Calculus is one of the most powerful tools in mathematics, used for everything from physics to economics. But what is its origins? The development of calculus was a long process, shaped by many brilliant mathematicians over thousands of years. This article explores the key moments in the history of calculus, from ancient methods to the groundbreaking work of Isaac Newton and Gottfried Wilhelm Leibniz.

### Ancient History of Calculus

#### Eudoxus and the Method of Exhaustion

Imagine trying to figure out the area of a circle by filling it with different polygons. If you use a triangle there are some big gaps but if you use a square, then a regular hexagon, then a regular octagon those gaps start to shrink and the areas of the two shapes get closer. This is the technique that a Greek mathematician named Eudoxus discovered in the 4th century BCE and called the “method of exhaustion.”

#### Archimedes and Tangents

Fast forward 100 years, and we meet Archimedes. He built upon Eudoxus’s work and began exploring methods to calculate the tangent line to shapes other than the circle. Using polar coordinates, he was able to calculate the slope of a line tangent to a spiral (known as an Archimedean spiral), becoming the first mathematician to do so. Nowadays, these computations would be done using derivatives, but since calculus had not yet been invented, Archimedes had to use more primitive tools. Still, Archimedes was a trailblazer, paving the way for future developments.

#### Other Calculus Discoveries Throughout the World

Around the mid-3rd century AD, the Chinese mathematician Liu Hui came up with his own version of the “method of exhaustion,” independently of Eudoxus. Other developments came later from the Middle East and India, in the 11th and 14th centuries, respectively, but the study of calculus was a slow march that did not capture the world’s attention. In the 17th century, however, all of that changed.

### Calculus Takes Shape

#### Kepler’s Integral Calculus

In 1615, German mathematician Johannes Kepler published a groundbreaking work that laid the foundation for integral calculus. His interest was sparked when a merchant who was selling him wine used an inconsistent method to calculate how much wine was in each barrel. Kepler thought that there must have been a better way and so he set out to study volumes and areas. Through his studies he derived a rule that served as an approximation of integrals, and while he did not have our understanding of integral calculus his work advanced it by leaps and bounds. Here is the rule he discovered for approximating definite integrals, also known as Simpson’s Rule:

$\displaystyle\int_a^bf(x)dx\approx\frac{b-a}{6}\left[f(a)+4f\left(\frac{a+b}{2}\right)+f(b)\right]$

Apart from mathematics, Kepler’s work in astronomy, especially Kepler’s Laws of planetary motion, inspired a young man named Isaac Newton.

**Isaac Newton: Father of Modern Physics**

Newton, famous for his work in physics and the story of the falling apple, is one of the key figures in the development of calculus. As the story goes, Newton sat under a tree one day taking a nap, and was hit in the head by an apple. Where most people would walk away with nothing more than a bump on their head, Newton walked away with the beginnings of a major scientific realization: all objects obey the rule of universal gravitation. While this story is more based on folklore than reality and it is likely that he merely witnessed an apple falling, his breakthrough itself is historically factual. Following this incident, Newton developed the core principles of Newtonian Mechanics. In 1948 the unit of force called the “Newton” was officially adopted in his honor. However, Newton was faced with a big obstacle – the mathematics necessary to describe the science of motion had not yet been developed. Newton found an elegant solution to this problem – he developed the mathematics himself! He called his theory “fluxional calculus,” which was very effective in describing motion. During the 1660s and 1670s, Newton made significant advancements, but he wasn’t alone in his discoveries.

### Newton vs. Leibniz

#### Two Minds, One Discovery

Ever thought you had a brilliant idea, only to find out someone else had it too? That’s exactly what happened with Newton and his German contemporary, Gottfried Leibniz. Both discovered key aspects of calculus around the same time.

