The Emergence of Calculus: A Mathematical Journey of Human Thought

Feb 15, 2020 · 26 min read
William Blake’s Newton (1795)

Ten years ago, I was sitting by the window of a library, the same one I still attend today. I purposely sat there because it was raining, and I enjoyed watching the movements of the raindrops cluttering against the window. I opened my book to the page where I had left off, and after a while, I read a sentence that sparked my curiosity immediately. The author, who called himself “a curious character,” said, “I have to understand the world, you see.” This sincere sentence was so profound for a young, curious math student like myself. It was such an “Aha” moment for me that I felt as if I should buy a cup of tea for everyone sitting at the library, and even have a long discussion about the book that would last until the next morning.

“I have to understand the world, you see.” Surely you’re joking, Mr. Feynman!, p.231

I immediately paid attention to the author’s name. His name was Richard Feynman, a well-known physicist, and theorist. After I finished his book, “Surely you’re joking, Mr. Feynman!” then I started reading some of his other works such as, “The Pleasure of Finding Things Out,” and even watched every documentary and interview about him I could find. He had fascinating ideas about life. For instance, in one interview on BBC for the Horizon, he shared a conversation he once had with his artist friends about ‘The Pleasure of Finding Things Out.’ Today, we call this monologue “Ode to the Flower.

“I have a friend who’s an artist that has sometimes taken a view of which I don’t agree with very well. He’ll hold up a flower and say ‘look how beautiful it is,’ and I’ll agree. Then he says, ‘I, as an artist, can see how beautiful this is, but you as a scientist, take this all apart and it becomes a dull thing,’ And I think that he’s kind of nutty. First of all, the beauty that he sees is available to other people and me, too, I believe. I can appreciate the beauty of a flower. At the same time, I see much more about the flower than he sees. I could imagine the cells in there, the complicated actions inside, which also have a beauty. I mean, it’s not just beauty at this dimension, at one centimeter; there’s also beauty at smaller dimensions, the inner structure, also the processes. The fact that the colors in flower evolved to attract insects to pollinate it is interesting; it means that insects can see the color. It adds a question: Does this aesthetic sense also exist in the lower forms? Why is it aesthetic? All kinds of interesting questions which the science knowledge only adds to the excitement, the mystery, and the awe of a flower. It only adds. I don’t understand how it subtracts.

A cool designer, Fraser Davidson, made a lovely short animation for that part of the interview above.

Richard Feynman was not only a curious character but also a unique personality. When he decided to go to Brazil to teach for a year, he started learning Portuguese to teach in the students’ native language. Feynman recommended the Pulitzer Prize author Herman Wouk to learn calculus when he learned that his friend didn’t know did not know it. Feynman believed that calculus is the language in which God talks. His words remind us of Galileo’s words; the laws of nature are written in the language of mathematics. Feynman was sharing Galileo’s faith and following his steps.

Richard Feynman. Wikimedia Commons

Calculus means “pebble, little stone” in Latin. In ancient times, people were using pebbles or little stones to make calculations.

I firmly believe that Feynman’s recommendation was not only for Herman Wouk but also for all human beings. He was right! In the modern-day, it is not beneficial that one does not know anything about calculus, even if you are a physiologist, engineer, philosopher, or musician.

Calculus was the best thing I have ever encountered in my life. It was a tool that taught me the abstract idea and showed me an easy way to make the problems in my life more manageable. It was calculus that made possible what Kennedy said, “We choose to go to the moon!”, when Armstrong said, “One small step for a man, one giant leap for mankind.”, and when Felix Baumgartner said, “I’m going home now!

Left: On Sept. 12, 1962, President John F. Kennedy tells a crowd of 35,000 at Rice University Stadium in Houston, Texas, “We choose to go to the Moon.” | Right: Neil Armstrong’s famous first words on the Moon.
Felix Baumgartner’s supersonic freefall from 128k! | Watch on Youtube

It was also calculus when Lucius Fox designed the Batsuit for Batman. It is this branch of mathematics that makes people’s favorite animation act like that of a human. It is the calculus that informs mothers about their babies’ gender or health. It is that calculus allows us to compare how fast our heartbeats are at 6:00 am to 11:00 pm. It is the calculus that will enable us to warm something up in the microwave. It is the calculus that helps a person to get to their destination on Google Maps. It was calculus when Einstein put his equations on his notebook to change the world. It helped create the world modern in which we live by combining mathematics, science, and sociology. That’s why Voltaire called calculus “the art of numbering and measuring exactly a thing whose existence cannot be conceived.”

