Before beginning this story, a little background. There are two really basic ways to think of calculus:

1. The study of the infinite (extremely large) and the infinitesimal (extremely small).

Or

2. The study of limits. Imagine a gnat that is flying from the middle of a room to the doorway. The gnat first moves halfway to the door. Then he takes a little breather and moves half of the remaining distance. Another breather, another jaunt half of the remaining distance. And so on and so on. Will he ever get to the door?

(Okay, most mathematicians might hate me for boiling things down to this very basic level, but for the average Joe or Jane, these explanations will do the trick. And due to space issues, I need for you to just trust me on why these things matter. Some day, I’ll write about the applications of calculus and other higher-level math.)

If you think math history isn’t very exciting — in a Batman meets Joker or Clint Eastwood make-my-day kind of way — you’re pretty much right. There are a few life-and-death situations, like Galileo’s (okay, it was his soul in peril, not his physical body), but for the most part, mathematicians were either revered or went unnoticed. Except for Sir Isaac Newton and Gottfried Wilhelm Leibniz.

I wish I could say that this was an actual duel, not because I love violence or wish ill on one of these fine mathematicians, but because it would make this story even more interesting — especially to high school students or grown ups who think math is BOR-ring. But in the end this story is still pretty fascinating, especially given the fact that these men never met or spoke on the phone or Skyped (because cell phones and the internet didn’t exist).

It was 1666, around the time of the Apple Incident (you know, when a fallen apple prompted Newton to develop his theories of gravity) that The Sir thought up his ideas of fluxions. Don’t worry, you shouldn’t know what that word is, as it’s never used in modern mathematics. Instead we call his development differential calculus.

Leibniz was just 20 years old at that time. Sure, he was a genius — he had already earned degrees in philosophy and law, and that year he published his first book, De Arte Combinatoria or On the Art of Combinations. While this expansion of his philosophy dissertation is obliquely related to mathematics, it was well before Leibniz began formally dabbling in the Queen of the Sciences.

This timing is pretty darned important. Trust me.

So Newton farts around with this idea of fluxions, finally getting around to publishing Method of Fluxions in 1736. But along he published a few manuscripts on the subject, sending early copies to some colleagues. Meanwhile, in Germany, Leibniz was jotting down his own discoveries in his journal. In 1675, he noodled around, finding the area under a the graph of y = f(x) using integral calculus.

In other words, the two men were discovering calculus at the same time and in completely different parts of the world. (Okay, Germany and England weren’t too distant from one another, but in the 17th century, they may as well have been on different planets.)

I’d bet that given Newton’s stereotypical absent-minded-professor approach to the world around him, he might never have even noticed Leibniz’s publications, which came in 1684 and 1686. Or at the very least, he might have simply acknowledged the great coincidence and moved on. (Apparently, the man could barely be trusted to keep a dinner date, much less worry about a rival in a different country.)

In fact, it was neither Newton nor Leibniz who lit the fire of the great calculus war. In 1704, an anonymous review of Newton’s fluxions suggested that he borrowed [ie stole] the idea from Leibniz, which of course infuriated Newton. Letters flew back and forth between the two mathematicians and their surrogates. Newton was behind the publication of these letters, called Commercium Epistolicum Collinii & aliorum, De Analysi promota. (I am not kidding.) A summary of this publication was published anonymously in 1714 in the Philosophical Transactions of the Royal Society of London. But everyone knows that Newton wrote it.

The Swiss mathematician Johann Bernoulli — who later made his own contributions to infinitesimal calculus — attempted to defend Leibniz, but Newton pretty much took him down. In the end Leibniz meekly defended himself, refusing to look through his “great heap of papers” to prove that he had independently discovered calculus at the same time as Newton. When he died in 1716, Leibniz had been pretty well beaten up by Newton and his buddies (metaphorically speaking, of course).

It wasn’t until much later that everyone came around to the accepted and logical — though really coincidental — truth of the whole ordeal. Both Newton and Leibniz discovered calculus at the same time, using slightly different approaches. To many of us math folks, this is a truly wondrous event.

But there’s more. Even though Newton enjoys (and did enjoy) a bit of celebrity for his genius, he largely wrote for himself, while Leibniz was a bit obsessive about notation, wanting to be sure that his discoveries could actually be used. This is one of the big reasons that today’s calculus is pretty much Leibniz’s discovery. Newton’s approach turns out to be a bit too clunky for everyday use.

So whether or not you possess a general (or specific) understanding of calculus, you can certainly appreciate the 17th-century-style drama surrounding the discovery of this critical field of mathematics, right? At the very least, we can thank Newton and Leibniz for that.

Did you know about the great calculus controversy? What questions does it bring up for you? Ask them in the comments section!

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