ Opus Chronologicum and Livy’s History of Rome outside of school—together with his secret study of the rest of the Janua in school—had already enabled him to master basic Latin grammar and vocabulary as well as all the science and history presented in Janua. Why, he wondered, was the teacher so intent on wasting his time? Finally one day, young Gottfried Wilhelm stood up to ask a question in carefully constructed Latin: “Nobis dixisti stellas esse similes luminibus in caelo [You have told us that the stars are like lamps in the sky].” Continuing in Latin, he then asked, “Can’t we move on to something else?” “Leibniz,” the teacher replied, fingering the small whip that he always had in his hand for disciplining his unruly charges, “where did you find that question?” “Where did I find it?” Leibniz asked suspiciously, once again in Latin. “What I would like to know,” the teacher said (also in Latin), “is who taught you how to ask that question in Latin.” “Oh, no Sir,” Leibniz replied, again in Latin. “You are mistaken. No one dictated the question to

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me. I constructed it myself. I believe you said that Latin is the only language that we are allowed to speak in school. Isn’t that correct? Or, is there a problem with my grammar, Sir?” “No, your Latin was correct,” the teacher admitted, forgetting to answer Leibniz’s question in his bewilderment. How could this child have already learned so much more than he had presented in class? Never before had he had an actual conversation with a child of Leibniz’s age using the Latin language. It was obvious that this child had been so bold as to read ahead in the textbook. Either that, or someone was teaching him Latin outside of school. Outrageous! What was to be done? The teacher immediately decided to investigate. No child in his class could be allowed to progress beyond the standard pace of instruction. Comenius was a beautiful work that should be adequate for any child of his age. From the teacher’s perspective, all children who went to a Latin school in Leipzig should fit in with the program, and that meant that Comenius’ text must be read and memorized slowly and carefully.

“Leibniz,” the teacher called to him that afternoon as he prepared to leave school. “Come here, boy! I would like to know what you do in the afternoons when you get home from school.” Young Gottfried Wilhelm, sensing the teacher’s challenge, answered vaguely, “Well, Sir, first I always eat dinner with my mother and sister, and then I like to read if I have the time.”

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“I see. Could you please tell me what you read in that time?” the teacher asked suspiciously. “What I’m asking you is what books do they allow you to read at home?” “I have read Wroclaw’s edition of Aesop’s Fables,” Gottfried Wilhelm answered carefully, digging deep in his memory for an example of what the teacher might consider appropriate for a child of his age to read. He had read this work with his father several years earlier. Although some of the fables were interesting, he had long ago concluded that the morals were obvious and limited. What an insult to children! However, this teacher was a dull man. Gottfried Wilhelm was aware that the teacher would almost certainly not approve of his reading anything more interesting. “Wroclaw?” the teacher asked. “Where did you find that?” “It’s a book my father gave me,” the boy explained. “Maybe you didn’t know that my father, who was a professor of moral philosophy at the university, died last year. I miss him very much.” The teacher, ignoring the child’s painful loss, then asked bluntly, “Can you tell me what you have learned from that book?” “Well,” Leibniz said, “there’s the story of the crow who wanted to drink from the urn, but he couldn’t because the opening at the top of the urn was too small and the level of the water was too low. The thirsty crow’s solution was to drop pebbles, one after another, into the urn, causing the water level to gradually rise so that he could finally drink.” “And what moral do you draw from that fable?” the teacher demanded.

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“It seems to me that the crow was pretty clever,” Leibniz said. “Solving a problem creatively is a good thing. I like it.” “Do you not see that the crow stands for the devil— the evil in this world?” the teacher asked. “How could that be?” Leibniz asked. “Using your intelligence is not evil. My father taught me to use my head—to think before I acted so that my actions would be more effective. He taught me that Charlemagne— the King who brought peace and Christianity to Saxony—was brilliant as well as strong and articulate. Did you know that although Charlemagne could not read or write, he spoke both his native language (Frankish, I think) and Latin fluently. He brought Christianity to our land, forcefully (and I guess sometimes almost cruelly) through his own strength. I would like to grow up to be an intelligent man like Charlemagne—not a fool who simply does what others expect me to do, Sir.” This discussion was not going the way the teacher had intended, and he was growing more and more skeptical. “You have been reading other books too, haven’t you?” he asked, as if accusing the child of deviltry of his own. Then consciously trying to adopt a more positive tone, he asked, “Have you read Orbis Pictus—the more elementary book by Comenius, the author who wrote our Janua Linguarum?” “No, Sir,” Leibniz replied. “That wasn’t one of the books that my father and I read together.” “That’s a pity,” the teacher said. “That would have been appropriate material for a boy of your age to read on your own. Please tell me what else you have been reading.”

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“Well,” the honest boy replied, “I have been reading a book of history too.” “And what is the name of that book?” his teacher asked. “It’s called Ab Urbe condita,” Gottfried Wilhelm admitted. “It’s just a book that I found at my house.” “All right,” the teacher said, “I guess we’ll discuss this later. I would recommend that you spend more time on Wroclaw and less time on whatever history book you have found.” “Yes, Sir,” Gottfried Wilhelm said politely, eager to escape from this interview.

The teacher, who was not willing to give up on the child so easily, resolved to call on young Leibniz’s mother the next day. However, when the teacher was shown in, he found that the lady was deep in conversation with an elegant gentleman. The teacher, who was acutely aware of his own shabby suit and lowly status, decided it would be best to put off his talk with Frau [Mrs.] Leibniz until another time. “I can return tomorrow if it would be more convenient,” the teacher volunteered. “I don’t want to interrupt.” The nobleman generously replied, “Oh, no. Please go ahead and talk with Frau Leibniz now. I can wait.” “That’s all right,” the teacher said. “I will come back later at a more convenient time.” “Herr Lehrer [Mr. Teacher],” Gottfried Wilhelm’s mother said, “please tell me what you have come for.

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I will be glad to include this gentleman in our discussion. Is there a problem with my son? Has he been misbehaving in school? A boy who has lost his father needs guidance from someone like this kind gentleman, so please go ahead.” “The problem,” the teacher began hesitantly, “is that your son is impatient. He seems to be unwilling to stick with the curriculum that has been developed here in Leipzig over the years. I have been shocked to discover that he must have been reading ahead in our classroom textbook on his own. In fact, I am suspicious that he may have actually finished reading the entire book by himself. Can you believe it? Not only that: I believe he is also reading other books that are too advanced for him. I can’t imagine where he could have found such books.” At this point, the nobleman interrupted the teacher’s tirade. “Herr Lehrer,” he began, “I believe that young Gottfried Wilhelm is a remarkably bright child. I have to admit, Herr Lehrer, that I’m impressed.” The nobleman then turned to the child’s mother, “Frau Leibniz, I am charmed by what your boy has done on his own. I don’t believe I would have had the wit to do that when I was his age. I believe we should not bow to the teacher’s demands. I know that my friend Friedrich Leibniz—your late husband and the father of this boy—would insist that his son be given ready access to all the books in his library as soon as he desires it. That was certainly his plan for the boy. Stifling a child’s curiosity is wrong, no matter what the professional educators say. I’m sure you agree, Frau Leibniz. I cannot believe that Comenius, the author of several excellent

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books on the education of children, would agree with this teacher.” Then turning to the teacher, he asked, “Am I correct in thinking, Herr Lehrer, that you use a Comenius text with Gottfried Wilhelm’s class?” “Yes,” the teacher replied. “We are using Janua Linguarum. Comenius produced excellent texts, but I firmly believe he would not have encouraged a child to progress too fast in his learning.” “That is not my reading of Comenius at all,” the nobleman said. “I believe he was the perfect teacher for a very bright child like young Gottfried Wilhelm, and I am sure he would never have tried to hold a boy back.” “I am sure you are right, dear sir,” Frau Leibniz boldly said. “I wanted to be careful, so I locked the door until I could consider it further. However, my husband loved his library even as he enjoyed sharing that library with his son. Therefore, I am sure he would have chosen to satisfy his son’s curiosity, wherever it might lead him. I will have the library door unlocked for him at once.” “Well then, Herr Lehrer,” the nobleman said in summary, “we agree that the child will have free use of his father’s library whenever he likes. Please do not interfere with his independent quest for learning, either in school or outside of it. Possibly the other children in the class need to be held back—but certainly not this one. Frau Leibniz, where is the boy?” Gottfried’s mother rang for the maid so that she might call her son and arrange for the opening of the library. When the boy appeared, he was surprised to find his teacher in conversation with his mother and

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her friend. The nobleman spoke kindly to young Gottfried Wilhelm, “Young man,” he began, “your mother and I have been talking with your teacher. We have concluded that there is no reason for you to be barred from your father’s library any longer. From now on, you will be free to read any of the books whenever you wish. Your mother has already asked the maid to unlock it for you.” “Oh, thank you Sir!” Gottfried Wilhelm exclaimed, bowing energetically to the gentleman, silently noting that his teacher was looking uncomfortable. “I have dreamed of this ever since my father died, but I feared it would never happen.” “I am sure you will use the library wisely,” the nobleman said. “However, if you find something you don’t understand or something that shocks you, I hope you will immediately ask your mother or me to help you with it. Herr Lehrer,” he continued, “I assume you won’t mind if he asks you as well?” “That would be all right,” the teacher reluctantly agreed, and quickly said his farewell. Later that day, the nobleman made an appointment to speak with the child’s official guardian— probably Heinrich Freiesleben, the husband of Gottfried Wilhelm’s much older half-sister—to formalize this agreement. Since Frau Leibniz was no longer under the legal protection of her deceased husband, the law of Leipzig required her to submit to a male guardian for approval of any change in her children’s upbringing. This permission was apparently granted, and from then on Gottfried Wilhelm read widely from his father’s books and rapidly began to

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fill out his knowledge of history and the Latin language. Sensing his teacher’s disapproval of the arrangement, Gottfried Wilhelm tried to be quiet in school, waiting patiently for the school day to end so he could go home and continue his learning there. The child began a serious study of the ancients as soon as he finished Livy’s History of Rome. First he read general descriptions of the works of Cicero, Seneca, and Pliny. Then he moved on to the ancient texts themselves. He diligently learned all that he could from these sources. True to his upbringing as a young Christian, he also delved into earlier texts on the Christian church as well as more recent developments in the Lutheran church. He knew that his father had been a pious scholar of great learning, and the boy assumed he should follow in his father’s path. When Gottfried Wilhelm was 12 or 13 years old, he learned in school that an older boy had been assigned to prepare an original oration in Latin verse for a church holiday—a holiday that was only three days away. Unfortunately, that boy was sick and would not be able to do it. What could the school leaders do? When young Leibniz heard about it, he volunteered enthusiastically to compose and speak a verse in the other boy’s place. It was his first truly demanding assignment in school. After school that day, as soon as he got home he headed for his room. He took out several pieces of paper, an ink pot, and a quill and set about the task. The Latin verses flowed out of his pen with no need to cross out errors or to pause and consider what should come next. The whole speech came to him fluently,

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and, when he read it aloud three days later, everyone applauded. “Tell me,” the principal, Johann Hornschuch, asked young Leibniz, “where did you find that excellent piece?” “I wrote it myself,” Gottfried Wilhelm replied, bowing quickly to the principal as he spoke. “I understood that it was supposed to be original. Isn’t that correct?” “Yes, that is correct, but surely you couldn’t have produced something so polished in just three days,” Hornschuch insisted. “But I did, Sir,” Gottfried Wilhelm explained. “I have read a great deal of prose and poetry in Latin in the last few years, and with that background the assignment was not difficult. I simply composed what seemed appropriate to me. I’m glad you found it acceptable.” “It was more than acceptable,” Hornschuch said. “Your presentation was the best that I have ever heard from a boy of your age. Congratulations!” This, coming from a man who also served as professor of Greek and Dialectic at the university, was high praise indeed. “Thank you, Sir,” Gottfried Wilhelm said, bowing once again. “My mother will be pleased to hear your opinion. She saw me writing it, but of course she couldn’t understand it since she doesn’t know Latin.”

In his final year at the Nicolaischule, Gottfried Wilhelm expected to study philosophy in depth with the

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help of an enlightened teacher. Unfortunately that did not happen. The teacher’s lectures were just as limited as his understanding of philosophy. Students should simply memorize lists in the same way that they had in the earlier grades. What an outrage! Gottfried Wilhelm had been reading philosophy for seven years, and he had questions, which would go unanswered as long as he remained at the Nicolaischule. He knew the only recourse in the meantime was to continue reading on his own, so that is what he did. Knowing that Aristotle was the most important philosopher among the ancients, he read many works about Aristotle and did his best to understand them. If only his father had been there to guide him! One book in his father’s library was a work of Ramon Llull (1232–1315). Llull had studied Aristotle carefully and had concluded that Aristotle’s categories for organizing all of human knowledge, while interesting, were far from complete. Llull was a philosopher whose goal was to advance philosophy. Llull used Aristotle as a starting point for his own revolutionary thinking. In his book Ars Magnus [The Great Art], he listed nine attributes—goodness, greatness, eternity, power, wisdom, will, virtue, truth, and glory, and claimed that by using only those nine criteria he could construct all of Christian doctrine. Gottfried Wilhelm was fascinated. As he played with Llull’s attributes, agreeing that they were a satisfactory vehicle for studying the Almighty, Gottfried Wilhelm took Llull’s work as a challenge. Could he come up with a set of categories that was better than either Llull’s or Aristotle’s? Young Leibniz was beginning a

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pattern that was to develop throughout his life. “This is something I know I can do,” he said to himself with pleasure. Then he remembered, “I really need to read the rest of Llull’s brilliant writing too, and I should probably do that first. I’ll try to finish up my own categories over the weekend.” As happened so often in his life, Gottfried Wilhelm had too many plans—too many projects. “I want to make my categories and I will, but they will have to wait.”

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1661–1666

A Student at the Universities of Leipzig and Jena

In 1661, at the age of 14, having completed his classical education at the Nicolaischule, Leibniz entered the University of Leipzig—officially called Alma Mater Lipsiensis—as a candidate for the Bachelor’s degree. This was the usual age for a young man to begin studying at the university, although he was undoubtedly better prepared than most. In fact, because of his late father’s status as a professor, Gottfried Wilhelm had officially matriculated at the university at the same time that he entered the Nicolaischule when he was only seven years old, but his early matriculation was only a formality. This time, he was ready for his advanced education. Leibniz found the atmosphere at the university far different from that of the Nikolaischule, with its inferior teachers and rigid curriculum. Among the university professors, by contrast, he found some genuine scholars. In his first semester, he studied under Jakob Thomasius—a marvelous intellectual with whom Leibniz corresponded for many years. This 25 © 2012 by Taylor & Francis Group, LLC

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scholar of Aristotle—the most revered philosopher of both ancient and modern times—quickly recognized Leibniz as a brilliant student who was ready to be challenged. They spent many hours together, deep in conversation. Leibniz’s intellectual curiosity, which had been interpreted as an unpleasant challenge by his school teachers, had suddenly become a marvelous advantage. What a pleasure it was for Thomasius to teach this eager and intelligent young man! “Would you please tell me, Leibniz, about your background,” Thomasius asked when they first talked. “I assume that you have studied Aristotle in some detail. Would you please tell me which of his works you have read?” “Well, actually, Sir,” Leibniz admitted in some embarrassment, “I have read very little of Aristotle’s own work. In school, the teachers presented nothing more than summaries of scholarship on Aristotle— and in fact, I wonder if they had actually read even those secondary sources! They apparently just took lists they had memorized and made us memorize those same lists. When I tried reading Aristotle on my own, I found it remarkably difficult, and I have to admit that I gave up. I suppose I really didn’t have any idea how to do it.” “Ah, that is a pity!” the professor sighed. “If you want to study Aristotle, you must go directly to Aristotle. Second-hand learning is a poor substitute for the real thing. I’m sure if you apply yourself, you can understand his works.” “That’s probably true,” Leibniz said. “But Sir, I have a question: I have read that the works of Aristotle

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we have today may not be as he actually wrote them. Could they really be nothing more than the notes his students wrote as they listened to his lectures?” “That may be,” Thomasius conceded, “but they are the best that we have and probably the best we ever will have. I think you have no choice. Regardless, I believe you owe it to Aristotle to give him a serious try. I would imagine you are more sophisticated now than you were the last time you attempted to read his works, so I would think that you would have more success with him now. You do read Greek, don’t you?” “Oh, yes Sir,” Leibniz answered. “I first learned to read Greek when I was much younger because I wanted to read Diogenes’s Lives and Opinions of Eminent Philosophers. My father described the work to me, so I was determined to read it myself. After Diogenes, I found that I could easily read many other works in Greek.” “Ah, yes,” Thomasius said, “I’d forgotten. You are the son of my late colleague Friedrich Leibniz, aren’t you? I wonder—do you have the use of his library?” “Oh, yes, Sir,” Leibniz said. “When I was young, he and I read together in his library, and soon after I began school I was allowed to read in the library at will. It has been wonderful.” “So you actually taught yourself to read Greek?” Thomasius asked. He was impressed. Then he continued, “I recommend you start with Aristotle’s Organon [Logic]. After you have read several pages of that, why don’t you come discuss them with me? I can provide some structure to your reading. Certainly Aristotle is

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not easy, but you will not be the first student who has mastered his works. I believe you will find them well worth the effort.” “The Organon presents Aristotle’s ten categories, doesn’t it?” Leibniz asked. Thomasius suppressed a chuckle. “I suppose your teacher required you to memorize those categories, didn’t he?” the professor asked. “That’s right, Sir,” Leibniz said, “but we scarcely discussed them.” “That’s not surprising,” Thomasius said. “However, I return to my original statement. I don’t think you should judge Aristotle’s works until you have read them on your own. Summaries are no substitute. Do you have them?” “Yes, my father’s library includes all of Aristotle’s works. Thank you for your advice, Herr Professor” Leibniz said. “I will return to talk with you, as you suggest, after I have made a serious start.” “Excellent! I look forward to our next meeting,” Thomasius said as Gottfried Wilhelm bowed in respect and left. Over the coming months Leibniz spent many hours working his way through Aristotle in his father’s library. The works were still very difficult for him, but Thomasius had challenged him to do it and he was determined. If the professor thought he could master Aristotle, then it must be true. When Gottfried Wilhelm grew tired, he sometimes went out for a walk in the Rosenthal, a lovely park just outside of Leipzig. As he strolled among the trees and flowers, watching a squirrel or a bird

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flitting about in the bushes, he mulled over what he had been reading. It was obvious that Aristotle had arrived at his views through contemplation alone. Gottfried Wilhelm considered that limitation carefully. Couldn’t there be other ways to study life? Couldn’t the world be studied through quantifying (i.e., measuring) rather than simply qualifying (i.e., analyzing) events and concepts. He began to wonder about more recent philosophers, like Galileo Galilei (1564–1642) and René Descartes (1596–1650), vowing that he must spend time on those writers as well, although they were never mentioned in his classes at the university. How odd that was! Nevertheless, Aristotle was his assignment for now, so the others would have to wait their turn.

In December 1662, less than two years after beginning his studies at the university, 16-year-old Gottfried Wilhelm completed his Bachelor’s degree. He had made remarkably quick progress for a young man. Six months later he defended his thesis, “Metaphysical Disputation on the Principle of Individuation,” in which he explored the concept of what constitutes an individual—what distinguishes one living being or one inanimate thing from another. Professor Thomasius was proud to write a preface to its published version. In this work, Leibniz argued that any substance is itself because of its innate composition. At this time he began his work on monads, the basic building blocks of any individual person or thing—the components that the philosopher Democritus had called

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atoms. Leibniz strove for years to regulate the process of breaking complex things and beings down into monads, thus allowing the analysis of an individual in the same way that one could identify each of the ingredients in a pot of soup. An entity was transformed from a formless thing into a combination of parts. Although his thesis won praise as a carefully reasoned work of philosophy, the most significant part of it is now seen as Leibniz’s use of monads, which evolved throughout his life in his attempt to construct an alphabet of human thought—a universal language that he hoped would function in the same way prime numbers do in mathematics. His plan was to allow people to resolve their controversies through the use of algebra rather than through arguments or wars. The Thirty-Years War—which had recently devastated much of central Europe—convinced Leibniz that finding a way to avoid wars was critical to the survival of the human species. After completing this first step in his university education, Leibniz traveled 100 kilometers to the University of Jena, where he studied for the summer semester. With his interest in mechanistic philosophy and his desire to quantify phenomena as well as qualify them, he had become acutely aware of his limited mathematical background. He had heard of the talents of the scholar Erhard Weigel (1625–1699), whose brilliant mathematics lectures were well known throughout Saxony, and was eager to study with him. Although Weigel’s knowledge of mathematics was limited (by French and Dutch standards), he was the best mathematician to be found in Saxony.

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Gottfried Wilhelm was delighted by Weigel’s modern approach to the broader field of natural philosophy—what we now call science. Although Weigel was a firm believer in Aristotle’s philosophy (as would be expected of a professor in seventeenth century Lutheran Saxony), he did not view it in the limiting way that Thomasius did. Leibniz eagerly read Weigel’s book of 1658, in which he proposed a radical reform of philosophy. This work was organized mathematically—developing his ideas logically from definitions and axioms in a way that was reminiscent of Euclid (300 bc). Leibniz was enthusiastic about Weigel’s approach and used it extensively as he developed his own philosophy. Leibniz was often amused during debates in Weigel’s class to see his fellow students struggle when Weigel demanded they express their opinions in plain German rather than in Latin. It was obvious to both Leibniz and Weigel that some of those budding scholars were simply mouthing gibberish—that although their statements sounded plausible in Latin, they were unable to explain clearly in their native language what they meant! “What fools!” Leibniz said to himself. “If they can’t explain themselves in German, it must be that they don’t know what they mean. I think they’d better go back and start over.” While in Jena, Leibniz joined the Societas Quaerentium [Society of Inquiry]—a student group that held weekly meetings to explore intellectual questions together. Involving himself in such activities was an exhilarating experience for Gottfried Wilhelm. At these meetings, he could hear other students and faculty

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members report on the week’s reading assignment as they discussed one fundamental philosophical argument after another. The students grappled with difficult questions, and the resulting intense discussions were fascinating. “I have read Plato’s justification of the study of Euclid’s mathematics,” a student named Schmidt reported at the first meeting that Leibniz attended, “but I wonder where he finds the beauty in it. It’s just geometry, isn’t it?” “Oh, it’s far more than that!” a student named Braun replied. “In Book VII he defines perfect numbers and then in Book IX he gives an algorithm for finding as many perfect numbers as you like. It is clearly beautiful!” “But didn’t Euclid think that all numbers are perfect?” Schmidt asked. “I think Euclid liked mathematics a lot.” “Well, he certainly found numbers beautiful, but he knew that only a few of them are perfect,” Braun said. “Let me explain how he taught us to find the perfect numbers.” “But wait!” Schmidt interrupted. “What is a perfect number anyway?” “Euclid defines it as a number that is equal to the sum of its parts,” Braun explained. “What does that mean?” Schmidt asked. “I thought it was obvious,” Braun said. Leibniz couldn’t keep quiet any longer. “Schmidt, let’s look at the number 10. Its factors besides itself are 1, 2, and 5. If 10 were a perfect number, those three numbers would add up to 10. Since 1 + 2 + 5 is not

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equal to 10, we know that 10 is not perfect. The proper factors of the number 6 are 1, 2, and 3. The sum of 1 + 2 + 3 is equal to 6, so that means that 6 is a perfect number. It is the smallest perfect number.” “That makes sense,” Schmidt said. “Thank you.” “So now shall I explain how we find more perfect numbers?” Braun asked. “I’m ready,” Schmidt said. “Great!” Braun said. “We’ll start with what Euclid called the unit, but which we call the number 1. We double it and then we double that result again and again, adding the resulting numbers as we go until we get a sum that is a prime number. For example, the very first result comes from 1 + 2 · 1 or 1 + 2 = 3, which gives us the prime number 3. Then we multiply 3 (our prime number) times 2 (the last number we added) to give us 3 · 2 = 6. Six is the first perfect number.” To get a prime Calculate number the perfect number

Perfect number

Test for perfection: the sum of the proper factors equals the number

1+2=3

2·3=

6

1+2+3=6

1+2+4=7

4·7=

28

1 + 2 + 4 + 7 + 14 = 28

1+2+4+8 + 16 = 31

31 · 16 =

496

1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496

1+2+4+8 + 16 + 32 + 64 = 127

127 · 64 =

8128

1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128

Perfect numbers: Add 1 to the consecutive powers of 2 until you get a prime number. Then multiply that prime number times the last power of 2 that you added in order to get the perfect number. To test for perfection, add the proper factors (all the factors except the number itself) to be sure you get the perfect number.

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“Okay,” Schmidt said. “Let me continue,” Braun said. “We get the second perfect number by first adding 1 + 2 · 1 + 2 · 2 or 1 + 2 + 4 = 7, which is a prime number. Once again, we multiply that prime number (7) times the last number we added (that would be 4) to get our perfect number. In other words, 7 · 4 = 28, which is the second perfect number since the factors of 28 are 1, 2, 4, 7, and 14, and those factors add up to 28.” “Didn’t somebody else—was it St. Augustine (354–430)—have something to say about perfect numbers?” another student asked. “It seems to me that I read something about that.” “That’s right!” Leibniz said. “St. Augustine took perfect numbers even further. He observed that the reason the number 6 is perfect is not because God created the world in six days. He believed that perfect numbers existed long before God created the world. He said the reason God chose to create the world in six days is because 6 is a perfect number—He chose the number 6 specifically because of its perfection. According to St. Augustine, God also chose to have the cycle of the phases of the moon 28 days because 28 is the second perfect number. God is the ultimate mathematician.” There was a moment of stunned silence as the group absorbed that new information. From there, the discussion went on to consider further the perfection and beauty of mathematics. Leibniz was impressed—even he had learned as he listened to the discussion. This had been a wonderful afternoon! “You had some interesting ideas for us, Leibniz,” Braun said to him warmly as the meeting ended.

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“I hope you will join us again for our meeting next week.” “I would like very much to do that,” Leibniz said. “This was fun.” Clearly you have read Euclid,” Braun continued. “The rest of us had prepared carefully for this discussion, but you just happened to be ready to discuss it. I commend you. I personally have read very little mathematics. Your explanations were impressive, Leibniz.” “I should tell you, Braun,” Leibniz said, “that I had heard that Euclid’s Elements was to be today’s topic, so I spent an hour or so in the library yesterday looking over Clavius’ version of Euclid. It was not just by chance that I was prepared for today’s discussion. However, I should say that I have not read St. Augustine’s comments for a couple of years, although I found them fascinating at the time.” He continued, “However, I must say that I like the atmosphere of this group very much. It seems that everyone here is learning from everyone else, and I like that! I also like the openness with which you accept one another’s comments—I felt no objection or disapproval. I personally have engaged almost exclusively in solitary learning—and that has its limits—but until today, I had seen few alternatives. Thank you for welcoming me to your group.”

When he returned to Leipzig in October 1663, Leibniz settled into studying and living in the house where he had grown up with his sister and his mother. He soon

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became an active member of Leipzig’s Societäs, and in 1665 he became treasurer of the organization. After completing his university studies he left his notes from those meetings in the care of his older half-brother, Johann Friedrich, because he thought he might find them important later. In fact, throughout his life Leibniz was always careful to preserve his writings on all topics, recognizing that he couldn’t predict what he might or might not find useful in the future. As he reentered his home university, it was time for Leibniz to make a major decision: what direction should he choose for his future studies? What career should he prepare himself for? Up until then, there had been no choices. Preparation for the first degree in philosophy was the same for all students. Now, however, Leibniz had to decide if he wanted to continue in philosophy (as his father had done) or to prepare himself for another career, in one of the three fields of law, medicine, or theology. Those were his only choices. He might have felt some pressure from within his family to pursue theology, which was clearly one of his principal interests and which would have prepared him well for a career in the Lutheran church or as a scholar at the university in Leipzig. However, Leibniz knew that if he were to do that, he might have to compromise his somewhat radical religious stance. He found that he couldn’t condone the rampant hatred among the various Protestant sects. For example, he was unwilling to condemn the Calvinists simply because they were not Lutherans. He also disagreed with several mainstream Lutheran views, which omitted anything resembling a modern, mechanistic science and philosophy.

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No matter what field he chose, however, Leibniz knew he would need to be able to support himself financially. He had no reason to expect a major inheritance. Despite the fact that in later life he sometimes signed his name as Gottfried von Leibniz—the von indicating nobility—in fact that von was fiction. He knew he would have to work intelligently and diligently to support himself, regardless of how much he might wish for noble privileges and independent wealth. Judging his abilities and preferences honestly—as well as his ambition to earn enough money—the young student decided against limiting his studies to either philosophy or theology. He knew he could continue to thrive within the Lutheran church without becoming a theologian. Dismissing theological studies, he boldly chose to follow not just one but two other curricula, and both at the same time. He continued his study of philosophy and pursued the standard degree program in law as well. The combination allowed him to affirm that the underlying basis of law must be philosophy. As he saw it, legal issues must come down in the end to basic questions of right and wrong. Therefore, Aristotle’s logic, along with its more recent applications and interpretations, were the only honest foundations for the law. His choice of two fields rather than one could have been perceived as arrogance. While most students found either one of those programs demanding, this brilliant young man proclaimed he could easily handle both at the same time. As his distinguished career showed, there is no question that he was correct in his judgment. Since he had been reading philosophy all

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his life, he already had a complete understanding of the major philosophical writers of antiquity and medieval Europe. He was also an accomplished scholar in both Greek and Latin as well as in his native German—he read and wrote in all three languages with the same ease that most of his fellow students were capable of only in the academic language of Latin. In addition, he already had a surprisingly complete background in legal studies, having read many legal books as he perused his father’s library over the years. He had already discovered that approaching law through an understanding of philosophy allowed him to grasp quickly even the most complex legal arguments. At no time did he appear to be overwhelmed by his studies— an academic challenge was exactly what the young polymath had expected. Such challenges were what he craved. In fact, at this time Leibniz did not even limit himself to his coursework and reading. On top of them, he spent hours working with a practicing lawyer in town, studying precedents and analyzing arguments as together he and the lawyer resolved the legal case at hand. However, although he was fascinated by the trials he read about and observed, he was never tempted to make a career as a trial lawyer. That life was loathsome to him. Leibniz’s goal in the law was not to win at a game—as he saw the events in a courtroom. He was far more concerned with basic moral issues, which more often than not were forgotten in the drama of the courtroom. He was determined to be an upright man, following his father’s prediction from 16 years earlier.