Left: A Postscript to Einstein’s 1936 Science Paper | Source: University of Pittsburg | Right: Letters concerning the English Nation by Voltaire, p.152

Also, according to Voltaire,

“It is not long since the ridiculous and threadbare question was agitated in a celebrated assembly; who was the greatest man, Cæsar or Alexander, Tamerlane or Cromwell? Somebody said that it must undoubtedly be Sir Isaac Newton. This man was certainly in the right;”

Everything that we want to understand in mathematical terms, we understand through calculus. Calculus is not just algebra and formulas! Unfortunately, the instructors at colleges make calculus seem hard and tedious. I remember that one of my friends attending college was going to become a psychologist one day. When his psychologist asked him the reason why he was there, he said that he had been taking calculus for four years. For many first-year students, calculus is an obstacle that allows them to enjoy their lives fully. At first, it is as if someone brought a brand new car that has an engine problem. However, once you get started on fixing it, it will become readily available to use. That’s why, if one talks to someone who passed calculus, the only advice you will get is, “No matter what grade you get, you pass, but don’t turn around and run away.”

Back in the day, I saw the most beautiful TV show, Dr. House. In one episode (Season 1, Episode 8-Poison), there was an interesting conversation between Dr. House and his apprentice, Dr. Foreman.

Dr. Foreman: The kid was just taking his AP calculus exam when all of a sudden, he got nauseous and disoriented.

House: That’s the way calculus presents.

Calculus is an apparatus to make the invisible, visible. It is the connection between curiosity and solution. In other words, it is the best tool to answer questions and reveal scientific mysteries. When mathematicians work on a project to build something new, they are also inspired by mathematics. Moreover, it may take years to apply modern mathematical ideas to the real world. However, calculus is one of those rare fields of mathematics that is derived from physics.

For instance, if you put a magnet on your table and shake the iron filling out over the magnet, you will notice that the fillings will begin to arrange themselves across distinct lines and make a perfect pattern around the magnet. You will also see that the magnetic field extends in all directions. Today, we know that this scientific beauty is a result of the magnetic field.

Iron filings on paper. A bar magnet sits underneath. | Source: Sciphile

However, almost 200 years ago, Michael Faraday did not know this. He simply approached this idea intuitively and believed that there should be an invisible force around the magnet because of the movement the iron filings were creating. Algebra, English, or other languages were not enough to explain or prove his exciting thoughts about the magnetic field; therefore, Faraday needed to use a different approach, such as mathematics. Although he was a good physicist, his math was not enough to describe his ideas. Furthermore, he did not have the faintest clue as to what he was going to see.

During this time, the number of physicists dealing with magnetic fields was increasing. The Scottish physicist, James Clerk Maxwell, was one of them. He decided to take a different route by using calculus to improve Faraday’s works on magnetic fields. Even so, how did Maxwell use calculus to explain something related to physics? First, he converted all his knowledge about magnetic fields into mathematical equations. Then, Maxwell started using differential calculus and achieved new equations. He originally got 20 equations. Finally, he combined them and succeeded!

Maxwell revealed the mystery of the magnetic force! The language of which he used was just calculus, his only voice.

Left: Michael Faraday, 1861 | Right: James C. Maxwell Biography Physicist (1831–1879)
James Clerk Maxwell, F. R. S. Philosophical Transactions of the Royal Society of London, 1865 155, 459–512, published 1 January 1865

Calculus was calling curious people and telling them, “use me if you want to understand the universe!” After a short time, Nikola Tesla followed Maxwell’s steps and made the first radio by using Maxwell’s equations. Edouard Branly invented the first real detector of radio waves, the coherer. Marconi sent a wireless message over hundreds of miles away.

