###### By Michael Starbird, PhD, The University of Texas at Austin

## Stretching from the days of ancient Greece, calculus was developed and refined 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 developing calculus. It is is an incremental development, as many other mathematicians had part of the idea. Newton’s teacher, Isaac Barrow, said “the fundamental theorem of calculus” was present in his writings but somehow he didn’t realize the significance of it nor highlight it. As Newton’s teacher, his pupil presumably learned things from him. Fermat invented some of the early concepts associated with calculus: finding derivatives and finding the maxima and minima of equations. Many other mathematicians contributed to both the development of the derivative and the development of the integral.

Ironically, the person who was so averse to it ended up embroiled in the biggest controversy in mathematics history about a discovery in mathematics.

Newton was, apparently, pathologically averse to controversy. Ironically, the person who was so averse to it ended up embroiled in the biggest controversy in mathematics history about a discovery in mathematics. It was a cause and effect that was not 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 controversy surrounds Newton’s development of the concept of calculus during the middle of the 1660s. Between 1664 and 1666, he asserts that he invented the basic ideas of calculus. In 1669, he wrote a paper on it but refused to publish it. He wrote two additional papers, in 1671 and 1676 on calculus, but wouldn’t publish them. In time, these papers were eventually published. The one he wrote in 1669 was published in 1711, 42 years later. The one he wrote in 1671 was published in 1736, nine years after his death in 1727. The paper he wrote in 1676 was published in 1704. None of his works on calculus were published until the 18th century, but he circulated them to friends and acquaintances, so it was known what he had written. 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’s Paper on Calculus

But Gottfried Wilhelm Leibniz independently invented calculus. He invented calculus somewhere in the middle of the 1670s. He said that he conceived of the ideas in about 1674, and then 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.” It was six pages, extremely obscure, and was very difficult to understand.

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

One consideration we take as modern readers is that at that time, what we today think of as absolutely fundamental to start thinking about calculus, was that some of those ideas simply didn’t exist at all, such as the idea of function. The concept itself wasn’t formulated until the 1690s after calculus was invented, so people’s understanding of it was a little vague.

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

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

A famous couplet from a poem by Alexander Pope helps to demonstrate the 17th-century view of Newton, for 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 latter part of the 1600s, in 1699. Leibniz statement of Newton, then as now, calls us to take notice of the importance of one great mind commenting on another, “Taking mathematics from the beginning of the world to the time when Newton lived, what he has done is much the better part.”

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 himself began to realize in the 1770s, Newton’s response to ensure that he received the credit for calculus was to write a letter to Leibniz. In the letter, he encoded a Latin sentence that begins, “Data aequatione quotcunque…” 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 sentence encapsulated Newton’s thinking about derivatives. He took that sentence and he took the 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 this was what he included in this letter to Leibniz to establish his priority for calculus. Even though you read the sentence, it 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.

He tried to establish his priority in that fashion, but what followed were accusations that Leibniz had read some of Newton’s manuscripts before he conceived his own ideas. But, since Leibniz had published first, people who sided with Leibniz said that Newton had stolen the ideas from Leibniz.

It became a huge mess, that, incidentally, led to the retardation of British mathematics for the next century because they didn’t take advantage of the developments of calculus that took place in 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 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.