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In his work both as a student and as an assistant to a lawyer, Leibniz argued that there were some major deficiencies in the standard legal education of his day. He was increasingly frustrated by the mindless tasks his professors asked him to do for his classes as well as by the scant emphasis his coursework placed on practical law. He asked himself, “Shouldn’t a young graduate in law know how to practice law?” Leibniz’s plan was to improve the fabric of his world radically and immediately through his own insightful reforms. In October of that same year, 1663, at the age of 17, Leibniz completed the requirements for the first degree (the Bachelor’s degree) in law. His thesis proposed a new way to approach legal cases through mathematics, although at that time his mathematical knowledge was still spotty. In February 1664, he submitted a thesis that completed his work toward a Master’s degree in philosophy. His thesis was entitled Specimen quaestionum philosophicarum ex jure collectarum [Specimen of the Philosophical Question in the Body of the Law]. In the thesis, he recommended the insertion of the study of philosophy into legal studies. Soon after completing his Master’s degree, Leibniz and his sister Anna Catharina were faced with tragedy: their 46-year old mother Catharina died of a respiratory illness. She had been the mainstay in their lives, and her death was difficult for Gottfried Wilhelm and for his 15-year-old sister. He said to his sister, “I’ve made an appointment to talk with Christian Freiesleben tomorrow to help us sort out our financial position. Mother seemed to trust

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him, and so I think we should allow him to continue to guide our affairs.” “Yes, I think she talked with him recently,” Anna Katharina said. “He’s a relative of Heinrich, our official guardian.” “That’s right,” Gottfried Wilhelm said. “But tell me, Anna Katharina, how do you get along with Heinrich?” “I find that I can deal with him easily,” Anna Katharina said, “and I’ve always found our half-sister Anna Rosine very pleasant.” “That’s good,” Gottfried said. “I suppose Heinrich will continue to be your official guardian. I wouldn’t be surprised if he gets it into his head to find you a husband pretty soon,” Gottfried Wilhelm said. “Actually, I suppose that is what he ought to do.” “Well, if he does, I hope the candidate will be one I like,” Anna Katharina said. One thing that Leibniz wanted to sort out immediately was the possible inheritance of a piece of property their mother had owned. Small though this property was, it formed a part of their mother’s modest estate, which he believed was rightfully theirs. However, his mother’s older sister, who was married to a distinguished lawyer in Braunschweig, was claiming the property for herself. Although the uncle claimed to be impressed with his clever nephew’s legal skills, he had no desire to lose any of his wife’s possible inheritance. Much to Leibniz’s dismay, the uncle easily won the case. Leibniz never had further contact with this aunt and uncle. Leibniz worked congenially with Christian Freiesleben, the administrator of their family estate, from

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that time forward. Freiesleben had helped him arrange the finances of his studies while his mother was still alive, and he continued in this role, making several loans to cover Leibniz’s expenses over the coming years. After he completed his Master’s degrees both in law and philosophy, Leibniz prepared a thesis for his habilitation in philosophy called Disputatio arithmetica de complexionibus [Arithmetical Disputation of Combinations]. This work was the beginning of his Dissertatio de arte combinatoria [Dissertation on the Art of Combinations], which he published in 1666, completing his formal studies in philosophy. Leibniz then was ready to go on and complete his doctorate in law, but accomplishing that was not as easy as he might have hoped. At the university in Leipzig, there were several other students—older than he—who also were ready to complete their doctorates in law. It soon became clear that Leibniz would have to wait for them to finish first. “After all, young man,” the authorities would have argued, “You will have plenty of time to make your career. Relax. No one achieves his doctorate in law at the age of 20!” Leibniz could see that this was a losing battle. He soon learned he could enroll instead at the university at Altdorf, a newer university in the Free Imperial City of Nürnberg—approximately 300 kilometers from Leipzig—where he would be able to complete his doctorate at his own accelerated pace. The professors there were delighted to allow the then 21-year-old scholar to submit a thesis—once again a legal treatise based on philosophy—almost immediately,

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since Leibniz had practically completed it before he arrived. His thesis dealt with the general question of how to solve difficult cases in law. He wrote that decisions in such cases should be based neither on drawing lots nor on the whim of the judge—both of which happened far too often at the time. Instead, if the law was ambiguous in a particular case, the decision should be based on natural justice and precedents from international law. The committee was perfectly satisfied with Leibniz’s carefully constructed argument. As he defended his thesis, Leibniz delivered an oration that his listeners praised as fluent and brilliant—in fact his delivery was so brilliant that the committee assumed either he had memorized it or he was reading it from the papers in his hand. Next, the extremely nearsighted Leibniz read a second oration he had written in Latin verse, holding the paper so close that his listeners couldn’t see his face. They were baffled. “Herr Leibniz,” one of the professors asked afterward, “could you please tell us why you did not bother to memorize your second oration as you did the first?” “But I didn’t memorize the first one,” Leibniz protested, “I just explained my thesis to you as I talked. I understood that that was the standard way to do it.” “Oh, come now!” the professor persisted. “No young scholar could have composed and delivered such a fluent presentation while standing in front of us. In fact, I’m not sure I could have done it myself with all my years of experience!” “But that is what I did, Herr Professor,” Leibniz said to his questioner. “I am comfortable with the topics in my thesis, so it was not difficult to explain them

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to you. Simply reciting a prepared explanation would have been tedious for all of us. I wanted to explain my points clearly through genuine communication, so that is what I did. I assure you, I memorized nothing.” “Astounding!” the professor replied. “This was an impressive accomplishment. I congratulate you, young man!” Then the professor in charge told Leibniz, “Our committee will meet early next week to discuss your candidacy, and we will inform you of our decision soon after that. Thank you, Herr Leibniz. We will contact you when we have come to a decision.” “Thank you, Sir,” Leibniz said as he bowed and departed.

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Dr. Leibniz Begins His Career

In Leipzig, Gottfried Wilhelm’s sister Anna Catharina was settling into life with her half-sister and brotherin-law, Heinrich Freiesleben. One evening, Heinrich turned to his young charge, “Anna Catharina, I had an interesting conversation today with the theologian, Simon Löffler. Do you know him?” “Yes, we have talked a couple of times,” she said. “He seems like an interesting and decent man.” “Yes, I believe he is,” Heinrich agreed. “At any rate, he has asked me if I would consider allowing you to marry him.” “To marry him?” Anna Catharina asked in surprise. “But, he’s old enough to be my father!” “Well, certainly he is older than you,” Heinrich said, “but I wouldn’t call him old.” “No, I suppose not,” Anna Catharina said. After considering it for a minute, she continued, “Well, Heinrich, what do you think?” 45 © 2012 by Taylor & Francis Group, LLC

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“I believe he could provide well for you. Do you like him?” “Well, I certainly don’t dislike him,” Anna Catharina said, “but I’d like to get to know him better before I make a decision, if that’s all right with you.” Heinrich turned to his wife. “Anna Rosine, could we invite Löffler for tea later this week?” “I don’t see why not. How about Thursday afternoon?” Anna Rosine asked. “Would you like me to write to him?” “I think that would be perfect,” Heinrich said. “Do you agree, Anna Catharina?” “Yes, that would be fine,” Anna Catharina said. Löffler joined the family for tea several times over the coming weeks, and he and Anna Catharina came to enjoy one another’s company very much. In September 1666, Anna Catharina and Simon Löffler were married in Leipzig, and within a year, Anna Catharina was the happy mother of a healthy son whom they named Friedrich Simon Löffler. Anna Catharina immediately wrote to her brother Gottfried Wilhelm telling him the news and encouraging him to come visit the family and make the acquaintance of his new nephew. “So, Anna Catharina, you have a brand new baby,” Gottfried Wilhelm wrote back to his sister. “What a delight! I congratulate you and your husband. May the sun always shine in young Friedrich Simon’s life and may he succeed in all his endeavors! I look forward to meeting him and your husband soon. However, right now I am eagerly awaiting the results of my doctoral examination, and I’m afraid I can’t think of traveling until that is decided. I believe I performed well, but I

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can’t rest until I have the official word from the committee. The decision is not mine to make.”

When the committee in Altdorf assembled to discuss the young applicant’s fate, they decided collectively that Leibniz was more than worthy of the title of Doctor of Law—yes, that would be only the beginning for this brilliant young man. They agreed that this scholar would be a superb addition to their faculty, and they wasted no time in offering him an appointment as professor on their faculty to complement his newly won Ph.D. “Herr Leibniz,” Johann Michel Dilherr, the rector of the university, said to him the following week when the professors called him back to present their decision, “given your outstanding qualifications, we have unanimously decided to offer you a professor’s teaching chair in our university’s department of Jurisprudence. You would, of course, have the honor and all the privileges of a professor at our fine university, in spite of your young age. You should know that we are all eager to work with you and support you in your promising academic career. We congratulate you, Herr Professor Doctor Leibniz,” he said, dramatically pronouncing Leibniz’s new title for the first time. Leibniz was amazed. “I had not expected anything like this,” he protested. “I am pleased to accept my doctorate from your fine university, but please allow me time to think about your offer of a teaching chair. I must confess that I am stunned.”

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“Do you realize what we are offering you?” Dilherr demanded. “I certainly do,” Leibniz said. “I am extremely flattered by your offer—I certainly don’t mean to appear ungrateful. It’s just that I need to think carefully about my career plans. I’m sure you can understand, Sir, that I did not expect an offer of this sort.” “It is not something we usually present to our new doctors,” Dilherr admitted, “but we believe you have real talent. This is a wonderful and rare opportunity for you. It is almost unheard of for a man of your young age to achieve a professorship. You have impressed us mightily! Personally, I am baffled by your hesitancy.” “I appreciate your confidence in me, learned gentlemen,” Leibniz said, carefully addressing the entire group. “I understand that you have made your offer with my best interests in mind, but I believe I cannot accept it. I have several other things in mind, and I am simply not ready to make a long-term commitment to you and your university. As you observed, I am young, and my career—whatever it may bring—is still ahead of me. I must consider my moves carefully.” Leibniz paused briefly to consider the effect of his words. “Please understand that I thank you heartily for your confidence in me. However, I must tell you, Herr Dillherr and most esteemed professors, that I truly have no choice but to turn down your generous offer. I cannot in good faith make such a commitment to your university today.” Leibniz then picked up his papers and prepared to leave. After they all had shaken hands and Leibniz had

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departed, the professors and Herr Dillherr looked at one another in astonishment. “The young man is a fool!” one of them announced. “We offered him the chance of a lifetime, and he turned us down! We have been insulted by that boy—and that is what he is: a boy. He is a perfect fool—perhaps a brilliant fool, but nonetheless a fool! Outrageous!” “No, he’s no fool,” another older professor said. “Perhaps we are the fools. He is a brilliant young man, and he will find other arenas for his career. I suspect we will see major accomplishments from him. Like you, I assumed he would accept our proposal, but I believe he has made a wise choice. It may also be best for us in the long run. It is unlikely that he would have been happy in our university for long. His kind of genius needs a broader setting than we can give him, and he is intelligent enough to see that.” Leibniz was confident in his decision. He had higher goals than simply to help prepare ignorant young men for careers in law, although he firmly believed that that was an important role for others to fill. He had decided he should work toward peace and enlightenment in the world on a grander scale—not just within the confines of a provincial university. He was still dissatisfied with the fragmentation of Christianity in Europe, and he thought that now he might be in a position to do something about it. He asked himself why couldn’t the Lutherans and the Calvinists learn to work together? For that matter, why couldn’t the Lutherans and the Roman Catholics work together?

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In the secular world also, he was baffled by the war-like attitude of so many apparently intelligent and responsible people who prided themselves on taking advantage of one another’s limitations. How could they call themselves Christians when they behaved that way? What had become of morality—the concept of right and wrong—in the modern seventeenth century world? He included his maternal uncle in that criticism, who had taken shameless advantage of him and his sister. Leibniz wanted to believe that this was—or at least it could be—the best of all possible worlds, but it seemed to need some help from him to make that dream come true. For his part, he was prepared to serve the world, and he knew he had the ability and skills to do it. He had mastered Western philosophy and law better than anyone else he had met, so it was obvious that he should take up this challenge. Natural philosophers before him like Galileo and Descartes pointed to a universe that was rational. Now he, Leibniz, must step forward and do his part. He sent a short letter to his sister, telling her of his success, and explaining succinctly why he had turned down the university’s offer. He then described in vague terms how he expected to proceed from there. He was surprised the following week to receive a letter—not from his sister but rather from his brother-inlaw Simon, saying, “Your sister and I congratulate you heartily on the completion of your doctorate. At her request I am including a promissory note for a loan of 1000 Taler, a sum that we hope will help you start your career.” Gottfried Wilhelm was relieved to accept the generous loan since he had very little money left.

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He viewed the loan as an expression of confidence in his decision to turn down Altdorf’s offer and strike out on his own. “How shall I begin?” he asked himself. “I need to promote my plans among the princes of Europe, but I have no noble title, little money, and no high-ranking contacts. Clearly I need to find a prince to sponsor me—only with that kind of help can I realize my plans. Ah, but how to find that prince and get his support? “Ah, money,” he continued grimly to himself. “The loan from Anna Catharina and her husband is nice but it’s not enough, so I guess I had better write to Freiesleben to get a bit more. I should also look around and see if I can make some contacts in Nürnberg.” Having come to know a few people there and hearing through them of a society of alchemists, Leibniz decided to investigate. The historical record is not entirely clear about how he began. According to some accounts, the clever young doctor of law composed a fake treatise on alchemy, using its jargon so skillfully that he was invited to join the society and to serve as its secretary, based on his obvious—if superficial and newly acquired—skill with the written language of alchemy! Alternatively, he may simply have presented himself at a meeting and stunned the members with his knowledge and passion. Whether or not he stooped to the former shenanigans, he was hired at a small salary for this position. His experience there gave him his first taste of laboratory “science,” a topic that had become increasingly popular among the learned classes of Europe and that would evolve into chemistry over the next century.

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Through this connection, Leibniz met noblemen, clergymen, medical doctors, and creative entrepreneurs within the Nürnberg community. During the several months that Leibniz served as the secretary to the society, he talked with many experimenters, concluding that at least some of them might be doing important research. Leibniz was careful to say little and to learn as much as he could. He heard about the scientific societies in London and Paris as he listened to the assembled professional men contemplating what experimental feats they might accomplish. Other times members talked about political events in the German states and about the outrageous policies of King Louis XIV of France. Several people discussed the impressive accomplishments of Johann Philipp von Schönborn, Elector and Archibishop of Mainz—who was one of the eight Electors of the Holy Roman Empire. The Holy Roman Empire had been the most important political organization in the German states for many years and was the obvious vehicle for any significant change in central Europe. Leibniz learned that Schönborn, with both secular and ecclesiastical influence, was the most powerful person in central Germany. “I understand the Elector would like to arrange for some sort of accommodation between the evangelical churches and the Roman Catholic Church,” one articulate, older man said one evening after the formal part of the meeting had ended. “He is also attempting to reform the antiquated and inconsistent legal structure of our state.”

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“Why does he want to do that?” another man asked angrily. “I say that if someone steals he deserves to be punished—and punished severely! I don’t think we need any watered down laws to coddle sinners. Let them be hanged! They deserve it!” “Oh, wait a minute!” the first man corrected him. “As I understand it, the Elector isn’t talking about letting criminals escape punishment. No, Sir! What he is concerned about is our general system of law. He says it is better suited to subsistence farmers than it is to our advanced society, in which some are farmers who produce our food, some are merchants who sell us food and other items, some are doctors who heal us, some are parsons and priests who look to our spiritual health, and so on.” Leibniz listened carefully to the discussion. It sounded to him as if the Elector’s projects were important. He had read of situations within the Empire in which the justice that was doled out was anything but just. Surely the German states could do better than that! He couldn’t help comparing the accounts of the Elector’s reforms to his own thoughts. “Wouldn’t I love to be the Elector with all his power!” Leibniz said to himself. “No, no,” Leibniz then scolded himself. “I should limit my dreams to what I can realistically do. I know that I have no chance of becoming Elector of Roman Catholic Mainz. That is a hereditary title. For that matter, I’m not even a Roman Catholic—and I have no intention of becoming one! I must be careful not to turn myself into a fool—as some of the men assembled here sometimes do!” Then he cheered up, asking himself, “I wonder if the Elector could use my help. Perhaps in a year or

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two I could offer my services to him. Surely I have some skills and knowledge that would be helpful to him. And if he is as intelligent as these men think, perhaps he would be glad to benefit from my knowledge and abilities. Surely with all his responsibilities, the Elector has not had the opportunity to acquire the superb education in the law and philosophy that I have.”

After living frugally for most of 1667, the 21-year-old scholar decided the time had come for him to move beyond Nürnberg and its alchemical society. First, he was determined to do something about an enormous gap in his education: his incomplete knowledge of mathematics. Although he had studied classical Greek mathematics at school and at the university, that subject (at least as he had learned it) had not been updated for at least 1500 years! He believed that even Professor Weigel’s knowledge of mathematics was woefully incomplete. He was convinced that mathematics was essential to develop his fundamental language of thought. Although he had used Pascal’s Triangle in his work on the art of combinations for his aptitude in philosophy, he knew that that was not enough. He guessed there was much more to modern mathematics than Blaise Pascal’s (1623–1662) work. “What about Galileo?… and Descartes? Where can I learn more about their work? And who is working in mathematics today?” he asked himself. He was appalled at his own ignorance.

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1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 Pascal’s Triangle.

The one thing he knew about modern mathematics was that almost nothing had been done in the field within the German-speaking world in recent years. To learn more, Leibniz decided he would have to travel to Holland, France, and England. His journey would take him first to Frankfurt on the Main River and then to Mainz, where he hoped to board a boat going down the Rhine to the Netherlands. He decided that this was the best plan. He first hired a horse to begin the ride from Nürnberg. He would need to change horses several times en route to Frankfurt, probably spending two nights at inns along the way. “It might be more comfortable to go by carriage,” Leibniz said to himself, “but it would be much too expensive for me. I can ride. I must conserve my funds.” When he arrived in Frankfurt, saddle-sore from three days on horseback, he listened in on conversations at the inn and in the rathskeller (the restaurant and beer hall in the cellar of the city hall). The favorite topic once again was Schönborn, the Archbishop and Elector. Everyone knew about his two big projects— reconciling the Catholic and Protestant faiths within

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the Holy Roman Empire and reforming the legal structure of the land. What a coincidence that the archbishop had chosen the very topics that Leibniz considered most important! He continued to wonder: “Is there some way that I might arrange to assist this powerful man? Like me, he is clearly ambitious and intelligent, but unlike me he has power and money. However, I have learning—learning that could be important to him. Clearly, he needs my help, but he can’t avail himself of my help if he doesn’t know who I am and what I am capable of. I must come up with a way to present myself to him.” Leibniz continued to think. The next morning, as he drank a mug of cider alone at a table in the inn, Leibniz said to himself bitterly, “Listen to them! These people talk and talk, yet they have no understanding of the real questions— let alone their solutions. I don’t dare speak because I don’t know these people, and I might not be able to control my tongue. Their ignorance is so obvious! What makes them think that they or their neighbors are qualified to comment on these important issues? I might be the one person here who actually knows anything about them—could I be so bold as to say that in all likelihood I know more about these topics than even the Elector himself? Come to think of it, could it be that I—Gottfried Wilhelm Leibniz, Doctor of Philosophy and Law—have a responsibility to share my knowledge and insights with the Elector? Perhaps I should spend some time in Mainz before I go to Holland.” A man sitting near him said, “I beg your pardon, Sir. Did you say something?”

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“Oh, no,” Leibniz said quickly as he paid his bill. “It’s nothing. I was just muttering to myself. Good day to you, Sir.” Leibniz finally saw what he should do—he found he had stumbled on a plan to make himself known to the Elector. Procuring a supply of paper, ink, and quills at a stationer’s shop, he took them with him to his room at the inn, commenting to himself, “I will write a treatise on how best to restructure the education of students in law within the Holy Roman Empire. I will call it Nova Methodus Discendae Docendaeque Jurisprudentiae [A New Method for the Learning and Teaching of Law]. Through it I will show the Elector that I am an informed and thoughtful scholar who has the knowledge and skill to solve problems like his. He will certainly recognize that education is an excellent way to effect a genuine change in the legal system, and through my treatise he will see that I am perfectly qualified—that I have the skills—to help him in many ways. He wrote all day, stopping to stretch every couple of hours, and then immediately sitting down again to continue. He knew what he needed to say—he had thought through the problem many times—and his only challenge was to put it on paper. In the middle of the day, he forced himself to go out and buy a loaf of bread and some cheese, knowing that he could not continue to work without sustenance. As it grew dark in the afternoon, he went out once again, this time to procure a steaming mug of warm milk. Each time, however, he quickly returned to his room to pick up his quill once again.

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“Am I being too bold?” Leibniz asked himself as he ate his meager bread and cheese later that evening. “No. I am clearly the best legal scholar in these fields, so it would be irresponsible for me not to present myself to the Elector.” He was able to compose his erudite essay without books or background papers because he had amassed in his head such an impressive knowledge of the law from Roman times up to the present and of classical philosophy. He demonstrated in his essay that current legal education in the German states was inefficient and misguided—it took too long, it wasted students’ time on trivia, and it gave them no practical legal instruction whatsoever. At the end of the current programs in law, graduates were clearly not prepared to practice law despite having mastered a phenomenal number of useless facts. What a waste! Leibniz knew he was much better prepared than most budding legal scholars because, unlike them, he had been careful to supplement his formal education in law and philosophy with practical work in the lawyer’s office. He was one of the few graduates who actually had the practical training he was recommending for all. After outlining in his essay the shortcomings of the current system, Leibniz forcefully presented his own plan for the intelligent education of students in law. He recommended reducing the current five-year program to two years, restructuring it so that it would include far more practical knowledge and hands-on experience and at the same time reduce its mindless rote learning to a minimum.

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When he finished writing the next morning, he delivered his manuscript to a printer, boldly dedicating the work to Schönborn and arranging for one copy to be sent to Mainz to the Elector himself. “When the Elector reads it,” he congratulated himself, “he is sure to be impressed. He will then inquire where he can find me, and everything will develop naturally from there. I must be sure to tell the printer the name of the inn where I am staying so that the Elector can find me.” Then Leibniz said a silent prayer that his plan would work.

When the Elector received the essay the following week, he was stunned. He asked his assistant, “Where did this essay come from? Who is this young scholar? Find him for me at once. I must meet him.” Through the printer, his assistant located Leibniz in exactly the way he had intended and invited Leibniz to present himself to the Elector. Leibniz did not hesitate. “I understand your name is Leibniz,” Schönborn began their interview. “I have read your essay and I would like to know more. I presume it is an essay that you prepared during your legal studies.” “Oh, no, Your Highness,” Leibniz said. “I wrote it in my room in Frankfurt last week. I could also bring you my academic theses on a variety of legal and philosophical topics if you would like.” “No, that isn’t necessary, but I must confess that I am baffled. Your essay makes references to several

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works of philosophy and law, yet surely you had no resources with you in your hotel room,” Schönborn argued. “I frankly don’t see how you were able to write such a learned essay under the circumstances.” “Please let me explain, Your Highness,” Leibniz said. “You should realize that I have read both philosophy and law all my life. Although I admit that forming my arguments in this essay was challenging, I have been working on them for several years, and finding the supporting material was easy enough. I learned long ago to organize information in my mind as I read so that I have no trouble finding it later when I need it.” “You say you have read philosophy and law all your life,” Schönborn said with a condescending smile. “Since you are only in your early twenties, that is not too long a time for the reading you describe.” “In fact, Your Highness, I spent my entire childhood reading both law and philosophy from my father’s library. I did little else.” “Were you able as a child to read those texts in Latin and Greek?” the Elector asked suspiciously. “Or did that reading have to wait until after you had been taught those languages in school?” “No, I didn’t have to wait. I taught myself to read both Latin and Greek before I was ten years old,” Leibniz said. “I could see no reason to wait.” “That is impressive,” the Elector said. “I’m not sure I would have known how to study like that as a young child. Please tell me about the rest of your background. You must have studied at a university.” “Yes, Your Highness,” Leibniz said. “I studied at the university at Leipzig for five years, completing the

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doctoral program in philosophy and the first two degrees in law. Then I went to the university at Altdorf to complete my doctorate in law.” “And why didn’t you do that in Leipzig?” Schönborn demanded, suspicious that there might be a flaw in this young genius after all. “Well, Your Highness, Leipzig has a policy of giving priority to older students—allowing them to complete that program first,” Leibniz explained. “The professors at Altdorf seemed pleased to allow me to submit my thesis to them almost immediately, and afterwards they astonished me by offering me a position on their faculty at once.” “And you chose not to accept that?” Schönborn asked in surprise. “Yes, Your Highness” Leibniz said. “Since the university at Altdorf is very small, I decided that I would look into other possibilities. I concluded that I would not be happy spending my entire life at a provincial university in a small town. I don’t want to sound arrogant, but I have set my sights higher than that.” “I can understand that. You may have heard that I have plans to reorganize our legal system,” Schönborn said. “I already have one man—Hermann Andreas Lasser, my lawyer—working on it, and yet I believe it is too big a job for him alone. Would you consider working with Herr Lasser on this project for a small weekly salary?” “Oh, yes, Your Highness!” Leibniz agreed, trying to conceal his excitement. He was ecstatic to accept the Elector’s offer—it was exactly what he had hoped for. Even if Lasser proved difficult to work with, this had

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to be a good move for him. Leibniz’s gamble seemed to be paying off. Leibniz spent the next three years working with Lasser. The two young men respected each other and were able to work productively together, day after day. They wrote many works for the Elector, and much of the actual writing was clearly the work of young Leibniz. The only awkward part of the arrangement was that the lawyers sometimes had to remind the Elector when their salaries had not been paid for several weeks at a time. They might be idealists, but neither of them was wealthy and they needed to eat.

At this time, Leibniz also came to know a diplomat named Baron Johann Christian von Boineburg, who was impressed with the young scholar’s intelligence and world view. The two men found they had much in common. The Baron, who was 20 years older than Leibniz, had wisdom, power, and money, all of which he was happy to use to promote Leibniz’s position. He—not Schönborn—was the sponsor who soon became the focus of Leibniz’s life. Leibniz accepted a position as Boineburg’s legal representative sometime before 1670, working with him on many political and diplomatic projects, and cataloguing the baron’s library. In these many roles, Leibniz was a happy man. At that time, Leibniz met Duke Johann Friedrich of Hannover, who was also interested in the promising young scholar. He immediately invited Leibniz to move to Hannover to work with him there instead of with

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Boineburg. However, when other prominent persons in Mainz heard of this offer, they strongly discouraged Leibniz from accepting it. If Leibniz was unwilling to turn it down, they recommended that he at least delay accepting it. Leibniz himself was beginning to wonder if Hannover might be as provincial as Altdorf. The bustling cities of Mainz and Köln—seats of two of the archbishoprics of the Holy Roman Empire—seemed to be more likely places for Leibniz to thrive than sleepy Hannover. Leibniz kept the Duke interested through frequent correspondence concerning his philosophy and theology but postponed an actual commitment year after year. In 1670, Boineburg received a letter from Heinrich Oldenburg, a German who lived in London and served as the secretary of the Royal Society there. Oldenburg wrote to Boineburg, complaining of the state of philosophy and science in Germany—his native land. Boineburg knew there was one young legal scholar and philosopher who could ease Oldenburg’s worries—24-year-old Leibniz. Leibniz was naturally thrilled to make this contact and began correspondence with Oldenburg almost immediately, preparing the way for his eventual travel to London in 1673. In addition to his own philosophical writings and his work on the development of a universal language, Leibniz worked on many projects for both Boineburg and the Elector at this time. One such assignment was the preparation of a tract arguing for the election of Philipp Wilhelm von Neuburg to be the next King of Poland, after a vacancy on that throne had been caused by the abdication of King Johann Casimir.

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Leibniz published his clever tract under a pseudonym Georgius Ulicovius Lithuanus, purporting to be a Polish nobleman, and using a smattering of mathematics to buttress his argument. The work arrived too late to influence the choice of King. Nevertheless, Boineburg was pleased with the work, and in later years Leibniz was proud of its mathematical reasoning. As they worked together, Leibniz and Boineburg became close friends who agreed on many basic questions. Boineburg, like Leibniz, had been brought up in the Lutheran church, although he had converted to Catholicism in his early years of working with Schönborn. Although Leibniz never wavered in his Lutheranism, he argued repeatedly for the reunion of the Catholic and Evangelical churches within the Empire. Boineburg and Leibniz were happy to work together on this mission, neither of them worrying about the other’s personal religious preference. Instead, they discussed threats to the greater Christian church both from atheists and from the Ottoman Empire. Early in their friendship, Leibniz sent Boineburg an essay he had written several years before that provided a logical proof of the existence of God. Boineburg forwarded it to others in the Catholic Church, and a slightly edited version was published in Augsburg in 1669. Another project commissioned by Boineburg was Leibniz’s new edition of Mario Nizolio’s 1553 work, Antibarbarus seu de veris principiis et vera ratione philosophandi contra pseudophilophos [Against Barbarians: How to Contest the Faulty Reasoning of Pseudophilosophers], a plea for rejecting pseudo-philosophy and cleansing philosophy of obscure reasoning. Leib-

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niz’s new version began with his own lengthy preface, in which he explored the use of logic in Nizolio’s work. In fact, Leibniz’s revised version of Nizolio’s work, which had been largely forgotten, turned out to be far more important than the original. Leibniz’s edition was featured prominently and to much acclaim at the Frankfurt Book Fair in 1670. In his preface, Leibniz argued not just for a cleansing of the corrupted academic Latin of the time but also for the increased use of the German language in official communications. Leibniz noted that in France and England, writers had been publishing in the vernacular (for example, Shakespeare in English and Molière in French) while most Germans were still writing only in Latin. Leibniz, unlike many of his contemporaries, wrote German clearly and well, and he considered its mastery essential to the progress of modern European society. A significant number of his writings— like those of his fellow Saxon Martin Luther—were published in German. Twenty years later, his mentor Thomasius at Leipzig was prosecuted for presenting lectures in German rather than in Latin. Clearly Leibniz’s reform was not yet complete—and it is ironic that his work arguing for the use of the German language was in fact written in Latin. While working for the Elector, Leibniz began to dream of founding scientific societies in the German states to serve as Germany’s answer to the Académie des Sciences in Paris and the Royal Society of London. After all, he argued, we Germans are an intelligent people. If we were to set up Académies in which intellectuals could flourish, we too could make

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important scientific discoveries. In his vision, all societies throughout Europe should collaborate on the important work of improving the human lot in every respect. The Nürnburg society of Alchemists had introduced him to the work of such societies at the local level, and he was determined to help the Germanspeaking world realize its potential. One of Leibniz’s great hopes for such societies was the formulation of a universal characteristic—a language that would allow all people everywhere to communicate. This would fulfill Leibniz’s dream that he had set forth in his Dissertatio de Arte Combinatoria. It would allow people to resolve their disagreements using mathematical processes rather than war. It was a far more comprehensive plan than L.O. Zamenhof’s future artificial language Esperanto, which was first published in 1887, two centuries later. Zamenhof wanted to improve communication among nationalities, but his goal was simply communication. Whereas Zamenhof grew up in a relatively peaceful Europe, Leibniz had grown up listening to eye-witnesses’ vivid descriptions of the nearby battle in Saxony at Breitenfeld, one of many bloody slaughters in the seemingly endless Thirty-Years War. Leibniz knew the brutality and long-term consequences of war, and the language he planned would make war obsolete. Leibniz’s universal characteristic would employ the languages of science and mathematics and use symbols in much the way Leibniz imagined the Chinese language did. He intended that people would be able to work collaboratively to carry out intellectual pursuits in such a way that there would be no misunderstandings.