There are still some people who believe that Marconi invented the first radio. However, the US Supreme Court decided that Marconi’s radio patents were deemed invalid, and instead awarded the licenses for radio to Tesla on June 21, 1943, 6 months after Tesla’s death.

Nikola Tesla (1856–1943)

Afterward, Alan Turing shortened World War II by as many as two to four years by breaking German Enigma, in which he saved millions of lives. Other uses of calculus can be seen with Philo Farnsworth, who invented the television. He made it possible for 2 billion people to watch the world cup final between Italy and Brazil on July 17, 1994. He made possible for me to watch “The Goal of the Centurywhich Maradona scored against England in the 1986 World Cup semi final. I still find myself murmuring Uruguayan commentator Victor Hugo Morales Peres’ commentary on Maradona’s goal.

“Ahí la tiene Maradona, lo marcan dos, pisa la pelota Maradona, arranca por la derecha el genio del fútbol mundial, y deja el tendal y va a tocar para Burruchaga… ¡Siempre Maradona! ¡Genio! ¡Genio! ¡Genio! Ta-ta-ta-ta-ta-ta… Goooooool… Gooooool… ¡Quiero llorar! ¡Dios santo, viva el fútbol! ¡Golaaaaaaazooooooo! ¡Diegooooooool! ¡Maradona! Es para llorar, perdónenme… Maradona, en corrida memorable, en la jugada de todos los tiempos… barrilete cósmico… ¿de qué planeta viniste? ¡Para dejar en el camino a tanto inglés! ¡Para que el país sea un puño apretado, gritando por Argentina!… Argentina 2 — Inglaterra 0… Diegol, Diegol, Diego Armando Maradona… Gracias Dios, por el fútbol, por Maradona, por estas lágrimas, por este Argentina 2 — Inglaterra 0…”.

The Goal of the Century” which Maradona scored against England in the 1986 World Cup semi final.
I also made a poster design and put Morales’ words on a canvas which made my room so beautiful.
A brilliant picture of Roberto Baggio’s penalty misses in the 1994 World Cup final. | Scraps From The Loft

Anyway, I need to get back to calculus. Today, the company Loon is designing balloons that will help bring free wireless technology for people around the world. All those discoveries and inventions have been telling us something unique about the universe. By the way, I am not suggesting calculus made Roberto Baggio miss the penalty to make Brazil the World Cup champion in 1994 here. Calculus simply made the invisible, visible. Otherwise, how am I a witness to those unforgettable moments?

In Faraday, Maxwell, Tesla, and Loon’s examples shown above, you may notice that intelligent people are interested in change. Either they want to understand it, or they want to pursue it. To accomplish their purpose or dream, each of those beautiful minds used calculus. We may say that calculus is concerned with how things change over time. Mathematics itself creates change.

The idea of mathematical change emerged 5000 years ago. The Ancient Greek philosophers were thinking about the concept of the change of things very deeply. For instance, the Ancient Greek philosopher, Zeno, was the first human who dealt with the idea of instantaneous speed. We may have heard him because of his famous Zeno’s paradox — the race between Achilles and the tortoise — however, his arrow paradox might be more important than the others because it is an introduction to calculus. Zeno claims that an arrow in flight is always at rest. You may ask yourself: “How can a moving arrow not move?” Nevertheless, if we take a snapshot of an arrow in space at that particular moment, it has to be motionless. Since time is a collection of many instances put together, we can say the arrow never moves, thus becoming paradoxical because the arrow is traveling.

After Zeno, the first person who did research related to calculus was one of Plato’s students, Eudoxus of Cnidus. During this time, almost everyone was able to find the area of regular shapes such as squares, rectangles, and triangles in Ancient Greece. They were responsible for developing our understanding of shapes and their characteristics. However, it was time for a revolution! They needed to find the area of a curved shape, such as a circle, but it was quite difficult for them. A circle there has no vertices from which to draw lines and later split up into triangles. Instead, they had to find something more sophisticated. Our historical resources say that Eudoxus used a method of exhaustion, which is precisely a calculus thing to do. He discovered that the volume of a cone was one-third of the volume of its corresponding cylinder.

