Figure 1. This particular dispute concerns the squabbles that Isaac Newton - arguably the greatest scientist that ever lived - had with Gottfried Leibniz. The basis underlying many of Newton's disputes was scientific priority; who made the claims and discoveries first ... and was plagiarism involved? For example, Newton's squabbles with Robert Hooke had, at its roots, the tussle between the two behemoths of physics in England at the time and the unwillingness of either to give any ground to the other. In fact, Newton, bored, irritated and tired of Hooke's criticisms, decided to withhold publication of one of his greatest books, Optiks until after Hooke's death. Newton's battle with Leibniz, which is the main theme of this article, was over the discovery of the calculus. Perhaps, like the controversy between Hobbes and Wallis over geometry and algebra, the vast majority of people might think that a squabble over the bragging right to the calculus would be of little interest. Certainly, in my experience most undergraduate students who have to struggle with calculus show little concern for who it was who discovered it first! However, the conflict that developed between Newton and Leibniz was not a battle between two mismatched enemies, nor, indeed, as I mentioned before, was that between Newton and Hooke. Both Hooke and Leibniz were highly accomplished and revered men. For example, Leibniz has been commemorated many times on stamps and coins, see figure 2.
So, what's so important about the calculus that people would want to argue about its origin? Well, basically there are two parts; differentiation and integration. The main point about differentiation is that it deals with the tangent or slope of a line or curve. Thomas Hobbes, would have argued that we could construct a tangent using geometry, and he would be quite correct. But his adversary John Wallis would have argued that the same result can be achieved using algebra with more accuracy, and he would be correct also. Today, differentiation uses algebra; but the calculus of Newton and Leibniz is a major advance over, say, what Wallis could do, because it allows us to determine the slope of any curve. Normally, a curve tells us how one quantity changes with another, like position and time. An example would be your journey from home to FAU to be here today, see figure 3; we can see how the distance you have traveled measured by your odometer changes with the passage of time. The reason that the tangent or slope is so important is because it tells us the rate of which something changes, i.e., in this example we could work out your speed at any instant without using a speedometer. Without going into great detail, the rate at which things change, is fundamental in science.
Integration allows us to measure the areas and volumes of complex shapes. For example, one problem that many mathematicians struggled for centuries with was how to measure how much wine would fit into casks and barrels, i.e., the volume of wine casks and barrels! For example, it was a problem that the famous astronomer Johannes Kepler had worked on around 1600 but hadn't managed a good solution. As we saw in the Wallis-Hobbes dispute, the Greek scientist Archimedes had tried to work out the area of a circle by using inscribed polygons of every increasing numbers of sides. Wallis, as we saw, used a somewhat different method, by splitting a quarter of a circle into an ever increasing number of strips whose areas we can calculate. Similarly, as shown in figure 4, the volume of a barrel can be approximated by adding the volume of a series of disks with different diameters. It is these ideas, of using infinitesimally small quantities, that are used in the calculus. To the non-mathematician this may all seem esoteric; even Voltaire in his usual acerbic fashion once described the calculus as
So, neither Newton nor Leibniz created their versions of the calculus from scratch for by the mid-1660's some of the basic ideas were in existence. However, the major difference and major advances made by Newton and Leibniz, was that their calculus literally works for any curve or any shape; prior to them only specific cases could be attempted using either algebra or geometry.
So, if this is what the argument was all about ... who actually discovered the calculus first ... who has the bragging rights ... Newton or Leibniz? Well, let's see ... but first, the background.
There had never been a scientist like Newton nor has there been one like him since, see figure 5. Nobody else has measured up to anywhere near his stature. Indeed, it is extremely unlikely there can ever be a scientist like Newton again, for the scientists of future generations will be able to draw on a wealth information from a wide range of sources; computers, books and libraries, and so on. Newton had virtually nothing, except for Galileo's mainly qualitative thoughts and ideas and Kepler's laws of planetary motion. With little more than that, he formulated three laws that govern all motion in the universe and the universal law of gravitation; these laws have withstood the test of time for over three centuries. Although some of the classical ideas and prejudices of other scientists from the classical period needed modification under the impact of Einstein's theory of relativity, Newton's laws have come through unscathed. In addition to his unique contributions to terrestrial and celestial dynamics, mechanics and gravitation, he made major and fundamental discoveries in optics and developed (and made) the first reflecting telescope. In fact, one can easily argue that on the scale of all he achieved, the discovery of the calculus was the least spectacular.
His adversary, Leibniz, see figure 6, was a world-class intellectual also, who had blossomed at an early age. He rose to become the leading natural philosopher outside England - the "Continental Newton" if you like. Although he is far less well known than Newton, in many respects he was both broader and deeper than Newton and, today, his reputation as a mathematician is still rising. Leibniz has been variously described by biographers as
He was interested (and wrote) on history, economics, theology, linguistics, biology, geology, law, diplomacy and politics, as well as mathematics, celestial and terrestrial mechanics and speculative philosophy. Frederick the Great of Prussia (Frederick II) described him as
Yet, as we will see, he wasn't an academic, he held no academic position; he had been educated in law and supported himself by acting as a librarian and court agent and by doing legal and diplomatic work - for example, tracing history and genealogy - for members of the nobility in his native Germany. Also, he had a deep interest in metaphysics, which was part of the reason he and Newton could never agree.