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Leibniz also planned that his scientific societies would both develop the Christian faith and work seriously to address important medical problems of the day. Although he approved of the increasing use of the telescope to improve human understanding of the heavens, he considered the use of the microscope to find treatments for ailments and suffering of the human body to be far more pressing. He would be glad to succeed in squaring the circle, he said, but he considered curing human disease more urgent, and he recognized that the state must help in that. Most individual scientists on their own simply didn’t have the resources to accomplish anything significant in that field. Furthermore, there simply were not enough physicians to care for the people of the German states, and Leibniz’s proposed institutes could be expected to prepare many more of them. He saw urgency of the project for the Germans, arguing that this would be a way to accomplish unity in their states. He proposed to the Holy Roman Emperor the founding of a new scientific journal, which would function in the same way as the French Académie’s Journal des Sçavans [Scholars’ Journal]. By 1669, Leibniz, now 23 years old, had also spent considerable energy exploring questions of physics. He had been contemplating motion and its origins as well as the results of the collisions of two moving bodies. On a trip to a spa with Boineburg, he wrote a short treatise outlining his theories, but after receiving some negative reactions he decided to consider it further before sharing it with anyone else. In a letter to Oldenburg in 1670, he summarized his discoveries on

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motion. Oldenburg, trying to support his young compatriot, presented the paper to the Royal Society, where it was met with guarded approval. As an ambitious young man, Leibniz decided his next move must be to present another treatise on physics, this time to the Académie in Paris, and in that essay he made the first mention in print of a marvelous new calculating machine he had devised. The French found this tantalizing—had this unknown young man come up with an improvement on Pascal’s calculating machine? Although the members of the Académie showed little enthusiasm for Leibniz’s theoretical work, they did not dismiss him entirely. Leibniz’s science interfered little with his job working with Boineburg, as together they made plans with Schönborn to solve difficult political and international problems in the Europe of Louis XIV. Leibniz had the responsibility of writing political pamphlets for Boineburg, and he eventually came up with a clever scheme to distract the French King from invading the Low Countries and German Borderlands: the King of France should invade Egypt instead. Leibniz developed several arguments for his proposal. Although the Crusades of medieval Europe were long past, Leibniz argued that protecting Europe from invasions from the Ottoman Empire would be far more important to European civilization than conquering France’s neighbors. That ploy—if it worked—would be a real help to the Holy Roman Empire, since Louis XIV’s France was far more powerful than the Empire. Furthermore, the Empire was truly vulnerable to an attack from the East—far more so than France was. The difficulty was

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to convince Louis XIV to agree. Leibniz, who was pleased that Boineburg liked the plan, was certainly aware that making a presentation of his plan to the French might involve a trip to France for him—what a delightful thought!—and that desirable possibility couldn’t have been far from his mind. On January 12, 1672, while Leibniz was happily working on many schemes with Boineburg, he received a letter from his sister, Anna Catharina. After filling her brother in on the escapades of her young son Friedrich Simon and then on her husband’s activities, she brought up her real worry: her brother Gottfried Wilhelm’s faith. “I know you benefited greatly from our pious upbringing in the Lutheran confession,” she wrote. “Now I fear you have strayed from that faith. As good Lutherans, we know that Martin Luther, not Calvin and not Zwingli, brought the true faith to us Saxons. Gottfried Wilhelm,” she implored, “don’t abandon our true religion! And won’t you please consider returning to your home in Leipzig? You cannot do better than that! Living in the heart of our faith would surely be best for you. Please, Gottfried Wilhelm! Please come home.” Religion was a subject that he was loath to discuss with his sister. He had believed for some time that the conflict within the evangelical churches was senseless. Although he rarely if ever attended services, he still considered himself a committed Lutheran. He sincerely hoped that all Christian faiths would come together as one, perhaps preserving some of their individual features but nonetheless becoming a united church. Having no intention of abandoning the Lutheran faith

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to join the Catholic confession as Boineburg had done, Leibniz was repelled by current petty religious quarrels. “Why is she so worried about my faith?” he asked himself. “As she said, we grew up in a strong Lutheran home, and I would never abandon our religion. She should know me better than that. Oh, Anna Catharina, don’t worry about me! Yes, I’ll write her a letter tomorrow.” Unfortunately, Anna Catharina died in Leipzig before she received that letter. She had hoped to assure her brother’s salvation before she died, but it was now too late.

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1672–1676

Paris, London, and Mathematics

Boineburg decided in 1672 (with Schönborn’s approv-

al) to send Leibniz to Paris on a diplomatic mission. He wrote to Pomponne, Louis XIV’s foreign minister, alluding mysteriously to a new and marvelous plan for a mutual project that would benefit both countries. In fact, Boineburg may or may not have been as enthusiastic about the “Egyptian plan” as Leibniz was. Boineburg also had a personal motive in sending Leibniz to Paris: he owned some property there, on which he was owed considerable rent. He hoped that Leibniz might be able to collect it while he was there. For his part, Leibniz was ecstatic. Although he was committed to the diplomatic mission and willing to attempt to collect the rent, he also wanted to learn more mathematics, and Paris was an excellent place to do so. Surely he would be able to find some time for mathematics among his other errands. Leibniz and one of Boineburg’s servants boarded a coach to Paris in March 1672, carrying a letter of 71

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introduction to Pomponne. The letter presented Leibniz as an extremely capable person, suggesting that the minister should not be put off by the young man’s outward appearance. Boineburg described Leibniz as rather short, with an overlarge head that was distorted even more by his generous but stylish wig and a large nose that seemed to dwarf his mouth. Nevertheless, Boineburg wanted his reader to know that, in spite of it all, Leibniz should be taken seriously. Here was a man who had mastered all of modern science and philosophy and who could draw on all that knowledge as needed. When he arrived in Paris, Leibniz rented an apartment that was a short walk from the university in the Latin Quarter and close to governmental offices at the Louvre. His first chore was to improve his mastery of French. Although he could already read French, writing and speaking in the language were difficult for him. He immediately hired a tutor, and within a few weeks the young man’s French had improved greatly. Since French was increasingly the language of the educated classes throughout Europe, this was essential, whether in Paris or elsewhere. Leibniz repeatedly submitted petitions asking to present the Egyptian plan to Pomponne, but each time he was forced to wait for someone else who might meet with him instead. Although he was never told explicitly that he should quit bothering the officials at the Louvre, in fact he never was able to meet with anyone official. As the delays continued, Leibniz followed the news of the French invasion of the Low Countries, realizing that his diplomatic mission had become

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moot—he could not avert the onset of a war that had already begun. He repeatedly amended his plan to fit the current international situation. While he waited for official replies, Leibniz challenged himself to improve his mathematics, for that was the tool he needed for his pet project: the development of a universal language. Beginning almost as soon as he arrived, he did his best to introduce himself to the scientists and scholars of Paris, although his limited mathematical skills hampered him. He simply didn’t have the background to understand what they were talking about. “Oh, if only I had grown up in France!” he sighed to himself. “If I had had access to the latest mathematics as I matured, think what I would have been able to accomplish by now! What a pity! Along with philosophy and history, I would certainly have read mathematics in my father’s library if I had found it there, and I would have learned it in school if it had appeared in my schoolbooks. And how wonderful it would have been if I had found a teacher who knew mathematics and was willing to share it with me! I am proud to be a Saxon, but it would have been better for my mathematics if I had been born a Frenchman! But, those thoughts are useless. I am here in Paris, and I will learn mathematics so that I can communicate with these scholars. However, first I must find someone who can help me—someone who can direct me to the appropriate texts. Then all I’ll have to do is work, and I have never been afraid of work.” Finally he was able to arrange to meet Christian Huygens (1629–1695), a Dutch scientist who had been

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living in Paris since 1666 and served as the director of the French Académie des Sciences. Huygens, recognized as the most important mathematician in Europe, was certainly an ideal contact for Leibniz. At their first meeting, Leibniz boldly offered to show Huygens a clever method he had devised for summing a finite series. He had decided it was essential that Huygens recognize him as a promising, if untaught, mathematician. “And what is your method?” Huygens asked politely. Was the young man simply presumptuous or was there some substance to him as well? “Well, Sir, I began with Euclid’s axiom that the whole is always greater than the part. Then I played around with it, using subtraction (instead of addition) to find the sum of the series. Shall I show you what I found?” “Please do,” Huygens said. “Well, when I considered the set of perfect squares (0, 1, 4, 9, 16, 25, …), I started by subtracting pairs of terms,” Leibniz began. “That means I said, 1 – 0 = 1; 4 – 1 = 3; 9 – 4 = 5; 16 – 9 = 7; etc. You see, Sir, the differences are actually the set of odd numbers.” “Yes, of course,” Huygens confirmed. Leibniz’s discovery was no breakthrough—it was a well-known fact. Undeterred, Leibniz continued, “I wondered if this fact might allow me to add a set of odd numbers in another way. When I simply add the first eight odd numbers, (1 + 3 + 5 + 7 + 9 + 11 +13 + 15), I get 64, which is the square of 8. Next I decided to work from my set of differences of the perfect squares, writing the same sum this way: (1 – 0) + (4 – 1) + (9 – 4) + (16 – 9) + (25 – 16) + (36 – 25) + (49 – 36) + (64 – 49). Next I

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dropped the parentheses, giving me 1 – 0 + 4 – 1 + 9 – 4 + 16 – 9 + 25 – 16 + 36 – 25 + 49 – 36 + 64 – 49. As you can see, in this expression I am simply adding and then subtracting first 1, then 4, then 9, then 16, then 25, then 36, and finally 49. That means that all the numbers except 64, the last number, disappear. And so, I am left with 64, which is the sum of the first eight odd numbers. I could do the same with the first 15 odd numbers. In that case, the first 14 squares, (1, 4, 9, 16, …) all the way to 196, will disappear, and what will be left is 152 or 225, and that is the sum of the first 15 odd numbers. What do you think?” “That was neatly done,” Huygens said. “I can see that you have been doing some good thinking.” Before Leibniz left for the day, Huygens posed a challenge to him: “I wonder if you could find an original way to calculate the sum of the reciprocals of the triangular numbers.” Perhaps young Leibniz would develop into a capable mathematician, but Huygens wanted further evidence before he invested too much time in the young Saxon. “Triangular numbers?” Leibniz asked. “Oh, yes! Those are the numbers you get when you add the consecutive natural numbers, aren’t they? The first triangular number is 1, the sum of 1 + 2 = 3 is the second, the sum of 1 + 2 + 3 = 6 is the third, the sum of 1 + 2 + 3 + 4 = 10 is the fourth, etc., so the set of triangular numbers is 1, 3, 6, 10, 15, 21, 28, …. Yes, I see. So you want me to find the sum of the reciprocals of the 1 1 1 1 1 1 1 triangular numbers: 1 + 3 + 6 + 10 + 15 + 21 + 28 + ... . I wonder if I could use a variation of my method for the series of odd numbers. Do you think it’s possible?”

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“I think it might be,” Huygens said. “But before you begin, you might want to take a look at Wallis’ Arithmetica Infinitorum and Gregroire’s Opus geometricum. Gregoire includes some information on summing series. You should be able to find both those works in the Royal Library (the library of the Académie des Sciences). I’ll be interested to see what you come up with.” “Thank you, Sir,” Leibniz said bowing deeply as he left. On his way back to his lodgings, he stopped at the Académie’s Library to borrow Gregroire’s book so he could begin to work at once. When Leibniz met with Huygens the following week, he had his solution. “Monsieur [Mister] Huygens,” Leibniz began as he bowed in respect, “I believe I have the sum of your series. Would you like me to show you what I have done?” “Yes, indeed,” Huygens said, sitting down at the table and indicating a chair for his earnest young acquaintance. “Well, Sir, I began by playing with the series a bit,” Leibniz said. After some false starts, I decided to 1 1 1 1 1 1 multiply the whole series 1 + 3 + 6 + 10 + 15 + 21 + 28 + ... by the 1 fraction 2 and then I multiplied the new series by 2 so that the value remains the same as the original series. I found those fractions easier to play with. That gave me 1 1 1 1 1 1 1  + + + + ...  .” 2 + + +  2 6 12 20 30 42 56  “Of course,” Huygens agreed. “Then,” Leibniz continued, “I expressed each of those fractions inside the parentheses as a difference, giving me this: 2 1 − 12  +  12 − 13  +  13 − 14  +  14 − 15  +  15 − 16  +  16 − 71  +  71 − 81  + .... Again, the amount is exactly equal to the original se-

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ries. Then, you see, I worked with it in exactly the same way I did with my other series. The halves, the thirds, the fourths, the fifths, and all the rest disappear. Ideally, I would work it all out all the way to infinity. However, it really wasn’t necessary to go very far because the amount to be added each time gets smaller and smaller until at last it is negligible. In the end, all that remains is 2 ∙ 1, which is the number 2. Therefore, I have concluded that the sum of the series is 2.” “Well, well, young man,” Huygens said, genuinely impressed. “You have succeeded. I congratulate you.” “Thank you, Sir!” Leibniz said. Over the next weeks, Leibniz continued to explore the problem, reporting to Huygens on his further discoveries: “I used a different approach this time, abandoning my system of differences. I began with the number 1, the reciprocal of the first triangular number. Then I found that the sum of the next two terms is 1 1 1 1 1 1 + = + =. When I added the next pair, I found that 10 15 6 3 6 2 1 1 1 and that the sum of the next pair is 21 + 28 = 12 . When I 1 added those last two sums together, I got 16 + 121 = . You 4 1 1 see, 2 and 4 are reciprocals of the powers of 2. Then 1 1 1 1 the sum of 361 + 451 = and the sum of 55 + 66 = 30 , and the 20 1 1 1 sum of those last two fractions is 20 + 30 = 12 , while if I take the next four fractions in the same way I get a sum 1 1 1 1 of 24 . When I add those together I get 12 + 24 = 8 , which is the next in the series of reciprocals of the powers of 2. Continuing in that way, I eventually reached the sum 1 1 1 1 1 1 + + + + + + ... . You see that it always results in the 2 4 8 16 32 reciprocal of the next power of 2. Because the sum of all the reciprocals of the powers of 2 after the number 1 is exactly 1, that demonstrates that the sum of

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the reciprocals of the triangular numbers is exactly 1 + 1 = 2.” While working on this problem, Leibniz had returned occasionally to Pascal’s Triangle, which he had first used as he dealt with combinations in his book Ars Combinatoria. He had already known of triangular numbers, which formed the second column in Pascal’s Triangle, but the more he considered Pascal’s Triangle, the more intrigued he became with it. The result of those musings was the construction of his own Harmonic Triangle, which easily allowed him to sum the reciprocals of figurate numbers (such as the triangular numbers, pyramidal numbers, etc.). When he added the first two entries in the second column of his 1 1 2 triangle, he got 2 + 6 =3 . When he added the next entry, 4 5 3 he got 4 , and next he got 5 , and then 6 , etc. Continuing adding the entries in this way, he always got increasingly larger fractions whose numerators were one less than their denominators, thus giving a sum closer and closer to 1. To get the series of the reciprocals of the triangular numbers from his Harmonic Triangle, he had only to double each of the entries in the second 1 1 1 1 1 1 column of his Harmonic Triangle 2 + 6 + 12 + 20 + 30 + 42 + , re1 1 1 1 1 sulting in 1 +1 + 3 + 6 + 10 + 15 + 21 + ... . Therefore, the sum of the reciprocals of the triangular numbers must be 2 ∙ 1—in other words, 2. While Pascal’s Triangle was the result of summing the two entries that are located above a given number in the triangle, Leibniz’s Harmonic Triangle was the result of subtracting the two entries above and to the left of the given number, although once the triangle was constructed the entries could be con-

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1 1

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1 1 1 3 6 3 1 1 1 1 4 12 12 4 1 1 1 1 1 5 20 30 20 5 1 1 1 1 1 1 6 30 60 60 30 6 1 1 1 1 1 42 105 140 105 42

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Leibniz’s Harmonic Triangle.

firmed by adding the two entries below the number instead. The connection between addition and subtraction and the fact that he had figured the sum of the triangular numbers by combining differences (rather than sums), set his mind to considering the use of inverses in approaching problems. This was an important step. It led a few years later to his understanding of the fundamental theorem of calculus. Leibniz may have named his triangle harmonic because the outer columns form the harmonic series, whose sum (despite appearances) is infinite, unlike the sum of the reciprocals of the triangular numbers. The sum of the 1 third diagonal in the harmonic triangle is 2 , the sum of 1 the fourth diagonal is 3 , the sum of the fifth diagonal is 14 —in other words, the sums of the columns are also the harmonic series.

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“Very nice,” Huygens said warmly. “I see you have discovered that mathematics is a most engaging activity.” “Oh, yes, Sir!” Leibniz said with feeling. “My greatest regret is that it took me so long to discover its beauty.” Encouraged by Huygens’ responses, Leibniz set to work, writing up his results and planning to send it to the Journal des Sçavans, the journal of the French Académie. In the end, for unrelated reasons, the journal did not publish any issues for the next several years—and Leibniz’s article was lost in the confusion. However, because his result was nothing new even though his approach was novel, he realized it was probably best that his explanation had not appeared in print.

At that time, Leibniz was also hard at work on his calculating machine, a significant improvement over Pascal’s of 1642—Pascal’s machine was limited to adding and subtracting, while Leibniz’s would multiply and divide as well. Leibniz claimed that his machine would also take square roots, although it never actually performed those calculations. His goal in developing his machine was to expedite the use of his universal language, which would reduce all of human thought to simple calculation. His machine was an essential feature of his plan. Leibniz called his calculator the Staffelwalze [step cylinder]. The example at the Landesbibliothek in

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Leibniz’s Staffelwalze [step cylinder], his mechanical calculator.

Hannover, Germany, is 26 inches long, made of brass and steel, and housed in an oak case. The calculator has two parts: the accumulator (where numbers are stored) and the input wheel (where digits are entered). The eight dials in the input section are operated using the crank at the front of the machine and a worm gear. After the crank is turned, the entered number appears in the accumulator. To perform addition the crank is turned in one direction and to perform subtraction the crank is turned in the opposite direction. A second crank allows the user to multiply and divide in the same way. On his second or third visit to Huygens, Leibniz brought with him the first prototype of his machine. He had hired a mechanic to construct it from wood, and when he turned the crank Huygens was impressed. “You see, Sir,” Leibniz began, “I enter the first number here, and then I turn the crank to place it in

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the accumulator. Then, after I enter the second number—the number that I want to add to it—I simply turn the crank and the accumulator shows the total.” “Remarkable!” Huygens said. “I assume you are thinking that it would be best to construct the machine of brass and steel when you have perfected your plans?” “Yes, I think so,” Leibniz said. “There is always the danger that wood will warp or swell, and the gears can be made much more precisely in metal. As I perfect it, the machine should also multiply and divide easily using the second crank.” “I think this has possibilities,” Huygens said. “I hope you will continue working on it. I believe it is a significant improvement over Pascal’s machine.” “I believe that is true, Sir,” Leibniz said. “The process of calculating is nothing but drudgery, and I would like to save people from that tedious chore.” For his calculating machine, Leibniz elected to use decimal arithmetic (base 10) instead of binary (base 2), which he had invented a few years earlier but had not yet published. From a modern perspective, we might ask why Leibniz chose not to use binary arithmetic. After all, computers and calculators today use Leibniz’s binary system. As Leibniz knew, calculation in binary is just as accurate and efficient as it is in a decimal system, and Leibniz’s dials on his calculator could have been much simpler, using only the digits 1 and 0 instead of the ten digits that decimal calculating demands. However, that will remain a mystery—we simply don’t know Leibniz’s thinking on that question. Although, if he wanted to im-

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press businessmen and noblemen with his machine, they might have been put off by a binary mechanism, which would have been utterly foreign to them.

With encouragement from Huygens, Leibniz continued to work on his machine. However, as usual, he had many other projects going as well. Officially, his assignment was to carry out diplomacy, with mathematics simply filling up his leisure hours. In reality, he was spending much of his time mastering any mathematics he could find—there was no doubt in his mind what his primary mission should be. In between his political duties and his mathematics, Leibniz also tried unsuccessfully to collect the outstanding rent on Boineburg’s properties. He contacted the authorities, was told the person in charge would not return for several months, waited those several months and then asked for the desired payments. Again he was told the person with the authority to make such a decision was still not available. Leibniz was powerless. He continued to ask and wait for the rest of the four years he was in Paris, but to no avail. Boineburg might justifiably want his money and Leibniz might honestly want to fetch it for him, but in the end nothing happened. In November 1672, Leibniz received a letter from Boineburg with a new assignment: he would tutor Boineburg’s 16-year-old son, Philipp Wilhelm. A few days later the young man arrived, accompanied by Melchior Friedrich von Schönborn, the nephew of

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Elector Schönborn, who was married to Boineburg’s daughter. Boineburg and the Elector had chosen Melchior to present a new plan to Louis XIV, suggesting that a peace conference take place in Köln to end the hostilities that had already begun in the Low Countries and Northern Germany—the war that Leibniz had hoped to avert by convincing Louis XIV to invade Egypt instead. Taking his cue, Leibniz composed a careful statement for Melchior to present to the French King. Unfortunately, Leibniz—who was again reminded that he had no noble blood and thus could expect no privileges in the upper levels of society—was not allowed to be present at the meeting and Melchior chose not to read Leibniz’s statement. The proposal failed, a development that should have surprised no one. Louis XIV was the most powerful ruler in Europe, and Melchior, who was young and inexperienced, did not make a compelling presentation. That failure was a disappointment to Leibniz, but it was only the beginning of his troubles. The following month, in December 1672, Leibniz received devastating news: his patron and friend, Boineburg, had died. The Baron, who was 50 years old, had been under considerable stress as warring troops destroyed his estate near Köln. His death closed both a vibrant friendship and a comfortable financial arrangement for Leibniz. “How am I going to pay my bills now?” Leibniz asked himself. “What will become of me? I was not meant to beg for money to support myself in my lofty projects. My wants are simple. All that I need is the time and resources to accomplish important things. Surely that is not too much to ask!”

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However, Leibniz was still a realist. Knowing that life would probably never be fair, he took stock of his situation and resolved to make the best of it. He immediately wrote to the brother-in-law of the late Baron: “Please accept my sincere condolences for the death of your brother-in-law, the Baron. I should tell you that your nephew, young Boineburg, has borne up well in spite of the depressing news of his father’s death. For my part, I promise to support the young man in any way I can. By the way, are you aware that I am still owed salary for my diplomatic work for the late Baron over the past few months? In addition, I have had considerable expenses in my attempts to collect the past due rent on the Baron’s properties here in Paris. Also, dear Sir, I hope I am correct in assuming that my position as tutor to the young Baron is still in effect. I will appreciate any help you can give me. Once again, kind Sir, I am so sorry for your painful loss.” Leibniz also wrote to the Baron’s widow, once again expressing condolences and also asking for reassurances that he would be allowed to continue to serve as the young Baron’s tutor. To Leibniz’s relief, she wrote back, confirming his job as tutor and through it her modest financial support of Leibniz. Neither she nor her brother mentioned Leibniz’s past salary or compensation for his attempts at collecting rent. Leibniz had no recourse—as a dependent he could submit requests, but the Baroness was free to do as she wished with him. Since the French King had rejected the proposed peace conference in Cologne, the Elector decided to send his nephew on to England to try his luck with the English King instead. The Elector, who had absolute

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respect for Leibniz, requested that Leibniz accompany his inexperienced nephew, young Melchior Schönborn, on this important mission. “What wonderful news!” Leibniz said to himself. “I have wanted to go to England for some time. Certainly there are important diplomatic issues I will be glad to help solve in London, but there should also be time for me to meet with my contact Oldenburg (the expatriate German who was the effective head of the Royal Society) and the mathematicians there. What luck!” When Melchior Schönborn and Leibniz arrived in London, they made appointments to meet with officials, carrying out their assignment as well as they could. However, their reception in England was little better than it had been in France. Once again, they were forced to wait for possible developments. In his free time, Leibniz made arrangements to meet on his own with Oldenburg. Leibniz and Oldenburg had exchanged many letters, and Leibniz was eager to meet Oldenburg in person. Because Huygens had written to his friend Oldenburg about Leibniz’s visit, including the news of a calculating machine that Leibniz would be carrying with him, Oldenburg was eager to meet him. Their first meeting on February 1, 1673, went well. After that promising beginning, Leibniz was invited to demonstrate his calculating machine to members of the Royal Society. “What does the young German gentleman have?” the ultimate tinkerer, Robert Hooke (1635–1703), asked. “Here, let me have a closer look,” Hooke said, pushing Leibniz out of the way so that he could examine

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the machine more carefully. Without asking permission, Hooke removed the cover of the machine so he could look more closely at its inner workings. “Aha! I see,” Hooke said smugly. “It seems very straightforward.” Two weeks later at a meeting of the Royal Society, Hooke reported on Leibniz’s calculator with a sneer, indicating that it wasn’t as great as Leibniz had indicated. Two weeks after that, Hooke presented his own calculating machine, which was suspiciously similar to Leibniz’s in some ways, although the mechanism was less sophisticated. Leibniz complained to Oldenburg about Hooke’s dishonesty, but there was little Oldenburg could do. Hooke was a master contriver, and, as Oldenburg knew, a difficult man. Hooke’s contraption was not as good as Leibniz’s, but Leibniz’s machine was not yet perfect either. A few days later, Oldenburg arranged for Leibniz to meet with Samuel Morland (1625–1695), who had constructed several calculating machines over the past ten years. Morland demonstrated his adding machine, which Leibniz was critical of because it could only add and subtract and did not have a carrying function—the user had to add with pencil and paper the remainder of any sum beyond 9 in a given column. However, Morland’s adder was innovative in that it was designed for use with English money, having dials devoted to base 4, base 12, and base 20—at the time, English currency included the pound (that was equal to 20 shillings), the shilling (that was equal to 12 pennies), and the penny (that was equal to 4 farthings). Morland’s second calculating machine, which could multiply and divide, used a system similar to Napier’s

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Chapter 4 Wooden model of Napier’s bones.

bones. Like his adder, it also had no carrying function. Leibniz also criticized this machine, saying that it was no more than a mechanical version of Napier’s bones. Leibniz left the meeting satisfied with the superiority of his own machine, which he promised to perfect very soon. Unfortunately, this task stretched for years, and the English grew weary of Leibniz’s many delays. Although Leibniz and young Schönborn made little progress in international relations, Leibniz continued to make important contacts for himself in London. He was able to meet John Pell (1611–1685) at the home of Robert Boyle’s (1627–1691) sister, Lady Ranelagh. Pell was a respected mathematician who, although he had done no impressive work in recent years, prided himself on keeping up with current literature in mathematics. He listened with interest to Leibniz’s explanation of his work on series using differences. “But Mr. Leibniz,” Pell asked disingenuously, “you have seen Gabriel Mouton’s (1618–1694) book, haven’t you?”

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“Mouton?” Leibniz asked. “No, I’m afraid I have never heard of him. Is he a mathematician?” “Well, yes! I would say so!” Pell replied, looking smugly at the others in the group for confirmation. “Mouton wrote an important book two years ago, explaining, among other things, that the Frenchman François Regnaud made the same discovery you seem to be taking credit for now.” “Really?” Leibniz asked in alarm. “What is the name of that book?” “I’m surprised you don’t know of it. It is Mouton’s De diameteris apparentibus solis et lunae [Concerning the Apparent Diameters of the Sun and the Moon],” Pell said. “You might want to read it.” “Yes, yes, I will certainly do that,” Leibniz stammered in embarrassment. The following day, Leibniz borrowed that book from Oldenburg, found that indeed Pell had been right, and quickly wrote a formal letter of explanation to Oldenburg, as secretary of the Royal Society, denying any possibility of dishonesty or plagiarism. “I assure you, Mr. Oldenburg,” Leibniz wrote, “I had no idea my technique had already appeared in print. Those were my own discoveries; I was unaware that someone else had made them before me. Since I had no knowledge of Mouton’s book when I made my discoveries, clearly I was not borrowing from someone else. However, you may be sure, that in the future I will always try to scan the relevant literature before I present any discoveries I may have made.” In April 1673, responding to Leibniz’s request to be made a fellow of the Royal Society, that honorable body

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voted unanimously to elect him as a member based on his promising calculating machine and probably also based on Huygens’ support. Although English mathematicians did not like Huygens as a person, they had no choice but to respect his mathematics. Leibniz was delighted and wrote a quick note in appreciation, but he learned a month later that his simple note was not adequate. Because the Society had given him a great honor, courtesy required that he compose a formal letter of acceptance. Oldenburg informed him that he must write that letter, and Leibniz did so, apparently frustrated at the necessity of replying to his election yet again. Leibniz had a busy schedule with all his demanding projects, and taking the time to write a formal letter was inconvenient. Leibniz had already met with many important mathematicians and scientists in London, learning as much as he could and seeing how he needed to proceed from there. However, there was one Englishman whom he tried in vain to meet—Isaac Newton (1643–1727). Leibniz had heard of him and had certainly hoped to make his acquaintance, but Newton was not available. In fact, Newton was rarely available to anyone. Leibniz was aware that Newton had been making interesting discoveries in mathematics and mechanics. Although he saw some summaries of Newton’s work, in fact Leibniz understood little of what he saw and left England realizing that he simply had more work to do if he was ever to understand Newton. Although Newton was only four years older than Leibniz, his mastery of mathematics was far superior.

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With diplomatic negotiations in England in limbo, the official part of the trip was not as successful as Leibniz had hoped. It appeared the English were no more enthusiastic about young Schönborn’s diplomacy than the French had been. In mid-February of 1673, Leibniz and Melchior Schönborn received word of yet another catastrophe: the Elector himself was gravely ill. As a result they were forced to pack their bags and hurriedly leave London. Leibniz tried to meet once again with Oldenburg but failed in that attempt. Instead, Oldenburg left an apologetic note along with a letter for Leibniz to deliver to his friend Huygens and the latest issue of the journal of the Royal Society. Leibniz left a note for Oldenburg in reply, apologizing for his hurried departure and returning the books he had borrowed. By the time the delegation had crossed the English Channel and reached Calais, the Elector had already died. Within three months, Leibniz had lost both his noble sponsors. Without them, Leibniz’s financial situation was precarious. He sent word with Melchior, who returned directly to Mainz, offering his services to the new Archbishop and Elector. The reply from Mainz gave him permission to remain in Paris but offered no salary to support him there. “What good is that?” Leibniz asked himself. “What am I supposed to live on? I have worked tirelessly to help the causes of the Empire, and this is what I get! I deserve better.” In the meantime, Leibniz’s young charge, the young Baron of Boineburg, was chafing under the demanding program of instruction that Leibniz was imposing on

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him. Rising at dawn, taking just an hour off for each of three meals, and completing his studies late in the evening, the young Baron resisted. He had far more interest in the social life of Paris than in an intellectual education. After all, did his tutor actually expect him to hide inside his rooms all day and all night when Paris was out there waiting for him? Leibniz had no sympathy. When he complained to the boy’s mother about his young charge’s laziness, she replied curtly, “My dear Leibniz, I am sorry that you are frustrated in your work with my son. However, I believe the best solution is for me to make other arrangements for his education. Thank you for working with the boy, but we will no longer have need of your services. I’m sure you understand.” Thus, on September 13, 1674, after a year and a half in Paris, Leibniz lost his last regular source of income. Although his friend the late Baron would have applauded Leibniz’s ambitious program for his son, clearly Boineburg’s wife did not. In Leibniz’s eyes, the mother and son were wrong, but he was powerless to change their minds.

Over the next two years, Leibniz supported himself as well as he could. As an accomplished lawyer, he could find freelance work, drawing up documents for people as needed. It was an uneasy existence, but it allowed him to remain in Paris where he could pursue mathematics, his principal interest by that time. He didn’t starve, but his finances were uncertain at best.

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Leibniz had continued his correspondence with Duke Johann Friedrich von Hannover, who had always been eager to engage Leibniz as his resident intellectual. Over the years, the Duke had repeatedly asked him to come work in Hannover, and each time Leibniz had delayed making a commitment. However, Leibniz had always been careful to keep the Duke informed of his accomplishments through regular correspondence, always implying that a commitment to the Duke would be coming soon. Although Leibniz feared the prospect of real poverty, Hannover was not Paris—nor was it even Mainz. In response to a newsy letter from Leibniz a year earlier in March 1674 about his latest activities and discoveries, Johann Friedrich von Hannover replied on April 25, 1674, that he was prepared to offer Leibniz a generous annual salary of 400 Thaler to serve as councilor at his court in Hannover. Although Leibniz respected the Duke and still considered him his most likely sponsor, he waited to respond to the offer. He was not hopeless yet. He might be in serious financial straits, but he had mathematics to do and Paris was obviously the place to do it. Hannover? How could he be expected to survive in such a provincial town? Leibniz knew that he deserved better. So far, he had only one other possibility: he had been offered a post as secretary to the chief minister to the King of Denmark. Denmark? What could the Danish court possibly offer him? He saw the language spoken there as nothing but a wretched perversion of German—no self-respecting German would master

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that gibberish! He never considered the Danish offer even remotely acceptable. Leibniz had been in sporadic contact with his older half-brother, Johann Friedrich Leibniz, in Leipzig since his departure from Saxony in 1666. Johann Friedrich had written to Gottfried Wilhelm twice since his sister’s death—the first time in 1674, criticizing Gottfried Wilhelm for never having come to pay his respects at his sister’s and his parents’ graves in the cemetery in Leipzig. The second letter in 1675 upbraided him for not having acknowledged the death of his sister’s husband, Simon Löffler. Gottfried Wilhelm replied to each letter, explaining in his first that the journey to Leipzig was simply too far for him with his many responsibilities in Paris and London, and in his second that he hadn’t heard of his brother-in-law’s death until his half-brother’s most recent letter. He then expressed great sorrow over his brother-in-law’s death and also genuine fondness for his nephew, Friedrich Simon Löffler. He indicated that he wished to remain in contact with his nephew. Neither of Gottfried Wilhelm’s letters were answered—and it is possible that they never reached their intended recipient since postal services at the time were unreliable. Moreover, worry in Leipzig concerning Leibniz’s faith continued to fuel general mistrust of his motives—why would a good Lutheran choose to work in the service of the Roman Catholic Elector of Mainz? Why didn’t he pursue his career in the court of the Elector of Saxony, a Lutheran prince of the Holy Roman Empire? And why would he choose to spend so much time in the Catholic city of Louis XIV’s Paris? His rela-

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tives could only conclude that Leibniz had converted to Catholicism without informing them of that dire move. In the family’s view, he might as well be an infidel as convert to Catholicism. Leibniz protested in his letters that he would not abandon his religion simply for financial gain. They could be sure he remained a resolute Lutheran. How could they suggest that he wasn’t? Furthermore, he had always spoken in favor of the German states, and of Saxony in particular, in his dealings in Paris. He explained optimistically that his work in Paris had been successful, and that he hoped to stop for a visit in Leipzig on his way to Italy in the coming year. He promised he would pay his respects at the family graves then. Nevertheless, in spite of what he said, Leibniz’s situation was becoming desperate. In the fall of 1675, in spite of his assurances to his family about his many successes in Paris, Leibniz decided that he must ask for some financial help and wrote once again to Christian Freiesleben, the administrator of the family estate, to procure some funds. Because Freiesleben was in regular contact with Leibniz’s half-brother, the response was not enthusiastic. Freiesleben sent Leibniz some money, but the implication was clear that more funds would be difficult to arrange. However, with the newly arrived money, Leibniz was able to continue in Paris a little longer. He continued to apply himself to his mathematics day after day, having no intention of repeating his mathematical embarrassment at Pell’s hands in London. By now, mathematics had become his passion, and he knew that Paris was the place to study it.

