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the indexIn the history of mathematics and science, few conflicts have attained the notori-ety of the Newton/Leibniz dispute. The outlines of the debate between the two pro-tagonists, and their supporters, are familiar. The major question at the heart of the priority conflict dealt with the issue of which of the two, Sir Isaac Newton or Gottfried Wilhelm Leibniz, deserved credit for the first (prior) invention of the differential and integral "calculus". In the course of this debate the mathematicians John Keill and Johann Bernoulli drafted letters that defended their masters, Newton and Leibniz, respectively, and attacked the other party [Ha 3, pg.144-145, 116-117]. According to Newton scholar Alfred Rupert Hall, the priority question has been settled. In the introduction to his work, Philosophers at War: The Quarrel Between New-ton and Leibniz, Hall writes: "It was certainly Isaac Newton who first devised a new infinitesimal calculus and elaborated it into a widely extensible algorithm, whose potentialities he fully understood; of equal certainty, the differential and integral calculus, the fount of great developments flowing continuously from 1684 to the present day, was created independently by Gottfried Wilhelm Leibniz [Ha 3, pg.1]." Although Hall has studied Newton's life and work in great detail, and the present author takes his judgment on the dispute as such as definitive, other sources independently verify the claim that both Newton and Leibniz deserve credit for the invention of calculus. An important point should be made about the sources. In the sources we have examined, the possibility that Newton himself might have stolen his ideas does not come into consideration. There is a good reason for this. Such modern scholars as A. Rupert Hall and D.T. Whiteside, unlike Leibniz and Bernoulli, had access to the product of cen-turies of scholarship. As Hall shows, a carefully reconstructed chronology reveals that Newton had formulated the essentials of his calculus by 1666, years before Leibniz had attained the mathematical knowledge necessary to develop his own point of view on the calculus [Ha 3, pg.15]. All the sources we have examined are in accord on this point.. Thus it seems to the present author that these scholars all approach the priority issue with the attitude that the burden of proof lies wth Leibniz. With the exception of Joseph E. Hofmann, other scholars substantiate the essen-tial truth of Hall's statement with respect to Leibniz, although not as explicitly as he does. In his treatment of the feud, V.I. Arnold does not follow Hall, but he does acknowledge Leibniz's originality, which was based on his ideology of finding "all possible universal methods for solving all problems straight away [Arn, pg.46-48]." Hofmann, in his work, Leibniz in Paris: 1672-1676, takes the same view, writing in the conclusion, "That Leibniz here (in the field of calculus) wholly uninfluenced by others, gained his crucial insights unaided, is beyond all doubt [Ho, pg.306]." Hall takes exception with Hofmann's claim that Newton, in his correspondence with Leibniz in 1676, sought to conceal his work on fluxions (roughly analogous to the concept of derivatives), a claim that would seem to substantiate the thesis of Leibniz's originality [Ha 3, pg.65-66]. However, as indicated by the quotation cited above, and his treatment of Leibniz throughout, Hall believes that Leibniz did not plagiarize his ideas concerning the calculus, as Newton and his Britsh followers alleged. Given the consensus among modern scholars who examine the Newton/Leibniz dispute, one is moved to ask -- how could such a misunderstanding have developed between these two brilliant men? Hall proposes an answer when he writes, "Among the Newtonians, as a mong the Leibnizians, there was a natural ignorance of the genuine and highly personal origins of the calculus in the mind of the 'other' claimant to the intervention; partly for this reason, there was a tendency among the followers of either side (far less noticeable in the principals) to assume that there was a unique true calculus, which the 'other' inventor had not discovered...[Ha 3, pg.118]" Hall's assertion bears closer examination. What he seems to be saying is that the conflict occurred because each side could not truly fathom the manner in which the other side formulated its version of calculus. In part, this occurred because neither Newton nor Leibniz, despite their awareness of each other's results, could "see" the mental processes at work in the other party. As Hall notes, this led each of the two men, and especially their followers, to believe they alone had made the genuine contribution to the development of calculus [Ha 3, pg.118]. Beyond an inability to read the other side's thoughts, however, there seems to have been a genuine difference in the world views of Newton and Leibniz. Such world views do not spring up in a vacuum; they are often a result of the personalities and the surrounding environment. This raises some interesting questions: "Did Newton's and Leibniz' personalities and the scientific climate of the late seventeenth and early eighteenth centuries have any impact on the dispute? If they did, to what extent?" An examination of Newton's and Leibniz' dealings with other scientists and mathematicians, and the environment of their dispute may be useful in this context. The Newton/Leibniz debate has already been covered in great detail in many sources, such as Hall's book, and will thus be considered only insofar as it bears on the questions raised above. Furthermore, while the author cannot analyze the nuances of the competing versions of Newton's and Leibniz' mathematical theories, an attempt will be made to address them adequately as the occasion arises.