Perhaps, then, confrontation was inevitable; for how could two such geniuses live contemporaneously without arguing with each other? There is no doubt that at the time, Newton got the better of Leibniz, but maybe, in all fairness, and with the luxury of hindsight, some might say that, today, it could be approaching a tie.
So, let's take a closer look at the men themselves, and their feud. Isaac Newton was born prematurely on Christmas Day, 1642 in Woolsthorpe, Lincolnshire, England, the same year that Galileo died. Figure 7(a) shows a sketch of the house - drawn in 1727 - where Newton was born, and figure 7(b) shows what the house looks like today. He was not expected to survive; it was said that he was:
Occasionally, you may see his birth date as January 4th, 1643 as here figure 8, but the latter date is the Gregorian calendar date, which was adopted in 1752, some 25 years after his death.
Newton's life can be divided into three distinct periods. The first is his boyhood days up to his graduation at Cambridge University. The second period, from 1669 to about 1690, while he was Lucasian Professor of Mathematics at Cambridge, was his most highly productive period although the foundations had been laid a few years earlier while he was still a student. In the third period, from roughly 1690 until his death in 1727, which was almost as long as the other two combined, Newton was a highly paid government official in London and the time when he became more interested in the "business of science" as an administrator.
During the 17th century Woolsthorpe was little more than a collection of small farms and humble dwellings clustered around the manor house. His family were farmers but his father died, at the young age of 36 years, before Isaac Newton was born. There was considerable civil strife in England at the time, between Parliament and King Charles I. It is not clear what side the Newton's were on; most biographers suggest they had Royalist leanings because their class and social aspirations would have them favor the status quo. However, there were strong religious issues as well; the Civil War was as much to do with the ideological clash between the Roman Catholic Church and the relatively recently formed Church of England. King Charles' sympathies were with the Catholics, but that view would have not been shared by the Newton's. In fact, in later life, Isaac Newton became positively anti-Catholic. Perhaps, then, the Newton's, like so many others, simply "bent with the wind".
In late January 1646, when Newton was 3 years old, his mother married Barnabas Smith, the rector of a nearby hamlet, who was well into his 60's; Hannah Newton was probably around 30 years old at the time. To us today, the marriage looks very much like a business transaction; part of the deal was that Hannah would move to live with her new husband but Isaac would remain in Woolsthorpe to be looked after by his grandparents.
When his step-father died in 1654 his mother sent him a grammar school in nearby Grantham. Up to then his performance at school was lackluster and unexceptional. (Indeed, there was nothing in his background to suggest that he would become the world's greatest scientist for his father had been illiterate, signing his will with an "X"!) Grantham was 7 miles away from his home - too far to travel each day - and so he lodged with the family of the local apothecary, the Clark's, next to the George Inn. It is certainly true that Newton's knowledge of "primitive chemistry" and his later obsession with alchemy was formed during this time, while he watched Mr. Clark produce "cures". Newton was also able to carry out his own experiments. Although, again, he was not an exceptional student, his schoolmaster, Mr. Stokes, realized he was certainly a likely candidate for university entrance. However, his mother wanted him to manage the family farm and in late 1658, when he was nearly 16 years old, and against the advice of his schoolmaster, his mother removed him from school. His surviving notebooks - meticulous in their detail - attest to the fact that this was clearly an unhappy time for him. Stokes, the schoolmaster, appealed to Hannah, his mother, a second time and he also obtained the support of Newton's uncle, the Reverand William Ayscough - himself, a graduate of Trinity College, Cambridge. She relented and the young Newton returned to Grantham in late 1660 to prepare for Cambridge University.
He entered Trinity College, Cambridge a year later, in 1661, see figure 9. Entering at age eighteen years he was three or four years older than most undergraduates. His aim was to get a law degree. His instruction was dominated by the philosophy of Aristotle but in his third year he was allowed some freedom of study. Newton chose to read the philosophy and the mathematics of, amongst others, Descartes and Wallis and the Copernican astronomy developed by Galileo. His true genius was beginning to emerge; and it was given a significant, positive jolt with the arrival of Isaac Barrow as the first Lucasian Professor of Mathematics at Cambridge in 1664, see figure 10(a) and figure 10(b).
Even though he brushed up on his Aristotle and Euclid, Newton clearly did little in the way of formal preparation for his Bachelor's examinations in the Spring 1665. He did graduate, but what is laughable now, he only achieved a second-class Bachelor of Arts degree. (In fact, undistinguished undergraduate careers among great scientists is not unusual; for example, Charles Darwin's father - we'll meet Darwin in a later essay - removed his son from medical studies in Edinburgh because he thought he was wasting his time, Albert Einstein barely scraped through his degree and then found it difficult to get a job, and more recently, Stephen Hawking, who spent more time on the river in Oxford than in the lecture rooms, was essentially forced to leave Oxford to go to Cambridge.) Newton had decided that his vocation was to unravel the laws governing the universe rather than passing exams. And after gaining his degree he had the official sanction to continue his studies.