The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. — Wikipedia

After Eudoxus, Archimedes took the calculus flag. Archimedes was obsessed with mathematics that he would often forget to eat. Moreover, when he was in the midst of dying by the hand of a Roman soldier, he told the Roman soldier not to disturb him because he was drawing a circle in the sand. Even his tombstone is marked with a figure of a sphere enclosed by a cylinder with the ratio of their volume being 2:3 as indicated. When Galileo mentioned Archimedes, he always said, “Suprahumanus Archimedes, inimitabilis Archimedes ve divinissimus Archimedes.”

Archimedes, painting by José de Ribera, 1630 | The Death of Archimedes, by Thomas Degeorge, 1815
Cicero Discovering the Tomb of Archimedes (1805) by Benjamin West | Nineteenth-century photograph of the tomb known in Syracuse as Archimedes’ Tomb/photo Chris Rorres

Two thousand two hundred years ago, Archimedes’ passion was curved lines. He found a way to calculate the areas and volumes of objects with curved surfaces and called it “the Method.” He copied his notes on a papyrus scroll, which were later put on a piece of parchment paper. After the transfer of his notes, a very interesting event happened. Somehow, 700 years ago, a monk needed paper to write his prayers on something and choose a book on the shelves at random. Unfortunately, he was holding Archimedes’ notes in his hand and used the book like his own. After 2000 years, however, mathematicians discovered the book decided to work on it. It led to the revelation of Archimedes’ method, “the Method.”

Inside the Archimedes Palimpsest | PBS

Like every two-dimensional shape, a circle also has an area. Archimedes was the type of person to come up with the conclusion for an area of a circle by inventing a style of mathematics called heuristics, which accelerates the process of reaching a satisfactory solution. Although the heuristic method is not perfect to prove something mathematically, it was practical and sufficient enough to conclude his work. Archimedes also further developed Eudoxus’ method of exhaustion to calculate areas under a parabola, the surface area and volume of a sphere, or prove that the area of a circle is equal to πr²(pi times r squared).

Reminder: “pi” is defined as the ratio of the circumference of a circle to its diameter. The circumference is the distance around the circle, and the diameter is the distance across. Also, “r” is the radius, which is half the diameter or the distance from the center of the circle out to the edge.

Interesting Fact: For Euclid, who had one of the greatest minds, pi wasn’t a number. Euclid said the area of a circle is proportional to the square of the radius and therefore doesn’t mention pi.

After Archimedes’ discovery, we were taught that the formula of an area of a circle with the radius “r” is pi times r². The formula for an area of a circle may be tedious for a student or a homemaker, but the logic of Archimedes’ approach behind it is utterly fascinating.

Archimedes’ first method to find the area of a circle was so simple, but it could only come from a genius mind! Only gifted people can find simplicity in any given situation. Like Johan Cruyff said, “Soccer is simple, but it is difficult to play simple.” Regardless, Archimedes inscribed regular polygons inside a circle until the regular polygon had so many sides that they practically became the circle itself. This way, the area of the polygon is getting closer and closer to the exact value of the area of the circle. However, the polygon would need to have an infinite number of sides to have an area that is identical to that of the circle. Today, we say that in the limit of infinity, the area of the polygon equals the area of the circle, where n represents the number of sides in the polygon. However, the Greeks did not fully grasp this concept of limits during this period.

Johan Cruyff by David Diehl

Archimedes used the same method to find the area of a parabolic segment. He turned the curved shapes into a combination of triangles. Because of this method, Professor Steven Strogatz claimed Archimedes as a first cubists artist like Picasso in his last book, Infinite Powers: How Calculus Reveals the Secrets of the Universe (p.37).

Infinite Powers: The Story of Calculus — The Language of the Universe by Steven Strogatz | Buy the book on Amazon.
Pablo Picasso The Bull (Le Taureau), state VII 1945

First, Archimedes put the largest triangle under the curved area. Then, he put two smaller triangles on the left and right sides. When there was a bit of space under the curved line, he tried to put even more tiny triangles. He was able to convert the curved area into a combination of triangles since he knew how to find the area of that shape. This method introduced him to an interesting fact; the ratio of the area of that parabolic segment compared to the area of the first big triangle was 4/3. The ratio 4/3 is extraordinary because, in music, it is called “a perfect fourth.

