Who invented the integral sign? Leibniz, on 29 October 1675

The integral sign ∫ was invented by Gottfried Wilhelm Leibniz, who first wrote it on 29 October 1675 in an unpublished manuscript, Analyseos tetragonisticae pars secunda. It is an elongated S for the Latin summa, “sum”, because Leibniz treated an area as the sum of infinitely many thin slices.

Until then he had been writing omn., short for omnia (“all”), in front of the quantity to be summed. His manuscript records the switch:

“It will be useful to write ∫ for omn. so that ∫ l = omn. l, or the sum of all the l’s.” (Leibniz, 1675, as translated by MacTutor)

Leibniz was living in Paris, where he was given access to unpublished manuscripts of Descartes and Pascal; he later said reading Pascal’s mathematics set him thinking about what became his calculus.

Why do some sources say 21 November 1675?

Because the two dates describe different steps. On 29 October Leibniz wrote the bare sign; in a November manuscript he first wrote the full form ∫ f(x) dx, with the differential attached. The symbol reached print only in 1686, in a Leibniz paper on integral calculus in the Leipzig journal Acta Eruditorum.

Why did Leibniz’s ∫ win out over Newton?

Because the notation Newton used was easily misread and awkward for printers. Newton developed calculus independently, but his integration notation was a small bar above a letter or a box drawn around a term, and a barred x looked like x-prime. It never became popular, even in England, which is why students everywhere now write Leibniz’s ∫ and dx.

The sign has kept evolving. Leibniz and his successors described limits of integration in words; the modern definite integral, with limits a and b written at the bottom and top of the ∫, came from Joseph Fourier, who used it in his 1822 Analytical Theory of Heat.