Then suddenly, in 1665, the university was closed because of the Great Plague (or more correctly, the bubonic plague). Although not the first in English history, but coming as it did straight after the Civil War, and taking the lives of almost 100,000 people - some 70,000 of them in London alone, representing over 14% of the City's population - it was seen by many as yet another fulfillment of the prophecies in the Book of Revelation. Accordingly, Newton left Cambridge for Woolsthorpe, probably at the end of June or beginning of July, and did not return to Cambridge, except for a brief spell in early 1666, for almost two years. And it was during these two "miraculous" years that he began the groundwork for most of his revolutionary work in physics, optics, astronomy and mathematics that we are so familiar with today. In particular, he laid the foundation for differential and integral calculus although his notes indicate that he started to think seriously about it in February 1665, even before he left Cambridge. He termed it "the method of fluxions" and it included his crucial insight that integration is merely the inverse of differentiation. Taking differentiation - or, as Newton would have said, "finding the fluxion of a fluent" - as the basic operation he produced simple ways that unified many separate ideas and techniques previously developed to solve such apparently unrelated problems such as finding areas, tangents, the lengths of curves and the maximum and minimum values of functions. In 1669, he wrote a short manuscript on the method entitled De Analysi per Aequationes Infinitas (On analysis by Infinite Series), which he showed to a few people, including Isaac Barrow, the Lucasian Professor, who urged him to publish it. But, he would not agree. Why? ... because of his almost pathological fear of criticism. He wrote De Methodis Serierum et Fluxionum (On the methods of series and fluxions) in 1671 but again, he didn't publish it; indeed, it did not appear until an English translation was produced in 1736. So, at age 23 years, and still a student, Newton had surpassed all the leading mathematicians in Europe - and almost no-one knew it except a few close colleagues! He returned to Cambridge early in 1667, and in March 1668 he obtained a MA degree and become a major Fellow of Trinity College. By then, however, he had developed his version of the calculus and he had turned to other scientific endeavors, principally continuing his studies of light and optics. On October 29 1669, thanks in part to his unpublished work on the calculus, he was selected to succeed Isaac Barrow as the Lucasian Professor of Mathematics at Cambridge; a position held currently, incidently, by Stephen Hawking. The ambitious Isaac Barrow had resigned to become the Royal Chaplain to King Charles II, but had recommended to the Lucas estate that Newton succeed him. Newton's rooms were on the second floor, between the Great Gate and the Chapel, see figure 11. Today, an apple tree stands on the lawn outside the College and to the right of the Great Gate, and is overlooked by Newton's former rooms, see figure 12.
So now that we have looked at the background to Newton's development of the calculus and some of his reasons for not publishing it, let's turn our attention to Leibniz.
Gottfried Wilhelm von Leibniz, see figure 6, was born on July 1st 1646 in Leipzig. His father, Friedrich Leibniz, who was a professor of moral philosophy, died when Gottfried was only 6 years old. He was brought up by his mother, Catharina, who was Friedrich's third wife. He learnt his moral and religious values from her and he grew up in strict Lutheran piety; something that was to be with him throughout the rest of his life. He went to school in Leipzig and although he learned some Latin in school he was so highly motivated that by age 12 he had taught himself advanced Latin and Greek, possibly driven by the desire to read and understand his father's books. During his early schooling he was taught Aristotle's logic and methods of categorizing knowledge. But he was not satisfied; he started thinking about and developing his own ideas, even at this young age. He read widely; books on metaphysics and theology by both Catholic and Protestant authors.
In 1661, aged 14 years, he entered the University of Leipzig to take a degree in Law (like Newton, as we saw). It may sound to us today that this is a young age, which is was, but it is also likely that there would have been others of similar age. He studied philosophy, which was well-taught at Leipzig, and mathematics, which was not! He also studied rhetoric, Latin, Greek and Hebrew and graduated with a bachelor's degree in 1663, after just two years, with a thesis entitled De Principio Individui (on the Principle of the Individual). During his time as an undergraduate, Leibniz developed a deep interest in, and an understanding of, mathematics principally through his appreciation that proof, an essential ingredient in mathematics, could also be applied in logic and philosophy.
In October 1663, aged 17 years, he started his studies for a doctorate in Law. He received a master's degree for a dissertation that combined aspects of philosophy and law with mathematical ideas; you'll note a parallel here with Thomas Hobbes. Incidently, a few days after he presented his master's dissertation, his mother died. In his research, he continued to try to reduce all reasoning, logic and discovery, to a combination of basic elements such as numbers, letters, sounds and colors and eventually produced a doctoral dissertation entitled Dissertatio de arte combinatoria (A Dissertation on the Combinatorial Art) in 1666. Some scholars see this as providing an early theoretical model for the modern computer. However, despite his growing reputation and acknowledged scholarship, he was refused a doctorate at Leipzig. It is not entirely clear why he was refused, perhaps he was considered too young (at 20 years old) although there is another story that, for some reason, a Dean's wife had persuaded her husband to vote against Leibniz. Whatever the reason, he left Leipzig immediately and went to the University of Altdorf-Nuremberg, where he did indeed receive a doctorate in Law in February 1667, less than a year after leaving Leipzig, submitting another dissertation, de Casibus Perplexis (On Perplexing Cases). It was a brilliant piece of work and he was offered a professorship, but he declined. During the next few years he undertook a number of different projects, scientific, literary and political. He also continued his law career taking up residence at the courts of Mainz. He proved adept at administration and was employed for a short time as a diplomat and lawyer for the Elector of Mainz, Johann Philipp von Schönborn, working to improve the Roman Civil Law code for Mainz. At the same time he also worked as secretary, assistant, librarian, lawyer and advisor to Johann Christian von Boineburg. As you can see, Leibniz was truly multi-talented.