According to a recent biographer of Newton, Gale E. Christianson, Sir Isaac Newton was by nature a solitary person [Ch, pg.51], who did not travel much, and never outside England [Ch, pg.112]. Moreover, Newton did not relish involvement in open and public quarrels [Ch, pg113]. As this biographer notes, "Clearly, Newton's conception of face-to-face contact with other human beings is that it should be as dispassionate and as harmless as possible...Newton knew well his own quickness to anger, and he decided early on never to let his temper get the best of him in public, which it rarely did [Ch, pg.113]." Although Newton preferred to work quietly in seclusion, he also seems to have been a proud man who could not bear to have his meticulous work criticized. This characterization is borne out by considering his feud with Robert Hooke, the Curator of the Royal Society, an organization with which Newton was involved, and over which he would later preside. To understand the feud, it is important to understand the nature of the Royal Society in that era. As Michael Hunter notes in his monograph on the Royal Society, the organization was open to all kinds of viewpoints and its members were not systematically sanctioned for advocating scientific ideas that sharply conflicted with widely held beliefs [Hu, pg.9]. Newton in the early stages of his career had been conducting experiments to better understand the nature of light. Newton had made the important discovery that "white light is composed of the primary colors [Ch, pg.149]." Although he "was the first to measure the wavelength of light," Newton held the view that light was not composed of waves, but of particles [Arn, pg.13]. This theory conflicted with Hooke's view on light which had been publicized in 1665, a view which Newton was aware of [Ch, pg.95]. Hooke, based on his own experiments, believed that light was made up of waves [Arn, pg.13]. After his election to the Royal Society, however, and perhaps reassured by the generally tolerant nature of the Society, Newton composed a letter to Henry Oldenburg, the Secretary of the Royal Society, in which he made his research on light available to the Society, an uncharacteristic action for Newton [Ch, pg.146-151]. Hooke quickly issued a report on February 15, 1672 that seemingly praised Newton's work, but declared, "For all the expts [experiments] & obss [observations]: I have hitherto made, nay a nd even those very expts which he (Newton) alledged, doe seem to me to prove that light is nothing but a pulse or motion propagated through an homogeneous, uniform and transparent medium [Ch, pg.156-157]." A few years later, Hooke wrote in his diary, "We (a Royal Society Club) began our first Discourse about light upon the occasion of Mr. Newtons Late Papers. I shewd that Mr. Newton had taken my hypothesis of the puls or wave [Ho, pg.206]." Hooke was basically saying that Newton's experiments actually provided greater support for the wave theory of light that Hooke subscribed to. More seriously, the diary entry shows that in later years, Hooke charged Newton with basically appropriating his work. Hooke had published his studies on light in the Micrographia of 1665 [Ch, pg.95]. Newton was familiar with Hooke's 1665 work [Ch, pg.101]. This familiarity does not mean that Hooke's charges were justified. In the initial letter to Oldenburg, Newton had meticulously laid out the reasoning and observation that led him to conclude that white light was a mixture of the other colored lights [Ch, pg.150-151]. However, when Newton stated his belief that light was not made up of waves, he did not have the same level of observation to back up his ideas that he had with his findings regarding the decomposition of light [Ch, p.161]. Christianson believes that by promulgating a thesis with little concrete evidence, Newton violated his principle "not [to] mingle conjectures with certainties [Ch, p.161]." Such a statement exposed Newton's other, more carefully drawn observations to Hooke's attack. According to his biographer, Newton was so affected by the incident with Hooke that he promised himself to never make such a mistake again [Ch, pg.162]. Newton wrote