The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time (2006)

Bibliographical Essay

TWO SUMMERS AGO, when I was first starting to seriously work on this book, my wife and I were not yet married and were living what would turn out to be our last carefree summer before she became pregnant. One night we had an overnight guest at our place in the Bankers’ Hill section of San Diego—an old friend from graduate school I hadn’t seen in years. After a few beers he began asking me what I was working on, and I did my best to give him a synopsis . . . Newton, Leibniz and their famous fight.

My friend looked puzzled. “How do you become an expert in something like that?” he asked me. Though I was loathe to call myself an expert, my answer was basically good source material and an extraordinary amount of scholarship by generations of writers and academics who were interested in every aspect of their lives and work.

After Leibniz and Newton died, they both left large piles of papers, books that they had bought, and their correspondence, and these papers have been well preserved through the years because of their obvious importance as the life’s work and thoughts of these two great men—from their boyhoods to their deathbeds and every stage in between.

This perception was especially true of Newton’s papers, and since he was so famous in England, his collection was instantly regarded as the treasure that it was. Ironically, as these papers were the embodiment of Newton’s intellectual legacy, this legacy may have suffered somewhat because of his fame. Newton had carefully gone through and ordered his papers before he died, but in the years following his death, his legacy was shuffled, reshuffled, reordered, and finally divided.

At first, these papers became the property of John Conduitt, the husband of Newton’s niece Catherine Barton, who was Newton’s favorite relation. Shortly after Newton died, a Dr. Thomas Pellet was appointed to examine the papers and select those that were publishable. Almost none of them were, according to Pellet, and some of the papers today bear the legacy of this examination in the form of a note on their covers warning, “Not fit for publication.” The only items in the entire mass that he selected for publication were short works on the chronology of the ancient kingdoms and a work called The System of the World, which Conduitt published soon after.

After Conduitt, the papers passed to him and Catharine Barton’s son Lord Lymington, and from there they passed to a Mr. Saunderson in London, and eventually on to the Portsmouth family. Later, one of the Earls of Portsmouth allowed the university access to all the papers, which by this time, were not in the best of conditions. Some were water stained, others partially burnt, and many pages were not numbered and had fallen out of order. Besides that, some of the papers were on mixed subjects. There were theological papers, for instance, with mathematical notes in the margins. The decision then was to classify the papers into subjects like alchemy, chemistry, mathematics, chronology, history, and theology, and so the entire collection was reordered accordingly. Then it was split, and the earl donated those papers that related to mathematics to Cambridge University while keeping Newton’s work on theology, the chronology of ancient kingdoms, and alchemy for himself.

From the nineteenth century onwards, Newton’s biographers have more or less all been able to draw upon his papers and correspondence to aid in their work, and in the twentieth century, this primary source material became especially accessible when the publication of a set of seven volumes of Newton’s correspondence was printed with notes and translations. The letters in this collection range from interesting historical texts to completely banal messages, such as the letter Newton wrote to Humfrey Ditton, March 16, 1714—right in the middle of the calculus wars. The letter reads, in its entirety: “Sir, If you please to call on me Friday morning next about ten of ye clock you will find me at home. I am Your most humble Servant Is. Newton.” Other letters were much more valuable to me in the writing of this story, since they dealt directly with the calculus wars, and I have referred to these letters and in many cases quoted them directly throughout my book. Another useful work for a few of the early letters written by and about Newton was The Correspondence of Henry Oldenburg, Volume IX.

I should say that in many cases, I have taken the liberty of modernizing the spelling of certain words when I quoted from these letters. Words like “philosophicall,” “concerne,” “planetts,” “centrall,” and many more were altered to get rid of the extra vowels and consonants and others such as “ye” and “wch” were replaced with their obvious modernization. I also Americanized certain spellings like “favor.” I’m sure some would bristle at the arbitrariness of my decision, but I felt that these spellings detracted rather than added anything, and so with “aapologies” to the editors of Newton’s correspondence . . .

In addition to the seven volumes of Newton’s correspondence, the Principia and Opticks are still in print and readily available. There are also numerous books, some of which can be found in the bibliography that follows, that excerpt passages and comment extensively on these texts. The most comprehensive and useful commentaries I found on Newton’s great works were one by A.R. Hall on Opticks called All Was Light and an Introduction to Newton’s Principia by I.B. Cohen.

These works are just the beginning. So much has been written about Newton, and so many times have his old writings and notes been gone through that there seems to be no end to Newtonian scholarship. People have read and printed and psychoanalyzed lists of words he wrote as a boy practicing his Latin grammar, and I once read a study by a top scholar looking at the way that books in his personal library were dog-eared—and what those dog ears reveal about his thought on important passages in books that he owned. And then there are the biographies—several of which I can mention.