One of his lifelong aims was the reunification of the Christian Churches and he drafted a number of monographs on religious topics, mostly concerned with points at issue. Another of his aims was to collate all human knowledge; as part of this ambition he tried to bring the work of various learned societies together to coordinate their research. He was fully aware of the pioneering work of Galileo, Kepler and Descartes and he had, himself, made some studies of motion and collisions. In 1671 he published Hypothesis Physica Nova (A New Physical Hypothesis) in which he claimed that movement depends on the action of a spirit; this was similar to Kepler's argument that it was angels that kept the planets orbiting the Sun. He sent letters to the Royal Society in London and even dedicated some of his work to the Royal Society and the Paris Academy of Science. But above all, he wanted to visit Paris to make scientific contacts. In 1672 he started constructing a calculating machine that he thought might help him make those contacts. He had also formulated a political plan to try to persuade the French to attack Egypt, and that provided him the means to visit Paris, on behalf of his friend, Johann Christian von Boineburg. von Boineburg was trying to divert Louis XIV from attacking Germany and instead to set up a peace congress. Although Leibniz made a number of scientific contacts, the peace mission in France failed.
In January 1673 Leibniz traveled to London with a similar mission. He visited the Royal Society and met some of the leading English scientists of the day, like Robert Hooke - who as we will see in another essay, proved to be a particularly sharp thorn in Newton's side - and showed them his incomplete calculating machine. Although he did not meet Newton, Leibniz learned of a certain John Collins, a book publisher, and someone who had maintained a sporadic correspondence with Newton. As a book publisher, Collins was fully aware of Newton's (unpublished) works, particularly his recently conceived calculus, having urged Newton, unsuccessfully, to have it put into print.
Leibniz was elected a Fellow of the Royal Society on April 19, 1673. After a short stay in England he returned to Paris only to find that his principal patron, the Elector of Mainz, had died. So, suddenly, he was out of a job! Instead of returning to Germany and seeking new employment, he decided to stay in France choosing to live on the verge of poverty while following his intellectual pursuits.
And so began Leibniz's "two miraculous years"; working in almost complete isolation between 1673 and 1675 he mirrored Newton's own development a decade earlier. When he visited England he was simply an ordinary mathematician but after his two miraculous years he had become a creative genius; one of his developments was his version of the calculus. As he sat doing his work he needed contact with the broader scientific community and he was in regular correspondence with the Danish scientist Christiaan Huygens and through Collins and Henry Oldenburg, the secretary of the Royal Society, he felt he could tap into the rich network of European mathematicians and philosophers who corresponded with the Royal Society. As his ideas flowed Leibniz sent them to Collins who, although no mathematician himself, could understand well enough their importance. In return Collins sent Leibniz the latest ideas circulating within the Royal Society - a well meaning idea that, as we will see Newton later interpreted as an act of betrayal. This correspondence was to prove crucial later in their public battles when Newton accused Leibniz of stealing his ideas. Yet, in Leibniz's defense, for the most part the material consisted mostly of the non-technical details of current mathematical debates. Indeed, one document, sent by Collins in April 1675, on recent advances by Newton among others, didn't reach Leibniz until after he had developed his own method. It was clear that in November 1675 and later in the Fall 1676 he had developed most of his ideas on differentiation and integration.
As I mentioned just now, Newton did not meet Leibniz when the latter was in London in 1673, and the Lucasian Professor was probably not aware of him until 1675. However, within months of Leibniz's visit both Collins and Oldenburg began to appreciate that potential trouble was in the air. Both were well aware of Newton's discovery of the calculus some 10 years earlier and although the Newtonian and Leibnizian approach and notation were different - actually, we tend to use the less-cumbersome Leibniz notation today - both Collins and Oldenburg were fully aware that their techniques produced the same results. Furthermore, the latter pair knew just how touchy Newton was through his angry reaction to Robert Hooke's criticisms of his work on Light and his threat to resign from the Royal Society. Convinced that Leibniz was about to publish his discoveries the two of them tried to pressure Newton into publishing his version first.
Collins the publisher, in particular, knew he was caught between a rock and a hard place. The only solution as far as he could see was to get Newton to publish his 10-year old results straight away. Firstly, he knew he couldn't admit that he had passed information to Leibniz without Newton's permission for that would have produced outrage. Secondly, it was he, Collins, who had essentially forced a reluctant Newton to allow him to publish his work on the Theory of Light and Colours that had later caused great conflict with Robert Hooke. Collins knew therefore that Newton would be unlikely to give him permission to publish his work on the calculus. (In addition, Collins had been one of those who had urged Newton - unsuccessfully - to publish his calculus over 5 years previously.)