Making the parabolic segment out of triangles is a unique idea since it is a remarkable example of the invisible presence of calculus. Engineers and graphic designers at Pixar use Archimedes’ method daily. They make all our favorite characters out of triangles. When we go to a theatre to watch a movie, we see characters act like a real person, but in reality, they are made out of millions of regular polygons. We just don’t happen to notice the calculus that’s present. There is even more calculus in computing the right triangulation to make whatever you want at Pixar. It has become an Archimedean idea that we can represent any smooth surface with triangles, of which is another excellent representation behind calculus (differential), the approximation of curvy objects through straight lines.

Archimedes’ second remarkable approach was about finding the area of a circle. At the very beginning, finding the area of a circle was a headache for him. He needed to find different methods to solve this issue. Fortunately, in ancient times, when people dealt with any type of problem, they tried to make it smaller by breaking them into parts to later work on them individually. Thereby their problem would be much more manageable than the original. Afterward, when they solved the problem for all the tiny pieces, they would put the answers back together and form a whole. This mathematical approach was one of the most fabulous layouts in human history. Usually, it might be taught to young pupils.

To picture Archimedes’ method in our minds, let’s first take a circle with the radius “r” and chop it into four pieces. Now we have four equal quarters. Incidentally, the circumference of our circle will be 2πr (2 times pi times r). If we rearrange the quarter ones like in the figure below, we will get a new shape. Besides, the length of the scalloped edge on the bottom will be half of the circumference, which is πr (pi times r) because the other half will be on the top, and the width will be the radius of our circle, “r.”

From the article “Take It to the Limit” by Steven Strogatz | NY Times

Here we have a simple plan; if we could figure out the area of that new shape, we would then know the area of the circle. However, our new form might be seen as more complicated, so we should try with more chopped pieces to change the circle into a shape whose area we know.

The use of shapes, such as a parallelogram or a rectangle, is considered a “beautiful” idea.

We can try this circle construction with eight equal pieces and rearrange them to get a better form of a parallelogram below. If we look closely, we will see that it’s trying to turn into a shape that we recognize. The width is becoming more vertical, and the scalloped edge at the bottom is becoming straighter. However, the length will be πr, and the width will simply be r.

How to show the area of a circle is π x r2 | Bitesize

Therefore, if you approach this idea intuitively, you would notice that we can construct a circle with smaller pieces. For instance, if we build another circle with 32 segments and repeat the process, we realize that the circle slowly starts looking a bit like a rectangle, which makes finding the area of it utterly easy.

From the article “The Visual Proof of the Area of a Circle” by Waldo Otis | Medium

We can now conclude that as we break up the circle into more and more pieces, the shape becomes a rectangle. If you do this infinite times, you will get infinitely many pieces that form a perfect rectangle when rearranged. The shape becomes more accurate as of the number of pieces increases. Therefore the length of the width is still πr, and the length of the side, which is equivalent to the radius of the circle, remains “r.” It gives us the area of a circle with the radius “r,” which is 2πr (two times pi times r). It is fundamentally the real reason the area of a circle is pi r squared.

From the article “The Visual Proof of the Area of a Circle” by Waldo Otis | Medium

Today, in modern calculus, we slice problems into infinite tiny pieces and add them together as Archimedes did 2200 years ago. In other words, calculus is about making difficult problems more manageable.

Although almost all textbooks about calculus are 1,000 pages long, what calculus truly wants to achieve is simplicity.

Slicing a problem, however, is not the main idea of calculus. The main idea of calculus is that actions are always going to be done. We do operations continuously, whether differential or integral, which is probably the most critical mathematical technique ever devised. Both concepts involve the idea that we can do something infinitely to achieve a finite answer. Since calculus is the mathematics of change, by definition, calculus has to be continuous. Continuity is the essence of calculus.

Integration is finding the area under a line above the horizontal axis. For example, the area under a velocity-time graph would be the actual distance traveled. Integration can be achieved by breaking the area into infinitesimally thin rectangles and then adding up all the areas of the rectangles to get the exact area under the curve. Through the limit of infinitely small rectangles, we can find the precise area under the curve.

