# The History of the Priority Dispute between Newton and Leibniz

Priority disputes among mathematicians are from all times, but the one between Newton and Leibniz about the discovery of calculus is notorious. Many authors, and historians have written about it. Even during the lifetime of the protagonists, the Royal Society had a commission to investigate the matter. Their conclusion was that Newton was the first, but since at that time Newton was the president of the Royal Society, this conclusion may have been a bit biased.

The supporters of Leibniz whose home base was Hanover were mainly from the continent, while most of the British defended their national hero. In those days mechanics, mathematics, optics, chemistry, alchemy, astronomy, and history were all part of the job of a prominent scientist. In Newton's case certainly also theology, history, and monetary politics. While Leibniz started as a lawyer and published on palaeontology. So the whole scientific (and political) community was involved.

Newton introduced his fluxions inspired by physics. A fluxion is the instantaneous change in a fluent. We now say that it is the time derivative of a function of time (the fluent). The problem was that the notion of limit was still unknown, so his peers had problems with computations that used infinitesimal small (but nonzero) quantities, that seemed to vanish when appropriate and remained nonzero at other instances. This was directly connected to the construction of a tangent and what was called a quadrature, which is the computation of the area under a curve, thus what we now call an integral. Newton's great insights happened mainly during the period of the Great Plague in 1995-1667 when he retreated to Woolthorpe Manor to live with his mother. In that time he also developed his theory of gravitation, laid the foundation of classical mechanics, and explained the planetary motion. None of this was however published until much later. The mechanics were published for the first time in his *Philosophiæ Naturalis Principia Mathematica* in 1687 and two other editions in 1713 and 1726. His book *The method of fluxions* was only written in 1671 and published in 1736.

Leibniz was educated as a lawyer ans got only interested in mathematics later in 1672 when he visited Paris and meets Huygens. He was mainly concerned with quadrature. The approximate length of a curve $ds$ could be considered as the hypotenuse of a rectangular triangle with sides $dx$ and $dy$. Using geometrical arguments and similarities of triangles he obtained a method to compute the quadrature of an arbitrary curve. This was around 1674, but it was not published before 1684. He used the notation $dy/dx$ for the derivative, which was conceptually much easier to work with than Newton's fluxion notation which used the dot atop the fluent variable. This of course becomes problematic for higher order derivatives. Leibniz also introduced the integral sign ∫ as a elongated 'S' for sum, that we are still using today and which is included in the title of this book by writing "Dispute" as "Di∫pute". It is clear, and generally agreed by now, that Leibniz and Newton developed their theory independently by following different methods. However in the heat of the controversy Leibniz was accused of blatant plagiarism. Strangely enough, it were not Newton and Leibniz that stood in the barricades most of the time. In fact they exchanged polite and friendly letters. It were their followers, friends, and believers who did all the fighting on the front line, although they were of course backed up and sometimes directed by the protagonists. Newton remained more on the background, but when accusations became too direct, Leibniz had no choice but to protest against an open insult by a warrior from the opposite camp.

Among the historical defenders of Leibniz were Jacob and John Bernoulli. Among Newton's warriors were John Collins, John Wallis, and Nicolas Fatio de Duillier, which is called Newton's monkey by Sonar. This Fatio has put the fuse that lit the powder keg by openly accusing Leibniz of plagiarism. At a later stage John Keill became the `army commander' of the group defending Newton. Some of the problems arose because the first correspondence was not directly between Newton and Leibniz but passed via others like Henry Oldenburg, the secretary of the Royal Society, who was not a mathematician. Oldenburg was advised on matters of mathematics by Collins, an outspoken nationalist, who was naturally opposing anything that came from the continent. There were misunderstandings, half spoken truths, and hesitation to disclose results that oxygenated the fire. The war went on, even beyond the grave. Clearly the new calculus found applications, and because Leibniz's formalism was easier, his calculus was the eventual winner. In fact it caused a drop back of the English mathematical scenery. While they were at a comparable level with the mathematics on the continent when the controversy started, they were not able to keep up with the development of calculus and analysis for a while in the eighteenth to nineteenth century post-Newton era.

This fight may be well known, but disputes in those days were very common among others as well. Newton and Hooke became personal enemies over a priority dispute in optics (Newton did not want to publish his *Opticks* until after Hooke died), Huygens rejected Newton's corpuscular theory of light. He also fought with Heuraet over the rectification of curves, and he quarrelled with Hooke over a clock mechanism. Newton and Flamsteed, the Astronomer Royal, were fighting over the trajectory of the Great Comet of 1680, which Newton explained with gravity. And there were other such disputes that are also described by Sonar in this book.

Thomas Sonar is from Hanover and before he engaged in the study of this history, he was rather convinced that it was a good-hearted Leibniz that was the one who was maltreated and unjustly accused by a quarrelsome and short-tempered Newton and his disciples. Sonar may have started his research with the idea of defending Leibniz, when he finished the original German version of this book in 2016, his conclusion was much more mollified. Leibniz also had not always told the truth and he wasn't the saint attacked by the devil Newton. He also had his pawns in the war and used them. This conclusion becomes clear only after meticulously investigating all the original correspondence of the seventeenth century and of all the books and papers that were published about the matter. This is the most thorough discussion of the matter that has been published so far and that still remains very readable with a minimum of mathematical knowledge, hence available for a general readership. In fact Sonar starts with an elementary introduction like a modern introductory calculus book would, so that the reader should know what calculus is about, or at least grab the meaning of derivative and integral. Then he introduces the `giants on whose shoulders Newton claimed to stand': John Wallis, Isaac Barrow, Blaise Pascal, Christiaan Huygens. So we find a biography of these people, and what they did for mathematics. In retrospect it is clear that calculus was on the doorstep, and that it only took some great minds to bring it in the open.

But Sonar also gives a detailed description of the political situation and events of those days in England, France, Spain, and the Netherlands. Of course these are not really essential for the mathematics, but it sketches the framework in which scientists were working. It were usually political leaders that employed the top scientists and they made the start of academies financially possible. This part of the book has several very useful timelines, and there are many beautiful pictures throughout the book. Just reading this political prequel to the main dish is already a wonderful experience. Then of course we meet both Newton and Leibniz, how they grew up and studied and how they arrived at the discovery of their new calculus. At first there is some friction (Sonar calls it a cold war) between the two, then there is a period of relaxation, but when things get published the smoldering fire becomes a real war. Sonar includes many quotes from the letters that go back and forth about the matter with precise dates of when they were written, whether it was as an impulsive reaction to a previous message or it was written only after a long time of postponing it, possibly who was the messenger, and, not unimportant, when the letters arrived. With every new player we are given his or her background and some biography.

Fortunately the excellent and smoothly reading English translation comes so shortly after the German original and was done by Sonar himself with the help of Keith Morton, his Oxford thesis advisor and later by his advisor's wife Patricia Morton. I can highly recommend this book if you have just a slight interest in history and/or mathematics. Perhaps the professional mathematical historians may not find much new or innovative material, since this 'cold case' has long been settled and solved, I believe they will still enjoy reading this book.

**Submitted by Adhemar Bultheel |

**29 / May / 2018