Eventually Oldenburg persuaded Newton to write to Leibniz on the pretext that Leibniz had some mathematical queries that only he, Newton, could answer. Reluctantly, Newton wrote two major letters through Oldenburg to Leibniz. The so-called First Letter - written in June 1676 - was 11 pages long and the so-called Later Letter - written in October 1676 - was 19 pages long. Together, they summarized Newton's mathematical discoveries and were designed to show Leibniz that he had made a number of breakthroughs many years ago. But even then, nervous that others would steal his ideas, Newton did not mention the calculus specifically; instead, he added a coded sentence - what today we might interpret as a patent establishing his priority. He wrote
This encryption defined the meaning of the calculus (explained later). In a covering note to Oldenburg with the Later Letter Newton wrote:
Two days later Newton wrote again to Oldenburg saying:
Both letters took a long time to reach Leibniz - the Later Letter took about 8 months to reach him by which time he had moved to Hanover. He had wanted to stay in Paris at the Academy of Science but no invitation came and so he was forced to leave Paris in October 1676 to take a position with the Duke of Hanover as librarian and Court Councillor. He traveled via London, where again he met Collins, who somewhat unwisely allowed him free access to his collection of papers and correspondence. Again, when Newton was to discover this later, he accused Leibniz of theft and Collins of complicity. However, there is some evidence that Leibniz ignored Newton's papers on the calculus, adding weight to later arguments by Leibniz's supporters, that he had already devised his own methods by then.
All was quiet for a while; there was no correspondence between Leibniz and Newton since they were both busy. Newton, in Cambridge, was was concentrating on alchemy and the writing of his Principia Mathematica. Leibniz, in Hanover, described his duties as
In 1679-80 Leibniz undertook a project involving the draining of mines in the Harz mountains; he tried using water and wind power to operate pumps, he designed windmills, pumps and gears, but it ended in failure. However, he did accumulate enough geological evidence to help him form the hypothesis that the Earth was first molten. He also worked on the binary system of numbers and his metaphysical system to reduce reasoning to an algebra of thought. Furthermore, he was asked to produce a history of the Guelf family, of which the House of Brunswick was a part, which, despite involving an incredible amount of work, was never published.
Then in October 1684, after a delay of some 9 years, Leibniz delivered his first paper on the calculus in Acta Eruditorum, a learned journal produced by the University of Leipzig, and Newton's name wasn't mentioned! Surely, he must have known about Newton's work! Newton's reaction is not recorded but the amount of duplication with his own, as yet unpublished, work must have come as a complete shock. Obviously, he had felt quite secure that his work was unique, and years and years in advance of anything anyone else could achieve. Furthermore, he must have been especially surprised that it was by Leibniz, for the latter had asked him for help some 8 years earlier and he had responded with two long letters. However, the fact that he was still working feverishly to finish the Principia - which he completed in 1686 - may have helped to cushion the blow, at least, initially.
Newton had been scooped but even this event did not trigger him to go into print himself; although he did give a hint of his calculus in his Principia and John Wallis mentions it in 1693, but the first, main reference to it was in an appendix to Newton's other great book, Optiks,, which was published in 1704, 20 years after Leibniz's publication.
His first move to claim priority was to add a passage in the Principia, (the Scholium to Book II, Section II, Proposition VII) referring to the letters he sent to Leibniz, with the coded reference, about the calculus. (Interestingly, in a later edition of the Principia published after his battle with Leibniz, Newton refers the reader not to the letters he sent Leibniz, but to a letter he had sent to Collins dated December 10, 1672, announcing his version of the calculus.
Now, Leibniz, probably spurred on by the pending appearance of Newton's Principia, published a second paper on the calculus in 1686, again with no mention of Newton!
Leibniz was totally oblivious to the furor that was brewing in Cambridge. Indeed, when the prospect of priority was brought to his attention he said he thought it would be possible to announce it as a parallel invention ... little did he know Newton! The concept of two or more collaborators working together was completely alien to Newton. And that others, quite independently, could acquire the same insights and accomplish the same breakthroughs that he had, was simply unacceptable to him. He was obsessed by his own uniqueness and was convinced that he, and only he, had been chosen to be the interpreter of divine knowledge. So, quite simply, he saw Leibniz not only as a thief, but worse than that, as someone who had then displayed the material to the world. The publication of the Principia, see figure 13, brought Newton immediate success and his reputation blossomed. So the issue of the calculus became a true cause célèbre, very much the sort of thing the newspapers of today would relish! What it came down to then was this: Newton did indeed discover the calculus first (between 1665-1666; Leibniz: between 1673-1676) but Leibniz published it first (in two papers 1684-1686; Newton, eventually, in publications between 1704-1736).
Newton's reaction was to attack and undermine his enemy although to be fair to both Newton and Leibniz, much of the ensuing battle, at least initially, was stirred up by their followers. For example, after reading Leibniz's 1684 paper on the calculus, the Bernoulli brothers, Johann and Jacques, figured out the method and saw its power. Within a short time they were using calculus to solve mechanical problems and teaching others how to use it. They became two of Leibniz most outspoken supporters; Johann Bernoulli even suggested Newton had built his calculus out of Leibniz's work! Also, he referred to one of Newton's colleagues, John Keill, as
Although he did not use Keill's name specifically, there was no doubting who the "certain individual of the Scottish race" was. In many ways Bernoulli played a rather less than gentlemanly role; continually, he urged Leibniz to fight but he endeavored to remain anonymous himself and later he even tried to establish friendly relations with Newton.