a seemingly mild letter to Hooke that took issue with his claims, trying to conceal his anger over Hooke's charges [Ch, pg.161]. Newton's reply saw print in Philosophical Transactions, the journal of the Society, but Hooke's February 15 report did not [Ch, pg.161]. Although Newton had been able to answer Hooke, the incident seems to have had an impact on his willingness to publish. Newton's comprehensive treatment of the subject that he touched on in the Oldenburg letter, Opticks, was circulated after Hooke's death, to avoid the Curator's criticism [Ha 3, pg.135]. Hooke eventually sought to make peace with Newton, sending him a letter in November of 1679 that set out some problems on which Hooke ostensibly desired New-ton's opinion [Arn, pg.14-15]. As Arnold notes, this letter led Newton into the studies that culminated in the Law of Universal Gravitation [Arn, pg.14]. Ironically, Hooke's attempt at reconciliation failed because his letter precipitated correspondence about "Hooke's own theory that the motions of planets are compounded of a tangential motion and a n attractive motion towards the sun [Ch, pg.267]." Hooke angered Newton when he read some of Newton's preliminary work on the subject, which was contained in a private letter to Hooke, before an audience at the Royal Society and also when he discovered that Newton had made a mistake by not accounting for "orbital motion" in one of his calculations [Ch, pg.271]. In one of these letters Hooke wrote, "But my supposition is that the Attraction always is in a duplicate proportion to the Distance from the Central Reciprocall (Hooke's spelling), and Consequently that the velocity will be in a subduplicate proportion to the Attraction...[Ch, pg.272]." Hooke later used this phrase as proof that he had come up with the gravitation law before Newton [Ch, pg.27 2]. However, Christianson states that Newton was already aware of the inverse square law and that Hooke did not have the mathematical skill necessary to substantiate his intuitions [Ch, pg.272]. Newton's disputes with Hooke and Leibniz were not his only scientific disputes. He also quarrelled with John Flamsteed, the Astronomer Royal at the Greenwich Observatory, who refused to provide observational data that Newton needed, and that Flamsteed had carefully recorded [Jo, pg.99]. However, the Hooke feud is important because it preceded the Leibniz feud, and gives hints of Newton's personality. From the account above, Newton seems to have had a hard time taking criticism, although Hooke's criticism of his entire theory of light seems unjustified. Newton also avoided public quarrels with Hooke when he could. Even before the Hooke feud, Newton h ad a reluctance to publicize his results if he knew that they would be subject to scrutiny. The Hooke quarrel only served to reinforce this trait. Finally, after his experience with the particle theory of light, Newton strove to take greater pains to make sure that everything that was published was accurate and defensible. This was the legacy that Newton brought to the calculus priority dispute. Leibniz also brought his own experiences to his quarrel with Newton. Leibniz' Personality and Conflicts Gottfried Wilhelm Leibniz was a diplomat and historian for the House of Hanover [Ar, pg.vii]. In the course of his travels, Leibniz learned about the Royal Society in England. As a traveling diplomat, Leibniz had been posted to Paris to try to convince the French to stop flexing their muscles on the continent, something which made his employers nervous [Ma, pg.21], and to convince them to invade Egypt instead [Ma, pg.21]! When Leibniz got to Paris, however, the French were on the brink of war and there was little that he could do [Ma, pg.21]. However, he decided to remain in Paris.
It would be a wise choice. In 1672, Leibniz met the mathematician Chritian Huygens in Paris [Ha 2, pg.131]. It was here that Leibniz began to reveal his talent for mathematics for the first time [Ha 3, pg.48]. However, Leibniz was handicapped to some extent by his great pride. Hall writes of Leibniz, "Yet Leibniz was by no means a modest man. He was really different from Newton in this respect [Ha 3, pg.48]."
