The differential and integral calculus — the mathematical theory of instantaneous rates of change and accumulated areas under curves — was the foundational tool of 17th- and 18th-century European mathematical physics. Two men developed it substantively independently between the mid-1660s and the mid-1680s: Isaac Newton at Cambridge, working from approximately 1665, calling his version the method of fluxions; Gottfried Wilhelm Leibniz at Mainz and Hanover, working from approximately 1672, publishing his version under the name calculus differentialis in the journal Acta Eruditorum in October 1684.

The substantial subsequent priority dispute between Newton and Leibniz — and between their respective British and Continental partisans — substantially poisoned European mathematics for approximately a century.

The chronology

The substantial accepted modern chronology runs approximately as follows:

1665–1666: Newton, then 22 and at his mother’s farm at Woolsthorpe during the Cambridge plague closure, substantively developed the basic ideas of the fluxional calculus and applied them to several geometrical problems. He showed the work privately to his mathematical mentor Isaac Barrow and to a small Cambridge circle but did not publish.

1671: Newton wrote the Methodus fluxionum substantively as a comprehensive treatise on the calculus. He circulated it in manuscript among the Cambridge mathematical community. He did not publish (Newton’s reluctance to publish was a professional habit through his entire career and produced subsequent priority disputes with multiple other contemporaries).

1673: Leibniz visited London on a diplomatic mission and met members of the Royal Society. He was shown some of Newton’s mathematical correspondence by the Society’s secretary Henry Oldenburg. The historical question of whether Leibniz learned the basic fluxional method from this exposure has been the central question of the priority dispute since 1699.

1675: Leibniz substantively developed his own version of the calculus at Paris, with different notation (the dy/dx and the integral sign that we still use today) and different conceptual foundation (Leibniz conceived of the calculus as the manipulation of infinitesimal quantities; Newton conceived of it as the geometric analysis of instantaneous motion).

1684: Leibniz published the calculus in Acta Eruditorum. The publication was substantively the first European publication of the calculus.

1687: Newton published the Principia, which used fluxional methods throughout but expressed them in geometric form rather than in the fluxional notation.

1693–1699: The Genevan mathematician Nicolas Fatio de Duillier, a young protégé of Newton, published a accusation that Leibniz had plagiarised the calculus from the 1673 Royal Society correspondence. The accusation was the opening of the priority war.

The Royal Society verdict

The dispute escalated through the next decade. Leibniz appealed in 1711 to the Royal Society for an arbitrating judgement. The Society — by then under Newton’s personal presidency — appointed a commission of inquiry, which produced the Commercium Epistolicum in 1712: a published collection of the relevant 1670s and 1680s correspondence, edited and arranged to support the Newtonian priority claim.

The Commercium Epistolicum was Newton’s own work. Newton had drafted the commission’s report personally; he had selected and arranged the included correspondence; he had written the subsequent anonymous review of the Commercium in the Philosophical Transactions of the Royal Society. The substantive procedural irregularities of the 1712 inquiry were and were known to European mathematicians at the time.

Leibniz refused to accept the verdict. He continued to defend his priority claim through correspondence and through the mediation of the Hanoverian court (he was substantively the court librarian and historiographer at Hanover for the last forty years of his life). He died at Hanover in November 1716, aged 70, with the priority dispute unresolved.

The consequences

Modern mathematical-historical consensus is that Newton and Leibniz substantively independently developed the calculus — Newton earlier in private, Leibniz earlier in publication — and that no plagiarism was involved on either side. The 1712 Commercium Epistolicum is considered a procedurally compromised document. The Royal Society itself acknowledged the procedural irregularities in a commemorative reassessment in the 20th century.

The damage to European mathematics from the dispute was lasting. The British mathematical community — loyal to Newton — refused to adopt the Leibnizian notation through approximately the next century, with the result that British mathematics diverged from the more successful Continental mathematical tradition. The Cambridge undergraduate mathematical curriculum used the less efficient Newtonian fluxional notation until approximately the 1820s, when the Cambridge Analytical Society finally imported the Leibnizian framework. The century-long gap produced the subsequent dominance of French, German, Swiss, and Russian mathematicians in the European mathematical mainstream and relegated British mathematics to a secondary position that it did not recover until approximately the late 19th century.

The priority dispute also substantively reproduced for subsequent priority disputes: the form of bitter national-and-personal mathematical priority war became a recurrent feature of European mathematical history, with later examples ranging from the 19th-century non-Euclidean geometry disputes to the 20th-century quantum mechanics priority claims.