This Day in Tech 1675- Leibniz sums it up

Image: Bernhard Christoph Francke

Perhaps Gottfried Wilhelm Leibniz was inspired by his hairstyle when he first wrote the integral symbol ∫ in an unpublished manuscript.
Image: Bernhard Christoph Francke

1675: Gottfried Leibniz writes the integral sign ∫ in an unpublished manuscript, introducing the calculus notation that's still in use today.

Leibniz was a German mathematician and philosopher who readily crossed the lines between academic disciplines. He had a doctorate in law, served as secretary of the Nuremberg alchemical society and fancied himself a poet.

He also conducted diplomatic missions in London and Paris. While visiting those cities, Leibniz acquainted himself with such scientific luminaries as Christiaan Huygens, Robert Boyle, Robert Hook, John Pell and Jacques Ozanam. He showed an unfinished calculating machine to the Royal Society, which elected him a fellow.

Leibniz discussed with his English colleagues his interest in summing series and the geometry of infinitesimals, and he corresponded with them when he was back in France. They apprised him of books in the field and also told him about Isaac Newton's yet-unpublished work on the subject.

Newton wrote to Leibniz through an intermediary, and they began an exchange of letters that often took weeks or even months to reach their recipient. The muddled back-and-forth eventually led to bad blood, with Newton claiming that Leibniz had stolen his work in founding the science of calculus.

Newton's letters, however, described results, not methods. The legal and philosophical formalism in which Leibniz had been trained allowed him to create his own symbolic system, including not just the integral sign but the same notation of differentials we still use. Newton published his system slightly before Leibniz, but the German's notation was superior.

Continental and English mathematicians would spend decades arguing over who invented the calculus, but it seems yet another example of simultaneous discovery. The two scientists were of the same era, associated in the same circles, read the work of the same precursors, and shared some of their own ideas. It should amaze no one that they came to the same results in slightly different mathematical language at nearly the same time.

Leibniz contributed mightily to our knowledge of differential equations. He discovered the method of separation of variables, first reduced homogeneous equations to separable ones, and figured out how to solve first-order linear equations. He also worked on the multinomial theorem.

The math department of St. Bonaventure University in western New York state celebrates Integral Day on Oct. 29 to honor Leibniz. The mathematics suite is decorated with integral and summation ornaments, and students and faculty eat "calculus cookies" and imbibe "summation cider," presumably with infinitesimal nibbles and sips. Students compete in a calculus contest to win a gift certificate at the college bookstore.

Does Newton deserve more credit? Maybe, but it's Leibniz's language you learned in your calculus class. And ol' Isaac gets his props for many other discoveries, so don't overestimate the gravity of the situation. Happy Integral Day, Gottfried!

Source: Eric Weisstein's World of Math, MacTutor History of Mathematics