Newton had supporters too although none were at the same intellectual level as the Bernoulli's. Leibniz referred to them as Newton's enfants perdus, or "lost children". One of them, John Wallis, was certainly a first-rate mathematician - indeed Newton had built on many of Wallis's ideas - but he was past his prime. But Wallis, who held little love for the Germans, was deeply concerned that they would pull ahead of the English in mathematics and science, which, indeed, was the case. He urged Newton to publish his calculus, calling on his sense of national duty. In 1695, for example, he told Newton that his discoveries were passing everywhere
Newton gave permission to allow Wallis to publish his two letters to Leibniz, but they did little good to resolve the dispute. Newton did have some supporters in Europe; perhaps the most flamboyant was the French mathematician Fatio de Duillier, who also had cause to dislike Leibniz. de Duillier announced
Although Newton may have agreed completely with Fatio, he was embarrassed because he felt the attack was too forthright and too soon; Newton had decided to bide his time. As it was, it was a battle that was to continue for 4 decades, until Leibniz's death in 1716, and we can see how it was to grow into a partisan battle of ideologies between Newton's defenders in London and their rivals in Europe. What began as a dispute between two men over a question of priority led eventually to a schism in philosophical thought and mathematical practice that lasted for generations. Leibniz's superior notation was adopted in Europe but was deliberately ignored by British scientists, until the early 19th century when Leibniz's notation replaced fluxions. So it is the Leibniz notation that is used by students and scientists today; as I said earlier, it is much less cumbersome than Newton's notation and therefore much more convenient.
Eventually, Newton learned of the correspondence between Collins and Oldenburg and Leibniz. So, given the way the dispute had developed, Newton felt the only way he could respond was to show that either Leibniz's formulation was inferior to his - which would be difficult - or, to put it bluntly, that Leibniz had plagiarized, that is had stolen, his ideas and portrayed them as his own. According to Newton and his followers the most likely time of the robbery was when Leibniz visited London in October 1676 and Collins had shown him some of Newton's unpublished papers. Not only that, they claimed, Leibniz had made no mention of the meeting nor of Newton's two letters; he had, then, posed as the sole inventor of the method. (As an aside, modern analysis of Leibniz's notes, however, seems to indicate that he gained little from October 1676 meeting.) When Leibniz was to learn of this accusation he referred to Wallis's
But still Newton waited.
In 1704 his book Optiks appeared; it is worth noting that Newton had withheld the publication this book until after the death of Robert Hooke who was one of his greatest scientific adversaries and critics. The first edition of Optiks contained the first, proper published account of Newton's calculus although that particular section was removed by Newton in later editions. In the first edition Newton refers also to the letters he wrote to Leibniz in 1679.
The dispute took a further turn in January 1705 when an allegedly anonymous review of the book appeared. The review fooled no-one; clearly, it was written by Leibniz, and in it Newton was accused of using Leibniz's version of the calculus. Worse than that it equated Newton with a certain Honoré Fabri, a notorious plagiarist of the time - a charge Leibniz later dismissed as
In October 1708 John Keill, a young admirer of Newton, upped the anti- when he published an article in the Philosophical Transactions of the Royal Society in which he claimed
When he finally saw a copy of the article, late in 1710, an angry Leibniz immediately wrote a letter to the Royal Society demanding an apology. Accordingly, Leibniz's grievances were discussed at a meeting held on April 5, 1711 and Keill was asked to write an apology.
It is important here to backtrack a little on Newton's career. Well before this dispute started, in 1672, he had been elected a Fellow of the Royal Society after donating one of his reflecting telescopes to the Royal Society. Following the highly negative reactions of Robert Hooke and Christiaan Huygens towards his first paper in the Philosophical Transactions of the Royal Society, Newton only rarely attended meetings of the Royal Society (and as we've seen decided not to publish most of his work). The decade following the publication of the Principia was probably the most difficult of his life. In 1687/88 he became somewhat more involved in the politics at Cambridge, due mainly to his opposition to the attempt of the pro-Catholic King James II to repudiate the oath of allegiance and supremacy then existing at the university, and at the end of 1688 he was elected as a Member of Parliament, a position he held for about a year. The recluse was becoming much more sociable and seeking new challenges; he was on the verge of giving up his life as a scientific researcher. He had a mental breakdown of uncertain origin in 1693 that kept him indisposed from almost 2 years, but in 1696 he was appointed Warden of the Royal Mint. In 1699 he became Master of the Royal Mint, an important position he carried out with expertise, conviction and zeal. He was also re-elected to parliament in 1701 and shortly afterwards, he resigned from the Lucasian Chair at Cambridge. Along with the metamorphosis came an increasing interest in the Royal Society. However, Robert Hooke had been the Society's mainstay and President for many years so Newton didn't play a truly active role until Hooke's death in 1703. He became the new President that same year and remained as such until his death. Within a very short period of time, he stamped his own authority on the Society. In 1705, he was knighted by Queen Anne, the first scientist to be so honored. So, during his dispute with Leibniz, Newton had made the transition from a scientific researcher to what we would call today, a scientific administrator, that is from one of those who "does" science to one who "runs" science.