Leibniz’s breakthroughs came just several years after Newton’s as the young German mathematician was convinced to study calculus by Dutch Mathematician Christiaan Huygens in 1672. Leibniz embarked on a fierce self-study of calculus in the following years and soon he really began to make headway. He reasoned that the slope of a line was a ratio of the change in the y-coordinate (‘ordinate’) to the change in the x-coordinate (‘abscissa’), or in other words, the motion of the y-coordinate per the motion of the x-coordinate. He then reasoned that an integral was the sum of all the infinitely small rectangles beneath a curve. This is where Leibniz had his real breakthrough as he realized that the slope of a tangent and integral were inverse operations. He knew his discoveries were special and took pains to create a notation to convey them.

Between October and November of 1675, Leibniz formalized his notation creating much of the common calculus we use today such as the large s-shaped integral sign and using ‘dx’ and ‘dy’ to describe the infinitely small ordinates and abscissas:

$\displaystyle\int_a^bf(x)dx \qquad\frac{dy}{dx}$

His work was published first, but both he and Newton made incredible contributions that together formed the calculus we learn today.

#### Controversy

As people recognized the profound significance of the discoveries made by Newton and Leibniz, one question persisted: who discovered calculus first? Initially, this question did not seem to matter. Both Newton and Leibniz had independently discovered the fundamental principles of calculus, and they appeared to respect each other’s achievements. Although they had some disagreements over the details and methods of calculus, they did not question each other’s integrity.

All that changed in 1695 when a British mathematician, John Wallis, started a fire he couldn’t put out: he accused Leibniz of plagiarizing Newton’s work! What ensued was a dirt-slinging insult battle with drama to rival a Shakespearean play.

Newton embarked on a smear campaign to silence Leibniz to make sure that he, and only he, would be remembered as the “father of calculus.” He told sources that Leibniz had some of his early calculus manuscripts and that he had written letters to Leibniz discussing these topics. When Leibniz’s friends tried to defend him, Newton turned up the heat and pressured them into taking back anything they said.

Soon the mathematical community lost its trust in Leibniz and hailed Newton as the true father of calculus. In 1713 the Royal Society, which Newton was the president of (talk about bias!), published an official paper proclaiming that Newton was the first in the calculus race and that Leibniz had been influenced by Newton.

Sadly, after being slandered by Newton and his cronies, Leibniz died poor and dishonored without much to his name, but since then history has been kinder to him. Mathematicians eventually realized that Leibniz invented much more effective notation. It was widely adopted and it is now used worldwide. Newton’s notation for integrals was inconsistent and unintuitive, which introduced significant difficulties. Here is Newton’s notation for derivatives and integrals with the dot notation referring to derivatives and the prime notation to integrals.

$\text{Derivative: } \dot{y} \qquad \text{Integral: } \int y \, dt \text{ or } y’$

Source: https://en.wikipedia.org/wiki/Notation_for_differentiation#Newton’s_notation_for_integration

While his notation works, Leibniz’s is more specific as it identifies both the variable of integration and differentiation and it emphasizes the ratio relationship between dy and dx. In addition to this, Newton’s notation for integrals would be confusing to use because of its overlap with the derivative notation of Joseph-Louis Lagrange. Lagrange was the first to write a derivative using the prime that Newton used for his integral, and to not confuse the two, most of the mathematical world uses Leibniz’s integral notation.

Historians now agree that despite the controversy, Newton and Leibniz independently developed calculus without any wrongdoing. However, due to the debate, England clung to Newton’s cumbersome notation for over 100 years, while much of mainland Europe advanced mathematically. As a result, England fell behind in math and science. Morris Kline captured this, saying, “The English settled down to study Newton instead of nature.” This focus on Newton’s methods led to stagnation, and it wasn’t until the 1820s, when modern notation was adopted, that England began to catch up scientifically.

### Conclusion

You’ve now explored the fascinating journey of calculus, from its early beginnings to its development by Newton and Leibniz. The history of calculus shows how a collective effort of brilliant minds over the centuries shaped this powerful tool.