Differentiation is concerned with how fast things move or change (rate of change). It is used to find the velocity, speed, and tangents of curves. A curve you can think of as changing direction, and motion you could think of as changing position.

Both concepts involve dealing with things that are infinitely small or infinitely close together.

It is very interesting that, after the soldier killed Archimedes, the development of calculus had stopped immediately and stayed static for more than 1500 years. Humanity had to wait until the 17th century to go any further. Calculus was officially discovered in this era and would allow for mathematicians and engineers to truly make sense of motion and dynamic change in the world around us, such as the orbits of the planets and the motion of fluids. It is not a coincidence that after the invention of calculus, the scientific revolution officially started. Many great mathematical ideas, formulas, and proofs were discovered during this era.

In the 1630s, German Mathematician Johannes Kepler and Italian mathematicians Bonaventura Cavalieri and Galileo Galilei improved Archimedes’ method of exhaustion separately to make a modern version of it. This technique was called “the method of indivisibles.”

The basic idea of the method of indivisibles is determining the size of any figure by drawing infinitely many parallel lines and getting an infinite amount of rectangles until the width of the rectangles cannot be further subdivided. Afterward, the sum of the areas of each rectangle will be equal to the size of the figure at the beginning. This method is very similar to the method of integration.

After Cavalieri, a long line of mathematicians including Rene Descartes, Pierre de Fermat, Blaise Pascal, Isaac Newton, and Gottfried Wilhelm Leibniz began to work on calculus. None of those mathematicians had any idea that they were about to make one of the most incredible milestones in the history of all time. However, only Newton and Leibniz were able to finish their work and publish it. These two genii changed math and science forever by introducing calculus to the world. It also led to the creation of an average of 20 more math courses in universities. Students were now exposed to a much more diverse setting involving math.

In 1994, calculus was rediscovered by another so-called “genius,” Mary M. Tai. She claimed that she found a ‘new’ way of measuring the area under a curve by adding up areas of rectangles and triangles. Her method was named “Tai’s method” (very original). Later some academicians found her paper fabulous and published it. More interestingly, her article was indexed by a Google Scholar and has ever since been cited by 375 people.

In 2014, John Canning wrote a beautiful blog post about Mary Tai’s awkward discovery, ‘the 20th anniversary of the rediscovery of calculus.’ In one part of his blog post, he says, “perhaps the most disturbing thing is that twenty years later the paper is still being cited, and not just by people pointing out that this paper is nothing new. This paper from 2009 reads’ glucose and insulin areas were determined using Tai’s model’” (p.1046).”

After Leibniz and Newton published their findings in the 17th century, mathematics received the most significant increase in its power since the time of the Greeks. Thankfully, their original writings recorded the discovery of calculus still preserved in the university library in Cambridge, and we have a chance to see the journey of their mathematical rediscovery of Calculus.

Mathematical Treasure: Leibniz’s Papers on Calculus | Source: MAA
Newton Papers: Philosophiæ naturalis principia mathematica (Adv.b.39.1) | Source: Cambridge Digital Library

Today, when we put our phones away and manage to talk about some physics stuff, we will most probably mention the name of three scientists; Einstein, Feynman, and Newton. Since we mentioned Newton, we have to say something about Leibniz as well. However, Newton and, of course, Leibniz were the ones who opened the doors for future physicists and mathematicians.

On the one hand, Newton wanted to explain the astronomical system of Copernicus, Kepler, and Galileo, to describe how gravity works. On the other hand, Leibniz wanted to formalize the rules of logic and systematize mathematical reasoning. He worked to mechanize all reasoning processes in his life. Both Newton and Leibniz were going to succeed with the help of calculus.

Newton was the type of guy who wanted to understand everything. He wanted to see the truth behind mysteries and explain them to people all around the world.

An apple had never fallen on Newton’s head, yet he wondered why the moon was standing in the sky and wasn’t falling as well.