Let us return to our story. It was during the Keill/Leibniz incident that Newton saw an opportunity to enter the debate personally. He helped Keill write the letter of apology to Leibniz; naturally, it was no apology at all, for it simply reiterated that
Leibniz would have none of it and in early 1712 he wrote angrily again to the Royal Society. Newton then decided to bring things to a head. Since both Keill and Leibniz were Fellows of the Royal Society, Newton decided to create a committee in late 1712 to investigate the affair once and for all, for he could not have Fellows squabbling. Although he was stretching his authority as President somewhat, he insisted that the panel was impartial, numerous and skillful and composed of gentlemen of several Nations. In truth, it consisted, with one exception, entirely of Newton's supporters. Indeed, its makeup was so transparent that the names of the Committee members were withheld when the original report was made. In fact, the final 3 members were appointed one week before the report was published. (Interestingly, the identities of the committee members were only unearthed from the Society's archives in 1846!)
The odds were stacked against Leibniz. The report, a long and detailed review of the situation, was produced in January 1713, after just 50 days, and it contained information that could only have come from Newton himself. (In fact, despite Newton's denial of any influence on the committee there is a draft of the committee's report in existence in Newton's own handwriting!) Not surprisingly, it favored Keill and, therefore, Newton. The report concluded
It was distributed to academic centers throughout Britain and Europe. Despite a few more exchanges the battle was essentially over. Here was Newton, the "Perpetual Dictator" of the Royal Society, the most highly esteemed scientific society in the world, upheld by a large group of supporters and young disciples who hung to his every word. In stark contrast, there was Leibniz, neglected in his position as archivist for the Elector of Hanover.
In January and February 1715 a long account of the proceedings was published in the Philosophical Transactions of the Royal Society written anonymously by Newton; it's an interesting piece because it really indicates the depth of his determination and the lengths he would go to in order to destroy Leibniz. He wrote
Although he never succeeded in breaking Leibniz's resolve, Newton later confessed to his dooctor that
Leibniz continued to molder at the Hanoverian court and was forced to earn a meager salary by writing an interminable history of the House of Brunswick-Hanover. Nothing seemed to go right for him; he was denied superior postings and ironically, in 1714, when the Elector of Hanover, his employer, became George I of England, he even lost favor in his own court, probably as a result of his feud with England's favorite son, Sir Isaac Newton. The calculus had become a factor in the diplomatic maneuvering between Britain and Hanover. Leibniz was clearly on the losing side of the feud and who wants to be associated with a loser? Leibniz also tried unsuccessfully to have Galileo's Dialogue removed from the index of prohibited books. When he died in 1716 his funeral was attended by no one other than a former secretary. A friend later noted that Leibniz
In contrast, Newton continued to be respected, admired and feted. In 1708 he sanctioned a second printing of the Principia and a third in 1726. He died on March 20, 1727 in London and was buried in Westminster Abbey on March 28, after lying in state. Voltaire attended the funeral and is reported to have said:
Newton's own view of his achievements was more modest:
He left a considerable sum of money to his relatives who used some of the money to build a monument in the Abbey, see figure 14. The memorial, which was executed by the sculptor Michael Rysbrack (1694-1770) to the designs of the architect William Kent (1685-1748) and dates from 1731, is in a prominent position in the nave of Westminster Abbey, along with other British "greats". Newton's grave is in front of the choir screen, close to his monument. The Latin inscription reads:
So, is there a difference between Newton's and Leibniz's calculus? In fact, conceptually, there is, although both provide the same results. Let's apply them both so the same problem ... we'll look at Newton's "fluxions" first. Today, we think of a graph - of the function y = x2, say, - as a complete entity, figure 15. On the other hand, Newton imagined such a graph as a curve that was produced by a point, P(x,y), that was moving with time, so, as time progressed, it generated the curve. (We should recall that one of Newton's "interests" was motion!) Therefore, Newton thought that both the x- and y- coordinate changed with time; he called x and y fluents, because they flowed. The basis of Newton's calculus was to find the rates of change of x and y with time; he called those quantities their fluxions. What he did was to think of two instances of time, very close to each other, and to find how much x and y changed in that time, and to divide those differences by the time interval. The crucial step was to set the time interval to zero, see figure 15. Let's see how it works for y = x2. Take a small time interval; we'll use the same symbol as Newton, i.e., O. In that time, the change in the x coordinate is Ox', where x' is Newton's symbol for the rate of change, or fluxion, of x. (Today, we call it x-dot.) The change in the y coordinate is Oy'. So, the "new" x and y coordinates after the time interval are:
But these two quantities must satisfy the original equation so:
But y = x2 so,
Dividing both sides by O, we get
If the time interval O => 0, then
which shows the relation between the fluxions of x and y. (Science and engineering students may recognize x' and y' as simply the velocities of x and y.)
But, we (and Newton!) can get much more than this. Look, if we divide the fluxion of x by the fluxion of y we get:
which is both (a) the rate of change of x with y, and (b) the slope of the tangent to the curve at the point x. The essential point is that although Newton thought of x and y as changing with time he ended up with a quantity - the slope of the tangent to a curve - that does not depend on time; time simply allowed him to develop and formulate his ideas. In the17th century this was called the tangent problem, today we call it differentiation. It allowed Newton to study numerous curves; to find their slopes, positions of maxima and minima, points of inflexion and curvature.