The one biography I relied upon the most was Never at Rest by Richard Westfall, which was extraordinarily complete. Frank E. Manuel’s Portrait of Isaac Newton was a very interesting read, particularly for his take on Newton’s 1693. I also liked the shorter and earlier work Sir Isaac Newton by Andrade. I found a useful book about Newton’s time at the mint in Craig’s Newton at the Mint. Of the older works, I enjoyed the great 1855 two volume Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton by Sir David Brewster. Another early book I read was Birch’s History of the Royal Society, which provided some of the specific details for Chapter III.

There were a number of sources of information regarding Newton’s growing fame and celebrity—the most obvious manifestation of which was his elaborate funeral, ornate tomb, and the explosion of poetry and art that invoked him. Some of the most interesting reading on Newton’s influence of the worldview was written by Alexander Koyré, which I found in his Newtonianism. Koyré also expounds the clash between Newton’s and Leibniz’s metaphysics in an essay in the book by Frankfurt. This essay deals largely with the Leibniz-Clarke Correspondence, which is itself readily available in print in nicely translated and annotated editions.

Also helpful to me for understanding Newton’s place in the world at the time that he died was the book by A.R. Hall called Newton: Eighteenth Century Perspectives, which contains some interesting biographies that appeared about him shortly after he died. Another book by Hall titled The Revolution in Science 1500-1750 has a chapter devoted to Newton’s Legacy, and yet another useful book that contains this sort of commentary is Let Newton Be!, which was edited by Fauvel et al.

A very visual presentation of Newton’s influence, as seen in the art and writings of many in the eighteenth century, was the two-part show at the Huntington Gardens and Museum in Pasadena, California called All Was Light. This show, along with the companion book, The Newtonian Moment, by the show’s curator Mordechai Feingold, were both very helpful to me because they presented copies of some of the original documents of the calculus wars, such as Newton’s famous 1676 letters, and they focused on the growth and general acceptance of Newtonianism following his death.

Leibniz, too, left a pile of books, papers, and hand-written materials after he died, and because he spent his final years in the court library at Hanover, his collection of books and papers was naturally kept here as well. This created an interesting dilemma for King George and his family because Leibniz’s papers were not solely important for their intellectual content. He had written numerous memoranda on subjects of courtly interest, political intrigue, and the goings-on at Hanover. As the new king of England, George was worried that these might shed a bad light on him or his family. When Leibniz died in 1716, George had barely been on the throne of England for two years, and the enemies to his reign were numerous. While Leibniz had been a loyal subject, his papers in the wrong hands might have provided some form of ammunition against George, so he took possession of everything.

This created a minor controversy as Leibniz’s relatives had expected to inherit his books and papers. This was no insignificant inheritance—books were valuable items in those days, and Leibniz was famous so his papers were not without value either. The family took George to court, and the trial stretched on for years, decades, and was not decided for fifty years. Eventually the heirs were compensated for the value of the books, but the delay and the ultimate decision in the lawsuit meant that the pile of Leibniz’s writing were kept essentially in one collection.

What a pile of papers it was. Leibniz left an overwhelming glut of papers, notes, and especially correspondence. By his own estimate, Leibniz wrote some 300 letters a year, which means that in the course of a decade, he would have written some 3,000, and over the five decades of his adult life, he would have written about 15,000—so much material, in fact, that by one estimate if a person sat down to read everything Leibniz ever wrote, assuming they read about 8 hours a day, it would take more than 20 years just to read all this writing—assuming, of course, that they could read the Latin, German, French, and the occasional Dutch and English, that Leibniz corresponded in. “It would seem, indeed,” one nineteenth-century biography put it, “as if these writings were a mine which could not be exhausted.”

In today’s world of email and text messaging, it may seem like a simple matter to send 300 letters in a single year—sometimes one might send three hundred emails in a single week. But there was a profound difference in what Leibniz was writing. Leibniz did not just dash off messages fit only for a chatroom (“LOL CU L8er loser”) the way that people do today. Many of his letters were more like scholarly papers—the sort of which were fit for publication then and continue to be published today.

This is apparently not the easiest collection of papers to work with. Read Leibniz from the original sheets and you are not merely reading the words you are reading the scratch outs and the additions—all of which combine into a complicated fabric of a genius mind spilling forth, sometimes uncontrollably so. Copies of a few of his original letters are on display in the Leibnizhaus museum in Hanover, Germany. They are impressively detailed. His writing is tiny and exact, though in a script that was no doubt as thick as his accent was. In the tradition of the time, he writes over the entire surface of the page, sometimes writing additional comments vertically across the margins.

Perhaps because Leibniz’s legacy was an unfinished encyclopedia instead of an opus, a large book for which he is primarily remembered the way that Newton is for the Principia, it was somewhat difficult to assemble a complete picture of his views. Some might argue that such a complete picture still does not exist anywhere, since despite nearly two centuries of intense study of his work, his complete works are still not published.

For years, a number of scholars have been undertaking the Herculean task of compiling the complete writings of Leibniz. The first attempts at this were made more than a century ago when a librarian in Hanover named G. H. Pertz tackled the history as his part. His colleague C. L. Grotefend helped him with the philosophical work, and C. I. Gerhardt helped him with the mathematical works. These mathematical works encompassed seven volumes, which were published in the mid-nineteenth century. And a few decades later Gerhardt contributed another seven volumes of philosophical works. Another eleven volumes of historical and political writings were produced by an O. Klopp, and an L. A. Foucher de Careil came out with seven volumes of history, politics, and church reunification.