It is useful at this point to know something about Leibniz' philosophy. "...Leibniz was always interested in anything general and had all sorts of general ideas [Arn, pg.45]." Leibniz believed that all mathematical problems could be solved by these general methods [Arn, pg.46]. Leibniz' philosophy seems to have manifested itself in his search for his Lingua Philosophica, a language to capture the ways in which the mind worked [Ma, p.183]. Leibniz believed that the rapid pace at which mathematics was expanding its horizons was largely contingent on its use of superior forms of notation [Ma, p.183]. To Leibniz, unlike many of his contemporaries, notation was the key. He believed that if this notation was sufficiently developed, it could help "(guide) people to knowledge [Ma, p.185]." In Leibniz' mind, a great achievement in mathematics could only lead to further insights if accompanied by the correct notation [Ma, pg.185]. This point is important in understanding Leibniz' stance in his dispute with Newton. Newton's follower, John Keill, would level the charge that Leibniz made only cosmetic changes to Newton's calculus theorems before presenting them as his own [Ha 3, pg.1 45]. This was not the case, but it is not hard to see Leibniz arguing that even if his and Newton's methods had sprung from the same sources, the differences in their notation, and the assumptions carried by that notation, would mark the uniqueness of each man's contributions. In January 1673, Leibniz made his way to England for the first time [Ha 2, pg.131]. Before the journey, Leibniz had been exchanging letters with Henry Oldenburg, the Royal Society's Secretary [Ha 2, pg.131]. As Newton scholar A. Rupert Hall notes, howev er, Oldenburg was not a mathematician, and his reports to Leibniz touched on subjects more like Isaac Barrow's work in optics [Ha 2, pg.131-132]. Although Olden-burg did fill his letters with news of mathematical advances at the Society, he himself was not a good judge of the quality of the work being done in the field at the time [Ha 2, pg.132]. Moreover, Oldenburg relied heavily on John Collins, a government official, for the mathematical information that he passed on to Leibniz [Ha 2, pg.133]. Collins, in Hall's words, was a man of "limited intellectual horizons. [Ha 2, pg.133]." Although these letters to Leibniz certainly contained some important information, it is not clear that Leibniz realized this, and in any case he does not appear to have put a great deal of effort into applying his considerable skills to extracting from them the information that would have been most valuable to him.
Whe Leibniz made his first visit to England in 1673, he sought out the mathematician John Pell and showed him what he believed was an original discovery. Leibniz believed that he had found "...a method of forming the terms of any continually increasing or decreasing series whatever from a certain sort of differences that I call generative...[Ha 3, pg.54]" Leibniz seems to have been very boastful in making this claim. He was thus shocked when Pell informed him that his "discovery" was no such thing, a nd that it had been advanced already by a man named Frangois Regnauld [Ha, pg.134]. Hall states that while Leibniz had definitely seen a copy of the book containing Regnauld's discovery, his method was sufficiently different from Regnauld's that he should not be judged too harshly [Ha 2, pg.134].
This incident was painful for Leibniz, who at this time, was in many ways still quite ignorant of the developments in various fields [Ha 3, pg.54]. To avoid the suspicion that he had stolen Regnauld's idea, Leibniz wrote a letter explaining his position and the development of his ideas to Oldenburg [Ha 2,pg.133]. Someow a copy of this letter came into Newton's hands [Ha 3, pg.55]. In his later dispute with Leibniz, Newton would seize upon this event as proof of Leibniz's dishonesty [Ch, pg.504].
Leibniz was developing at this time into a formidable mathematical talent, as evidenced by his method of "arithmetical quadrature of the circle" [Ha 2, pg.135]. It is crucial to note that when Leibniz made this discovery, after his first trip to England, he had exchanged no letters with anyone in England and was working in isolation from developments at the Royal Society [Ha 2, pg.135]. What this seems to show is that Leibniz possessed substantial mathematical creativity, in contrast to charges later leveled against him.
