## Happy Integral Day

At St. Bonaventure University in southwestern New York state, today is Integral Day.  It commemorates the invention and first use of the familiar integral sign, ∫, in a note written by Gottfried Leibniz on this date in 1675.  Leibniz was, with Sir Isaac Newton, a co-discoverer of the calculus.  Leibniz, who was elected a Fellow of the Royal Society, was something of a self-taught polymath:

﻿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.

There was for many years a disputation between Newton (and his English supporters) and Leibniz (and his continental supporters) over who had come up with the basic concepts of the calculus first.

﻿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.

What is not in dispute is  that Leibniz made significant contributions of his own.  He did fundamental work in differential equations, and invented the method of separation of variables.  And his notation for the calculus, not Newton’s, is what we almost always use today.  Say we have a function y = f(x).   Leibniz’s notation for the first derivative is:

$f'(x)= \frac{dy}{dx}$

whereas Newton’s notation for what he called “fluxions” just placed a dot over the variable:

$f'(x)= \dot{y}$

Those who teach or have taught introductory physics courses sometimes like Newton’s notation, particularly for use on a black- or white-board, since one can integrate just by erasing the dots.