Since this initial effort, a longer and more comprehensive effort has been underway to collect the complete works of Leibniz. The effort has been proceeding for several years without interruption in Germany at the library known as the Niedersachsische Landesbibliothek, a low-rise modern glass and concrete structure in the center of Hanover, which I visited during my research. Here and elsewhere scholars are collecting his letters, papers, and manuscripts into areas like law, politics, theology, history, philology, logic, geology, mathematics, and physics, and the effort is underway.

To date, more than half of what Leibniz wrote has been edited and published in some form or another, and as of March 2005, some 42 volumes of these writings had been collected in this definitive collection. Each volume comes in around 800 to 1,000 pages, and this is something less than half of the total. I have read that this work started in 1923, and one scholar estimates that when everything is finally collected, there may be perhaps 110 volumes in all. They are not yet halfway done with the effort, though it is estimated that they may reach the halfway point in the next decade.

Why was there so much writing? Leibniz traveled so extensively throughout Europe and maintained contact with the outside intellectual world through his extensive correspondence. He was willing to enter into correspondence with almost anyone. Many of these letters have been translated into English in individual books, which I purchased and read in the course of my research. Most notable were the translations by Leroy Loemker of several hundred pages of philosophical papers and letters. Also important for my efforts was a 1925 book entitled Early Mathematical Manuscripts by J. M. Child.

In addition to these “primary” sources, I drew heavily on a few biographies of Leibniz for this book. In the nineteenth century, there was an explosion of Leibnizian scholarship and a rediscovery of the value of his old papers and letters—at least a portion of them. A definitive biography, by a German scholar named Dr. G. E. Guhrauer, appeared in Germany in 1842 and drew heavily on the old papers. A biography based on Guhrauer’s work appeared in English in the mid-nineteenth century and was an enjoyable read for me. I refer to John Milton Mackie’s Life of Godfrey William von Leibnitz, and this work provided many translations of Leibniz’s letters from which I was able to pull quotes. Another mid-nineteenth century sketch that was useful to me was a review of Guhrauer’s work that appeared in the Edinburgh Review in the mid-nineteenth century.

Worth noting as an aside is that there are many instances, especially in the older literature, in which Leibniz’s name is spelled with a “t.” Indeed, Newton, Keill, and many of Leibniz’s contemporaries preferred to spell it that way, and the spelling persisted in English language works for more than a century after Leibniz died. In my book, I chose to use only the spelling without the t, and for the sake of avoiding confusion, I removed the alternative spelling where it appeared in the quotations of others.

A modern treatment of Leibniz’s life can be found in Aiton’s 1985 book Leibniz which is perhaps the best English language treatment of his life and works. Curiously, Aiton largely ignores the controversy over the invention of calculus, touching on it only incidentally. Nevertheless, without Aiton’s thorough scholarship, it would not have been possible for me to penetrate the character of Leibniz nor assemble the facts that are represented in this book’s narrative.

There were several other biographies that were also helpful to me. Hofman’s book Leibniz in Paris was a thorough and excellent examination of the years 1672-1676. Another interesting work, though much shorter, was Ross’s Leibniz. Also helpful was a brief biosketch on Leibniz in Benson Mates’s The Philosophy of Leibniz and a similar chapter in Jolley’s Cambridge Companion to Leibniz.

In addition to these, there were a number of other books I read on Leibniz’s work in other areas that I only paid cursory attention to in my text. Leibniz’s philosophy, political writings, and his writings on China, to name just a few areas, are rich and interesting, and while I read a few books on these subjects with interest, I was not able within the limited confines of my narrative to include everything—since my primary concern was the calculus wars.

The fight between Newton and Leibniz was so legendary that nearly every bio-sketch I found about either man touched on the calculus wars in part. And where some biographers, like Aiton, seem to consciously ignore the dispute, others, like Newton’s biographer Westfall, devote considerable attention to it. To my knowledge, mine is the first book to tell the story of the calculus wars in a popular form, although Hall’s Philosophers at War is an excellent scholarly history of the fight. For readers who want to know more about the details contained in this book, Philosophers at War is a great place to start.

Finally, suffice it to say that nobody can approach the writing of a story like this, which took place in the late seventeenth and early eighteenth centuries, without also becoming familiar with those times—the general political history of Europe in those days and the scientific revolution as a whole. I spent many afternoons perusing in the stacks of the central branch of the San Diego public library, and I have listed several books in the bibliography that helped me get more of a handle on those times. The books most useful for research on the House of Hanover were Redman’s The House of Hanover and Black’s The Hanoverians. Useful biographical information on some of the other mathematicians from the seventeenth century came from A History of Mathematics by Carl Boyer. Boyer’s History of Calculus was also a useful read.