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Also in 1675, Leibniz decided to seek election to the Académie des Sciences in Paris—something that could provide him with funds to continue living in the city. He felt that he had finally gained the respect of most of the mathematicians of Paris, and, with Huygens’ support, an appointment seemed likely. Unfortunately, that is not the way the Académie viewed it. Leibniz was a foreigner, and there were already too many foreigners in the Académie. Furthermore, contrary to his family’s worries, Leibniz was not a Catholic—a fact that did not help his cause among the Académie’s largely Catholic membership. It was an honor this great mathematician would not receive for another 25 years. Despite his difficult financial position, Leibniz still continued to meet with philosophers and mathematicians in Paris as he struggled with his ideas. The philosophers there were engaged in a debate about Descartes and his philosophy—had Descartes been an atheist or not? By now, Leibniz had read Descartes’ works and was happy to participate in these discussions. However, Leibniz’s major focus was on mathematics. Descartes’ analytic geometry was allowing seventeenth century mathematicians to approach curves algebraically, thus opening up new ways of finding the areas of curved figures and the slopes of their tangent lines. Those were questions that Leibniz felt compelled to answer. Leibniz met several times with Jacques Ozanam, a self-taught mathematician and author of at least 14 books on mathematics. He was a few years older than Leibniz and took great pleasure in proposing thorny problems in number theory to his acquaintances.

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When the two men met one evening, Ozanam challenged Leibniz, “Can you find three numbers that add up to a perfect square, and whose squares add up to the fourth power of some number?” Leibniz breezily replied, “Oh, that shouldn’t be too difficult.” Ozanam then demanded, “Well then, find me the solution!” It was harder and took considerably more time than Leibniz had predicted, but finally he succeeded. He wrote to Ozanam several days later with his solution: 64 + 152 + 409 = 625 = 252, and 642 + 1522 + 4092 = 194,481 = 214. Presumably Ozanam enjoyed watching Leibniz struggle to solve his puzzle. This was one of Leibniz’s first forays into number theory, a subject that would intrigue him for years. He was delighted with his solution to this problem, although he failed in his attempts to solve some other problems from Ozanam. Leibniz had to admit to himself that he was not—or at least, not yet—the most successful number theorist in Paris. Leibniz continued to meet with Huygens, who had come to respect the young German. His little diplomat was clearly mastering mathematics at a remarkable rate! When Leibniz had asked a naïve question about the location of the center of gravity of a plane surface a few months earlier, Huygens had laughed and sent him to an appropriate source so he could correct his error and move on. During that time, Leibniz read a wide variety of sources on Huygens’ recommendation. With no other responsibilities, he was free to spend as much time on mathematics as he wished. The schedule

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he voluntarily set for himself was far more demanding than the one he had asked of young Boineburg. Leibniz studied Frans van Schooten’s (1615– 1660) excellent textbook Geometria, learning Descartes’ method for representing geometrical curves algebraically. In other sources, he saw that an algebraic equation must have at least one solution and that the power of the highest variable in the equation tells the number of solutions (both real and imaginary). Additionally, he read Pascal’s work on finding the moment of part of a circle (the product of the mass and the distance from the center of gravity of the figure from a line or plane), and suddenly realized it could apply to other curves as well. Leibniz was studying the masters of mathematics not just to understand their works but to use them as points of departure for his own investigations. Because his mathematical learning up to that point had been haphazard, Huygens was providing the structure he needed. Like many scholars before him, Leibniz was fascinated with the classical question of how to square the circle—or more generally how to find the exact area of a curved figure. Archimedes (ca. 272–212 bc) had approached finding the area of a circle through his method of exhaustion, in which he found the areas of more and more, thinner and thinner rectangles that were caught inside a circle or curve. In another approach, Archimedes found the areas of pairs of polygons with more and more sides that trapped a circle between them, coming closer and closer to the area of the given circle. However, those were methods that involved hours of tedious calculations for each

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individual figure, and they did not provide a general solution. Leibniz read James Gregory’s (1638–1675) work on infinitesimals with great interest and learned Gregory’s technique for constructing tangents (the line that touches the curve at precisely one point without cutting through it) and normals (the line that hits the curve perpendicularly at the tangent) to a variety of curves. Pierre de Fermat (1601–1665) had identified the extrema of curves—finding the highest and lowest points—although he had no way of identifying which extreme it was: the highest or the lowest. Johannes Kepler’s (1571–1630) and Galileo’s methods of infinitesimals and indivisibles were analogous to Archimedes’ method of exhaustion, but they also provided no general technique that could be universally applied. At this time, Leibniz received a letter from Oldenburg alluding to a discovery by Isaac Newton at Cambridge University for a general method of finding the area of a curved figure. Oldenburg’s letter provided no specifics, but it piqued Leibniz’s curiosity. What could that general method be? Leibniz wondered if perhaps he might arrive at such a method himself. change in y rise Slope is defined as run or change in x —in other words, the ratio of the change in the vertical height to the change in the horizontal width. The slope of a line shows how steep it is. The calculus addresses two questions: differential calculus provides a general method to find the slope of the tangent to a curve (thus the rate of change at any specific point). We call that curve the derivative—it allows us to see the curve of the rate of change. Leibniz’s

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Tangent to a curve: the tangent is the only line that can be drawn at any one point on a curve.

y = x, or y = (1)x + 0 slope = 1 =

1 = up one 1 right one

y

run rise run rise

x

rise

y

y = (–2)x + 4

left 1

−2 = down two or 2 up two slope = (–2) = = –1 left one 1 right one

up 2 left 1 down 2

up 2

right 1 down 2

x

right 1

Slope: In a linear equation of the form y = mx + b, m is the slope and b is where the line crosses the y-axis.

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rise = 0 = 0, in other words perfectly horizontal. run = 1

slope = rise =1 = 1 run =1 The line that is tangent to the curve tells the slope of the curve at that point—that is, the derivative of the curve at that point.

symbol for the derivative is dy/dx, where dy is the infinitely small differential (change) in y and dx is the infinitely small differential in x. By contrast, integral calculus provides a method to find the total area beneath a curve—it allows us to see the curve of the measurement of the area. We call that curve the integral. When we deal with an equation in calculus, we are concerned with those two curves—the curve that is called the derivative and the curve that is called the integral. Leibniz used the symbol ∫ for integral as an exaggerated letter S for the Latin word summa [sum]. The integral gives the sum of the areas of all the infinitely thin rectangles that make up the area. In his earlier work, Leibniz used the abbreviation omn—short for the Latin word omnis, which means all or the whole thing. Leibniz’s ∫ is the

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only symbol that has been used for the integral since his publication of the calculus. Leibniz’s calculus allows us to work with those two curves, seeing that when we differentiate the curve that is the integral, we get the curve that is the derivative. When we integrate the curve that is the derivative, we get the integral. Leibniz’s genius was to see that both the derivative and the integral were inverses and to develop a method for calculating them easily. When Leibniz considered a curve such as y = x2, he soon realized that the slope of the tangent to this line at any point (x, x2) was 2x. With a curve such as y = x3, the slope of the tangent at any point (x, x3) was 3x2. To find the area of the curve, all that was necessary was to do the reverse operation. The area under the curve whose tangent slope is 3x2 must be x3. Leibniz’s skillful use of Descartes’ exponents helped him in his development of the calculus. Both Fermat and John Wallis (1616–1703) were able to calculate the slope and the area bounded by specific curves, but neither had a general method for doing it in such a way that it could be applied to any equation. Newton had such a method in his fluxions, but he did not publish his results until 1704, decades after Leibniz’s publications in 1684 and 1686. Like Leibniz and Newton, Fermat apparently had such a method, but he never saw the inverse relation. Because Fermat was an amateur mathematician, he never felt the need to publish any of his work. He satisfied himself as he played at his hobby, and that was enough for him. Although Isaac Barrow (1630–1677) discovered the inverse relation between derivative and integral, his

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approach was too geometric and not sufficiently analytic to be truly useful. In other words, only Leibniz and Newton deserve credit for formulating the process that we call the calculus. With Huygens’ support, Leibniz was able to figure out a way to calculate area and slope and to articulate the fundamental theorem of calculus. In many ways, it resembles the inverse relationship between addition and subtraction—each process undoes the other—and that reminds us of Leibniz’s process of taking sums of infinite series by adding the differences in the terms. Despite the controversy that raged at the end of the seventeenth and throughout the entire eighteenth century, credit for the discovery of the calculus should go to both Leibniz and Isaac Newton. Although Newton made his discovery well before Leibniz, he published it too late and thus he cannot claim exclusive credit. Newton had no intention of sharing his method with anyone but a chosen few. Although Leibniz saw and heard occasional references to Newton’s work, his construction of the calculus was entirely original. Furthermore, Leibniz published his calculus, and, with the help of the Bernoullis (Jacob Bernoulli, 1654–1705, and Johann Bernoulli, 1667–1748), he broadcast it to all of continental Europe. It is Leibniz’s calculus that we use today. Leibniz discovered the differential calculus in 1674 but first published it ten years later in 1684 in the Acta Eruditorum. The title of his 1684 article was “Nova Methodus pro Maximis et Minimis, itemque Tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus [A New Method for Maxima and Minima as well as

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Tangents, Which Is Neither Hindered by Fractional nor Irrational Quantities, and a Remarkable Type of Calculus for Them].” This was the first published explanation of the calculus, although even the brilliant Bernoulli brothers found it extremely difficult. Jacob described it as an enigma rather than an explanation. The name calculus (the Latin word for pebble, as in the pebbles on an abacus) comes from this article. Leibniz published his invention of the integral calculus, also in the Acta Eruditorum, two years later in 1686. In the May 1690 edition of the Acta Eruditorum, Jacob Bernoulli published an article further explaining the integral calculus, using the word integral for the first time to describe this process that allows us to find the area under a curve. No one has used any other word for it ever since. The discovery of the integral calculus was a product of the genius of Leibniz, but its name is due to Jacob Bernoulli. With the serious work that the Bernoullis did to publicize Leibniz’s calculus, it became—and still remains to this day—the standard method for calculating the rate of change and the areas of irregular figures. Because the English persisted in using Newton’s fluxions for many years, their narrowness served only to retard the growth of mathematics in the British Isles. In the nineteenth century Charles Babbage founded an Analytical Society aimed at bringing the English up to date on analysis. His Society argued that “the principles of pure d-ism as opposed to dotage at the university” must be adopted in England immediately. The d referred to Leibniz’s calculus with its dy/dx for the differential, intended as a play on

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the word deism (an unconventional religious movement loosely related to Christianity). He was contrasting Leibniz’s calculus to Newton’s fluxions, with its dots over the number to indicate the fluxion. Babbage meant to suggest the dotage (the decrepit, weakness) of the method that Newton had discovered. He knew that the English needed to abandon Newton’s fluxions in favor of Leibniz’s calculus if they were to compete in the modern world.

All this time, Leibniz was still afflicted with the need to perfect his calculating machine. He had promised early delivery of it to Oldenburg when he left London, and yet, many modifications and several years later, it still didn’t satisfy him. In his correspondence, Oldenburg repeated his request for the calculating machine. “Herr Leibniz, this is a situation that causes me much embarrassment at the Royal Society. I have received repeated promises from you about the delivery of this machine, but still I have nothing. Please, dear Sir, follow through on your commitment immediately. You must relieve me of my embarrassment.” Leibniz’s solution was to stop answering Oldenburg’s questions about it, putting off further correspondence until he was satisfied that it was ready. Although this was not a noble way to handle the problem, he justified it by vowing he would fulfill his promise as soon as he could find the time. As usual, Leibniz was hard at work on many other things.

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Nevertheless, Leibniz, who was eager to improve his standing among the intellectuals of Paris, reluctantly demonstrated his almost perfect machine to the Académie early in 1675, with the result that Colbert, the secretary of the French treasury, ordered three machines—one for the King, one for the royal observatory, and one for Colbert himself. Desperate to remain in Paris, Leibniz yearned for a position at the Académie, and that was an attempt to remind them of his accomplishments At that time, Leibniz still continued his correspondence with the Duke of Hannover in 1675, boasting that his marvelous calculating machine could multiply and divide numbers even faster than one could write the numbers on paper preparatory to multiplying or dividing them by hand. He suggested in his letter that he would be pleased to provide one for the Duke. Leibniz knew his machine was good, even in its imperfect form. In fact, Leibniz’s real motive in writing to the Duke was not to sell his calculating machine, although money from any source was welcome at that point. Leibniz could see that accepting the Duke of Hannover’s job offer was becoming inevitable. In his letter, he implied that he was ready to accept the Duke’s offer, although he made no firm commitment. At the same time, Leibniz wanted the Duke to understand the importance of his mathematics. Leibniz was a man with a mission, and he wanted to be sure the Duke—his probable future employer—understood that. In August 1675, Leibniz met Ehrenfried Walther von Tschirnhaus (1651–1708), a nobleman and scholar from Germany, whom Oldenburg had met in

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London and whom he had recommended to Leibniz as a possibly congenial associate. As a result, Leibniz and Tschirnhaus studied the works of Pascal together in Paris. By this time, Leibniz had already devised his differential and integral calculus, which he apparently described to Tschirnhaus in some detail, although Tschirnhaus gave no indication of grasping their significance. In the later controversy about Leibniz’s alleged plagiarism of Newton’s fluxions, the British suggested that Tschirnhaus might have given Leibniz some crucial information about Newton’s work. However, it is unlikely that Tschirnhaus would have been able to explain Newton’s fluxions to Leibniz even if he had tried—Tschirnhaus was not a mathematician of either Newton’s or Leibniz’s caliber. When Leibniz finally resumed his correspondence with Oldenburg late in 1675, it was to explain his methods for finding the area of a curved figure—part of his integral calculus. When he saw it, Oldenburg realized that Leibniz had done some original work on a topic similar to Newton’s. As a result, Oldenburg was willing to communicate the results—if not the methods—that Newton and Gregory had achieved and to put Leibniz in communication with Newton. When Newton then sent some information about his method of fluxions to Leibniz through Oldenburg in June of 1676, in what is now called his epistola priori, he deliberately encrypted it in such a way that Leibniz would be unable to read the essential parts. Newton was still jealously guarding his discoveries from ten years earlier. Although the letter presented enough of Newton’s work on series to show Leibniz

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that Newton was indeed an important mathematician, it stopped short of explaining his method of fluxions. Leibniz replied to Newton’s epistola priori, giving a few details about his own calculus and showing some of his own discoveries. Leibniz, a sociable man and consummate correspondent, then requested more specific information from his esteemed correspondent on his method of fluxions. Newton’s reply to that letter is now called his epistola posterior. Because of the convoluted way in which the letters were passed to Oldenburg and John Collins (1625–1683) and back again, eight months passed after Newton wrote his response before it was delivered to Leibniz. During that time, Newton erroneously assumed Leibniz was avidly studying his letter. In his reply, Leibniz praised Newton’s discoveries and then explained his own calculus in some detail, suggesting that he would be pleased to continue the correspondence. However, Newton concluded that Leibniz had constructed his calculus only during those months, using Newton’s own fluxions as his model. Because Oldenburg, who had been the medium for the correspondence, died a few months later, the correspondence died with him. The two scientists had no further communication. Leibniz was happy to take credit for his method— his original algorithm—of the calculus. He knew that others had found area and slope before him, and he had no desire to compete with them on those results. Instead, it was his method that he presented as his important accomplishment.

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When Gilles Personne de Roberval died at the end of 1675, Leibniz celebrated, hoping he might finally be inducted into the Académie in Roberval’s place, since membership was strictly limited in number. Although Leibniz thought he had enough votes, at the last minute his support evaporated and he was informed that the Académie didn’t need any more foreign members. That ended Leibniz’s hope for a permanent position as a scientist in Paris. Leibniz, the continent’s greatest mathematician, was informed once again that he was not worthy of that honor. “Oh, Paris!” Leibniz moaned. “City of light! I have accomplished so much within your scientific environment, and now, alas, I must depart. Hannover? How will I survive in Hannover? Perhaps I can delay my departure just a little longer. Yes, yes! Surely I can continue here for a few more months. I cannot leave now. Johann Friedrich in Hannover will have to wait.”

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1676–1679

Librarian and Councilor to Duke Johann Friedrich of Hannover

In January 1676, Johann Karl Kahm, an official in the court of the Duke of Hannover, officially offered Leibniz a position as Councilor—but not Privy Councilor, as Leibniz had requested—for a generous salary. Leibniz, deciding that he must accept the inevitable, wrote a formal acceptance but still continued to hope for other possibilities. He had been corresponding with the new Elector in Mainz, asking if he might remain in his position as political emissary for that court in Paris, hopefully with the addition of a modest salary. However, those hopes were dashed when his friend Melchior Friedrich von Schönborn informed him that, much though the Elector might want to support Leibniz, the money simply was not available.The archdiocese had its limitations, even when the most brilliant man in the world was involved. On January 27, 1676, Leibniz was officially named Councilor to Duke Johann Friedrich of Hannover. Leibniz had once again accepted a position in 111 © 2012 by Taylor & Francis Group, LLC

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the court of a Roman Catholic Duke, where once again he was not expected to convert to the Roman faith. He knew that Duke Johann Friedrich, his new employer, had converted in order to satisfy his wife. Although he had made his commitment to Hannover honestly, Leibniz didn’t like it. He groaned, “I don’t want to spend my life in Hannover—that backwater of civilization! I need to be in Paris. It’s obvious that Paris is where I create my best work. Oh, if only I didn’t need money to live! I don’t want to leave Paris—ever.” Then he continued on a brighter note, “Surely I can delay my departure to Hannover a little longer. Yes, I’ve told the Duke that I am coming, but he can’t expect me to travel to Hannover immediately. Since he is so impressed with me, he will allow me some leeway. Yes, Your Highness, I’ll go to Hannover soon. However, you must understand that I am destined to be an amphibian. I expect to spend some time in Hannover, of course—I have agreed to that. Nonetheless, I will spend most of my time in world capitals, like Paris or London or perhaps Vienna or even Rome.” Of course, he didn’t actually inform the Duke of these things, but he saw this as the only possible solution. A month later, at the end of February 1676, Leibniz received another letter from Kahm, informing him that his salary would be paid retroactively as of the first of January and he should travel to Hannover as soon as possible. After begging for a few weeks to put his affairs in order, Leibniz asked for and was granted permission to remain in Paris until the middle of May. Kahm wrote to Leibniz again in July, urging him to hurry, indicating that Leibniz would be appointed li-

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brarian to the Duke in addition to his duties as councilor—a development that Leibniz welcomed. At the end of July, the Duke’s ambassador to Paris met with Leibniz personally, pressing him to depart for Hannover immediately and giving him money to cover his travel expenses. In September the Duke’s ambassador informed Leibniz that the Duke was more than impatient—Leibniz must make his way to Hannover without further delay. Leibniz agreed, saying he would depart immediately, although his trip would take him through the Spanish Netherlands (where he needed to consult with Huygens who had returned home to recover from illness) and England. The ambassador immediately replied that a detour through the Spanish territory would be impossible due to the war. Leibniz accordingly made his plans instead to travel only to London and from there to Hannover, leaving his beloved Paris on October 2, 1676. Leibniz never again visited Paris—the city of his inspiration. During his week in London, Leibniz demonstrated his improved calculating machine (which he had brought with him) to Oldenburg, who was delighted with its powers. At long last, Leibniz satisfied Oldenburg’s repeated requests for the machine. With that commitment met, Leibniz was then able to see and copy some of Newton’s recent writings at the Royal Society library, although those works included nothing about Newton’s method of fluxions and fluents—his work on the analysis that Leibniz called his calculus. While he was in London, Leibniz was invited to begin his trip to Hannover on the yacht of Prince

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Ruprecht, a German prince on the Palatinate (Pfalz) on the Rhine. The yacht (a sea-going ship that was adequate for all but the worst weather to be found in the English Channel) would be carrying wine for the Prince, and it was no problem for Leibniz to ride along as far as Rotterdam. Ruprecht was charmed with the thought of spending several days in the company of the great intellectual as they sailed across the Channel. Although the ship was delayed in London for most of a week by strong head winds, eventually she set sail for Holland. Two weeks later Leibniz arrived in Rotterdam (which was a safe distance to the north of the fighting in the Spanish Netherlands), exhausted and in need of serious rest on dry land before continuing his journey. In a letter to Kahm, Leibniz assured him that he would make his way to Hannover as soon as he had gotten some sleep and recovered his appetite. Leibniz was not a seasoned sailor. He wrote that he expected to arrive in Hannover no later than the end of November. While he was recovering from his sea journey, Leibniz stopped briefly in Amsterdam, where he met the mathematician Jan Hudde (1628–1704), who had made many interesting discoveries about maxima and minima, concepts that helped Leibniz in his discovery of the calculus. Hudde had also helped prepare the Latin translation of Descartes’ Geometrie, making it accessible to Europeans outside of France. After Leibniz left Amsterdam, he went on to The Hague where he met with the controversial philosopher Baruch de Spinoza (1632–1677), with whom he had been corresponding secretly since 1671. Spinoza

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was universally known as an atheist but also as a brilliant and original philosopher. The two men spent several days talking. Spinoza was a renegade, choosing to dress in simple, comfortable clothes and to pursue his philosophy in unconventional ways, whereas Leibniz looked like the courtier he hoped to be, and developed his philosophy along more conventional lines. Although Leibniz often criticized Spinoza in his correspondence with other scholars, his private notes indicate that he respected Spinoza greatly. The two men must have discussed Leibniz’s view that this is the best of all possible worlds in contrast to Spinoza’s view that we have to take life as it comes and make the best of it. Both men, however, believed there was a supreme being (not necessarily the conventional concept of God) who wanted people to be charitable and fair to one another and to accomplish useful work. Criticizing the heretic Spinoza was the acceptable reaction to him in the political world that Leibniz inhabited, but still Leibniz appeared to admire the man, cautiously but apparently sincerely. Leibniz was devastated when he learned of Spinoza’s death only a few months later. At the end of December 1676—almost 12 months after making his formal commitment to the Duke— Leibniz arrived in Hannover. An apartment had been arranged for him beside the ducal library in the Leineschloss, the Duke’s palace on the banks of the Leine River in the heart of the city. The language at Johann Friedrich’s court was French, in which Leibniz was now fully conversant. Although his status as a mere librarian and councilor was less prestigious than he thought he deserved, the

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location of the library and his personal apartment in the palace (very close to the Duke’s residence and dining room) provided him with daily access to the Duke. With that frequent contact, Leibniz and the Duke were able to converse regularly about theology and philosophy as well as intellectual and scientific developments throughout Europe. The two men also shared a desire to unify the Christian church in Germany. The well-educated Duke was delighted to work with his learned scholar, and the two men formed a warm and lively friendship as they worked together. Leibniz was not surprised to find the collection of books and documents in the ducal library sorely lacking in comparison to the libraries he had known in Paris and London. With his universal interests, he needed works covering philosophy, science, law, medicine, and all the rest. Leibniz immediately informed the Duke that he would order for the library more than 3,000 additional volumes, which only Leibniz was qualified to choose. In addition, he requested the services of a secretary to help him complete the orders for those new books. He also informed the Duke that he had devised a new method of cataloguing the collection that would expedite his use of the library as he corresponded with scholars throughout Europe. “You do want Hannover to be a center of intellectual discourse throughout Europe, don’t you, Your Highness?” Leibniz asked his patron. “That is the reason I brought you to Hannover,” the Duke agreed. “Please make the changes you consider necessary. I am sure you know what is best.”

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Within a month of his arrival, Leibniz reminded the Duke that his current title was not sufficient. He really should be designated Privy Councilor, based on his distinguished legal and academic record. He argued, “That title will allow me to serve you far better because of the prestige it implies. If my title is merely councilor and librarian, scholars will assume that I am an inconsequential worker on your palace staff. They will not take me seriously. Please remember, Your Highness, that I have accomplished many important things in law, in diplomacy, in science, and in mathematics.” The Duke listened carefully and decided that Leibniz was correct. His accomplishments were far greater than his title indicated. In October 1677, he was finally made Privy Councilor and was then able to hire a full-time assistant. Others in the court resented his rapid rise through the ranks, but Leibniz ignored them. Small-minded bureaucrats were inconsequential to him. Finally, his real work in the duchy could begin. Leibniz was not surprised to find very few individuals in Hannover with whom he could carry on intelligent discussions, although he recognized that the Duke tried to remedy the problem by bringing in erudite people from time to time. Leibniz enjoyed conversations with several theologians who lived in the area or who visited regularly, including the Catholic Bishop Rojas y Spinola (1626–1695) from the Bishopric of Wiener-Neustadt, whom he quickly learned to respect. He also met some confirmed disciples of Descartes, providing a refreshing contrast to more conservative

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Christians—both Roman Catholic and Evangelical. And again, the Duke himself was a kindred spirit— Leibniz and his Duke could happily converse for hours on a wide variety of topics. Finally in the summer of 1678, a year and a half after his arrival in Hannover, Leibniz was able to escape the confines of Hannover briefly with a trip of about 160 kilometers to the nearby city of Hamburg, where he hoped to arrange for the purchase of a library collection following the death of a scholar there. Leibniz described that collection as small but select—the perfect complement to the Duke’s modest library. The new purchase included many volumes on natural sciences with potential applications to medicine and economics, fields that Leibniz was eager to study in more detail. He was able to acquire the whole collection of more than 3,000 volumes at a price significantly lower than the asking price. This was a real coup. In Hamburg, he was also able to meet with several important scholars and diplomats with whom he had corresponded for several years. Their discussions often dwelled on questions of Descartes’ philosophy and theology. Descartes’ writings on God were matters of great concern to Leibniz, who was critical of Descartes for not recognizing that this is the best world God could have made. Leibniz suggested that perhaps Descartes should have spent less time theorizing about the mechanical operation of the world and more time studying it and experimenting. Soon after his return to Hannover from Hamburg, Leibniz learned of the discovery of the element phosphorus by an alchemist named Hennig Brand (1630–1710)

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in Hamburg, who had discovered it while attempting to change silver into gold. With the Duke’s approval, Leibniz invited Brand to Hannover and the two men together prepared a supply of the element, collecting large amounts of urine from a nearby compound of soldiers, then evaporating and distilling it to get some purified phosphorous. Exposing the new material to the air, Leibniz and Brand presented spectacular light shows for visitors to the palace. Johann Friedrich was impressed. “Please tell me, Monsieur Leibniz,” he asked, “what practical use do you think we will be able to put this to?” “Well, Your Highness,” Leibniz said, “you should realize that this discovery is very new. We haven’t had time yet to determine its practical applications, but you have to admit that there must be some use for this spontaneous source of light. I’m sure it will turn out to be useful.” Phosphorous has since been used in the manufacture of fertilizers, toothpaste, and detergents, but Leibniz could not have predicted any such uses in 1678. At that time, Leibniz was delighted to meet Princess Elizabeth (1618–1680), Abbess of Herford, when she visited Johann Friedrich and his court in Hannover. The Princess Abbess and Leibniz were pleased to find common ground in philosophy and theology. Although in failing health, she was a woman of great learning who had been a close friend of Descartes and had read many of Nicolas Malebranche’s (1638–1715) works in philosophy and mathematics, leading to deep discussions with Leibniz concerning the works of both Descartes and Malebranche.

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“Your Highness,” Leibniz said, “I am greatly impressed by your knowledge of theology and philosophy. I have rarely found such knowledge in a member of your sex.” “Well, dear Sir,” she quickly replied, “you should realize that I have spent much of my life exploring those subjects. I corresponded with Descartes, who for seven years was both my teacher and my friend. I was a mere girl—only 24 years old when we first met—and he was a mature scholar who was already 47 years old. In addition to philosophy, he and I also communicated on the subject of mathematics, which I believe is a subject that interests you as well.” “That is true, Your Majesty. Might I be so presumptuous as to ask what sorts of mathematics you discussed?” Leibniz asked. “Of course,” Princess Elizabeth reassured him. “At one point in our correspondence—I believe at that time he was testing me to see if I had learned my lessons well—he asked me to find a fourth circle that touches the circumference of each of three other circles. I believe he was pleased with my solution to that difficult geometry problem, although he wanted to be sure I understood that his own solution was far more elegant than mine.” “I am curious,” Leibniz said, “what is your opinion of Descartes’ proofs of the existence of God?” “I have to confess that I have not been satisfied with them,” she admitted. “I have the impression that Malebranche wasn’t either.” “It is my belief,” Leibniz said, “that in fact there is no need to prove God exists—all that is necessary is

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to prove there is a possibility He exists. If no one can prove He cannot exist, then that is all that is needed. The Cartesians cannot prove the existence of God because they have no logical concept of the essential building blocks of substance—like my universal characteristic. Without that, the concept of a perfect being is simply impossible. “What a charming thought!” the princess exclaimed. “However, will you allow me to change the subject?” “But of course,” Leibniz agreed. “I believe you are of the Lutheran faith,” the Princess said, “and yet your Duke Johann Friedrich converted to the Roman Catholic faith many years ago. Have you not come under pressure to convert to Catholicism?” “No,” Leibniz replied. “When we first met, I told him that I was unwilling to convert to his Church, and the subject has never come up again.” “Very interesting,” Elizabeth said. “Many years ago my family tried to arrange my marriage with Wladislav V, King of Poland, but it was with the assumption that I would convert to his religion, Roman Catholicism. I flatly refused, and that was the end of those negotiations. At that point, I decided I preferred not to marry at all, if it required such fundamental accommodations. I have found my life within the Evangelical church satisfying. Instead of having my own children, I am instead the mother of the Protestant convent at Herford, where I preside over a thriving family of Christian souls. That is enough for me.”

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Leibniz wrote three formal proposals in 1678 to Duke Johann Friedrich, outlining improvements he recommended for the duchy. First, he wanted a detailed survey of the duchy to identify resources, commerce, craftsmen in various fields, etc. He planned to include in that study recommendations for systematic improvements in medicine, because a healthy populace is essential for a productive society. Second, he urged the founding of schools for the promotion of business and languages and to foster the education of young people in the realm. Because ignorant people are a burden on society, it was up to the government to prepare all people to contribute positively. His plan also envisioned an archive of all important documents in the duchy, organized in much the way he had already organized the ducal library so that items could be easily located when needed. He strongly suggested the appointment of a director of the ducal archives to oversee all such matters, with the strong implication that he himself would be the best person to fill the position. His third proposal addressed the issue of farming—the major activity in the duchy. He stressed the importance of developing better agricultural practices at the same time that the cultural structures were improved. Farmers and their families should be encouraged to enjoy music and dance as well as the occasional glass of hearty beer. He also strongly recommended

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the construction of a brewery for the duchy—a source of good beer was essential in Leibniz’s view. In a letter accompanying the three proposals, Leibniz described his recent invention of a greatly improved windmill that would expedite the draining of water from the silver mines in the Hartz Mountains, approximately 100 kilometers from Hannover. His windmill would feature a horizontal wheel, rather than the traditional vertical one, and he predicted that it would be far more efficient. The Hartz Mountains, which provided most of the silver found in Germany, were already an excellent source of revenue for the duchy, but Leibniz proposed improving them. The difficulty had always been that the mines filled up with water, which needed to be pumped away in order to extract the silver. The current system drew water from nearby streams to power the pumps that would suck the water out. The method was inefficient at best, and in the heat of the summer when the streams ran dry, it provided no power at all. Leibniz proposed using a combination of his windmills, which he said would function in strong winds as well as gentle breezes, in addition to the water power already available, so that the mines would always be dry and therefore productive. Early in 1679, he proposed that he personally would guarantee the dryness of the mines in exchange for a significant additional yearly salary. Leibniz was sure he had finally arrived at a solution to all his financial problems. Because he knew his scheme would work, he was confident that both he and the duchy would thrive.

