Two operations, secretly one
The claim at the core of the fundamental theorem of calculus is that two operations, which mathematicians had studied separately for a long time, calculating the area under a curve and calculating how fast a quantity is changing, are actually inverse versions of a single underlying operation. One part of the theorem shows that building up a running total of a function’s values, a process called integration, and then asking how fast that running total is growing, gives back the original function exactly. The other part turns this insight into a genuinely practical shortcut: to find the exact area under a curve between two points, it is enough to find any antiderivative of the function and subtract its value at one point from its value at the other, avoiding the tedious approximation methods mathematicians had relied on before.
A geometric proof, a decade early
The theorem’s core insight did not arrive fully formed. James Gregory published the first statement and proof of an early version in 1668, notable for its geometric style, and Isaac Barrow later proved a more general form of the result, directly influencing his student Isaac Newton. Newton began developing his own broader theory, which he called the method of fluxions, starting in 1666 at the age of twenty-three, working through further manuscripts in 1669 and 1671 before eventually including some of the underlying geometric calculus in his 1687 book, Principia. Working independently and starting somewhat later, Gottfried Leibniz began developing his own version of differential calculus from 1674 onward, introducing differential notation in his private notebooks by 1675 and publishing his first calculus paper in 1684, apparently without full knowledge of how far Newton’s own unpublished work had already progressed.
Two men, working apart
What has held up, entirely uncontroversially among mathematicians, is the theorem itself and its practical value: the technique of evaluating a definite integral through an antiderivative rather than direct approximation remains the standard method taught and used today. What has also held up, on the historical side, is the now-settled scholarly judgement that both Newton and Leibniz developed calculus independently rather than one man copying the other, a conclusion modern historians reached despite decades of accusation to the contrary. Leibniz’s specific notation for differentiation and integration, built around symbols still recognisable to any calculus student today, proved more practical for actual use than Newton’s fluxional notation, which gradually fell out of use outside Britain even among mathematicians who accepted Newton’s priority.
An accusation, then an inquiry
What did not hold up well was the process meant to settle the dispute fairly. A public accusation that Leibniz had plagiarised Newton’s work was raised by Nicolas Fatio de Duillier in 1699, after having been made privately years earlier, and the argument escalated until the Royal Society convened a committee in 1712 to investigate. That committee never solicited Leibniz’s own account of events, and the resulting report, Commercium Epistolicum, was in fact secretly written by Newton himself, who was serving as the Royal Society’s president at the time, and who structured the report to reference the allegations against Leibniz and conclude decisively in his own favour. Leibniz himself did not even see the document until 1714, two years after its publication, and a supposedly incriminating discovery decades later, in 1849, of Newton’s own work copied out in Leibniz’s handwriting, could not be reliably dated and so settled nothing further.
The judge who wrote the verdict
Beyond the specific priority quarrel, both the mathematics and the institutional failure carry lasting weight. Calculus itself, whichever man is credited with inventing it, has been called one of the greatest advances in mathematics since antiquity, and the fundamental theorem at its centre underlies virtually every field that depends on modelling change over time, from physics and engineering to economics. The dispute’s resolution mechanism matters just as much as a cautionary case: a scientific society that allowed its own president to write, unchallenged, the very report meant to adjudicate his personal priority claim reveals how vulnerable seventeenth-century science was to exactly this kind of conflict of interest, in an era before standardised publication and independent peer review existed to prevent it.
A notation that won on its own merits
Yes, and it is worth reading as two stories rather than one. The mathematics itself, that integration and differentiation are mirror images of the same operation, is a genuinely elegant unification worth understanding on its own terms, well beyond memorising it as a formula. The dispute that followed its discovery is, separately, a genuinely instructive piece of the history of science, showing how a question of pure intellectual credit could curdle into decades of bitterness, institutional bias, and a report whose author had no business writing it. Together they make for a subject that rewards attention for its mathematics and for what it reveals about how badly even respected institutions can behave when one of their own has a personal stake in the outcome.