The Thrilling Story Of Calculus

The story of calculus is seldom told in schools. Most students learn calculus as a rigid block of algorithms. Because of its seeming abstractness and lifelessness, the majority of students never build any meaningful connection with this creative branch of mathematics. However, the real story behind calculus is anything but abstract and lifeless. If calculus students were to learn the subject from its roots (history), they would grasp the true life and beauty behind this ingenious human invention.

Just to set things straight: this article is not about the mathematics of calculus. It is about the people and the story behind its invention. The two major characters to be featured in this story will be Isaac Newton and Gottfried Wilhelm Leibniz. As we unwind through the twists and turns of the story, there will be a key cameo role played by Abraham Robinson. Without further ado, let us begin!

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The English Polymath

Our story starts with the birth of Isaac Newton. Newton was born to a middle-class, misfortune-struck Lincolnshire family on Christmas day in 1642. Newton had lost his father three months before his birth and was a premature child. As he grew up, he faced difficulties with bullying as well.

Despite his challenging background, Newton showed a liking for learning and a curiosity for natural phenomena. He went on to show great promise in academics and landed on Trinity College, Cambridge with a scholarship.

In 1666, soon after Newton’s graduation, Cambridge closed down in fear of the Bubonic plague breakout (a 17th-century pandemic). As a result, Newton returned to Lincolnshire and spent a year back home. This year — 1666 came to be later known as Annus Mirabilis (miracle year) simply because of the sheer volume of amazing things Newton was able to achieve in this period.

He worked out the laws of gravity, made crucial discoveries in the optics of light, made breakthroughs in motion and mechanics, and perfected his invention of calculus in the same year. All this was going on while a global pandemic was at large. Such was the genius and ingenuity of Isaac Newton.

The German Polymath

Our second protagonist, Gottfried Wilhelm Leibniz, was born in 1646 in Leipzig (the holy Roman empire back then) into a rich culture of philosophy. His father was a philosopher, and Leibniz was given access to his library from the age of 7.

This library consisted of a wide variety of advanced philosophical and theological works largely written in Latin. Leibniz mastered the Latin language by the age of 12. He later enrolled at the same university as his father at the age of 14 and graduated in Philosophy in 1662. He then went on to earn his master’s degree in Philosophy in 1664.

In early 1666 (yes, that fateful year again), Leibniz wrote his first book called De Arte Combinatoria (On the Combinatorial Art) at the age of 19. Such was the genius and sheer talent of Gottfried Wilhelm Leibniz.

Roads Meet

Newton took a particular interest in how things changed. He wanted to quantify change. That is how he landed on the invention of calculus. Calculus essentially describes the instantaneous rate of change of a physical quantity. This physical quantity can be speed, mass, weight, etc.

As it turned out to be the case, such a tool would be immensely useful to study and develop theories of planetary motion (among other things). Newton was covering ground thick and fast in all these fields, and calculus was a powerful tool he had developed to aid his progress.

In the meantime, completely independent of Newton, Leibniz had come up with a way of calculating the area under a function curve (integral calculus). Essentially, he had invented his own version of calculus independent of Newton.

In 1676, he visited London on one of his trips. Upon revealing his invention, he was accused by the English mathematical establishment of having taken glimpses of Newton’s unpublished work, and that his work was not original. This started a big dispute that went on for a long time.

Literature War

Leibniz published his fundamental theory on calculus in 1684. He then went on to define the inverse relationship between integration and differentiation (known as the fundamental theorem of calculus) in 1693. Newton, on the other hand, published his science-shattering book series: Philosophe Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) in 1687. This consisted of 3 books: Book 1 — the motion of free bodies in space (which consisted of derivatives and integrals), Book 2 — the movement of bodies in a restrictive medium, and Book 3 — celestial mechanics (which included his work on gravity).

Crucially, Newton published his work on calculus after Leibniz. Following the dispute that started in 1676, this did not sit well with the Royal Society of London. Back then, the Royal Society held a position of authority in science, catered to the public’s needs to solve technical problems, and offered salons of open discussion and debates.

The Royal Society Passes its Judgement

The Royal Society of London was one of the foremost and earliest established science societies. Furthermore, at the time of investigation into the authenticity of Leibniz’s work, Newton was the acting president of the Royal Society of London.

Needless to say, this did not work out well for Leibniz. The Royal Society used some circumstantial evidence from Leibniz’s travels against him, and he was stripped of all the credit for his work in inventing calculus.

Leibniz struggled to bring himself to challenge the mighty English science society and passed away in Hannover in 1716. Neither the Royal Society nor the Berlin Academy of Sciences honoured his death. His grave was unmarked for more than 50 years.

Newton went on to enjoy fame and further success in England until his death in 1727 at the age of 84.

Historians Vindicate Leibniz

Much after Leibniz’s lifetime, historians probed into his work as well as Newton’s. It was then established that Leibniz did indeed invent calculus independent of Newton. After this, slowly but surely, it was widely recognized that both Leibniz and Newton invented calculus (independently).

In fact, the common notation we use today in differentiation and integration (for example, d/dx) are from Leibniz and not Newton. If anything, Leibniz’s legacy lives on in this manner in the world of science and mathematics.

The ‘Infinitesimals’ Controversy

Both Newton and Leibniz used a notion of very small numbers that were very close to but not exactly zero. These numbers were later formalized as Infinitesimals. This notion was useful in practically applying calculus to solve real-world problems. But the mathematicians of the time did not think that the concept of infinitesimals was rigorous and consistent. After a lot of controversies, it was banned. You can read more about the ‘infinitesimals’ controversy in this article.

Later on, mathematicians and fellow Polymaths such as Augustin-Louis Cauchy and Karl Weierstraß invented the concept of Limits to formally fortify calculus. This way, calculus was mathematically rigorous and consistent.

The Modern Hero

In the final part of our story, we arrive at the 20th-century. Abraham Robinson was a German-born mathematician. He had a rollercoaster life, having been born to a Jewish family during the world war. He had to overcome difficult situations to survive. He eventually made his way through France into Britain.

In Britain, he taught himself aerodynamics and was involved in developing wings of fighter planes. Eventually, he was fascinated by the work of Leibniz in calculus and wanted to investigate the long-banned notion of infinitesimals.

He reportedly said that he wanted to get inside the head of Leibniz and understand everything about the first mathematician to articulate the concept of infinitesimal numbers. Through his work, he proved that the notion of infinitesimals can be defined rigorously and consistently. In the process of proving this, he popularized the branch of mathematics known as Nonstandard Analysis and Hyperreal Number System.

Until the 1960s, the mathematical world had accepted the calculus notations from Leibniz, but the concept of infinitesimals still remained banned. This was until Robinson published his book Non-Standard Analysis.

In the end, thanks to the efforts of Robinson, Leibniz now gets full credit for the entire work he had done to invent and use calculus. One can only wonder if things would have been different if he had been given more due credit when he was still alive.

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Further reading that might interest you: What Really Happens When You Invent Infinite Infinities? and How To Make Working With Squares Fun In Math?