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Model of Leibniz’s windmill on display at the Gottfried Wilhelm Leibniz Bibliothek.

Unfortunately, the mining project wasn’t as easy as Leibniz predicted. First, the miners protested. They quickly became impatient with the funny little man wearing the enormous wig who thought he could solve practical problems. What did he know about it? “Your Highness, we don’t like to criticize your judgment, but this man you have brought in to improve the workings of the mines simply doesn’t know what he is doing. It looks as if he has never seen a working mine before, and he certainly doesn’t understand the complexities of the job. You must admit, Your Highness, that the mines have functioned well for many years. Since the mines are not broken, there is no need to fix them.” Furthermore, during the interruptions in mining caused by the trials of Leibniz’s devices, the miners’ work—and thus their pay—was

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repeatedly interrupted. They soon came to despise him. There was another problem as well: the wind in the Hartz region was no more dependable than water for power, and it often happened that both wind and water failed at the same time. Sometimes the wind blew strongly, even for several days on end, but other times it was still, also for days at a time. Leibniz had assumed the wind would accommodate him, and he was astonished when it didn’t. At the court, there was a third problem. Complaints there concerned the cost of the project. Leibniz’s experiments were draining the duchy of vast sums of money. With limited resources, the Duke funded Leibniz’s projects by taking money from other departments, whose officials were incensed. To the Duke’s surprise, Leibniz also had a contradictory worry: the possibility that his plan might actually result in the production of too much silver! How would the duchy deal with a glut of raw materials? How would the duchy refine all those ores and transport them to markets? Although that was not Leibniz’s problem, it was still an issue he thought needed to be faced. Once the Duke agreed to Leibniz’s bold plans for the mines—and before they failed disastrously—Leibniz outlined an unrelated plan that was foremost in his mind. Mines and making money were certainly important, but they were only the vehicle that would allow him to undertake his major project—the one that he knew was truly important. “Your Highness,” Leibniz began one afternoon as they were talking, “now that I have solved the problem

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of the silver mines, I would like to return to one of my major goals. You will recall that I have spent some time in the past visiting the Académies in Paris and London. You would be amazed at the work that is done in those rarefied, intellectual environments.” “Oh, yes,” the Duke replied. “I know you have spent time in Paris in the last few years. Do you believe important work is being done in London as well?” “I’m sure it is,” Leibniz replied. “Scientists there have made some important breakthroughs. It is even possible that one Isaac Newton there has come up with a mathematical calculus that is similar to the powerful one I have devised.” “Newton?” the Duke asked. “I think I’ve heard of him.” “Yes, I believe Newton is important,” Leibniz admitted, “although I have not been able to meet with him personally. I think he’s rather odd, but he must also be very intelligent. Nevertheless, my point is that when you collect the finest minds and give them the freedom to develop their knowledge in whatever directions they think best, they can solve problems that have been seen as intractable for many years. With all the riches you will gain from your mines, Sire, you can found such a Societät [Society] of Sciences to rival the Académies in Paris and Rome.” “Yes, Leibniz, I agree that it is a grand idea,” Johann Friedrich said, “but we should wait to make firm plans for the Societät until we see results from your plans for the silver mines. Remember, the entire system remains untried. We have no solid proof that it will work.”

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“You are correct in that, Your Highness,” Leibniz admitted, “but you’ll see. Those mines will produce untold wealth for you. I plan to visit the mines next week, and after that I will be able to give you more specifics.” On his way home from his encouraging visit to the mines at the end of December 1679, Leibniz traveled to Herford to meet once again with the Princess Abbess Elizabeth—the formidable abbess whom he had first met in Hannover. Leibniz had heard that she was gravely ill, and he was eager to continue their earlier discussions of philosophy. “My dear Leibniz,” she said as he sat beside her bed, “you should realize that Descartes and I talked much about the human body and suffering and how we all must face the suffering that is a necessary part of human life. I am at peace with it. You mention the philosopher Malebranche. He is a fine man and a learned scholar. Our only argument concerns my Faith. He firmly believes that my evangelical faith will condemn my soul to hell, but I have told him that I have no worries on that count. I am sure that the Roman Catholic Church is not the only way to arrive in heaven. I approve of much of Malebranche’s writing, and that is enough. We don’t have to agree on everything.” Leibniz couldn’t help noticing that her arms were painfully thin and her stomach was grotesquely swollen. She also appeared to be in terrible pain. “Oh, Your Majesty,” Leibniz protested, “surely you are not contemplating departing this life!” “My dear Sir,” she corrected him. “The time comes for each of us to do that. I am 60 years old. I have led a

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good and long life. I don’t know if this is my time, but if it is, then so be it.” When Elizabeth closed her eyes to rest from the exertion of talking, Leibniz turned to her younger sister, Sophie von der Pfalz (1630–1714). She was the wife of the younger brother of his employer, Johann Friedrich. Elizabeth and Sophie’s grandfather had been King James I of England—Sophie’s children would later benefit from this heredity directly. It was through her that the House of Hannover would ascend the British throne. Although Leibniz didn’t yet know it, Sophie would become an important person both in the duchy of Hannover and in his own life. Suddenly there was a loud knock at the door as a servant burst in: “Excuse me, Your Highness,” he interrupted, “a messenger has just arrived from Hannover. I regret to tell you he reports that Duke Johann Friedrich has died at Augsburg.” “What?” Leibniz said in horror. “He was on his way to Italy! He seemed to be in perfect health when he left.” The women and Leibniz were stunned. “Bring in the messenger so that I may speak with him myself,” Sophie demanded. When the messenger appeared, Sophie said, “Please tell us all you know about the Duke’s death.” “Certainly, Your Majesty,” the messenger began. “Apparently the Duke took sick in Augsburg. I am told they called in the best physician in the area. He did all he could, but despite his best efforts, within a few days the Duke died. That is all I know. I am so sorry, Your Highness.”

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“I wonder what this means for me,” Leibniz mused. He knew better than to discuss his situation with the women. Their rank protected them from such insecurity, but he knew he had reason to worry. It was possible—even likely—that the new Duke, whoever that would be, would not share Johann Friedrich’s admiration for him. Perhaps he would not even respect the fruits of intellectual and philosophical study in general. Johann Friedrich had been his friend, and he had been pleased to serve him. Leibniz knew that there couldn’t be many noblemen of his quality. It was entirely possible that he would lose his position entirely. “Excuse me, kind ladies,” Leibniz said. “I must immediately depart for the court in Hannover. Your Highness, Princess Elizabeth, I do hope that you will recover your health very soon. Madame d’Osnabrück (Sophie’s title based on her husband’s position in the bishopric of Osnabrück), I must tell you once again how much I value your friendship as well. I hope we will be able to continue our conversations soon. I thank you both for making my visit here so congenial. Farewell, Your Highnesses, until next time.” Leibniz left immediately for Hannover. Someone who is not present cannot look out for his own interests. Certainly his earlier worries about living in the provincial city of Hannover had not been realized— he had sometimes even found himself enjoying his life in the duchy. He had even considered inviting his colleague Tschirnhaus to join him there. However, without Johann Friedrich, the court at Hannover was a complete unknown. His noble and congenial

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benefactor had created an environment where Leibniz could thrive, but who would succeed him? Although the women were sad to lose a dear friend, the news meant that Leibniz had lost considerably more. He had lost his third noble patron: first Boineburg had died, then Schönborn, and now Johann Friedrich. Leibniz’s livelihood depended on the patronage of a nobleman.

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1680–1690

Councilor and Librarian to Duke Ernst August

The succession to the dukedom in Hannover was not

obvious, even to an astute insider. Leibniz knew that the recently deceased Duke Johann Friedrich and his wife had four daughters with no direct male heir. This meant that the duchy would not be ruled by Johann Friedrich’s own offspring. Leibniz soon learned it was necessary to look to the late Duke’s brothers, in this case his younger brother Ernst August (1629–1698, husband of his charming and intelligent new acquaintance Sophie), for a proper heir. Leibniz also learned that Ernst August, unlike his wife, had no interest in the arts or the sciences. Apparently he was also not a deeply religious man. The new Duke would probably have no interest in exploring questions of theology or philosophy with anyone, probably least of all with the lowly ducal librarian and privy councilor. It looked as if the choice of the new Duke might disrupt Leibniz’s comfortable life. 131 © 2012 by Taylor & Francis Group, LLC

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Ernst August, Duke of Hannover (1629–1698).

However, he consoled himself with the thought that Ernst August’s charming wife, Sophie von der Pfalz (currently Madame d’Osnabrück), was clearly an admirable woman. She had fine noble blood, was apparently a devout Calvinist if not a Lutheran, and was both beautiful and intelligent. Leibniz had heard that she had produced six sons in addition to one daughter, Sophie Charlotte, who was now 11 years old. Up until this time, the new Duke had been the Prince Bishop of Osnabrück, a bishopric about 130 kilometers west of Hannover. It had been an appropriate role for the younger son of a Duke. Leibniz quickly decided that it would be wise for him to travel to Osnabrück to make the acquaintance of Ernst August and to let the new Duke see firsthand how impressive his privy councilor/librarian was. Much to his delight, he learned that the new Duchess Sophie would also

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Electress Sophie of Hannover (1630–1714).

be in residence, although he was saddened to learn that her sister Elizabeth had died soon after he had left Herford. During his month-long visit in Osnabrück, he and Sophie came to know each other better. “Your Highness,” Leibniz began boldly one afternoon as they were talking in the garden, “I am curious about your name. I have noticed that in general the nobility have two names, like your daughter Sophie Charlotte or your husband the Duke Ernst August. But if I may be so bold, how is it that you apparently have only one name? Or perhaps you have a second name that I don’t know.” “Well, you see, I was the 12th of my mother’s 13 children,” Sophie explained. “I am told that my arrival caused no great joy to my parents! By the time I came along, my parents had simply run out of appropriate godmothers, and so they had to be content with just

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one name for me. I have heard that in fact it was drawn at random from the possible available names.” “But I’m sure your mother prized you,” Leibniz said. “Probably not!” Sophie said. “My mother had no interest in raising any of us. She was far more enchanted with her pet dogs than with any children she might have borne. I was brought up by a governess, whose strict instructions were that I master several languages, history, law, and a modest amount of mathematics. My mother had no desire to see any of us until we were grown.” “I see, Your Highness,” Leibniz said, unable to think of a good response. The new Duchess was smart, she was witty, and he believed that she was already powerful—and soon to become even more so. Despite his worries about losing his position in the court, Leibniz soon learned that Ernst August would probably allow him to remain in his current position. The new Duke might or might not like him, but they would probably have an adequate working relationship. Leibniz set about presenting his various programs to Ernst August’s Prime Minister, Count Franz Ernst von Platen. Leibniz was eager for the Duke to confirm his many projects: his appointment as ducal archivist, the enlargement of the ducal library, the establishment of such cultural institutions as an art museum, the reorganization of the monetary system, the establishment of a modern mint to produce the coins of the realm, and the construction of a medical college within the duchy so the citizens’ medical needs could be properly addressed.

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Eager to ensure his continued livelihood in Hannover, he also suggested that it might be helpful if he were to study the genealogy and write a short history of the House of Brunswick-Lüneburg, the larger realm in which Hannover was located. That would establish its crucial connection with the important Houses of Guelf and Este. This modest suggestion would eventually become a proverbial albatross that would hang around Leibniz’s neck for the rest of his life—the “little” history would grow into a monster so vast that Leibniz would never be able to finish it. Leibniz conveyed all his plans to Ernst August through von Platen in a series of memos even before he actually met either of them. At the time, Leibniz received no assurances on any of these issues, but his requests were on record. As Leibniz talked in Osnabrück with officials in the new regime, one disturbing fact emerged: the Duke had decided to expand the ducal living quarters, and that necessitated moving the library and Leibniz’s personal living space somewhere else. Leibniz was made to understand that the Duke had a large family, and the library took up a major part of the palace. Although the need was obvious, Leibniz struggled to see what those arrangements meant for him. Receiving no response to his worries, he simply learned that the move would begin soon.

The new Duchess Sophie said to Leibniz one afternoon when they were both sitting in the garden in

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Osnabrück, “Please tell me, Monsieur, how you find living in Hannover.” “Well, Your Highness,” he said, “in all honesty, at first I fretted that I could never survive in such a provincial place, in contrast to Paris, where I had lived for the past four years. However, I have to admit that Hannover has its charms and is not as provincial as I had feared. The collections in the ducal library have been developing nicely, so that I am able to find the resources I need most of the time, and we have visitors from many interesting places, providing fresh perspectives from outside the duchy. I understand that the Duke plans to move the library, but I assume he will find a suitable location for it.” “I’m so sorry about that move,” she said. “I’m sure it will be inconvenient for you. Let us hope he will find a location that is satisfactory to everyone.” Then on a happier note, she continued, “I understand that you are a scholar of philosophy. Do you know much about Pascal?” “You mean Blaise Pascal, don’t you? Actually, I spent quite a bit of time in Paris studying some of his unpublished works,” he said. “Oh, that is wonderful!” she exclaimed. “He strikes me as a fascinating man, and I would like to know more about him. Wasn’t he a mathematician as well as a philosopher?” “That is true, Your Highness,” Leibniz said. “He did some brilliant work in mathematics, and he also invented the first moderately successful mechanical calculator.” “What a fine invention that must be!” she said. “Well, if you’ll pardon me for boasting a bit,” he said, “I have invented a better machine—that does all

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four arithmetic operations, not just addition and subtraction—which I would love to demonstrate to you once you have settled in Hannover.” Leibniz was also delighted to describe to Sophie his current researches in the field of probability, mentioning Pascal’s early work on the subject as well. He described to the Duchess how probability could allow a scientific prediction of the future. “You see, Your Highness, few things in life are actually guaranteed.” “Yes, that is true,” she said. “We all know that we will die—that is one of the few things that are sure. Even so, we have no way of knowing when our death will come.” “Yes, but when we consider uncertain things— those things that may or may not happen or the timing of things that will happen—there has to be a way to judge the relative likelihoods,” Leibniz explained. “I believe it must be a mathematical process. It’s a question that Pascal explored in detail in his correspondence with Fermat.” “Monsieur Leibniz,” Sophie said, “it seems to me that you work on so many different projects! How do you keep them all straight?” “That’s not a problem, Your Highness,” Leibniz said. “The real difficulty is finding the time to devote to each of them. All my life I have always had too many things to do at any one time. But, Your Highness, it must be that you too are always in the midst of several different things at once.” “I don’t deny that,” Sophie admitted, “and I think it will become even more difficult in Hannover. I understand I will have a staff of hundreds of people.

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I have learned that running a household is a major task. However, my real problem is finding the time for my study of philosophy. I hope that together you and I will be able to work on that.” “I would be honored to discuss philosophy with you whenever you have the time,” Leibniz said.

Back again in Hannover, on May 1, 1680, the day of Johann Friedrich’s funeral, Leibniz received a request from Count Ernst von Hessen-Rheinfels (1623–1693). He had written to Leibniz, in his role as the Duke’s librarian, to request the return of a copy of a book the Count had written and loaned to the late Duke Johann Friedrich several years earlier. Leibniz scanned the library’s holdings (which he could easily do using his new system of cataloguing) since the library had not yet been moved to its next location. “I regret to tell you, Your Highness,” Leibniz wrote back, “that I have not found your marvelous book in the ducal library. As a result, I must conclude that it is in the Duke’s private rooms—where it is unavailable to me. I must tell you, however, that I read your book several years ago when my dear friend the Baron of Boineburg loaned it to me. I was most favorably impressed. I congratulate you.” Reinfels sent his reply quickly: “My dear Leibniz, I was delighted to receive your letter. If the book is not found, it is no matter, and I am so glad you liked it when you read it. I understand that you are a philosopher of no small renown. I wonder what opinion you have on….”

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The two corresponded regularly until the Count’s death in 1693. At one point the Count tried seriously to convert Leibniz to the Catholic faith, arguing that his conversion would set an excellent example. Leibniz replied, “I am not prepared to leave the church in which I grew up. However, let me explain my campaign to reunify the Christian church. I believe there is no reason all the Christian faiths cannot come together as one. The Lutherans, the Calvinists, and the Catholics all believe in the same God and the same Jesus Christ. I think it should be a relatively easy matter for them to join together. Let me emphasize that I have nothing against your Roman church. However, as long as the churches remain separate, I plan to remain where I am.” Following a suggestion from Rheinfels, Leibniz wrote a satirical attack on the French King Louis XIV, who had been trampling Europe for several years. The Count assured Leibniz that he would hide the identity of the author if Leibniz could write something good. This pamphlet entitled Mars Christianissimus [Most Christian War-God] describes the King as the most powerful being in the world, with the possible exception of the devil. He explained that the reason the King had chosen to conquer Europe before moving on to the infidels in the East was because he followed the rules of the New Testament—he was killing off the Christians first! Contrary to Leibniz’s worries, Ernst August quickly recognized his librarian and privy councilor as a very capable man. Leibniz and the Duke discussed Leibniz’s plans for the pumps for the silver mines and

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Leibniz’s proposed invention of vehicles that might be constructed for carrying heavy weapons from place to place. Following up on Johann Friedrich’s agreement with Leibniz, the Duke approved a test of three of Leibniz’s windmills in the Hartz, and he committed himself to continuing the mining improvements there, assuming the test was successful. After the ducal family moved to Hannover, Leibniz had the opportunity to observe Ernst August and his family’s affairs—and perhaps more than he wanted to see. It didn’t take him long to learn of Ernst August’s relationship with his mistress—a situation painful to Leibniz. The Duchess, however, never let on that she knew of her husband’s infidelity and always acted as if her personal life were serene and happy. Leibniz was far too discreet to call her bluff. Instead, Sophie and Leibniz, sometimes accompanied by young Sophie Charlotte, whom her mother lovingly called Giguelotte [my little Fig], spent countless hours discussing philosophy and other subjects. Leibniz was not immune to the charms of attractive and intelligent women. What he couldn’t understand was why the Duke wasn’t similarly entranced by his wife. A few months after the ducal family’s move to Hannover, Leibniz watched in horror as Ernst August arranged for the marriage of Georg Ludwig, his oldest son, to the young man’s first cousin, the illegitimate daughter of one of Ernst August’s older brothers. Marriage to first cousins has always been controversial. However, the proposed marriage had significant advantages—it would allow the Duke’s oldest son to bring together in his jurisdiction the duchy of Han-

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nover and the larger duchy of nearby Celle, resulting in a much larger and thus more powerful entity. The Duchess Sophie, the boy’s mother, deplored the decision, but it was beyond her control—she was not the one who had inherited the title to Hannover. Ernst August later arranged for his daughter Sophie Charlotte to be married in 1684 to Frederick III of Prussia, who in 1701 would become King Frederick I of Prussia. At that time, she became Queen Sophie Charlotte. Their grandson would ultimately be known as Frederick the Great of Prussia. The Duchess Sophie was delighted with her daughter’s marriage. Sophie Charlotte’s husband Frederick might be old and lame from an early accident, but his blood was noble, he

King Frederick I of Prussia (1657–1713).

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was an intelligent man, and her daughter would be well provided for. Ernst August made other decisions as well. He ended the traditional Hanoverian practice of dividing his estate among all his sons on his death, which had given each a duchy of his own—a tradition that had resulted in smaller and increasingly weak duchies as time went on. Instead, he established primogeniture for the duchy, giving his oldest son Georg Ludwig the inheritance of the entire duchy and leaving his five younger sons to fend for themselves. The younger sons ultimately had no choice but to serve in the imperial army—a dangerous occupation in those unsettled times of war between the Holy Roman Empire and the Ottoman Empire. Although Ernst August was pleased when the emperor confirmed the change to primogeniture (a requirement for the desired status of electorate within the Holy Roman Empire), his younger sons were not. After Leibniz left Paris, he had kept up an active correspondence with his many contacts in that city. He wanted to know about current scientific discoveries as well as all noteworthy social events. He also did not give up on his hope of returning. In letters to Huygens and others, he explored whatever possibilities might still be for him there. Although his friends tried to help, nothing came of their efforts. Leibniz would have to stay in Hannover. After making careful arrangements for his work in the Hartz mines, Leibniz made a trip to Leipzig in July 1680—his first visit to his homeland since he had left it 14 years earlier. Leibniz paid respects at his

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family’s graves as promised and settled his inheritance with his half-brother, Johann Friedrich. As soon as he returned to Hannover, he received word that Christian Freiesleben, his financial advisor, had died. Freiesleben’s widow wanted his affairs to be settled immediately, including Leibniz’s debts to her husband, which she reckoned at 476 Thaler—a considerable sum. She also requested that he remove the Leibniz family library, which had been taking up too much room in her house for many years. That collection included Leibniz’s father’s library, many volumes left from his maternal grandfather and other relatives, as well as Leibniz’s own books and papers. The origins of many of the volumes from his father’s library were unclear, and finally Leibniz had to agree to give his half-sister’s husband Heinrich Freiesleben power of attorney to arrange an auction of the whole collection. Unfortunately, the proceeds from that auction were not adequate to pay off Leibniz’s debt to the widow, thus increasing Leibniz’s need to make a profit from the mines in the Hartz Mountains. In 1681 and 1682, work on his windmills proceeded sporadically. The weather was often a problem, and petty quarrels among the workers and between the workers and Leibniz slowed things down too. Although the project was costing Leibniz a great deal of money, nevertheless he remained optimistic about its completion. He was pleased that recent tests of the windmill had been satisfactory, but there were still problems with its operation. Despite his best efforts, by the middle of 1683 the costs of the project had reached alarming levels. Although Leibniz had originally predicted a cost of

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300 Thaler, the cost by that time had come to 2,270 Thaler, and still the project was not complete. In 1684 Leibniz agreed to take full responsibility for all costs from that time on. In the end, it became clear there was a fundamental problem: the wind blowing in the region was neither consistent nor sufficiently strong. Finally in 1685, the Duke’s minister informed Leibniz that his plans for improving the draining of the mines must be abandoned. Although this appeared to be the end of the project, Leibniz never conceded that his plans were not good enough. Instead, he blamed the combination of weather and cantankerous workmen. He firmly believed that his plan was fundamentally good.

In 1681, Otto Mencke, a professor at the university in Leipzig, visited Hannover to discuss many things with Leibniz. During that visit he informed Leibniz that he was preparing to launch his journal Acta Eruditorum [Journal of the Learned] in Leipzig if Leibniz would be willing to lend his name to the project. This journal would be the counterpart of the French Journal des Sçavants and the English Philosophical Transactions of the Royal Society, both of which came out regularly and were valued throughout intellectual Europe. Leibniz was enthusiastic. It was a project he had hoped to carry out on his own, but he had simply not had time to do it. He considered Mencke up to the task and was glad to support him in the effort.

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“Oh, Professor Mencke,” Leibniz gushed, “this is a most worthy endeavor! If I might be so bold, I would be honored if you would allow me to write an essay for your first volume.” “That, dear Sir, is one of the reasons I approached you,” Mencke agreed. “An erudite journal is nothing without the work of respected scholars, and you, Sir, are at the top of my list. Please submit your first manuscript to me as soon as possible. I hope there will be many more to follow.” “Of that you may be certain,” Leibniz said. “I am ready to write essays on a variety of scientific and philosophical topics. It will be no problem for me to prepare one each month for some time to come.” “Thank you, kind Sir!” Mencke said. “With your support, I shall proceed with my plans.” Leibniz’s initial essay, appearing in the first issue of the journal in 1682, was on finding the area of a circle, and it was followed by papers on optics, financial questions, philosophy, and many other topics. Two years later, his article in October 1684 introduced his differential calculus—“Nova methodus….” A month later he contributed an article on philosophy. He was happy to be seen as a regular contributor, although his scholarly production slowed somewhat as he finished up his work in the mines. In July 1686, he published the very important article that outlined his integral calculus. Leibniz was demonstrating to the world that he was a true polymath—a knowledgeable scholar who excelled in many different fields. His two articles in the Acta that presented the differential and integral calculus provided the first

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indication to the world that there was new mathematics to be learned. Arithmetic, algebra, and geometry were fine as far as they went, but they were only the beginning. Although his articles in the Acta Eruditorum were extremely difficult to read, Jacob and Johann Bernoulli managed to figure them out. Together the three men began a serious revolution in mathematics on the European continent. All of a sudden, a whole range of problems that had been impossible for most scholars were within reach of any diligent and intelligent person. With the Bernoullis’ help, Leibniz provided algorithms and notation that would allow ordinary people to solve problems too difficult for anyone but an Archimedes of earlier generations. This was one of the crowning achievements of the scientific revolution. It is worth noting that although Leibniz and probably also the Bernoullis were aware that Newton had been working on a similar mathematical engine, none of their writings at the time mentioned the Englishman or his work. Significantly, however, neither Newton nor his compatriots had published any of Newton’s work on the subject either. Publication of Newton’s method of fluxions and fluents could have caused the same radical shift in the development of mathematics, but Leibniz is the one who provided the scholars of Europe with the tools needed to carry out the effects of the scientific revolution. In another article in 1682 Leibniz identified a group of irrational numbers that he called transcendental. Such numbers are not algebraic—they cannot be the roots of algebraic functions. There are infinitely

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many transcendental numbers, although most people know little about them. The numbers π (3.14159…) and e (2.71828…) are the most famous. Ten years after his original article, in 1692 Leibniz was pleased to point out that Descartes’ mathematics didn’t allow the solution of equations that resulted in transcendental solutions, although Leibniz’s calculus did. He often took pleasure in pointing out such inadequacies of Cartesian mathematics. In 1684 Leibniz wrote an essay for the Acta in which he defined what we now call determinants. These can show whether a system of equations has a unique solution. A system of equations with the same number of equations as unknowns forms what we now call a square matrix, and Leibniz realized that the determinant of such a matrix is critical. Although he did not use the words determinant or matrix, he was the first mathematician in Europe to use the concept. In fact, Chinese scholars in the third century bc had explored the essence of the determinant, while two Japanese mathematicians also recognized it at about the same time as Leibniz did. Although Leibniz had a romantic fascination with Chinese culture, in fact he knew nothing of Asian mathematics and was probably unaware of this coincidence. While Leibniz’s time was being monopolized by finishing up his projects at the Hartz mines, he published only one article—on the physics of an inclined plane. However, to compensate for the lack of scientific articles, he wrote more than 13 reviews of books on a wide variety of scientific subjects, including a review of John Wallis’ book Algebra and a review of a work

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by John Craig on quadratures, using the differential calculus that Leibniz had published in an earlier issue of the Acta. Leibniz’s only criticism was over the fact that Craig had credited Tschirnhaus with the invention of the calculus rather than Leibniz—Leibniz wanted to be sure he got the credit that was due him.

Although Leibniz considered himself principally a philosopher and mathematician, he was always fascinated with inventions. In his spare time, he read about an invention in 1679 of a device by Denis Papin, a Frenchman who had traveled to England and worked closely with Robert Boyle. Papin called his device a bone digestor—it is what we would call today a pressure cooker. Papin’s device had a valve at the top that allowed it to build up pressure in the pot and thus raise the boiling temperature well above 212°F. Papin’s plan for his digestor was that it would be able to cook bones, making them brittle enough to be ground up and become edible for humans. Reacting to that invention, Leibniz wrote a satirical essay in which dogs complained loudly that bones were their rightful property. Papin should not steal the bones that everyone knows belong to the dogs!

Duke Ernst August had little interest in Leibniz’s private little realm, the ducal library. Leibniz already knew it would be removed somewhere out of the Duke’s way. However, since the Duke hadn’t been able

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to think of anywhere else to put the books, he simply ordered them to be put into storage in a distant corner of the palace. The ducal library in Hannover remained closed from 1680 to 1684. Leibniz was frustrated. Johann Friedrich and he had spent so much energy and money arranging for the additions to the library. Some of the new works had arrived but had not yet been catalogued; on other orders, only the first shipments had arrived in anticipation of the rest of the volumes sometime later; and others had not yet come at all. Nevertheless, all were essential to the center of learning that Leibniz had envisioned. When Ernst August cut the funds for the library, Leibniz was forced to pay some of the bills out of his own shallow pockets in order to avoid embarrassment among the literati of his world. Ernst August was deaf to Leibniz’s protests. After all, they were only books! At that time, in desperation Leibniz explored the possibility of taking the position of imperial librarian in Vienna—a post that had recently become available due to the death of the last librarian. However, Leibniz was careful. He had the status of privy councilor in addition to librarian in Hannover, and he was loath to lose that. He saw the position in Vienna of mere librarian—even librarian to the Emperor himself—as too humble for him since he was such an accomplished scholar. In the end, Leibniz’s candidacy came to nothing, probably in part because he had made it clear he had no interest in converting to the Roman church, which was a likely if unspoken requirement for the job. His religion hadn’t interfered with his work for Roman Catholic barons

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and dukes, but in the seat of the Holy Roman Empire it could have been another matter. Without a library, Leibniz’s duties as librarian were obviously curtailed, but the Duke found other projects to occupy Leibniz’s time. “Leibniz,” Ernst August said one day, “I seem to remember you proposed writing a little history of the duchy.” “Yes, Your Highness,” Leibniz agreed. “And you think there is a possibility of establishing Hannover and the Guelf House as the rightful heirs to the duchy of Sachsen-Lauenburg? As you know, Duke Julius Franz of Sachsen-Lauenburg has no male heirs” the Duke said. “Yes, Your Highness,” Leibniz said. As a legal scholar, he could reliably answer that question. “If I can find the documentation, I think that could be done.” Ernst August was eager to annex Sachsen-Lauenburg to his own duchy, even though the new addition would be significantly smaller than his own territory. One of its virtues was that it had the right to tax traffic on the Elbe river—a highly lucrative operation. Any Duke welcomes the possibility of more wealth, and Ernst August was as acquisitive as any. He was also ambitious. “You see, Leibniz,” the Duke continued, “what I want to do eventually is to elevate my duchy to the ninth electorate of the Holy Roman Empire.” “Yes, Your Highness,” Leibniz agreed, “that is an excellent plan. I believe the religious imbalance within the Holy Roman Empire is a real problem. Of the eight electorates, only three are Protestant.”

