History of calculus
Leibniz's earliest surviving calculus: ∫ on 29 October 1675
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Original prompt & settings (JSON) ↗The first integral sign was a stretched letter S for summa, and Leibniz wrote it into a private notebook simply as a shorter way of saying "all of them".
29 October 1675: Analyseos tetragonisticae pars secunda
On 29 October 1675 Gottfried Wilhelm Leibniz used the integral sign ∫ for the first known time, in an unpublished manuscript titled Analyseos tetragonisticae pars secunda. He proposed that ∫ stand in place of omn. — omnia, "all" — and wrote ∫l = omn. l, the sum of all the l's.
The page belongs to a sequence of October–November 1675 quadrature notes in which Leibniz was trying to express systematically the inverse relationship between summing infinitesimal pieces and taking differences. This is the earliest of those notes to carry the notation that survived.
Two weeks later he took the next step. In a second private manuscript dated 11 November, Methodi tangentium inversae exempla — "Examples of the Inverse Method of Tangents" — he first placed dx after the integral sign, replacing his earlier x/d-style notation; MacTutor gives 21 November for the first manuscript use of the combined ∫f(x)dx. The November memorandum sits in the critical edition of Leibniz's Sämtliche Schriften und Briefe at series VII, volume 5, pages 321–331.
The notation ran ahead of the rules behind it
The November manuscript worked an inverse-tangent problem: rather than finding a tangent from a curve, Leibniz tried to recover a curve from a prescribed tangent property, one whose subnormal was inversely proportional to its ordinate. He treated summation as the inverse of taking differences.
The method was unfinished. A Mathematical Association of America reading notes that he was still testing incorrect possibilities for the differentials of products and quotients, and that dx apparently functioned as a constant equal to one in part of the argument.
He reached a correct product rule later in the same notebook sequence, on 27 November 1675.
What 'earliest' rests on
The word earliest depends on what qualifies as infinitesimal calculus. Newton had formulated essential ideas of his fluxional calculus before Leibniz's 1675 manuscripts, though publication and priority became contentious.
Popular accounts describe the event as the first use of integral calculus to find the area under y=f(x). Specialist histories call that a modernised description: the October manuscript concerned quadrature and summation notation, the November one inverse tangents, and Leibniz did not yet possess a clear modern concept of functions, limits or area as an integral.
The work stayed private and caused no immediate public change. His first paper on differential calculus appeared in 1684, and ∫ appeared in print in 1686.
Sources
Researched 23 Aug 2026 5 sources 2 audit passes, 2 issues open Date disputed
How this was checked
Researched from the web into a fact sheet, rewritten from that sheet, then audited against it by a different model. How the pipeline works →
Date disputed. Sources disagree; the research places it at 1675-10-29.
What the sources leave uncertain
- The supplied event gives only the year and the unclear phrase “breaking down of a function.” The identifiable manuscript event is dated 11 November 1675.
- Popular accounts call this the first use of integral calculus to find the area under y=f(x). That is a modernized description: specialist histories emphasize that the manuscript concerned inverse tangents and that Leibniz did not yet possess a clear modern concept of functions, limits or area as an integral.
- The priority word “earliest” depends on what qualifies as infinitesimal calculus. Newton had developed related methods earlier, while Leibniz’s importance here lies particularly in the surviving manuscript sequence and notation.