###### By Michael Starbird, Ph.D., The University of Texas at Austin

## Calculus was an invention of many people over centuries. There were ideas of calculus in ancient Greek times, and it proceeded to be developed throughout the centuries up until the time of Newton and Leibniz. But when it comes to who gets the credit for “discovering” one of the most revolutionary concepts in all of mathematics, the matter is a little unclear.

Sir Isaac Newton was a mathematician and scientist, and he was the first person who is credited with actually developing calculus. As I say, it really is an incremental development, and many other mathematicians had part of the idea. In fact, Newton’s teacher, by the name of Barrow, actually said “the fundamental theorem of calculus” in his writings but somehow didn’t realize the significance of it and didn’t actually highlight it. But he was Newton’s teacher and presumably Newton learned things from him. Fermat invented some of the early concepts associated with calculus, finding derivatives and finding maxima and minima of equations. And other mathematicians, many mathematicians contributed to both the development of the derivative and the development of the integral.

Newton was, apparently, pathologically averse to controversy. He really didn’t like to be involved in controversy. And because of his aversion to controversy, he was involved in probably the biggest controversy in the history of mathematics about a discovery in mathematics.

Well, Newton was, apparently, pathologically averse to controversy. He really didn’t like to be involved in controversy. And because of his aversion to controversy, he was involved in probably the biggest controversy in the history of mathematics about a discovery in mathematics. So it’s ironic that the person who was so averse to it actually ended up being embroiled in the biggest one in history. It was cause and effect. It wasn’t just an accident. It was his aversion that caused the controversy.

Learn more about the study of two ideas about motion and change

## Newton’s Published Papers on Calculus

The reason that it caused it is that Newton actually developed the concept of calculus during the middle of the 1660s. And in 1664, ’65, ’66, in that period of time, he asserts that he invented the basic ideas of calculus. And in fact, in 1669, he wrote a paper on it but wouldn’t publish it. In 1671, he wrote another paper on calculus and didn’t publish it; another in 1676 and didn’t publish it. In fact, these papers were actually published. The one he wrote in 1669 was published in 1711. That’s, what, 42 years later? The one he wrote in 1671 was published in 1736; nine years after he was dead. And then the paper he wrote in 1676 was published in 1704. So none of his works on calculus were published until the 18th century. But he did circulate them to friends and acquaintances, so it was known that he actually had this. This wasn’t just hearsay, and he used the techniques of calculus in his scientific work.

This is a transcript from the video seriesChange and Motion: Calculus Made Clear. Watch it now, on The Great Courses Plus.

## Leibniz Paper on Calculus

But Leibniz, Gottfried Wilhelm Leibniz, independently invented calculus. He invented calculus somewhere in the middle of the 1670s. So he said that he thought of the ideas in about 1674, and then actually published the ideas in 1684, 10 years later. His paper on calculus was called “A New Method for Maxima and Minima, as Well Tangents, Which is not Obstructed by Fractional or Irrational Quantities.” So this was the title for his work. It was six pages and was extremely obscure and was apparently very difficult to understand.

Learn more about the first fundamental idea of calculus: the derivative

And one thing that you have to understand, by the way, is that at that time, what we today think of as absolutely fundamental to even starting to think about calculus, some of those ideas simply didn’t exist at all, for example, the idea of function. We’ve talked about function—you know, at every time we know where we are, at every time there’s a speed—those are examples of function. That concept itself wasn’t actually formulated until the 1690s, after calculus was invented. So people were a little vague on these concepts.

“Taking mathematics from the beginning of the world to the time when Newton lived, what he has done is much the better part.” This was Leibniz talking about Newton.

Well Newton and Leibniz didn’t understand it in any more formal way at that time. This was a big problem for all of the people in that century because they were unclear about such concepts as infinite processes. That was a huge stumbling block for them. They were worried about infinitesimal lengths of time. Both Newton and Leibniz would think about infinitesimal lengths of time. How far does something go in an infinitesimal length of time? Well that kind of thinking leads to all sorts of paradoxes, including Zeno’s paradoxes.

I wanted to read you a very famous couplet from a poem by Alexander Pope. These are the kinds of things one would like to have written about oneself. “Nature and Nature’s laws lay hid at night; God said, Let Newton be! and all was light.” So this was Alexander Pope on Newton.

## The Controversy Between Newton and Leibniz

The controversy between Newton and Leibniz started in the later part of the 1600s. 1699 was a date associated with a start of a tirade, which just went downhill. It was a tremendous controversy. But Leibniz had this to say about Newton. And I’ve served on many committees reading letters of recommendation for mathematicians, you know, for positions, and I can assure you that there are huge numbers of mathematicians who are the best three mathematicians in the world in any given field, but if a sentence like Leibniz’s sentence on Newton appeared in a letter, one would take notice. Leibniz said about Newton, “Taking mathematics from the beginning of the world to the time when Newton lived, what he has done is much the better part.” This was Leibniz talking about Newton.

Even a mathematician wouldn’t know from the actual translation of the sentence exactly what it was that he had done.

But when Newton began to realize that Leibniz had the ideas of calculus, which he began to realize in the 1770s, Newton’s response to make sure that he got credit for calculus was to write a letter to Leibniz in which he encoded a Latin sentence and I will—well, I won’t attempt the Latin, but I’ll attempt just a few words of the Latin. It starts out, “Data aequatione quotcunque” and so on. It’s a short Latin sentence whose translation is, “Having any given equation involving never so many flowing quantities, to find the fluxions, and vice versa.” This was a sentence that encapsulated his, Newton’s, thinking about derivatives. And what he did is he took that sentence and he just took the letters, individual letters, a, c, d, e, and he put them just in order. He said there are six a’s, two c’s, one d, 13 e’s, two f’s. He put them in order and that was what he included in this letter to Leibniz to establish his priority for calculus. And I read you the sentence, which means very little to anybody. Even a mathematician wouldn’t know from the actual translation of the sentence exactly what it was that he had done.

So he tried to establish his priority in that fashion, but then there were accusations that Leibniz had read some of these manuscripts of Newton’s work before he got the ideas. But since Leibniz had published first, people who were siding with Leibniz said that Newton had stolen the ideas from Leibniz. And it was a huge mess, which incidentally led to British mathematics being very retarded for the next century because they didn’t take advantage of the wonderful developments of calculus that were taking place on continental Europe.

Learn more about the derivative and the integral

### Common Questions About the Invention of Calculus

**Q: What exactly is calculus?**

Calculus is a specialized mathematics that allows one to calculate the behavior of functions as they near points close to infinity. It is the study of the relationships of limits, integrals and derivatives.

**Q: Who ultimately is responsible for inventing calculus?**

While Newton came up with many of the theorems and uses prior, the final conclusion is that Gottfried Wilhelm Leibniz invented Calculus.

**Q: How has calculus changed the world?**

Calculus has made possible some incredibly important discoveries in engineering, materials science, acoustics, flight, electricity and, of course, light.

**Q: Can calculus be used in real life?**

Yes, calculus is used predominantly in chemistry to predict reaction rates and decay. Calculus can predict birth and death rates, marginal cost and revenue in economics as well as maximum profit to name but a few practical uses.