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“That’s right,” the Duke said, “and you know that the Protestant Elector of the Palatinate has no male heir so his seat might be taken by a Roman Catholic as well.” “Yes, Your Highness,” Leibniz said, “and that would make the imbalance even worse. If the Brunswick-Lüneburg duchy could be made an electorate, it would be a step in the right direction. Furthermore, I think you can expect support from Saxony, and surely relations with Brandenburg will improve now that your lovely daughter Sophie Charlotte has married the Elector of Brandenburg.” After a slight pause, Leibniz confidently added, “All that is necessary now—beyond securing the cooperation of the Dukes who rule Wolfenbüttel—is to establish that you are descended not only from Charlemagne but also from the House of Este.” “Yes indeed!” the Duke agreed. “I would like you to provide that evidence.” This new Protestant electorate would resist the powers of the Catholic French King better than the Catholic electorates. With this move, Hannover would become a genuine power in Europe and elevate the Duke to a role as a formidable prince. Following this discussion, Leibniz began serious research into the genealogy of the Guelf dynasty and its connection to the house of Este. Since Leibniz found the current genealogic study both flimsy and inconclusive, he was eager to undertake a thorough study. The Duke was enthusiastic. He decided to make this project a major priority and announced in 1685 that he would broaden Leibniz’s title from librarian and privy counselor to the additional and more prestigious

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title of court historian. As evidence of his support of the project, he proposed a pension for life to support Leibniz as he carried out this all-important research, as well as an extension of his title to privy councilor for life. Since it would obviously involve significant travel, the Duke also committed himself to reimbursing Leibniz for those expenses too. With a dependable salary, Leibniz was once again allowed to hire a secretary to help him in his research. Leibniz began the project with gusto and spent the first two years poring through dusty documents in the archives and libraries of Lower Saxony. In between library and archive visits, however, he still spent considerable time in the Hartz Mountains, ostensibly finishing up arrangements related to his failed mining project. Even as Leibniz pursued his genealogical work, he found many other distractions. Like a child who first discovers the joys of a large dictionary and is unable to resist other entries, Leibniz just happened to find unrelated, fascinating material wherever he looked. In the Hartz Mountains, for example, he had stumbled on some fossils and minerals a few years earlier. When he returned, he wanted to see what else he could find. Although geology and paleontology were new to him, they were irresistible. Because the Duke had no interest in closely supervising his work, Leibniz followed whatever paths his curiosity led him on. Beginning in the 1680s, Leibniz began a serious study of the German language as it had evolved in the Hartz Mountains. In one essay, he heralded language as the vehicle for thinking, indicating that as a nation matured its language should also mature. Primitive

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man might get by with grunts and gestures, but more advanced societies use increasingly sophisticated forms of communication. Leibniz still believed the German language needed to include increasingly abstract concepts to match its already impressive practical language. Leibniz also suggested that it was possible to explore the movements of a people by analyzing the languages spoken in various parts of the realm. With this argument, he justified his linguistic studies as a critical part of the history of the Guelph House. Although he did all his writing in either French or Latin, this essay did serve to increase the usage of the German language in scholarship. By the middle of May 1687, Leibniz happily concluded that he would have to travel beyond the limits of Lower Saxony for his genealogical research. He decided that sources in Augsburg, Munich, and elsewhere in the southern German territories would be critical. He finally departed on this trip at the end of October 1687, taking with him two helpers: Friedrich Heyn, the superintendent of mining from the Hartz project, to serve as his secretary, and a servant to look after his personal needs. Experience had shown him that travel provided the opportunity to meet with scholars wherever he went. In many ways, Leibniz saw himself as a free agent, doing research but pursuing his many other interests as well. This was the life he had dreamed of. Although the original plan was for the trip to last only a few months, in fact it stretched to almost three years. Leibniz was pleased to spend so many months away from his “prison” of Hannover.

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His travel took him first through Hildesheim, a town 40 kilometers southeast of Hannover, where he was able to see a collection of curious and amazing natural phenomena. A collection of fossils similar to what he had found in the Hartz Mountains provided Leibniz with a thrill. With such knowledge, he was convinced he could continue his discovery into the origins of the earth. The Duke might not approve of such a digression, but a man—or at the very least Leibniz—could not limit himself to only one thing at a time. From Hildesheim, he traveled to Rheinfels (near Marburg) where he stayed for two weeks with his correspondent Count Ernst. The two men talked about the need to reach reconciliation between the Evangelical and the Roman churches, together deploring Louis XIV’s repeal of the Edict of Nantes that had resulted in the persecution and murder of so many Protestants in France in the bloody Massacres at St. Bartholomew and other places. “Your Serene Highness,” Leibniz said, “I recognize that tolerance alone won’t solve problems, but it is certainly a step in the right direction.” “I can’t disagree with that,” the Count replied. “And don’t you think,” Leibniz continued, “that the differences between the Protestant and the Roman Catholic religions are minor? The evangelical practice of allowing our priests to marry and our disagreement with several other minor principles of Roman Catholicism shouldn’t make us heretics in the eye of the Pope. We all believe in one God and one Jesus Christ, don’t we? Our differences are not fundamental.”

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In the midst of these discussions, the Count and one of the councilors to the Elector at Cologne decided that this brilliant man really ought to be nominated for the position of chancellor to the Catholic Bishopric at Hildesheim. Surely Leibniz’s many talents were being wasted in the service of the Duke of Hannover! “Well, Monsieur Leibniz,” the Count asked, “don’t you think this is a wonderful idea? You have the skills necessary, and think how much you would be able to accomplish toward the reunion of our faiths!” “That is very kind of Your Highness and esteemed Councilor and noble friend,” Leibniz replied, “but I would suppose that to serve in that position it would be necessary for me to convert to the Roman Catholic faith. Don’t you agree?” “Well, yes, of course that is true,” the Count replied, “but would that be such a problem? You have already indicated that you see the differences in the faiths as minor.” “But I think it would be a problem,” Leibniz protested. “It is true that I yearn for the rejoining of the churches, but that has not yet happened, and I’m not so sure it’s going to happen anytime soon. In the meantime, I intend to remain a Lutheran, true to my Saxon upbringing. As a result, I fear I am not a good candidate for the position you propose.” That was the end of the discussion. From Rheinfels, Leibniz went on to Frankfurt am Main, the city he had visited many years earlier after leaving his native Saxony and Nürnberg. By Christmas, he was in Würzburg, delving into its libraries and archives. A week later, he was in Nürnberg, where

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he was glad to renew his friendships there. He also stopped in the town of Sulzbach, where he discussed the relations between the Jewish and Christian faiths. While there, he took the opportunity to examine fossils and minerals as well as to tour some mines. Yes, he had officially given up on the silver mines in the Hartz, but his fascination with mining continued. At this time, Leibniz began to consider the coins of the Holy Roman Empire. Although each duchy produced its own coins, the silver content of the Hannover coins was much higher than the silver content of the coins of the other duchies. Unfortunately, it was a relatively simple process for the other duchies to “mine” the Hannover coins for some extra silver to their own benefit. Leibniz suggested that the duchy publish a standard weight for coinage in order to fight that problem. He then continued his journey south, venturing into Bohemia, where he hoped to meet up with his colleague Johann Daniel Crafft (1624–1697), who worked for the Count in Choden castle. In spite of ferocious winter weather, he kept on and finally met up with Crafft in the town of Graupen at the end of January. The two scientists had much to catch up on, as each tried to find out what new and interesting discoveries the other had made or learned of. After visiting with several other interesting people along the way, he eventually reached Munich at the end of March 1688, five months after his departure from Hannover. There he sat down and wrote his first letter to the court at Hannover. He chose to send that letter not to the Duke or another official in

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the government, but instead to his friend the Duchess Sophie. She was relieved to learn that Leibniz had not died on the trip—as she had greatly feared. Leibniz, not eager to have the officials at court know how indirect his travels had been up to this time, had chosen Munich as a safe place for his correspondence, since it was on his original itinerary, unlike most of his other stops. In Munich and Augsburg, Leibniz had hoped to establish that Albert Azzo II (996–1097) was the ancestor who would tie the Hannover family to the Este family. After he found the manuscript Historia de Guelfis principibus [History of the House of Guelph] in the archive in Augsburg, he asked for permission in the ducal library in Munich to view additional documents that would allow him to firmly establish the connection with the Este family. “Herr Leibniz,” the librarian said to him crisply, “I’m sorry but those documents are closed to you.” “My dear Sir,” Leibniz said, “I am not asking to remove anything from your collection. I only want to look through them and copy critical information. I will handle them carefully with clean gloves, and I will return the documents to their box in the same order that I find them. Please, kind Sir, this is extremely important. His Majesty, Ernst August, the Duke of Hannover, has sent me on this mission, and I must accomplish it.” “I’m so sorry, Herr Leibniz,” the librarian responded. “I don’t have the authority to show you those particular documents. Would any of these other documents suffice?”

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“No, Sir. I know what I need,” Leibniz argued. “What I seek is not sensitive, current information. It is merely genealogy, but it is most important to His Highness, the Duke.” “As I said,” the librarian said, “I am not allowed to give you access to those documents. Goodbye, Sir.” Leibniz was stymied. He wrote to Ernst August, telling of his amazing success in Augsburg but reporting his total frustration in Munich. He concluded his letter to the Duke by informing him that Leibniz had no choice but to travel to Italy, a country he knew the Duke himself was fond of visiting. He would explore the archives of the Este family in the city of Modena, hoping to have better luck with the authorities there. He immediately wrote a letter to the archive in Modena and then resigned himself to wait for permission from Modena to research there. He had to wait almost a year until February 1689 for that permission to arrive. At the end of April 1688 Leibniz boarded a boat that would take him down the Isar River to Deggendorf on the mighty Danube and from there past Linz, bringing him finally to Vienna on May 8. Leibniz preferred to travel by river boat rather than by coach whenever possible, since in a boat he could read comfortably on the deck in his traveling chair (which he had bought before he left Hannover) or work quietly in his stateroom as he floated through the countryside. In a coach, he was cramped in a small vehicle that was likely to bounce about between ruts, making any activity—even reading—difficult. It’s true that travel by boat was slower, but Leibniz didn’t feel pressed by time. He was simply waiting for word from Modena,

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Leibniz’s travel chair.

and he could do that as well on a boat as in a hotel room. When he arrived in Vienna, he was enchanted by that marvelous city. It wasn’t Paris, but it certainly wasn’t Hannover either! Furthermore, it was the seat of the Holy Roman Empire. While he was waiting in Vienna, he was able to meet with many interesting scholars and even with the Holy Roman Emperor himself, Leopold I (1640–1705), in October 1688. “Your Royal Highness,” he said bowing deeply in deference to the emperor, “I have an essay here in which I explain my proposal for an encyclopedia of all knowledge. It is something to which I have devoted my whole life—that is more than 30 years of serious study. I hope you will peruse it at your leisure.” “Yes of course,” the Emperor replied. Hurrying on, Leibniz continued, “I also have plans for some industrial devices that will cut hours of work

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from our laborers’ long days, allowing them to produce more and better products for the residents of the Empire.” “I thank you Herr Leibniz for all your work,” the Emperor said. “I will read it with interest. Will that be all?” Since it was clear that the interview was at an end, Leibniz bowed and withdrew. He was satisfied that he had been heard. An audience with the Emperor was a rare privilege, and he knew his ideas would be important for the Emperor if he were ever to find the time to read them. Leibniz knew that was the most he could hope for. While he was in Vienna, Leibniz was also able to work seriously on his philosophy, exploring how changes in the properties of a substance affect the essence of the substance itself. As something decays, is it still fundamentally the same substance? When it is heated or frozen, is it still the same substance? He spent considerable time discussing such topics with his acquaintances there. Leibniz’s traveling chair continued to serve him well as he moved from place to place, settling in comfortably wherever he landed. He wrote several articles for the Acta Eruditorum on questions of physics, in reaction to Newton’s Principia, which he was able to study in Vienna for the first time. Although it provided no detailed information on Newton’s fluxions and fluents, it was an important work of science that Leibniz was eager to study. He had to admit that Newton was a genius. While he was on this journey, those left behind in Hannover still knew little of what Leibniz was up to. Several of the councilors, who had watched Leibniz in action, were suspicious.

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“Aha!” they whispered among themselves. “So Leibniz has used his research as an excuse to leave our duchy and play around with the scholars of Bavaria and Austria! That’s no surprise. He obviously doesn’t like Hannover very much, and his assignment is so vague he probably feels he can get away with doing whatever he pleases. What an opportunist! If I were the Duke, I’d fire him!” The Duchess was also uncomfortable with his long absence, but for different reasons. She had grown fond of her favorite companion in philosophy, and she was simply worried. She had heard very little news of him since he had left Hannover a year ago. “Where is he now?” she wondered. “What is he doing? Clearly my dear friend is not cut out to survive in the world at large. He is probably carrying a goodly sum of money with him. What if he is beset by robbers? He is an impractical intellectual who needs to be cared for. He is also clearly the most brilliant person I have ever met. I don’t want anything bad to happen to him. Oh, dear! The only people he has with him are Friedrich Heyn and his servant. I wonder if either Monsieur Heyn or the servant is any more practical than Monsieur Leibniz. Surely they are not experienced travelers—probably not as experienced even as Leibniz himself. Oh dear! I wish I could get more news.” The Duchess might write letters and beg him to return, but there was nothing she could actually do. While Leibniz valued her friendship greatly, he was on his own mission, of which his researches into the origins of the House of Hannover were a small part. He

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would return when he was ready, and then they would continue their discussions. Until then, Sophie would have to wait.

In Vienna, Leibniz asked for and received permission to visit the imperial library, which he knew contained an impressive collection. He admired several different German Bibles, one written on parchment for King Wenceslaus of Bohemia—“Ah, won’t the Duchess Sophie, daughter of the Queen of Bohemia, love to hear about this!” Leibniz murmured to himself—and a Lutheran Bible, printed in Latin but annotated in German. He also found two volumes of Chinese mathematics, with many figures printed on silk. Chinese mathematics was a subject Leibniz was always eager to explore. When he finally wrote to Sophie, he was pleased to report on his findings in the imperial library. He also inquired after her health and expressed good wishes for her daughter, the young Electress Sophie Charlotte in Brandenburg, who was expecting her first baby— would that the child might be the new Electoral Prince of Brandenburg! While he was in Vienna, Leibniz was able to act for the duchy of Hannover in several important ways. Christopher von Weselow, the Duke’s representative in Vienna, requested Leibniz’s help in filing a claim for the Guelf house to the duchy of East Frisia. It was a complicated legal question about which Leibniz’s knowledge of international law and the history of the

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area allowed him to write an effective document. Weselow was grateful. Another project for Leibniz was in response to a request from Duchess Sophie. She asked him if he might write a suitable document requesting that her second son Friedrich August be promoted from Colonel to General. He was fighting against the Turks in the army of the Holy Roman Empire. Leibniz was glad to help the Duchess in any way he could. He was dismayed to learn later that young Friedrich August was soon killed in that war. Leibniz was discouraged to receive word from Hannover that his library, as well as his own personal possessions, had been moved again, this time out of the Leineschloss entirely in order to provide room for the construction of an opera house. The writer assured Leibniz that the move had been carried out carefully in order to avoid damaging either the books or Leibniz’s belongings, but that was little consolation. Perhaps the duchy needed an opera house—as a supporter of the arts, he had no quibble with that—but his rooms and his workplace were holy to him. How could the authorities do this to him in his absence? Here he was, traveling on a mission for Ernst August, and his precious domain in Hannover had been violated! He realized that, far from home, there was nothing he could do about it, but still he resented it. In December 1688, Leibniz was ill and confined to bed for several weeks. He was miserable, with a headache and congestion that were only intensified by going out into the winter air. As a result, he simply remained in his rooms, working quietly on his many

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projects. Because his primary assignment was to wait for permission to visit the archive in Modena, that was not a problem. In mid January 1689, Leibniz finally received the long-delayed permission to visit the archives in Modena, Italy. He decided he must depart as soon as his health would allow. He chose to travel through Venice, reporting to Hannover that this was necessary in order to avoid mud season in the Alps. He reminded the court that if a carriage should be mired in mud, it would be unable to travel farther for weeks at a time. Surely the authorities in Hannover would not want that to happen! When Leibniz reached Venice, he was charmed with its lovely canals. He enjoyed reclining in a gondola as it glided through the city, he looked in wonder at the beautiful palaces he could see from the Grand Canal, and he talked eagerly with intellectuals as they sipped coffee at a café in San Marco square. Ah yes! He was far, far away from Hannover! Although he had no official business there, he couldn’t be blamed for savoring the sights for a few days…or weeks. He was tired after his long journey, so he needed to rest. He must preserve his health, after all. He also accomplished much writing, sitting comfortably in his traveling chair, and looking up occasionally to gaze at the beauties around him. From Venice, he arranged to travel by boat down the Adriatic coast to the town of Mesola. By then Leibniz was heartily tired of traveling by coach, and this seemed like a good alternative. However, soon after they left port, a storm erupted, forcing the crew to work particularly hard to control the boat.

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Suddenly, one of the sailors had an idea. “Antonio, I have a plan,” he whispered hoarsely to his friend in Italian. “Let’s throw this funny little foreign man and his servants overboard so the boat is not so heavy. What do you think?” “That’s a great idea, Luigi,” his friend said chuckling. “After he drowns, we can clean out his well-lined pockets for him. This storm is great news for us!” Leibniz, however, was not the fool they took him for. He immediately reached into his pocket, extracted some charming rosary beads he had bought for the Duchess in Venice on a whim, and immediately began to pray devoutly: “Credo in Deum Patrem omnipotentem, Creatorem caeli et terrae. Et in Jesum Christum, Filium eius unicum… [I believe in God the Father Almighty, Creator of heaven and earth, and in Jesus Christ, His only son…].” The sailors knew the Apostle’s creed, and they knew the only reason Leibniz was saying it in Latin with the rosary beads was that he must be a good Catholic. Because they couldn’t murder someone of the Faith, particularly while he was praying—a sure way to send themselves to hell—they returned to controlling the boat and soon delivered Leibniz to Mesola, from where he would continue his trip to Modena. Leibniz may or may not have believed in the power of the beads, but that time they served him well. After five weeks in Rome, he traveled on to Naples, where he couldn’t resist climbing Mt. Vesuvius one beautiful spring day, explaining in his letters to Hannover that he needed to visit Naples before the heat of the summer made that city too hot for his

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fragile Northern constitution. After another detour to visit Florence (who could deny himself a detour to that beautiful city?), he finally arrived in Modena, where he planned to spend five weeks doing his research, although quite naturally he pursued his other interests as well. Leibniz was not a one-dimensional man. In Modena, Leibniz dug into the archives that had been opened to him and was able to establish beyond a doubt that the Lüneburg House was related to the House of Este. He even had copies of the documents in his own handwriting to prove it. While he was there, he was delighted to make an important contact for the Hannover family. He learned that the heir to the Este Duchy was available for marriage to a suitable young woman of appropriate rank. The Duchess Sophie informed Liebniz that one of the younger daughters of the late Duke Johann Friedrich might be the perfect choice. To Leibniz’s delight, the marriage was soon arranged. Duke Ernst August was gratified at this singular accomplishment—his court historian was earning his salary this time. Any additional connections between the two houses would help him in his attempt to elevate his house to Elector status. The Duchess Sophie approved of the arrangement for her charming niece. Once again, she was pleased with her favorite intellectual, although she continued to be frustrated by his long absence. In Modena, Leibniz received another good letter from Duchess Sophie, providing news of her communications with the English court. “It appears, dear Leibniz, that the English are sorting out the succes-

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sion to their crown. Did you know that my Protestant sons are possibly in line for the throne because of my ancestry and the fact that we are not Roman Catholic? That means that my son Georg Ludwig could be crowned King of England. Isn’t that a bizarre thought? I have to admit that I am doing what I can to improve his chances. I ask you, Sir, what self-respecting mother would not do that much for her son?” The Duchess’ letters to Leibniz were always written in this charming, half-joking manner, although the subject this time was clearly serious. Leibniz, Heyn, and the servant finally left Italy in March 1690, having accomplished Leibniz’s official mission. He had done well for the Duke, and he was confident that the Duke would approve. Surely no one could criticize him for taking a little—or a lot—longer than initially planned. After all, when traveling he never knew what he was going to encounter, and he couldn’t be expected to focus only on a narrow assignment. Leibniz needed to see the world and meet with scholars everywhere. He had had a wonderful trip. On his way back from Italy, he received word from Ernst August that he should stop again in Vienna to do an errand for him. This was a detour that Leibniz was glad to make because he had left some important papers and luggage in that city, on the assumption that eventually he would make his way back there. Leibniz finally returned to Hannover in the middle of June 1690, two years and eight months after his departure. He had seen major parts of Europe, had met with many fascinating people, and had been celebrated as one of Europe’s most important scholars.

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He had even succeeded in locating the genealogy that would allow the Duke to petition the Emperor for his status as an Elector of the Holy Roman Empire. Leibniz felt justifiably smug, and was glad at least temporarily to be back in Hannover, where he could catch up on his sleep and his correspondence, and resume his delightful conversations with Duchess Sophie. She was pleased to have him home again. However, he didn’t revel in being home for long. Soon he wrote to Huygens deploring once again his life in Hannover. “Why,” he asked himself, “did I volunteer to write that history? I should have known that it was a monumental chore. Oh well, I’m sure the worst of it is behind me now. All that remains to be done now is to write it. That shouldn’t take too long.”

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1690–1698

Writing and Not Writing the History

Leibniz had been pleased to accept a formal document

signed by Duke Ernst August on August 10, 1685, specifying his position as not just librarian and privy councilor but also as court historian. It had committed him to the production of a history of the noble house of Celle-Hannover and Guelph. He had been officially relieved of most of his other duties so that he could devote himself exclusively to the research and writing of this important work. His research in Southern Germany, Vienna, and Italy had been a marvelous vehicle for collecting the documents necessary for writing the history and incidentally meeting many fascinating people who had nothing to do with the historical project. His challenge now was to write the history. The sheer volume of information he had collected was overwhelming. He had documents about the genealogy of the Guelphs and Hanoverians, which were certainly important—and admittedly they were the only documents Ernst August had expected him to accumulate. 169

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In addition to those, however, he also had much extraneous information he had picked up as he traveled; he thought that some of that information might also turn out to be important to the project in the end. In addition to all the material he had assembled in his travels, now that he was back in Hannover he also had to catch up on his correspondence with countless scholars of Europe. He wrote many letters a day, and a significant number of those letters and their replies provided even more material for him to digest and more distractions. He was also delighted to resume his wonderful conversations with the Duchess Sophie and her daughter Electress—and soon-to-be Queen— Sophie Charlotte on the occasions when she visited in Hannover. He was a busy man. “Oh, Monsieur Leibniz,” Duchess Sophie said to him the day after he returned to Hannover, “it is such a pleasure to have you once again at home.” “Your Highness,” Leibniz said bowing and kissing the air above her hand, “you may be sure that the pleasure is entirely mine. I have seen many things and talked with many people, but I am always happy to return to your salon where I know I can expect wonderful discussions of philosophy and theology.” “No, Monsieur Leibniz,” she laughed, “make no mistake. Don’t confuse me with my sister Elizabeth, the Princess Abbess. She was certainly a philosopher and scholar. Unfortunately, I am not. I can appreciate the brilliant things you say, but I am hardly your conversational equal. I thank you for the compliment, but I cannot accept it. Probably my daughter is more worthy in that respect than I am.”

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“You underestimate yourself, Your Highness,” Leibniz replied. “Although you are possibly not as well-read as your sister the Princess Abbess, you have many demands on your time. Supervising all that goes on in the duchy is an enormous job. I wonder how you have time to read anything. Nevertheless, the fact remains that there have been occasions when you have said things that have radically altered my outlook on philosophical topics. Your intellect is wonderful.” “If that happens in our conversation, dear Sir,” she said, “it is only because as we talked something novel came to your mind.” “Au contraire!” Leibniz said. “The other nobles of Europe pale by comparison to you. I salute you, Your Highness!” “You are too kind,” the Duchess replied. “But, please tell me, Monsieur Leibniz, are you familiar with Pellisson’s new book on the religions of the world?” “Pellisson?” Leibniz asked. “Oh, yes, I have heard that he has published such a book. I know he converted to Catholicism many years ago. There was much talk of his views at the Académie when I was in Paris.” “That doesn’t surprise me,” Sophie said. “His argument in the book is that we need to rejoin the Christian churches, but I believe he sees it as a little too simple: he says that all Protestants should simply convert to the Roman Church.” “Oh, there’s nothing like a convert!” Leibniz sighed. “From our side, we wish the Romans would convert to our church. That would be easy too! Do you have the book, Your Highness?”

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“Yes, I do. I’ll be glad to loan it to you if you like,” she said. “I would love to read it so that I can write to him about his ideas,” Leibniz said. “I’ll get it for you right now,” Sophie said. “If you decide to correspond with him, I believe the easiest way for you to contact him is through my sister Louise Hollandine, the Abbess of the convent at Maubuisson. She converted to the Catholic faith many years ago and is in close contact with Pellisson.” After that conversation, Leibniz and Pellisson corresponded for years, together exploring the many difficulties and advantages of a possible reunion of the churches. When Pellisson published part of their correspondence years later, Leibniz prevailed on him to include some of his later letters so the book would present his views more fairly. Beginning in 1690, the one regular refuge that Leibniz found from Hannover was the library at Wolfenbüttel, a dukedom 80 kilometers east of Hannover. Leibniz had developed a good relationship with Duke Rudolf August and Duke Anton Ulrich there, all three enjoying one another’s company greatly. As Leibniz spent more and more time there, their discussion eventually included a proposal that Leibniz should become the official librarian of the Wolfenbüttel library—a role that Leibniz convinced Ernst August to approve. He was named director of that library in 1691. Although Ernst August didn’t like his peers in Wolfenbüttel, it didn’t hurt to be a little generous with them on some things so that he could be more demanding on others. Leibniz soon arranged for an apartment

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in Wolfenbüttel so he would have a convenient place to stay when he was there. In January 1691, Leibniz finally had his first interview with Ernst August more than four months after his return. He had much to explain since his trip had taken so very long—at least two years longer than he had planned—and had cost a great deal more money than anyone had expected. “Your Highness,” Leibniz said, bowing as he entered the Duke’s office, “you should realize that many of the documents I worked on were barely legible—in fact they were so difficult to read that it sometimes took me half a day or more simply to find out if a certain document had anything in it that I needed. Often it turned out to be worthless, but I couldn’t judge that until I had studied it carefully. I have to tell you, Sire, that this was long and frustrating work in archive after dusty archive. It was important research, and truly there was no way to hurry it.” “Yes, Leibniz,” the Duke interrupted him, “but am I correct in thinking that you got the documentation you needed?” “Yes, Your Highness,” Leibniz admitted, “but you must understand that it wasn’t easy.” “So where is my history?” the Duke asked. “You have had several months now to complete it. Is your work at the library in Wolfenbüttel taking up too much of your time?” “Oh, no, Your Highness,” Leibniz quickly replied. “It allows me to study important books and documents there, but it really takes no extra time. You see, the secretary in the library handles all the routine tasks, so that

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I am free to do my research. The collections are far more complete than what is available in Hannover, and I need to consult those documents from time to time. It also helps that I have an apartment where I can stay when I am there. It took me only a little time to figure out a better cataloguing system that the secretary has helped me implement in the library, and now I am able to find what I need surprisingly quickly.” “I thought you had finished your research!” the Duke said impatiently. “Isn’t that what you’ve been doing for the last two and a half years?” “I have finished my research in Augsburg and Modena, Your Highness,” Leibniz said, “but I still have to fill the gaps in order to put it all together into a unified history. I believe you do not understand how complicated this process is.” “You proposed a little history for me,” the Duke corrected him. “All I want is a little history. I don’t want a big history.” “I have already made two separate outlines of how I will approach the writing,” Leibniz explained, “but I am not yet satisfied. However, Your Highness, I expect to complete the history within two years.” “But Leibniz,” the Duke said in frustration, “how difficult can this be?” “You don’t understand,” Leibniz said. “It’s not that easy. A history has to be a unified work, and it has to be carefully constructed. I am working on it, Your Highness, but I still have much to do. I have found that I must go back to an earlier era than I had originally planned—even before the time of Charlemagne. Otherwise the history would not be complete. Remember,

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we are trying to establish the connection between the two houses. I can’t simply start with the moment that the families met. I must establish that connection so well that no one can ever contest it. There is far more to it than you know, Your Highness.” “I would like that history soon,” the Duke said grimly. “Surely that is not asking too much. Good day, Monsieur Leibniz.” Leibniz bowed as he took his leave. The Duke may have been justified in expecting results from Leibniz’s travel. The scholar honestly intended to write the history; however, it seemed unreasonable for the Duke to expect that Leibniz should complete the history to the exclusion of his more important scholarly work. Leibniz saw himself as a scholar—a scholar who was supported by a nobleman who respected his many gifts. He was happy to do specific jobs for the Duke, but he also felt compelled to do his own research and writing. Leibniz was reminded of the limitations of Ernst August and began once again to look for a more appropriate patron, but once more without success. He had had enlightened patrons in the past, but Ernst August fell far short of the mark. In spite of many frustrations, Leibniz’s research soon produced one important and tangible result. On March 23, 1692, Emperor Leopold of the Holy Roman Empire formally created the ninth electorate of the Holy Roman Empire, consisting of the Protestant territories of Hannover and Celle. The new Elector Ernst August commended Leibniz for this accomplishment—and he had to admit it was due in great part to Leibniz’s efforts.

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A formal ceremony in Vienna inaugurated Ernst August through his proxy, Otto Grote, as the ninth Elector. That made Ernst August a powerful man indeed. At the ceremony, Grote read a speech that Leibniz had written on behalf of the new Elector Ernst August. The elevation of the duchies into an electorate was by no means uncontroversial—ugly debate continued until 1710, including angry protests from Leibniz’s other employers, the Dukes in Wolfenbüttel. Nevertheless, the job was done and Leibniz was gratified. He saw his additional work on the unfinished history of the House of Guelf/Hannover as gilding the lily—clearly not worth the sacrifice of his other important work. He still planned to finish the history…eventually. Although Louis XIV of France had nothing but contempt for the inner workings of the loosely connected territories making up the Holy Roman Empire, Leibniz’s accomplishment was significant enough to attract notice within the Sun King’s court. In April 1692, Jacob Des Viviers wrote to Leibniz with a proposal: would the most erudite scholar Gottfried Wilhelm Leibniz consider accepting the position of librarian within the court of Louis XIV? Leibniz considered the offer seriously. After all, such a move would allow him to live comfortably in Paris, the most beautiful city in Europe. He had seen Vienna and Rome with all their “splendors,” and to him they were less impressive than his beloved Paris. The position would put him near the center of Europe’s greatest power, and might well allow him to make progress on his broad goal of uniting the Christian churches. It also would put him where he could

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regularly communicate with the scholars in Louis XIV’s court and the Académie in Paris. In many ways it looked like the answer to Leibniz’s fondest dreams. And yet, Leibniz turned it down. In his reply to Viviers, he explained his reasons. One part of his decision may have been the unspoken requirement that he would convert to the Roman Catholic faith if he accepted the position. That was, of course, detested by Leibniz. A second part was his long-held opposition to the politics of Louis XIV, who had been systematically encroaching on neighboring territories—including parts of the Holy Roman Empire—in order to increase his hold on power. The move would have been contrary to Leibniz’s loyalty to the German states, for which he had worked throughout his life. Furthermore, although his present employment was far from ideal, he was suspicious that his employment in Paris would also have been less than perfect. He may have decided it was better to remain in Hannover where he knew all the pitfalls rather than to immerse himself in the foreign universe of the Sun King. Working in the court of Louis XIV would not be the same thing as being an independent scholar within the King’s Académie. It was not an easy decision, but Leibniz believed he judged it correctly. In July 1692, Leibniz wrote in a letter to Sophie about a discovery made in Brunswick. Madame, An acquaintance has sent me an enormous tooth from an extraordinary animal that was found near his town. He has asked my opinion on the discovery. Common

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people might conclude that it is the tooth of a giant. If it were, based on standard proportions, that giant would be as big as a house. Descriptions that I have found of elephants make me believe the tooth belongs to one of these beasts. I have learned that an elephant has four large teeth above and a similar number of teeth below to allow it to grind up meat or vegetable matter. The dents in the tooth indicate it was used for that. It is interesting to me that we are beginning to find the remains of such animals in our area, as well as other large marine animals…. I knew you would be glad to hear of this interesting discovery. Leibniz

Sophie and Leibniz had had a warm friendship over many years, and both were pleased to share discoveries like this. She was no scholar, but she was interested in scholarship and she was always curious about his pursuits. An understanding of the natural world was growing at this time. The Electress saved this letter in her papers—an indication of her opinion of its importance.

