Plagiarism is usually associated with English papers, but this accusation was leveled at a prominent mathematical figure during the 17th century. Thus, we have an interesting point whereby an English scholar might feel comfortable entering the business of mathematics.
Gottfried Wilhelm Leibniz published his Nova Methodus pro Maximis et Minimis in 1684, a paper on differential and integral calculus. Despite that, Newton (who had developed his equivalent method of fluxions and fluents in the mid-1660's) seemed to show little inclination to publish or even to object to Leibniz's renown as the developer of the method of quadrature. The famous dispute between the two, known since as "The Calculus Wars", was actually catalyzed not by Newton himself, but by a Newton enthusiast Nicolas Fatio de Duillier.
Fatio: “I recognize that he was the first and by many years the senior inventor of the calculus . . . as to whether Leibniz, the second inventor, borrowed anything from him, I prefer to let those judge who have seen Newton’s letters and other manuscripts, not myself.”
Leibniz: "it may be the case that just as Mr. Newton discovered some things before I did, so I discovered others before him. Certainly I have encountered no indication that the differential calculus or an equivalent to it was known to him before it was known to me."
D.T. Whiteside: "Newton’s historical importance as the author of 'DeQuadratura Curvarum' is the minimal one of a lone genius who was able, somewhat uselessly in the long view, to duplicate the combined expertise and output of his contemporaries in the field of calculus. What is not communicated at its due time to one’s fellow-men is effectively stillborn."
Leibniz: Instead of the Leibnizian differences Mr Newton employs, and has always employed, fluxions which are almost the same as the increments of the fluents generated in the smallest equal portions of time. He has made elegant use of them both in his Principia Mathematica and in other publications. […] just as Honor´e Fabri in his Synopsis Geometrica substituted the advance of movements for the method of Cavalieri.
Newton: The sense of the words is that Newton substituted fluxions for the differences of Leibniz, just as Honor´e Fabri substituted the advance of movements for the [infinitesimal] method of Cavalieri. That is, that Leibniz was the first author of this method and Newton had it from Leibniz, substituting fluxions for differences.
Leibniz: having undeservedly obtained a share in this, through the kindness of a foreigner [that is, Leibniz], he [Newton] longed to have deserved the whole – a sign of a mind neither fair nor honest. Of this Hooke too has complained, in relation to the hypothesis of the planets, and Flamsteed because of the use of his observations.
Newton: If Mr Leibnitz could have made a good objection against the Commercium Epistolicum, he might have done it in a short letter without writing another book as big. But this book being matter of fact & unanswerable he treated it with opprobrious language & avoided answering it by several excuses, & then laying it aside by appealing to the judgment of his friend Mr Bernoulli & by writing to his friends at Court, & by running the dispute into a squabble about a Vacuum, & Atoms, & universal gravity, & occult qualities, & Miracles, & the Sensorium of God, & the perfection of the world, & the nature of time & space, & the solving of Problemes, & the Question whether he did not find the Differential Method proprio marte: all of which are nothing to the purpose . . . The proper question is: Who was the first Inventor?
Sir Isaac Newton
It all began in 1665-6 during the plague in England, while Cambridge was closed.
Newton is home from school and spends his time learning mathematics and thinking about the cosmos. In order to analyze the heavens, however, he recognizes that a mathematical tool is lacking, and so he develops a method of squaring (or finding lengths of and areas under) curves. This method involves dealing with infinitesimal objects. Newton manages to avoid the difficulties associated with contemplating such elusive objects by focusing not on the infintesimals themselves, but on the ratio between the tiny object and the quantity it's dividing. That ratio is known as the “fluxion,” in Newton's terminology, and is the same as what is known as the “derivative” in today's Calculus. (The fact that the ratio remains the same is what makes the derivative a function.) The “fluent” reverses that process, but in Newton's method of consideration, still is arrived at by “ignoring” the difficult infinitesimals. The fluent corresponds to the integral of modern-day Calculus (Meli 101).
Newton develops this mathematics between 1665 and 1666, but he doesn't publish it. In The Calculus Wars, Jason Bardi gives several reasons for the failure to publish this important development in mathematics.
- The Plague, which both helped and hurt his work
- The Fire of London
- Robert Hooke, who was prominent in the Royal Society and Daunting and wrong in his theoretical development. Newton had called him out.
Presumably, Newton intended to publish his mathematics as an appendix to his Opticks, but he was prevented from doing that by the very influential Hooke because of the demolition that Newton's ideas meant for his own proposed system.
Gottfried Wilhelm Leibniz
Leibniz is a jurist who, having little mathematical education, teaches himself mathematics and shows a remarkable aptitude for the subject. He develops a particular mathematical method and brings it before the Royal Society in 1673. However, during the event someone recalls that one Francois Regnaud has already developed the same method (Meli 5). There is a hint of plagiarism in the incident, but most likely Leibniz has simply not seen the method and therefore did not know the uselessness of bothering to solve that problem—since it has already been solved. This embarrasses him greatly and sends him back to his mathematical studies with vigor (Bardi). It is subsequent to this that he discovers his own method of calculating squares and finding tangents to curves, a task he completes, it is widely considered, by 1675 (Meli 1).
During the late 70's Leibniz developed an intense interest in the work Newton was doing, for it had come to his attention via Oldenburg, secretary of the Royal Society, that Newton had found a system of quadrature that had been instrumental to mechanistic work he was now doing. A series of letters went back and forth between Newton and Leibniz via Oldenburg in which the two offered vague descriptions of their ideas, steering clear of their exact methods, making sure that nothing was given away. This occurred between 1666 and 1667 (or thereabouts). Leibniz actually had a chance to look at Newton's text De Analysi, which describes his method in detail, when he (Leibniz) was in contact with John Collins who had a copy in his possession. Leibniz kept this fact hidden... perhaps understandably—Collins had not been given permission to divulge Newton's method to his main rival.
I offer another reason for Newton's failure to publish his DeQuadratura Curvarum. He was at heart a physicist, and though his remarkable intellect combined extraordinary brilliance in mathematics, theoretical physics, and experimental physics—perhaps more perfectly than anyone before or since—he was primarily a physicist. This meant his goal was to complete his universal theory of gravitation, and he subordinated his mathematics to that goal. He did, after all, develop the mathematics in service of his physics--i.e. as a tool, not as an end in itself. Therefore, even after developing such an important mathematical instrument, he still felt the urge to persist toward development and publication of this physical theory.
All photos are in the public domain.
Bardi, Jason Socrates. The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time. New York: Thunder Mouth Press, 2006
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