d and ∫ are dated to the day, in his own hand, in 1675. The fraction bar you write between them is not his — his own printed page reads it as a ratio, not a stacked fraction.
On 29 October 1675, in the private working paper Analyseos tetragonisticae pars secunda, Leibniz proposed writing ∫ — an elongated s, for summa, sum — for what he called “omnia,” all of a quantity taken together. Thirteen days later, on 11 November 1675, in Methodi tangentium inversae exempla, he placed dx after the integral sign and used the differential symbols dx, dy for the first time. CITED
Both were private. Neither symbol reached print for years.
dx first appears in a published work in Nova Methodus pro Maximis et Minimis, Acta Eruditorum, 1684 — nine years after the manuscript. But the now-universal stacked fraction dy/dx is not the form Leibniz set in print. His printed notation writes the relationship as a ratio — forms transcribed as dy ad dx (“dy to dx”) and dy : dx — not a fraction with a horizontal bar. The typeset stacked fraction familiar from every calculus textbook today is a later typographic convention layered onto his notation, not a page from his own hand. CITED
The integral sign has its own, later public date: ∫ first appeared in print in Acta Eruditorum, June 1686 — eleven years after the private manuscript that proposed it. CITED
Leibniz and Newton each built a working calculus independently — Newton's fluxions from roughly 1665–1666, developed privately and published later; Leibniz's differentials from roughly 1673–1676, published earlier, first via the 1684 and 1686 Acta Eruditorum papers. The dispute over priority that followed was bitter and, by the read of essentially every historian of mathematics since, badly handled: the Royal Society's 1712 report favoring Newton was produced under Newton's own presidency of the Society, and is not treated today as an impartial finding. CITED The modern consensus is independent co-discovery, not appropriation in either direction.
One practical fact outlived the dispute. Leibniz's ratio notation composes the way algebra composes — $\frac{dy}{dx}\cdot\frac{dx}{dt}=\frac{dy}{dt}$ reads like cancellation, because the chain rule is exactly the claim that it behaves like one. Newton's dot notation ($\dot y$, $\dot x$) does not carry the same variable explicitly and does not compose as transparently across several dependent variables. That is very likely why continental Europe's mathematicians, working from Leibniz's papers, moved faster on the calculus of several variables through the eighteenth century — a historical claim, offered here as a claim and not re-derived. OPEN
Book XVII, ch02 derives a distance, not a derivative — no differential appears in it. But every rate-of-change caption in this corpus that writes a ratio of small changes is writing in Leibniz's form, ratio and all, whether or not the page says so. ch-descartes, the previous chapter in this gallery, is the companion correction: the fixed grid Leibniz's differentials are drawn on on is not quite the grid Descartes himself printed either.
That Newton's calculus is derivative of Leibniz's, or that the priority dispute has a clean winner — neither is the historical consensus, and neither is claimed here. Nor is it claimed that Leibniz's superior notation makes his calculus more “correct”; both are formally equivalent accounts of the same mathematics. The claim is narrower and checkable: specific dates for specific symbols, private versus public, and the printed form of the ratio as Leibniz himself set it. OPEN The practical claim in Part III about why continental mathematics moved faster is offered as a historical judgment call from the sources read, not independently verified against primary eighteenth-century texts.
| source | accessed | tier |
|---|---|---|
| MacTutor-adjacent, “Earliest Uses of Symbols of Calculus” (Univ. of Hawaii mirror of Jeff Miller's page) | 2026-09-22 | WEB |
| Wikipedia, “Leibniz's notation” | 2026-09-22 | WEB |
Web sources, checked live on the date given, not held-and-hashed PDFs. book7/ch-leibniz-verify.py checks that the exact dates and quotes above still print on this page; it is a presence check against this page's own text, not a live re-verification of the sources. OPEN