In 1693, having decided to remain in Hannover, Leibniz decided to make another try at the silver mines in the Hartz Mountains. Although Ernst August had announced several years earlier that the project was dead and although Leibniz had officially accepted that decision, he couldn’t help dreaming about a change of for-

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tunes. After all, his plans for the mines were not fundamentally flawed. It was just that they hadn’t worked out yet. Furthermore, who could resist the appeal of more silver to fill the coffers of Hannover—and thus Leibniz’s—pockets? He was an incurable optimist. He had recently learned that the masters of the mint in nearby Celle had a new plan for extracting the ore—a plan that required only half as much water as had been needed before. Leibniz was suspicious that they had in fact stolen some of his ideas. A few weeks later Leibniz officially requested permission to restart his experiments. Although the Duke was loath to give Leibniz another opportunity to disappear into the mines for months at a time, Leibniz proposed delegating supervision of the operation to an underling so that it would not monopolize his time. Finally, Ernst August approved Leibniz’s plan, and the new trials began. This time Leibniz’s attention was directed at the efficiency of the pumps themselves. Once again he encountered serious opposition from those who were involved in the day-to-day operations of the mines, and the experiments necessary to carry out the work took far longer than Leibniz had predicted. The staff at the mines still had no patience with him. If an improvement didn’t work today (or at least tomorrow) it was useless to them. Also, Leibniz couldn’t resist getting involved himself, spending much time supervising the experiments. For Leibniz, the Hartz Mines were endlessly fascinating. As court councilor, Leibniz had many other responsibilities as well. Sorting out the legal dealings of the Duke and his peers within the electorate was

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a perpetual challenge. Georg Ludwig would inherit the entire duchy of Hannover and Celle, would be an Elector in the Holy Roman Empire, and was destined for great things. Such success is what Ernst August had hoped for, and now it was all coming true. However, as his mother Electress Sophie had feared, Georg Ludwig’s marriage to the illegitimate daughter of Georg Wilhelm—Ernst August’s brother— had fallen apart. Sophie was devastated. “Ernst,” Sophie began one morning after breakfast, “what are we going to do about Georg Ludwig?” “What do you mean, do about him?” Ernst August snapped. “I set him up with an electorate to rule for life. What more can he ask?” “Oh, I agree that you have provided well for him,” Sophie said. “The problem is how he has dealt with his wife. Sophie Dorothea is a very unhappy woman.” “That’s not my problem,” Ernst August said as he got up to leave the table. “Georg Ludwig has a legitimate heir. He will be an Elector of the Holy Roman Empire. That woman—his wife—is out of the way in a lovely castle that she has all to herself, and she will have no further contact with her children. I’m ready to move on to other things.”

Leibniz continued to enjoy his work in Wolfenbüttel, both working with the Dukes there and in the library. By now he and Duke Anton Ulrich were great friends, respecting one another and spending hours together discussing a wide range of topics. In 1688, Anton Ul-

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rich had begun the construction of his Salz-Dalem, a charming little palace to complement the opera house on the grounds of his estate. It was finally completed in 1694. In celebration of this accomplishment, Leibniz wrote a poem that he dedicated to the Duke: Nothing in this Transient Life is too Difficult What is beyond the power of human thought? A great duke can control both nature and time. He can set out for that noble land, And grow hyacinths at Yule-tide. When now roses should bloom only in Jericho. Salz-Dalem can bring forth the most beautiful blossoms. God allows that He also (according to my wish) Can always provide impossible fruits, as is His custom.

In the 1690s, Leibniz’s calculus attracted much attention in Europe. Jacob and Johann Bernoulli did what they could to present it in the Acta so that it was accessible to any scholar who chose to make the effort to master it. As time went on, both Bernoullis posed problems in the Acta—problems that could be solved only with the calculus. Those who could solve them were then recognized as Europe’s most brilliant mathematicians. In 1691, Johann, the younger brother of the two, entered into an arrangement with the Marquis de l’Hôpital in Paris whereby Johann agreed to teach the Marquis the calculus and give him all rights to his lessons in exchange for a handsome salary. At the time, Johann could see no disadvantages for himself. However, in 1696 when the Marquis published his textbook

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Analyse des infinement petits [Analysis of the Infinitely Small] with his own name on it (although with a note that Johann had taught him the material), Johann began to regret his selflessness. The book was in fact nothing more than his tutorials to the Marquis. Bernoulli should have been recognized as the author, but he was not. Nonetheless, that textbook taught Europe the differential calculus. L’Hôpital had planned to write a second volume presenting the integral calculus, but he died before he was able to write it. In reaction to the printing of l’Hôpital’s book, mathematicians in England began their protests. On the title page, they saw that l’Hôpital had credited Johann Bernoulli with providing much of the material in the book, in the preface he had stated that Leibniz had sanctioned his presentation of the material, and in the acknowledgments he had noted that Newton had apparently devised a similar calculus—with the added information that Leibniz’s calculus certainly had better notation than Newton’s. The British were livid. They wanted to know who were those continental mathematicians to promote Leibniz’s calculus, and who were they to say it was superior to Newton’s? It was Newton who had invented it first, and clearly Leibniz had stolen it and then presented it as if it were his own. The English gave several examples of situations when Leibniz could have seen and thus copied Newton’s work. Leibniz and his allies (led by Johann Bernoulli) met each assault with their own evidence, only to wait for the next attack. The two camps failed to reach agreement until long after Newton, Leibniz, and the belligerent Ber-

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noullis had died, although recent scholarship indicates that neither mathematician was guilty of plagiarism. The controversy was a cloud over Leibniz’s later years. Along with all his other projects, he had to reply to harsh attacks from England, knowing that his replies would not be sufficient in themselves to prevent the next outburst. Both Newton and Leibniz constructed the calculus independently, and both deserve credit. The petty war was a terrible waste of the brilliance of those two geniuses.

An unrelated mathematical project that Leibniz explored further at that time was his invention of binary or dyadic arithmetic, which he enjoyed describing to his part-time employers, the Dukes Rudolf August and Anton Ulrich in Wolfenbüttel in 1696. “Now Monsieur Leibniz,” Duke Rudolf August began one afternoon as the two were strolling through

Medal celebrating Leibniz’s invention of binary arithmetic.

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the library in Wolfenbüttel, “I wish you would tell me more about this new kind of arithmetic. You say it uses only the digits 0 and 1? How is that possible?” “That is correct, Your Highness,” Leibniz agreed. “You see, every number can be written as a sum of the powers of 2—1, 2, 4, 8, 16, 32, etc. You know that in base 10 the place names from the right to the left are ones, tens, hundreds, thousands, etc.—in other words, the powers of 10. In base 2, the place names are the powers of 2 instead: 1, 2, 4, 8, 16, etc. For example, the number 13 is the sum of 8 + 4 + 1. In binary I can write that as 1101; reading from left to right that would be one 8 + one 4 + no 2s + one 1. It’s all very simple.” “That seems all right,” the Duke conceded, “but are you telling me that you can write any number that way?” “Yes, Your Highness,” Leibniz replied. “What number would you like me to write?” “How about the number 41?” the Duke asked. “For that number, we would start with one 32,” Leibniz said, “and after we subtract that it leaves 9 more to go. We’ll need no 16s. Then we need one 8, no 4s, no 2s, and one 1. That means that the number 41 would be 101001—that is, 32 + 8 + 1. I could do as many numbers as you like. In fact, you could do them yourself, Your Highness.” “And are you telling me that it is possible to calculate with these funny little 1s and 0s?” the Duke asked. “Oh, yes, Your Highness,” Leibniz said. “Let’s suppose you want to find the sum of 13 (which we have just written as 1101) plus 6, which we can write as the sum of 4 + 2; in other words, 110. If we line

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up the columns as we would in base 10, the problem would look like this: 1101 110 When we add the column on the right (the 1s’ column), we have 1 + 0, which is 1. When we add the second column (the 2s’ column), we add 0 + 1, which is also 1. The third column (the 4s’ column) asks us to add 1 + 1, which gives us 2; this is too big so we have to carry it over to the next column, putting a 0 in the third column and carrying the new one into the fourth column (the 8s’ column). Now the fourth column also asks us to add 1 + 1, giving us 2 again, so once more we are forced to carry a 1 over to the fifth column (the 16s’ column). Here, Your Highness, is the result in binary: 1101 110 10011 That is the sum of 16 + 2 + 1 or 19, and if we add 13 + 6 in base 10, we also get 19. It is easy to do all four operations in binary, Your Highness, using only the digits 0 and 1.” “This is remarkable!” the Duke said. “I wouldn’t have thought it possible!” “Yes, we would have thought that only God could create something out of nothing, but this contrivance allows me to do it also,” Leibniz said. Then remembering the necessity of humility, he continued, “Well, we

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create something out of 0 and 1. The number 1 is not nothing.” Duke Rudolf Adolf issued a special seal demonstrating the wonders of binary numbers with the saying “omnia fecit unus ex nihilo [one makes all things from zero].” Both men were clearly delighted. Leibniz had never had such a discussion with Duke Ernst August, his primary employer in Hanover.

In July 1698, in a letter to Johann Bernoulli, his principal defender in the calculus wars, Leibniz mentioned a new piece of notation he had devised. “My dear Monsieur Bernoulli,” he wrote, “has it never occurred to you that our use of the symbol × to indicate multiplication has become too confusing? With the algebra we have learned from Descartes, we use the letter x to stand for an unknown quantity. I believe it is wrong to continue using that same x to indicate multiplication. In my work I have begun using a simple dot instead, and I find that it works remarkably well. I can write 7 ∙ 3 to indicate 7 × 3. What do you think?” Bernoulli was enthusiastic, and students of algebra have been using Leibniz’s dot ever since. Leibniz was still not making serious progress on his history, but he was delighted with the development of his mathematics. The history could wait.

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In 1696 Leibniz was promoted to Privy Councilor of Justice within the electorate. Now he had the seventh highest position in the Elector’s government, and he had a salary to match, particularly when he added the modest income he received for his duties at the Wolfenbüttel library and from Celle for his occasional work on the history of the Hanoverians and Guelphs. Furthermore, because Ernst August had decided to elevate the bishopric of Osnabrück to a major part of the House of Hannover, he still needed Leibniz’s legal skills for that undertaking. Leibniz had much to do even without spending much time on his history. In 1697 Leibniz found an opportunity to promote the establishment of an Académie in the German-speaking world. His dear friend Sophie Charlotte announced that she wished to establish in Berlin an observatory, similar to the one in Paris. Leibniz wrote to the Electress at the end of November strongly encouraging her to work toward that goal and suggesting it would be an important step toward the development of a serious Académie in Berlin. This was a project that Leibniz held dear. In the midst of all this work, Leibniz began to realize that his time on earth was limited. He was already 51 years old, and he had so much still to do. By that time, all his Roman Catholic patrons were dead, making his plan to unite the Christian churches far more difficult. It began to look as if he should satisfy himself with merely joining together the Evangelical faiths—the Lutheran, the Calvinist, and the Reformed. With the arrival in Berlin of many Huguenots from France, where their lives had been severely threatened,

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the electorate of Brandenburg was no longer exclusively Lutheran. With these diverse non-Catholic refugees from the brutality of the Catholic Church, acceptance of anything connected with Rome had become highly unpopular in Berlin. “What if I am unable to finish the work I have begun?” Leibniz asked himself. “I need to amend my goals. I think I should restrict myself to expediting the fusion of only the Protestant faiths. I need to complete my encyclopedia of the sciences and my universal characteristic. I also need to complete all my other intellectual projects in philosophy and mathematics and physics. “Ah, the history of the Guelf dynasty,” he continued unhappily. “That will have to wait. I have done enough on that project to satisfy myself. If I live long enough, I’ll finish it.” Fearing that he had some fatal disease that would end his life prematurely, he became a hypochondriac, interpreting every little ache or symptom as a sign of his imminent demise. Perhaps he wouldn’t be able to finish that history, but, at the very least, he needed to conclude his other important work.

In 1697, Czar Peter the Great made an extended trip to Western Europe, where he intended to learn about the whole of Western culture, from the theoretical to the practical. He worked in shipyards incognito, he learned practical trades, and he met with interesting people. Although Leibniz knew the Czar was there and

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tried to set up an interview with him, he failed to get an audience. Leibniz was greatly disappointed. He was as fascinated with the cultures of the East as the Czar was with the culture of the West. In fact, Leibniz saw Russia as the best way to open up communication with the Chinese, whom he still suspected of having a language that would serve as a perfect model for his alphabet of thought. That was something he had dreamed of since he was 20 years old. When he was visiting Berlin in 1697, Leibniz received word that his patron, the Elector Ernst August in Hannover, was seriously ill. He wrote to the Electress Sophie expressing his best wishes for the Elector’s speedy recovery. She immediately replied, thanking him for his good wishes, but wryly commenting that it was a pity such good wishes really didn’t help! By then, her husband was somewhat better. Throughout the summer and fall, as Ernst August’s health continued to fail, his wife the Electress Sophie served as his primary caregiver. That noblewoman was a noble wife as well. On January 23, 1698, the Elector Ernst August died. His oldest son, the new Elector, Georg Ludwig (1660–1727), immediately took charge. Although Leibniz had no expectation of more congenial treatment from the new administration than he had received from the old, he expected that he would remain in Hannover. At that time, however, Leibniz received an invitation from Sophie Charlotte to leave Hannover and come to Berlin to work with her instead. Nothing could have delighted Leibniz more. Sophie Charlotte’s

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mother Sophie was, of course, his dear friend, but the daughter was a more brilliant woman with far greater power at her disposal. Furthermore, in Berlin Leibniz might have more freedom to pursue his own research. Unfortunately, Leibniz was not able to disentangle himself from Hannover, either then or later. He was a servant of the Elector of Hannover, whoever that Elector might be, and he would remain so for the rest of his life. He also was reminded that he had a history of the House of Hannover to complete.

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1698–1713

Court Historian to Elector Georg Ludwig

In 1698, Leibniz became historian and librarian to the new Elector, Georg Ludwig. The Dowager Electress Sophie of Hannover, Georg Ludwig’s mother, continued to dote on Leibniz in the palace at Herrenhausen, but Georg Ludwig himself was in charge. Leibniz quickly realized that the new Elector was even less sympathetic to his librarian/historian’s irksome ways than his father had been. “Where is that man?” Georg Ludwig asked. “Is he writing that history? Highly unlikely! Has he run off to Berlin to talk endlessly with my sister? Or is he playing in the Hartz Mountains, trying once again to resurrect his impossible mining plans? My father hired him to write the history of the Guelfs, which he says is progressing nicely, but have I ever seen a single page of it? No! And, likely as not, I never will. I doubt that my father ever saw a page of that history either. Why did my father leave that obnoxious man to annoy me? I suppose I will have to continue to pay him while he 191 © 2012 by Taylor & Francis Group, LLC

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doesn’t write that history for the rest of his life! It’s my rotten luck that my mother likes him so much—otherwise I would gladly find some way to get rid of him.” It was true that Leibniz was spending a great deal of time away from Hannover, whether in Berlin or Wolfenbüttel or Vienna or Celle. Clearly Hannover was not the place where Leibniz chose to be, and the new Elector knew that. Some—but not all—of Leibniz’s activities in those far-off places were important for Georg Ludwig, and Georg Ludwig had no way of knowing which were for his benefit and which were not. For his part, Leibniz intended to keep it that way. While he knew that his repeated absences irritated the young Elector, Leibniz’s solution was simply to avoid him. If he couldn’t hear the Elector’s tirades, he could easily forget them, and he had many more important things to worry about than that bothersome history. In 1700 Leibniz learned that the ducal library and his apartment in Hannover would be moved yet again, this time to the home of Sophie Elisabeth von Lüden on Schmiedestrasse 10. His apartment, the biggest that Leibniz would ever occupy, was where he would live for the rest of his life. His rent for the apartment was 150 Thalers per year—not an exorbitant rate, but a significant increase over the free lodgings he had been allotted during his earlier years in Hannover. Following the move, Leibniz continued to work sporadically on the Guelf history as well as on all his other projects. He was pleased to be able to hire a new assistant to help him with copying and other routine tasks. Although Leibniz had visited Sophie Charlotte in Brandenburg several times in 1698 as part of his work

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Court Historian to Elector Georg Ludwig 193 Leibniz House am Holzmarckt, Hannover, where Leibniz lived during his later years.

to bring together the disparate Protestant sects, he was frustrated in 1699 in his attempt to get permission from Georg Ludwig to travel once again to Berlin. Sophie Charlotte had repeatedly begged him to come, he had repeatedly expressed his willingness, but Georg Ludwig had repeatedly said no. Leibniz tried to explain that it was not polite for him to fail to comply with the Electress’ kind invitations—after all she was a very important figure in the German states and within the Holy Roman Empire—but Georg Ludwig was not moved. That noblewoman was none other than his own little sister, whom Georg Ludwig often dismissed as a silly woman. Leibniz wrote a formal letter to Georg Ludwig dated January 19, 1699: Monseigneur, Her Highness, the Electress of Brandenburg, has made her intentions known through her prime minister and supposing the agreement of Your Highness, and having already written to Madame The Dowager Electress, her mother [Sophie], that she awaits with

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impatience the pleasure of seeing a certain man for a short time at her home. It is painful to this man to injure the heart of such a princess under these circumstances. In any case, this man dares to assure Your Highness of his voluntary good will and believes he has given proof. Thus, if Your Highness has reasons to doubt this man’s honesty, he will have to take this man with his faults. If Your Highness does not give other orders, it will require the man to excuse himself because of the bad weather that causes a variety of ailments, and to respond with unconvincing excuses in order to extract himself from obligations. I am with devotion, Monseigneur, the very faithful servant of Your Highness, Leibniz

Leibniz understood his subservient position in the court. Despite his erudition and many accomplishments, he was nonetheless a servant to the Elector. He could plead his case and explain his personal obligations, but in the end he was not a free agent. “My dear Leibniz, have you forgotten your primary assignment? Surely you are far too busy with your writing of the history of the Guelfs to make such a trip!” Georg Ludwig replied. “Oh, no, Your Highness,” Leibniz replied. “I promise you that the trip will not interfere with my work. I assure you that I am making excellent progress on your history. I will have it ready very soon. You may also be sure that I will spend several hours on it every day while I am in Berlin. I will fulfill my obligation to complete the history. You don’t need to worry about that, Your Highness.”

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One of Leibniz’s important assignments was his effort to ensure the Hanoverian claim to the British throne. Although both Georg Ludwig and his mother, the Dowager Electress, were not convinced of the advantages of such a move, Leibniz never doubted it, at least in part because of his determination to thwart the greedy ambitions of Louis XIV of France. Great Britain was the most powerful Protestant state in Europe, and arranging for the Protestant Hanoverians to take over that crown would guarantee that England remained so. Most of the other claimants to the throne were Catholic. Leibniz worked diligently with the English ambassadors to solidify the arguments that would place the Electress Sophie and her offspring on the English throne. In February 1700, while Leibniz was working on the Hanoverian claim and several other projects, he received exciting news of his own: he had been elected a foreign member of the Académie in Paris. After all his years of applying and asking, the Académie Royale des Sciences had finally decided to accept him.The Académie saluted him as Gottfried von Leibniz. “I accept the implied nobility attached to my name,” he said to himself. “I’ve earned it. With all that I have accomplished in mathematics and philosophy, the Académie really had no choice but to elect me. It would have been a scandal if they had not made this move. I can’t think of another scientist anywhere in Europe who has done as much important work as I

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Leibniz Statue in the Innenhof at Leipzig University.

have!” In response to his election to the Académie, he happily submitted an essay to the Histoire, an annual publication of the Académie, on his binary system of arithmetic. Although membership allowed him no formal position in Paris, it recognized him officially as one of Europe’s greatest scientists. The Elector in Hannover might not appreciate him, but the learned savants in

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Paris—as well as the Electress Dowager Sophie of Hannover, and the Electress Sophie Charlotte of Brandenburg—certainly did.

A month after his election to the Académie, on March 19, 1700, Leibniz was delighted to receive more good news. Sophie Charlotte’s husband, Elector Friedrick III of Brandenburg, officially established the Berlin Observatory as well as the Kurfürstlich Brandenburgische Societät der Wissenschaften [The Electoral Brandenburg Society of Sciences], naming Leibniz as the first president. Leibniz chose the name Societät instead of Académie, because he feared that in German the connotation of Académie was as a school for young people rather than as a research institute. In 1744, several years after Leibniz’s death, the name of the Brandenburg Societät in the Realm of Frederick the Great would be changed to Académie as it remains to this day. “My own Societät!” Leibniz congratulated himself. “My dream come true! I know what a proper Societät should do, and now finally I can carry out my plans! Thank you, Sophie Charlotte, my princess, my protégée! The time and effort I spent educating you for all those years was worthwhile. I knew it would be. You are a brilliant princess!” The Elector Frederick, who had once quipped that Leibniz was in fact an academy in himself with his grasp of many different fields, agreed to allow him to serve as president while still living officially in Hannover, often

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conducting business by long distance. The Societät’s major goal, according to Leibniz, was to improve the lot of the common man, with a strong emphasis on education and the development of such practical arts as agriculture and manufacturing and to foster scientific innovations and other scholarly pursuits. “Your Highness,” Leibniz approached the Elector Georg Ludwig as he bowed the day after he received the news, “I must ask your permission to travel immediately to Berlin.” “Again?” Georg Ludwig asked in aggravation. “You have major work to do here!” “Your Highness,” Leibniz persisted, “the Elector Frederick III has just established his Societät of Sciences in Berlin, and he has named me its president. I really have no choice. I must go to Berlin and see to the organization of the Societät. I won’t be gone long. The Elector has stated that I may perform most of my duties from here in Hannover, but first I must be there to establish the Societät properly.” “What about my history?” Georg Ludwig asked. “Or have you already finished it when I wasn’t looking?” he asked with a sneer. “I am working on it, Your Highness,” Leibniz said, “but can’t you see that I need to act quickly to carry out Elector Frederick’s plan? I’m sure you realize that this Societät is critical to the development of the Protestant electorates within the empire—that it will have many marvelous results throughout the empire.” “How long will you need to be away?” Georg Ludwig asked reluctantly, realizing that he shouldn’t offend his brother-in-law further.

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“Oh, I expect to return within a month,” Leibniz said. “All right, you may go,” Georg Ludwig said grudgingly, “but I must insist that you hurry back as soon as possible. I would like to have the history in my hands very soon.” “Certainly, Your Highness,” Leibniz said, bowing once again. “You may be sure that I will return within a month. Thank you.” Despite his promise, Leibniz was absent from Hannover for eight months, staying for four months in Sophie Charlotte’s lovely palace Lützenburg (now Charlottenburg in her honor) before traveling farther. He spent much of the time he was in Berlin socializing with Sophie Charlotte and her other guests, but managed occasionally to tear himself away from her charming conversation to work on the new Societät. Among other things, he arranged for the regular publication of the Gregorian calendar in Brandenburg, replacing the long outdated Julian calendar—a change that had been made in Catholic states more than a century earlier. By now, the Julian calendar was highly inaccurate, since it had lost ten days over the preceding millennium, placing the vernal equinox on March 11 instead of March 21, where it should have been based on the inclination of the earth. Many years earlier, Leibniz’s professor Weigel at Jena had campaigned for the adoption of this modern calendar, and Leibniz was eager to foster its acceptance in the largest of the Protestant electorates. It was clearly the correct thing to do. Because the Societät would have a monopoly on publishing the calendar, it was guaranteed to be accepted immediately, bringing the electorate into the same calendar as the

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rest of Europe and earning a substantial income for the Societät as well. Leibniz had other plans as well. He hoped to establish a lottery (an enterprise that might take shameless advantage of its simple investors although it would have brought in lots of money), and he planned to grow mulberry trees that would allow Frederick to produce silk (a valuable product) on the palace grounds in Potsdam. Although the calendar worked well, those other plans failed. Leibniz requested the mulberry trees to be planted under glass in the royal gardens, but the Elector delayed the project until the following year, saying that it was too late in the season to plant them immediately. Leibniz maintained that the cultivation of silk would be easy, since the trees needed to be nurtured only to produce good leaves—fruits were not necessary—and thus even very old or very young people could easily tend to them. The mulberry seeds were duly planted in 1704, but despite Leibniz’s detailed instructions to the gardeners, the trees never prospered. The harsh northern climate may have had something to do with that failure, but it could also have been due to Leibniz’s incomplete knowledge of the process of cultivating silk. After four months in Berlin, he moved on to Wolfenbüttel, Prague, and Vienna to carry out projects to which he was committed. Certainly Georg Ludwig would not have given permission for those side trips, but since Leibniz was away Georg Ludwig was helpless. Leibniz found these travels the perfect relief from his tedious work in Hannover. When he finally returned to Hannover, he carried a letter from the Holy

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Roman Emperor to Georg Ludwig, thanking him for sharing his brilliant scholar Leibniz for the benefit of the entire empire. Leibniz hoped the letter might mollify Georg Ludwig, who would undoubtedly be very angry with him by now. Back in Hannover in 1701, Leibniz continued his work on his many tasks, including the history of the Guelfs. In July, when the Electress Sophia Charlotte requested Leibniz’s immediate return to Berlin, he was forced to reply that he couldn’t travel at the moment— he had other responsibilities. Like the Elector Georg Ludwig, the Dukes of Hannover and Celle were also becoming impatient about what they saw as their historian’s primary job. They wanted their history, for which Leibniz had been handsomely paid for many years. In August, the Dukes called a meeting to discuss Leibniz’s progress. They named a copyist to help Leibniz and to report to them regularly on his progress—in other words, they appointed a spy. That put Leibniz under considerable pressure to finish the job.

The year 1701 was important to the noble family in which Leibniz was employed. It was then that the Elector Frederick III of Brandenburg was crowned King Frederick I of Prussia and Sophia Charlotte became his Queen. In the same year, Leibniz finally succeeded in establishing the Hanoverian right to ascend the throne of England—naming the Electress Dowager Sophie and her heirs the future Kings and Queens. Leibniz had worked on this arrangement for many years. In

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Sophie Charlotte, Queen of Prussia.

August 1701, a delegation from London brought to Sophie the Act of Settlement, which named her line as the legitimate future Kings and Queens of England after the deaths of William III and Princess Anne, unless Anne should produce an heir to the throne in the meantime—an event that was looking increasingly unlikely. Leibniz was pleased by this success and looked forward to moving to London with the court. Between 1700 and 1705, Queen Sophie Charlotte tried repeatedly to lure Leibniz back to Berlin. He was able to make the journey several times, but as soon as he returned to Hannover she requested his presence once again. He thoroughly enjoyed his visits with her—she was as well read and as scholarly as her Aunt Elizabeth had been—but her demands made it difficult for Leibniz

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to accomplish his assignments both on the history of Hannover and on the establishment of the new Societät. Admittedly, Leibniz was not eager to complete the history, but with the Queen’s demands, the job was more difficult than ever. In late January 1705, Leibniz was in Berlin on business. He and Sophie Charlotte had had their usual congenial discussions before she departed for Hannover, where she always went to celebrate Carnival— Mardi Gras—with her mother and friends. Leibniz had planned to follow her a few days later. However, during the journey, Sophie Charlotte came down with a bad cold that quickly progressed to pneumonia. Leibniz was concerned to hear that she was sick, but the young woman had appeared to be strong and so he gave it little thought. Then on February 2, 1705, a messenger delivered a devastating message to the palace in Brandenburg that the 37-year-old Queen Sophie Charlotte had died. Everyone was stunned. One story relates that on her deathbed she expressed some impatience with the pastor who came to

Street sign honoring Sophie Charlotte, Berlin.

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comfort her: “Do not torment me, for now I go to satisfy my curiosity on the principle of things that Leibniz has never been able to explain to me—on space, infinity, being, and nothingness. And I prepare for my husband the King the spectacle of a funeral, where he will have a new opportunity to display his grandeur.” Even when facing her own death, she retained her sense of humor. “It can’t be!” Leibniz said in horror when he heard the news. “She was truly my closest friend, and she was such a brilliant woman!” “Yes, Monsieur Leibniz,” his assistant said. “This is very sad indeed. Did you want me to copy this document next, Monsieur, or do you have something else that you want me to do now?” “But, you don’t understand!” Leibniz moaned. “I have lost noble sponsors in the past, but none who were so dear to me. With Boineburg and Johann Friedrich, I was greatly saddened and distraught as I thought of my future without their friendship and their noble benevolence. With the Queen of Prussia, it is different. She was not my employer—she was my dearest friend.” Leibniz paused as he brushed a tear from the paper before him. He didn’t return to Hannover for another month, and even after he had arrived there, he still was in a state of shock. He continually cried out, “Oh, Your Royal Highness! My Queen! You were so young! I thought I would have your wonderful friendship for the rest of my life. How can it be that you, who were so full of life, can have left this world? I never thought this would happen. How can I go on without you?”

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“Yes, Monsieur,” the copyist agreed, “she was a charming and wonderful woman, but I’m not sure what it is that you want me to do next.” With Sophie Charlotte’s death, Leibniz lost some of his spirit. At 60, he had begun to suffer from gout and chose to relieve his pain by dressing warmly (if bizarrely)—a fur-lined dressing gown, fur stockings, and large wool slippers. He still enjoyed conversation in Hannover with the Dowager Electress Sophie and sometimes even with the Elector himself. He was well aware he needed to finish the work he had begun, and he needed to do it quickly. In addition to the accursed history, however, he continued on his works of philosophy and mathematics, which gave him genuine pleasure.

Soon after his return to Hannover, Leibniz took the opportunity to travel to Wolfenbüttel, where Duke Rudolf August had recently died but where his brother Duke Anton Ulrich was moving forward with his plans to build a proper library building for the duchy. This was a project dear to Leibniz’s heart—the first building ever constructed in Europe specifically to house a library. Leibniz needed to be involved in this magnificent project! The building, which was completed in 1713, would be a lasting monument to Leibniz’s work. This was a fitting tribute to scholarship, and Leibniz enjoyed working on it with a nobleman who appreciated a library and all that it contained. His return to Hannover from Wolfenbüttel this time provided a rude awakening. In response to

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The library at Wolfenbüttel, which Leibniz was pleased to design.

Leibniz’s extravagant bow to the Elector Georg Ludwig, the Elector handed him a document dated June 6, 1705, forbidding him to travel anywhere outside of Hannover until his history was completed. Georg Ludwig had been frustrated with his historian for long enough. His many trips and projects always

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seemed to take priority over the major historical work he was supposed to have finished long ago. The new document required Leibniz to get formal permission from the Elector for any departure he might make from Hannover until the history was complete. Although Leibniz resented it, he knew there was no way he could formally protest. Also in 1705 Leibniz was pleased to learn of the proposed marriage of Georg Ludwig’s son and heir Georg August to Caroline of Ansbach, who had close ties to her new husband’s grandmother, the Electress Dowager Sophie of Hannover, as well as to the late Queen Sophie Charlotte of Brandenburg. A year earlier, when both Leibniz and Caroline had been in Berlin visiting Sophie Charlotte, he had helped Caroline write a letter to extricate her from an earlier agreement to marry the son of the Holy Roman Emperor Leopold I. She was unwilling to convert to Catholicism as that marriage would have required. Although Georg Ludwig was fond of neither his son nor his new daughter-in-law, the marriage appeared to be a happy one—far happier than Georg Ludwig’s marriage to his cousin had ever been—and Leibniz counted Caroline as a close friend as well. The Dowager Empress Sophie’s role as probable heir to the English throne continued to be complicated for her family, as Leibniz tried to establish the legal documents essential to that process. Georg Ludwig chose not to involve himself in the matter at all. He was far more interested in what was happening in Vienna with the Holy Roman Empire, where he hoped to play an important role. Therefore, Leibniz’s role in

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promoting the cause of the Hanoverian claim to the British throne may have been yet another bone of contention between the two men. English politics looked too messy to Georg Ludwig. In 1707, when Leibniz was required to be in Berlin to handle formal arrangements for the marriage of Georg Ludwig’s daughter—this time Georg Ludwig himself had provided the excuse to get around Leibniz’s imprisonment in Hannover—Leibniz used that trip as an excuse to spend six months in Berlin, working both in the interests of the marriage and in his role as president of the Kurfürstlich Brandenburgische Societät der Wissenschaften. The Societät’s meetings at the time resulted in the establishment and the preparation of the first issue of the journal of the society, which would come out in 1710 and which would include several articles written by Leibniz. However, his pleasure in that accomplishment was dampened by the fact that his presidency of the Societät seemed to be unraveling. The King had made several changes in the organization and leadership of the Societät without consulting Leibniz, including naming an honorary president to serve in Leibniz’s absence. Leibniz was irate, although the difficulty was partly the result of his misunderstanding of the situation. There was no doubt that his role in the Societät would have been easier if he had simply been able to stay in Berlin. When he returned to Hannover, he was finally ready to see through to publication the first volume of his history of the Guelfs in 1707. Although Georg Ludwig conceded that Leibniz had been accomplishing something, and although that first volume was later

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followed by two more, Georg Ludwig was not pleased with the results. Leibniz’s assignment had been to produce a small history of the family. Those first three monumental volumes brought the history up only to the year 1500. Although they provided a trove of evidence, they were much more detailed than the Elector wanted, and the job was still far from complete. Leibniz finally convinced the reluctant Georg Ludwig to reimburse him for his costs and to pay him the honorariums due. In 1708 Leibniz contrived a trip to Vienna with the help of Duke Anton Ulrich of Wolfenbüttel, in which Leibniz was supposed to negotiate the acquisition of parts of the duchy of Hildesheim into the duchy of Wolfenbüttel. On this visit, Leibniz was pleased to make the acquaintance of the new Emperor Joseph I (1678–1711), who had recently taken over the throne after the death of his father, Leopold I. Leibniz was also able to meet with the Empress, the daughter of his former patron (Duke Johann Friedrich of Hannover), whose marriage to the future Emperor Leibniz had been suggested several years earlier. He had fond memories of the charming girl and was pleased to see she had grown into a poised and accomplished woman. “Wouldn’t Johann Friedrich have been proud!” Leibniz contemplated. After completing his errands in Vienna, Leibniz traveled on to Leipzig, where he was able to meet with his nephew Friedrich Simon Löffler before stopping in Berlin to supervise the preparation of the Societät’s journal. With that job completed, Leibniz reluctantly returned to Hannover, where he knew he should expect a frosty reception from Georg Ludwig.