However, for all of his originality, it is hard to escape the conclusion that Leibniz often made things more difficult for himself by not heeding to those who tried to warn or to advise him. An example may be seen in the "circle series" episode. After his first trip to England, Leibniz began to publicize his discovery that "the area of the circle could be exactly determined by the sum of an infinite series [Ha 3, pg.57]." Henry Oldenburg tried to warn Leibniz that the famous English mathematician James Gregory had explicitly stated that the claim Leibniz was making about the area of the circle was wrong and that he could show this [Ha 3, pg.57]. When Leibniz asked for proof, Oldenburg and Collins sent a letter describing advances in series, but not showing how they were obtained. [Ha, pg.58]." After looking at what had been sent to him, Leibniz wrote back, dismissing the connection between his findings and those of the English mathematicians [Ha, pg.58]. Hall states that at this stage, Leibniz did not yet have the mathematical skill to understand what was sent to him. By casually dismissing those claims, Hall writes that Leibniz "was consistently guilty of underrating the opposition. And again like Newton he could never quite accept the notion that his own marvelous ideas, those tremendous illuminations, had been shared separately by others [Ha 3, pg.59]." Leibniz saw calculus as the study of curves and variables [Me, pg.68]. To him, as to Cavalieri, curves were finely drawn segments of regular polygons [Me, pg.68]. More-over, "differentiation and integration are operations on variables and change the order of infinity, not the dimension of a variable [Me, pg.68]." Leibniz also believed that "differentials are indeterminate because the sequences of the relevant variable or the associated polygon can be chosen in infinitely many ways [Me, pg.68]." What can be said about Leibniz' character? He seems to have been a remarkably broad-ranging thinker who tried to find the patterns that united various phenomena. He also seems to have been capable, once his level of mathematical expertise had increased, of finding clever solutions to problems that had seemed to him to be very difficult. Leibniz was unlucky in the sense that others accused him of plagiarism or solved problems that Leibniz was working on well before Leibniz had reached his own solutions. Compounding the problem was Leibniz's somewhat arrogant attitude and his tremendous confidence in his own abilities as demonstrated by Hall's account of Leibniz's dismissal of the other circle series solutions. These facets of Leibniz's character again manifested themselves in the feud with Newton. Conclusion There are a couple of important points to bear in mind about the state of mathematics in the late seventeenth century, the epoch of Newton and Leibniz. First of all, some of the more elementary techniques of calculus had been discovered before Newton and Leibniz arrived on the scene. Reni Descartes, Pierre de Fermat, and Blaise Pascal all knew how to perform the basic operations of the calculus in special cases [Arn, pg.47]. They "were able to differentiate polynomials and knew how to draw tangents to parabolas [sic] of all degrees ... [Arn, pg.47]" Huygens and Barrow had also made substantial advances [Arn, pg.47]. Hall writes, "The discovery of the calculus was more than a synthesis of previously distinct pieces of mathematical technique, but it was certainly this in part...[Ha 3, pg.7]." The implication is obvious - it was not all that unusual that priority disputes sprang up, given the fact that many had worked on these same problems, achieving overlapping partial results [Ha 3, pg.7]. The other point to remember is that scientific feuding was not confined to Newton and Leibniz. Indeed, Leibniz had problems with Hooke [Ha, pg.47], and Hooke made serious accusations against Oldenburg and Huygens [Hof, pg.122]. There seem to have been many mathematical conflicts in the era of Newton and Leibniz. What is interesting about them is that many, if not all, broke down along the lines of the English versus the Continental Europeans. Given the analysis of their personalities above, it may be possible to reach some conclusions about why the Newton/Leibniz conflict developed as it did. As Hall suggests, the two men saw calculus in radically different ways. It may have been inconceivable to either party that the other side really had come up with a version of the calculus single-handed. Both Newton and Leibniz seem to have shared this view, although one might argue that it may have been harder for Newton, who did not seem to place the importance on notation that Leibniz did, to believe that Leibniz's calculus had been created independently, than for Leibniz to believe a similar thing of Newton. The use of surrogates also becomes more natural in few of both men's prior his-tories. It seems to be Newton, who was more sensitive to public criticism, who resorted most frequently to the use of surrogates. Newton was also the moving forcebehind the Commercium epistolicum, a "committee" report that he actually authored in secret and which attacked Leibniz's claims to priority [Ch, pg.526-527]. Given his dealings with Hooke and his nature, such an underhanded step seems much less unusual for him. Leibniz does not escape blame. He knowingly let Bernoulli attack Newton in much the way that Leibniz was going to be attacked [Ha 3, pg.117]. Perhaps this was a result of his overconfidence and his belief in the worthiness of his cause, but it would cause Bernoulli many problems after Leibniz's death [Ch, pg.554-555]. Leibniz also was plagued by the bad luck that he had faced earlier in his career. It seems that although Leibniz was a brilliant mathematician, he was always behind in the race to solve specific problems, whether in competition with Gregory or with Newton. There is much that can never be known about such a feud. This feud is peculiar in that it erupted late, and was both sparked and carried on to a large degree by the followers of the men involved. There are scientific reasons for it ( the divergences in their interpretation of "the calculus" itself; personal reasons ( a history of suspicion, not only between the two principles but between each of them and other rivals; nationalism, never a negligible factor; and the bitterness associated with disputes on related matters, notably the ongoing rivalry between the Newtonian and Cartesian theories of gravity. At a personal level, Newton's pride, suspicious character, and reluctance to publish collided with Leibniz' naive optimism, arrogance, and his belief in "systems" as more valuable than inspiration, in a long-delayed but virulent explosion.
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