Now, let's look at Leibniz's approach. In figure 16 we have the same curve as before, i.e., y = x2. He wanted to determine the slope of the tangent at P, i.e., the ratio RT/PR. He called the distances PR and RT dx and dy, respectively. He argued that as dx and dy are very small quantities, the tangent line to the graph at P would be almost identical to the graph itself in the neighborhood of P or, more precisely, the line PT will very nearly coincide with the curved segment PQ. So, to find the slope of the tangent at P, he needed only to determine the ratio dy/dx. His argument has a flaw ... although the tangent line is very nearly identical to the curve at P, it does not coincide with it; that only happens when P at T coincide, and then dy/dx = 0/0, which is indeterminate! Today, we get around this problem by letting PR (= Dx) and RQ (= Dy) both approach zero. Their ratio (Dy/Dx) will approach a certain value, which we identify with dy/dx. To see how this works, let us find dy/dx for the curve y = x2. If x is increased by Dx and y is increased by Dy, then:
y + Dy = (x + Dx)2 = x2 + 2xDx + (Dx)2
i.e., Dy = 2xDx + (Dx)2.
Dividing both sides by Dx, we get
Dy/Dx = 2x + Dx.
And so, as Dx approaches zero,
Dy/Dx => dy/dx = 2x.
This is the slope of the tangent at the point x. Note that this is identical to the result obtained by Newton using fluxions!
Their approach to integration was rather different also, although the results were the same. Newton thought of integration as the inverse of differentiation, i.e., using his terminology ... "finding the fluent if you know the fluxion". In fact, this was the basis of Newton's famous anagram of 1676 that we saw earlier:
which gives the number of different letters in the (Latin) sentence:
Given an equation involving any number of fluent quantities, to find the fluxions: and vice versa.
Leibniz, however, started with the problem of finding areas. He thought of area as the sum of many thin strips - integration - and later "unified" integration and differentiation.
I remarked earlier that a clash between these two geniuses was almost inevitable. In some sense their battles over the the calculus was only part of the story. A comparison of their characters is illuminating. Newton and Leibniz were totally different philosophically so that if it hadn't been the calculus, it would have been something else. Leibniz had a deep interest in metaphysics, which was part of the reason he and Newton couldn't agree. In fact, that's what made him, at least conceptually, beyond Newton; for example, he played around with symbolic logic and binary arithmetic, now the basis of modern computers. In fact, Leibniz was searching for a unified system of knowledge that could, perhaps, unlock the secrets of human behavior. He even thought that his calculus might prove useful in this quest.
One biographer in 1888 wrote:
One problem was that Leibniz had little influence or power. Furthermore, as we have seen, many of his schemes failed. He had worked on plans for unifying the Catholic and Protestant Churches; he had tried to get Louis XIV to attack Egypt and weaken the Ottoman Empire as a way of deflecting French aggression towards Germany; later he was unsuccessful in developing a system to extract water from mines in the Harz mountains; and his attempts to set up scientific academies in Berlin, Dresden and Vienna all failed, although one was later set up during his life time in Berlin and he was to serve as its President.
In contrast, for Newton, calculus was simply a way of dealing with the physical problems he confronted, simply another mathematical weapon in the armoury of the physicist. It is clear to us that he used the calculus to work out many of the difficult problems he addressed in the Principia. But, then, he reworked the problems so that they could be presented in a more traditional, geometrical fashion. This, in itself, shows the genius in him. Also, in contrast to Leibniz, Newton gained ground with each of his disputes, not only because of his scientific genius but also through his power and influence. For example, throughout his 20 year battle with Flamsteed, whose data on the Moon he wanted in order to update his later printings of the Principia, Flamsteed stood his ground against Newton, his superior at the Royal Society. However, because of Newton's influence with both Prince George, Queen Anne's husband, and later the Queen herself, Flamsteed was eventually forced to publish the data through the Royal Society, and Newton got what he wanted. And we know that he got the better of Leibniz.
Newton religious convictions were Christian although he followed Arianism, a sect that did not believe in the Trinity. He was an empiricist and had provided in the Principia a single physical explanation for phenomena as diverse as the movement of planets, the tides, the swing of a pendulum and the fall of an apple. The single underlying force was gravity, that is a force that acts not through direct contact like pushing or pulling, but at a distance. Continental philosophers found so-called "action-at-a-distance" an anathema. Leibniz felt a need for some subtle matter or spirit to explain the motion of the planets; just as Kepler had considered that angels propelled the planets through space. To Leibniz, action-at-a-distance smacked of the occult.
Although Newton and Leibniz were both deeply religious men, their feelings of the part played by God were quite different. Newton thought of the universe as a clock that God wound up at the beginning of creation. He felt that the clock could not run forever, for, if that was the case, why did God continue to exist? Rather, he felt that some attention was necessary from time to time and that God, the clockmaker, intervenes to set things back in order. Leibniz thought the idea of God as an astronomical maintenance man as absurd. He believed that God had carefully chosen among an infinity of possible worlds, the one He felt the most suitable. So that although we may not have a perfect world, it was the
This was an idea that many, including Newton, thought was far too pessimistic a view with no opportunity for improvement.
Some while ago I saw the question posed ... could these two men have collaborated? Surely, the argument went, if they had worked together just imagined what they could have achieved. My opinion ... no way, they were two, unique individuals, although, perhaps, in a way, feuding did help them to think more deeply about their life's work.