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Arriving at the Herrenhauser Palace, Leibniz chose to call first on his friend Sophie. As he bowed deeply to express his reverence for her, he suddenly saw that he had been duped: the Elector Georg Ludwig stood there in her place. “Where have you been, Monsieur Leibniz?” the Elector demanded to know. “Oh, I didn’t expect to meet you here,” Leibniz said with as much poise as he could gather. “I asked, where have you been!” the Elector demanded once again. “I beg your pardon, Your Highness,” Leibniz stammered as he bowed once again and quickly disappeared around the corner. He didn’t wait for a response. He later wrote a detailed memo to the Elector, making up facts as necessary in an attempt to disguise the real nature of his work in Vienna. With that taken care of, Leibniz immediately set himself again to work. In 1710 Leibniz published his most ambitious and most popular philosophical work, Theodicy—using for the title a word he had coined for the purpose. The book had originated in his discussions with the late Queen Sophie Charlotte at her palace Lützenburg in Berlin. He had been grappling with these issues since his university days, and his discussions with the Queen had allowed him to develop them brilliantly. Although he had originally planned to publish the work in Latin with the intended audience of the intellectuals of Europe, he eventually decided to publish it in French instead, in the language that any educated European of the day could read.

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The major issue that Theodicy explores is the presence of evil on earth—why did God choose to create a world in which there is evil and suffering? Leibniz addressed other questions as well: why does man have the freedom to sin? Is an all-powerful God truly just? In the book, Leibniz also presented God as not only omniscient and fair, but also as a truly good being. It was in this work Leibniz argued that we live in the best of all possible worlds. This must mean that not only is God perfectly good, but that God wants us to be good as well. Leibniz argued that if this were not the best of all possible worlds, then God would not have made it this way. After all, God could have made any world He wanted to make, so it is obvious that He chose what is best. Voltaire later found much to mock in that statement when he wrote about it in his 1759 satire Candide. The character Pangloss is a thinly disguised parody of Leibniz. Candide was wildly successful throughout Europe. Leibniz had tried unsuccessfully to meet with Peter the Great in 1697. However, in 1711 he was finally able to meet with the Czar in the resort city of Carlsbad following the marriage of the granddaughter of Duke Anton Ulrich of Wolfenbüttel to the heir of Peter the Great—another event that gave him permission to escape from Hannover for a time. “So you are Gottfried Leibniz, the great scholar from Hannover,” the Czar greeted him. “I am pleased to meet you.” “Yes, Your Royal Highness,” Leibniz said while executing an extravagant bow to the Czar. “I have been eager to make your acquaintance for some time.”

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“I believe you are the man who set up the scientific Societät in Berlin, isn’t that right? Is that what you wanted to discuss with me?” the Czar replied. “Yes, Your Highness. I am eager to set up in Russia an Academy of Sciences similar to the Académies in Paris and Berlin,” Leibniz explained. “You see, Your Royal Highness, the Societät will serve two major purposes in Czarist Russia. It will provide an environment where scholars can work together profitably in many fields, making important discoveries that will improve the quality of life in Russia. It will also establish an educational system within your empire so that the population will become well enough educated to evolve into the progressive state that I understand Your Highness envisions.” “Yes, Monsieur Leibniz,” the Czar replied, “I agree that those are worthy goals. Is there anything else I need to know?” “Oh, Your Royal Highness, I have many other plans,” Leibniz explained. “I would like to set up a geography department that could produce reliable maps of your kingdom, perhaps allowing for the exploration of a sea passage between North America and Asia on your far eastern boundary. I also envision the production of a modern judicial system in Russia as well as serious economic reform. I have many other plans as well.” “Do you have all these proposals in writing for me so that my ministers can study them in detail?” the Czar asked. “Yes, Your Royal Highness,” Leibniz replied, “here are all my proposals.” The Czar’s assistant accepted the proffered stack of papers from Leibniz with a nod.

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“Good. Thank you for your excellent proposals,” the Czar replied. “After I look them over with my advisors, I will communicate my reactions. Good day, Sir.” Again, Leibniz bowed as he retreated from the Czar’s presence. Leibniz was always impressed with the formalities of the royal lifestyle, and the Czar, who looked down at him from his extraordinary height of two meters (six and a half feet) impressed him particularly. Leibniz had the sense that this was a man who could easily rule the entire world, and he was eager to help him in any way he could. The Czar was clearly no small-minded bureaucrat like the Elector Georg Ludwig—the contrast was striking. Viewing Russia as his best connection with East Asia, Leibniz was eager to use his meetings with Peter the Great to his advantage. In addition, Leibniz knew that Russia was a largely backward country but one with tremendous potential if the Czar could mobilize it properly. If Leibniz could set up a Societät that would promote the study of sciences and provide a good educational system, he believed Russia’s possibilities were limitless. Peter the Great agreed with Leibniz on all of that, and he was eager to carry out Leibniz’s plans for the modernization of Russia. The Russian Academy of Sciences would be a dream come true for Leibniz, because he saw that it would allow him to carry out many of his own plans—including perhaps the creation of a universal characteristic—stemming from his ambition as a student in Leipzig many years earlier. After the meeting, Leibniz quietly made his way by stages to Vienna, where he stayed for two years, avoiding a speedy return to Hannover simply by

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keeping out of the Elector’s way. Leibniz knew that his communication with scholars all over Europe was critical to the development of his philosophy and mathematics, regardless of whether Georg Ludwig approved. In 1706 Leibniz contrived to travel again to Berlin, carrying on some important work for his Societät although, even at this time, the Societät had no actual building. Meetings were held in whatever space could be arranged, and the Societät was in reality no more than an idea—a good idea, but still far from being realized. In 1707, when the Societät formally requested a building to call its own, the King issued a proclamation to construct it. Seven years after the beginning of the Societät it would seem the time had come. In 1711 Leibniz made another trip to Berlin, during which he called a meeting of the Brandenburg Societät and carefully outlined his goals for the Societät. By that time he was also heavily involved in the formation of another such society in Vienna, so he had little personal contact with the Societät in Berlin. Unfortunately for Leibniz, the 1711 meeting was the last one he attended in Berlin.

During all that time, Leibniz had maintained his contacts with the mathematical world, where the calculus wars were raging. The English were determined to prove that Leibniz had stolen Newton’s fluxions and presented them as his own, and the Germans and Swiss were convinced that if there was any dishonesty

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it was on the part of Newton and his English defenders, who had waited several years after the publication of Leibniz’s calculus before Newton presented his own version. Could Newton prove he had not read Leibniz’s calculus in the Acta and then created his own? In 1700 there was an assault on the calculus from within the French Académie. Pierre Varignon, who was called upon to defend Leibniz’s work, argued that those who had attacked clearly did not understand the calculus. As a result he could not take their criticisms seriously. In a second incident in 1701, Leibniz’s calculus was criticized, saying that it didn’t have a firm foundation— that the proofs were not rigorous. After all, if you make a fraction for the slope of the tangent to a curve where the differences in both the y and x values are 0, doesn’t 0 that give the meaningless fraction 0 ? How could the Bernoullis and Leibniz justify such a flimsy basis for their calculus? In 1702 Leibniz wrote an explanation in the Journal des Sçavants, explaining that infinitesimals should certainly not be interpreted as real things—that they were merely ideas one could work with to solve real problems. No one was suggesting justifying it with the 0 nonsensical fraction 0 . In spite of all, he argued, everyone agreed that the resulting calculus was a powerful tool scientists all over the continent were using successfully. Because the results were valid, they should just accept it and move on. In 1704, reacting to another English essay that totally botched an attack on Leibniz’s calculus, Newton decided to provide a detailed explanation of his own

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fluxions in an appendix to his book Opticks. Leibniz, not content to let Newton have the last word, wrote a review of Newton’s essay, implying that Newton had merely started with Leibniz’s own calculus and adapted it to his fluxions. Predictably, Newton was outraged by that review and declared total war. From that time on, Leibniz and Newton were sworn enemies, fighting openly within the learned journals of Europe, with plenty of help from their various proxies. This was a war that would not be resolved during either of the contenders’ lives—or even in the next 100 years. When Jacob Bernoulli died in 1705, Varignon wrote to Leibniz telling him that Fontenelle’s elegy of Bernoulli at the Académie included sincere praise for the calculus. What Varignon did not mention was that the elegy gave credit to the Bernoulli brothers— instead of Leibniz—for inventing it. When the elegy was printed in February 1706 in the Nouvelles de la republique des lettres [News from the Republic of Letters], it still contained the error. In their writings the Bernoullis always gave full credit to Leibniz for the discovery, but Leibniz was unhappy. In May the journal published Leibniz’s protest, indicating that he was the one who had invented the calculus in 1674, although he also credited the Bernoullis and L’Hôpital with extending the applications. In 1708 when Leibniz read an attack in the Philosophical Transactions of the London Society by John Keill, formally accusing Leibniz of stealing his calculus from Isaac Newton, Leibniz was outraged. It was all a lie! As a member of the Royal Society, he wrote directly to the secretary of the Society, demanding an

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apology. The secretary appropriately passed on Leibniz’s complaint to the president of the Society—Isaac Newton himself! Newton penned the response to Leibniz’s complaint, publishing it in the Transactions. That response was a brutal attack on Leibniz in spite of the fact that he was a member of the esteemed Society and had thought such an assault on one of its members was impossible. Needless to say, the exchange resolved nothing. The insults continued to fly from London to Basel to Berlin to Hannover and then back again to London. Each argument included new evidence proving the treachery of the other side.

Leibniz returned to Vienna at the end of 1712. The year before, when he had been there representing the Duke Anton Ulrich, Leibniz had proposed that the emperor might see fit to name him a Reichshofrat [councilor] of the empire. Although Georg Ludwig had deputized his emissary to do what he could to stop Leibniz’s candidacy, pointing out how Leibniz attempted too much and followed through on very little, Georg Ludwig reluctantly nominated Leibniz for the position, which was realized in 1713. The resulting pay was clearly one of Leibniz’s reasons for wanting the job, although it was in fact not a dependable salary, being paid in a haphazard way—if at all—over the years. In 1714 Leibniz, who had remained in Vienna too long in the eyes of the Hanoverian court, received a summons from Georg Ludwig to return to Hannover

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immediately or else. Because Duke Anton Ulrich had recently died, Leibniz couldn’t even expect to find a respite in Wolfenbüttel from Hannover on his return, but the threat of losing his income in Hannover was a sobering thought. Even the Electress Dowager Sophie was becoming impatient with Leibniz, urging him to return home immediately. When she also died on June 8, 1714, at the age of 83, Leibniz lost his last real friend in Hannover. “What,” he asked himself, “remains in Hannover for me to return to?” The answer was only Caroline, the wife of Georg August, whom Leibniz was indeed very fond of. The rest of the court had no use for its historian and its historian had no love for them. Leibniz grimly decided that he must make preparations to return to Hannover. However, before his return, Leibniz learned that Queen Anne of England had died on August 12, 1714, vacating the throne for the new King from Hannover. Georg Ludwig was immediately named King of England. The Queen was dead; long live the King! Suddenly Leibniz needed to return to Hannover immediately. He knew he couldn’t expect to be part of the court in London unless he was physically present in Hannover. Certainly Georg Ludwig was not particularly fond of Leibniz (and vice versa!), but the new King must recognize Leibniz’s many strengths. Leibniz eagerly looked forward to his life in London. King George I of England owed his new role to Leibniz more than to anyone else, and Leibniz confidently anticipated his reward.

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9 1714–1716

Alone in Hannover

When Leibniz arrived in Hannover on September 14,

1714, he learned to his horror that the Elector Georg Ludwig—now King George I of England—and his court had left for London three days earlier! Caroline, now Princess of Wales, was still in residence at Herrenhausen with her daughters and her younger son, but the rest of the court was gone. Leibniz considered hiring a coach to follow the royal entourage but decided it would be too difficult and might risk an unpleasant scene with the new King. Instead Leibniz wrote letters to the prime minister and other officials as well, telling them of his arrival in Hannover and of his wish to leave for London as soon as possible. A month later, when Caroline departed for London with her children, Leibniz found that he was truly alone. What a come-down for this sociable man! Then he learned that one person—the new King’s younger brother Ernst August—had remained in Hannover. Ernst August had one and only one assignment: to 219 © 2012 by Taylor & Francis Group, LLC

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guarantee that Leibniz would never again set foot in England. Ernst August was a stupid little man whom Leibniz had loathed as a child—and still loathed. For his part, Ernst August viewed Leibniz as a useless relic of the past. It was not a cozy relationship. How far he had fallen! Leibniz wrote to Caroline that he had nothing to do but sit in his room and write— which is exactly what King George I had planned. And so Leibniz wrote and wrote and wrote—about obscure dynastic history of interest to the Guelphs perhaps but to no one else. “What kind of life is this?” he asked himself. “After all that I have accomplished, here I am with nothing. I am universally respected as one of the greatest scholars of my time. I am a mathematician of great renown. The great Sir Isaac Newton feels threatened by my claims to be the originator of the calculus. In spite of himself, Newton apparently respects me as his peer. I am also a philosopher of great renown. And what is my reward? To slave under the eyes of a stupid man and the King he represents, trying to complete a minor historical work that is in no way worthy of my attainments. I never dreamed it would come to this.” Leibniz seriously considered returning to Vienna, where he had friends—Vienna was not yet officially off limits for him—but he had a conscience. He had promised to write the history of the Guelfs, and he supposed he ought to finish it before he died. He didn’t want the House of Hannover to complain that he had not been a man of his word. He would finish it, if it was the last thing he did.

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Leibniz kept up his correspondence with Bernstorff, the King’s prime minister, informing him of his progress with the history. In exchange, Bernstorff sent him a formal prohibition against traveling to London or anywhere else until his history was complete. Leibniz replied, boldly suggesting that perhaps he might be named historian to the court of Great Britain. Through Bernstorff, Georg Ludwig said no. Leibniz could expect nothing from King George or his court. In fact, with the controversy at that time over the origins of the calculus, Leibniz had no chance of establishing himself in any formal position in England, for Newton was widely revered there as the real inventor of the calculus. Certainly Leibniz should have been aware of it, but he probably had no idea how unpopular he actually was in England. As the late Electress Sophie had observed more than once, he was not a practical man and was not skillful at dealing with people. Caroline Princess of Wales tried to intervene on his behalf, but her efforts were rebuffed. Because the King didn’t like her very much either, it is doubtful that her word made any difference. She was, however, able to arrange for the payment of Leibniz’s overdue salary. That was small consolation. Meanwhile, back in Hannover Leibniz toiled away on his history. He had no time for mathematics, no time for philosophy, and no one to talk with. If Hannover had been his prison earlier, it was his dungeon now, complete with its jail keeper. Although his friends in Vienna were urging him to return there, where he could engage in stimulating conversations

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and an interesting range of work, Leibniz remained in Hannover, attempting to fulfill his responsibilities. He was an honorable man. In reality, Leibniz did not completely restrict himself to Hannover. Having been in further correspondence with Czar Peter the Great of Russia, he was able to meet him at a nearby spa several times, discussing the possible Societät in St. Petersburg. The Czar was intrigued with Leibniz’s plans, but so far had agreed to nothing concrete. Leibniz was also able to get away to Vienna, Wolfenbüttel, and other interesting places. King George I’s power extended only so far, and young Ernst August was no match for the wily scholar. At that time, Leibniz had financial worries as well. King George I was unwilling to provide his back salary for the time he had been away from both Hannover and Vienna despite Caroline’s best efforts. Furthermore, the Societät in Berlin decided first to cut his payments in half and then eventually to end them altogether since Leibniz had not been seen in Berlin for several years. Finally, in September 1716, Leibniz received word from the office of the Holy Roman Empire that his salary from that source was being terminated as well. As a result, Leibniz was forced to look around once again for other possibilities. Louis XIV seemed open to providing a position for him in Paris, although in fact the King was neither willing nor able to provide any money to support Leibniz there. Louis had his own financial problems. His many wars had bankrupted France by 1714, and the Sun King was in failing health and died the following year.

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In spite of all these distractions, Leibniz was making remarkable progress on the “little” history. He had amassed numerous documents and was hard at work, pulling all that information into a coherent history. Nevertheless, he was also growing old. On July 1, 1716, his 70th birthday, Leibniz was genuinely depressed. “What is there to celebrate?” he asked himself. “I have done much, but what does it matter? Will I ever be able to finish this cursed history? I am beginning to doubt it. And what of my other projects? Who will continue them? What a wretched state I am in. God, forgive me! I have done so much, but there is so much still left to do. And I am tired.” On July 26, 1716, King George I of England invited Leibniz to dine with him during one of the King’s visits to Hannover. Leibniz remembered sadly the often obnoxious young man, whom he had watched grow up in Hannover and who now had great power. While they ate, the King noticed that Leibniz was somewhat subdued, in contrast to the King himself, who was visibly pleased with his lot in life. Toward the middle of September, Leibniz admitted in a letter to Caroline that he didn’t expect to be able to travel to England anytime soon, and that as a result he might never be able to do it. He had become an old man. By November of that year, he was so crippled by gout and arthritis that he took to his bed, giving up on his writing at least for a short time. His secretary prevailed on Leibniz to allow a doctor to be called. The doctor gave him some medication, which mercifully allowed him to sleep. When his condition visibly

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worsened the following day, the doctor recommended calling his pastor and lawyer, but Leibniz declined. “I’ll take care of those details tomorrow,” he assured the doctor. Around 10 o’clock in the evening of November 13, 1716, Leibniz died in his sleep—his tomorrow never came. His funeral and burial took place on December 14, 1716. His body had been laid in an ornate casket, on which was engraved his motto “Inclinata resurgit [when forced down, it rises again]” with the symbol of a spiral, the number 1 inscribed inside a 0 (signifying his binary arithmetic), a picture of a phoenix, which is born again from its ashes, and the words, “The ashes will retain the honor.” The service took place in the Neustädter chapel with the Court Chaplain officiating. The school choir sang and his few friends paid their respects. The King and his court, who were vacation-

Neustädter Hof-und Stadtkirche St. Johannis, where Leibniz is buried.

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Leibniz’s grave, in the Neustädter Kirche St. Johannis.

ing near Lüneburg, chose not to attend. Some have theorized that the King chose not to honor Leibniz’s funeral because he had never known Leibniz to take communion, but the likelier reason is that the King simply didn’t like Leibniz very much. It wasn’t until 50 years later that a suitable stone was placed above his grave. King George I may have seen Leibniz’s life as a failure. From his limited perspective, perhaps it was. After all, Leibniz never completed his history of the house of Braunschweig/Hannover. However, Leibniz accomplished much that was far more important than that inconsequential history might have been. He earned the respect of all the scholars of the European continent in his own time. He wrote extensively on philosophy, articulating his carefully reasoned principles brilliantly. He made

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Gottfried Wilhelm Leibniz.

many important discoveries in mathematics, the most important of which was his invention of the differential and integral calculus. Although he failed to construct a universal language by his own definition, that monumental project was beyond the ability of any one person. Nevertheless, it is not an exaggeration to see his calculus as the universal language—even if it is not a language that can save mankind from war—so in that sense he succeeded in developing a universal language as well. The calculus has certainly been one of the most important tools in the scientific progress of civilization. It is also the language that scientists, mathematicians, engineers, and many others use daily in their creative and practical work. Yes, Gottfried Wilhelm Leibniz was one of the greatest scholars of all time.

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It was Leibniz’s fate to have been born in a warravaged Germany that recovered only slowly from the Thirty-Years War. In fact, he was the only great German intellectual of the seventeeth century to overcome that disadvantage. We can criticize him for spending too much time and energy on his appearance and deference to his noble princes, but he needed their financial support to survive, and the only way he could do that was to join their often vapid society. We should be grateful that as a young man he refused to accept a professorship in the law school at Altdorf—choosing instead to pursue his career with all of Europe as his stage. His accomplishments were far greater than that former setting would have allowed. Sir Isaac Newton was knighted in England and buried in a prominent place in Westminster Abbey, honoring his impressive accomplishments in science. Although Leibniz was not honored in that way, his universal genius earned him respect both as a supreme mathematician and as a philosopher. Furthermore, what greater respect could we give than to use his calculus—not Newton’s fluxions? It is Leibniz’s notation that we use today. In the end, we must honor this extraordinary man and his many accomplishments. He was a genius who overcame social and economic adversity to accomplish remarkable feats. He was indeed one of the truly great men of science, and he deserves our highest respect.

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Index

A Ab Urbe Condita 9–10, 12, 16, 20 Académie des Sciences 65, 68, 74, 96, 109, 171, 177, 187, 195, 212 Acta Eruditorum 103–104, 144, 146–147, 160, 181, 215 Act of Settlement 202 See also English throne Aesop’s Fables 14 alchemy 51, 66, 119 alphabet of human thought see universal language Altdorf 41, 51, 61, 63, 227 Anne, Princess 202 Anne, Queen 218 Antibarbarus seu de veris principiis et vera ratione philosophandi contra pseudophilophos 64 Archimedes 98–99, 146 Aristotle 22–23, 26–29, 31, 37 arithmetic 6 Ars Combinatoria 78

atoms 30 August, Ernst 131, 140–142, 148–151, 157, 163, 173–175, 189 August, Georg 207 August, Rudolf 172, 183, 205 Azzo II, Albert 157

B Babbage, Charles 104 Bachelor’s degree 25, 29, 39 baptism 1 Barrow, Isaac 102 Bernoulli, Jacob and Johann x, 104, 146, 181, 186, 215, 216 best of all possible worlds 50, 115, 211 binary arithmetic 82–83, 183–186 von Boineburg, Johann Christian 62–64, 67, 71–72, 83–85, 91–92, 130, 138, 204 Boyle, Robert 148 Brand, Hennig 118

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230 Index

C

E

calculating machine 68, 80–83, 86–88, 90, 105–106, 113, 136 calculus xi–xii, 79, 99–105, 107, 108, 145, 181, 226, 227 calculus wars 182–183, 214–217, 220–221 calendar 198 Calvin, John 69 Calvinists 36, 49 Candide 211 Caroline of Ansbach 207, 218, 219–220, 221, 223 Catholics see Roman Catholics Charlemagne 15, 174 Chinese 66, 147, 162, 189 Christianity 49, 64, 67, 139 Chronological Work see Opus Chronologicum Collins, John 108 Comenius, John Amos 10–12, 17 Crafft, Johann Daniel 156 Craig, John 148 curves 98, 99

eclipses 11 Eclipses 8 Edict of Nantes 154 Egypt 68 Egyptian plan 71–73 electorate 150, 175 Elizabeth, Princess and Abbess of Herford 119–121, 127–129, 133, 170, 202 English throne 167, 195, 207 epistola posterior 108 epistola priori 107–108 Esperanto 66 Este 135, 151, 166 Euclid 31, 32–35, 74 Euclid’s geometry 6 Evangelical see Protestant extrema 99

D Democritus 29 Denmark 93 derivative 99–102, 101 Descartes, René 29, 50, 54, 96, 98, 102, 114, 117–118, 119–121, 127, 147 determinants 147 differential calculus 99 Diogenes 27 Disputatio arithmetica de complexionibus 41 Dissertatio de arte combinatoria 41, 66 Doctor of Law 41, 47, 61

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F Fermat, Pierre de 99, 102 fluxions 102, 104–105, 107, 108, 146, 160, 214, 227 Frankfurt 55, 155 Frederick III of Prussia 141, 201 Freiesleben, Anna Rosine 40, 46 Freiesleben, Christian 39–41, 95, 143 Freiesleben, Heinrich 19, 40, 45–46, 143 French 72, 115 Friedrich, Johann 62, 93, 106, 109, 111, 115, 128–130, 138, 149, 204, 209 funeral 224

G Galilei, Galileo 29, 50, 54, 99 geometry 6, 32 German language 6, 31, 38, 65, 153

Index 231 Greek 5, 27, 38, 60 Gregory, James 99, 107 Gregroire 76 Guelf 135, 162, 176, 188

H Hannover 93, 111, 136, 138– 142, 156, 167 Harmonic Triangle 78–79 Hartz Mountains 123, 152, 154, 178 von Hessen-Rheinfels, Ernst 138 Heyn, Friedrich 153, 161, 167 Hildesheim 154–155, 208 History of Rome see Ab Urbe Condita history of the House of Brunswick-Lüneburg 135, 150, 168, 174, 188, 190, 191, 194, 198, 201, 203, 205, 209, 220–223, 225 Holy Roman Empire 52, 56, 63, 68, 94, 142, 150, 156, 159, 175 Hooke, Robert 86–87 l’Hôpital, Marquis de 181 Hornschuch, Johann 21 Hudde, Jan 114 human thought 80 Huygens, Christian 73–78, 86, 90, 96, 97, 113, 142, 168

I individuation 29 infinitesimals 99, 215 integral calculus 101–102 inverses 79, 103

J Janua linguarum 10–11, 18 Jena 30

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Joseph I 209 Journal des Sçavants 67, 80, 144, 215

K Kahm, Johann Karl 111–112 Kepler, Johannes 99 Köln 63, 84

L laboratory science 51 language of thought see universal language Lasser, Hermann Andreas 61 Latin 5, 7, 12–13, 31, 38, 60, 65 Latin school 5–6 law 36, 37, 38, 39, 42, 50, 53, 58, 60 Leibniz, Anna Catharina 4–5, 39, 45–46, 50–51, 69–70, 94 Leibniz, Catharina 4–5, 16–19, 39, 40 Leibniz, Friedrich 1–4, 10, 14, 17, 27, 38 Leibniz, Johann Friedrich 36, 94, 95, 143 Leipzig, University of 1, 25, 36, 41, 60, 144 Leopold I 67, 159, 200, 206, 208 librarian 112, 115, 176 library 7, 17–19, 27, 62, 116, 118, 122, 134, 143, 205–206 linear equation 100 Livy 10, 12, 16, 20 See also Ab Urbe Condita Llull, Ramon 22–23 Löffler, Friedrich Simon 46, 69, 94, 209 Löffler, Simon 45–46, 50, 94 logic 5, 6, 37

232 Index London 65, 113, 126 Louis XIV 52, 71, 139, 154, 176–177, 222 Ludwig, Georg 140, 142, 189, 191–195, 198–199, 200– 201, 209, 217–218, 225 Lutheran 6, 36, 37, 49, 69–70, 94 Luther, Martin 65, 69

gentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus 103, 145 number theory 96 Nürnberg 51, 55, 66, 155

M

Oldenburg, Heinrich 63, 67–68, 86, 89–91, 99, 105, 106–108 Opus Chronologicum 7, 11, 12 oration 20–21, 42 Orbis Pictus 11, 15 orbits 8 Osnabrück 132 Ottoman Empire 64, 68, 142 Ozanam, Jacques 96–97

Mainz 52, 63, 91, 94, 111 Malebranche, Nicolas 119, 127 Mars Christianissimus 139 Master’s degree 39 mathematics 6, 39, 54–55, 66, 71, 73–80, 95–105 maxima 114 Mencke, Otto 144 method of exhaustion 98 minima 114 Modena 158, 164, 166, 174 Molière 65 monads 29–30 money 41, 50–51 moral issues 38 Morland, Samuel 87 Mouton 89 multiplication symbol 186

N Napier’s bones 87–88 Newton, Isaac ix–x, 90, 99, 102, 107, 126, 146, 160, 182, 214–217, 220, 227 Nicolaischule 5, 25 Nizolio, Mario 64 nobleman 16–19 Nova Methodus Discendae Docendaeque Jurisprudentiae 57 Nova Methodus pro Maximis et Minimis, itemque Tan-

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O

P Papin, Denis 148 Paris 65, 71–92, 94, 109, 126, 136, 142, 159, 176–177, 212 Pascal, Blaise 54, 68, 98, 107, 136 Pascal’s Triangle 54–55, 78 Pellisson, Paul 171–172 Pell, John 88–89, 95 perfect numbers 32–34 Peter the Great 188, 211–213, 222 von der Pfalz, Sophie 128, 131, 141, 161, 162, 166, 170, 178, 195, 210, 218 von der Pfalz, Sophie Charlotte 132, 140, 141, 151, 162, 170, 189, 193–194, 199, 202–205, 210 philosophy 22, 31, 36, 37, 41, 50, 60 See also logic

Index 233 phosphorus 118–119 physics 6, 67–68, 147 von Platen, Ernst 134–135 Plato 32 Poland 63 Pomponne 71–72 prediction 1–3, 4 Privy Councilor 111, 117 professor 47 Protestant 52, 55, 69, 118, 121, 150, 154, 187, 199

Q quadrivium 6 qualify 29, 30 quantify 29, 30 quantitative science 36

R Rheinfels 154 Roberval, Gilles Personne de 109 Roman Catholic Church 52, 127 Roman Catholics 49, 55, 64, 112, 118, 121, 154, 165 Rome 112, 165 Royal Society 65, 68, 86–87, 89, 105, 144, 216–217

S Saxony 1, 15, 30, 66, 73, 95, 151, 152 von Schönborn, Johann Philipp 52–54, 55–62, 62, 71, 84, 130 von Schönborn, Melchior Friedrich 83–86, 91, 111 van Schooten, Frans 98 scientific journal 67 Shakespeare, William 65 silk 199 silver mines 123–125, 142–144, 152, 156, 179

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slope 99–101, 100, 108 Societas Quaerentium 31–36 Societät of Sciences 126, 197– 199, 208, 212–214, 222 See also Académie des Sciences Specimen quaestionum philosophicarum ex jure collectarum 39 Spinola, Rojas y 117 Spinoza, Baruch de 114–115 squaring the circle 67, 98 Staffelwalze 80 See also calculating machine St. Augustine 34 St. Bartholomew 154 summa 101 summing a series 74–78

T tangent 96, 99, 100, 215 teacher 11–12, 13–19, 26 Theodicy 210–211 thesis 29, 39, 41–43 Thirty-Years War 1–2, 30, 66, 227 Thomasius, Jakob 25–28, 29, 31, 65 transcendental numbers 146–147 traveling chair 158–159 trial lawyer 38 triangular numbers 75–78 trivium 5 von Tschirnhaus, Ehrenfried Walther 106, 148

U Ulrich, Anton 172, 180, 183, 205, 209, 217, 218 universal language 30, 54, 63, 66, 73, 80, 189, 213, 226

234 Index

V Varignon, Pierre 215 Venice 164 Vienna 112, 149, 158–163, 167, 176, 200, 213, 217, 222 Voltaire 211

W Wallis, John 76, 102, 147 Weigel, Erhard 30–31, 54, 199

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windmill 123–125, 140, 143 Wolfenbüttel 151, 172–173, 176, 180, 183, 200, 205, 209, 218, 222 Wroclaw 14

Z Zamenhof, L. O. 66 Zwingli, Huldrych 69