Until this time, millions of people had seen the moon in the sky over and over again, but only Newton asked why the moon didn’t fall on earth. This question was a turning point for him; it was a turning point for all humanity. It would push him to discover many things of which he was passionate about, in a way, it even made him obsessive. For instance, when he was obsessed with alchemy, he wasn’t interested in trying to turn lead into gold. He was pondering philosophically about the stuff by changing it.

His other obsession included gravity. When he realized the existence of gravity, he wanted to calculate the speed of a falling object at any given time. He knew that if you drop an object, its speed will increase every single instant until it hits the ground. Therefore, the object must have some definite speed at any given instant. He did not know of any type of mathematics that could adequately calculate these instantaneous velocities.

Consequently, he needed to come up with some kind of dynamic mathematical system to help explain his situation of gravitation. First, he mastered the method of Descartes’s for finding tangents. Then, he realized that as the secant of a curve becomes smaller and smaller, the slope becomes an exact point, and we can draw a tangent at this point. At that moment, he discovered the idea of the remarkable mathematical concept, the instantaneous rate of change, which is the differential calculus we see today.

He was delighted when he figured out his biggest obsession. He felt free for a while. However, one day, while he was sitting on his chair and taking a sip from his tea, the astronomer Edmond Halley specifically asked Newton how the sun invisibly controlled the planets. It was another “Let’s go!” moment for Newton. It would take him years to answer this question, but when he was finally able to explain it, he specified for the first time that gravity was the force holding all of the planets in their orbits around the sun. He completely understood Kepler at that moment.

Fortunately, he combined his notes in his book, “The Mathematical Principles of Natural Philosophy,” or simply “Principia,” in 1687. With this book, he unified the work of Descartes, Galileo, Kepler, and Copernicus into one mathematically sound system. It was the first time that natural philosophers in Europe had had a single system for understanding what and how things are since Aristotle. However, it was almost impossible to fully understand “Principia” because the math was so dense. He had to discuss calculus in terms of geometry because no one else had ever heard of calculus before! He called his discovery: “mathematics of motion.” His book was going to dominate the scientific view of the universe for the following three centuries.

Newton’s personal copy of the first edition of Philosophiæ Naturalis Principia Mathematica is displayed at Cambridge University Library.

As you can tell, Newton developed a new dynamic system of mathematics, calculus, to have the tools he needed to solve and explain problems in physics. It is evident that calculus emerged with the inadequacy of algebra. In the following process, algebraic methods would be used to solve differential equations for events to develop rapidly.

The story of another person who invented calculus is equally impressive. Gottfried Wilhelm von Leibniz was a remarkable master of mathematics. He worked on almost every area of natural philosophy — reshaping how libraries work, inventing the mechanical calculator, creating the binary notation that would centuries later be central to computer science, and becoming a significant figure in philosophy.

Leibniz made his independent discovery of calculus, and today, we use his version of calculus. Leibniz’s approach to calculus was from a metaphysical point of view, and that’s why Leibniz’s calculus is a system of reasoning.

He mainly worked on contemporary mathematical problems. The moment he found the idea of the sum of infinite rectangles, he experienced enlightenment. His feelings said that he just discovered the potential to form a whole new system of mathematics, which was going to be called calculus in the future.

Leibniz published his works independently from that of Isaac Newton in 1684. Mathematicians were able to quickly understand the idea of differential and integral calculus because he also invented a powerful and flexible notation for it. Leibniz used “the idea of integral” for the first time in history to find the area under the graph of a function. In doing so, he made up necessary notations, including the “d” for differentials and “long S — summa” for integral. That’s why we still use Leibniz’s notations today.

Furthermore, Leibniz described “the idea of change” very differently than Newton. For Leibniz, change was the difference ranging over a sequence of infinitely close values called infinitesimal. Infinitesimals were the small quantities, such as the tiny rectangles, that there isn’t any way of measuring them. Later on, mathematicians would describe this as a limit.

The definition of limit.

Greeks mathematicians were thinking about infinity and limits. Zeno, a philosopher, said that a person could never walk towards a wall and touch if they are going to approach it by half. First, they would have to go halfway across the room, then halfway there, and then halfway from that point, and so on. Because this remaining distance can be split in half infinitely many times, they will never get to the wall.

Of course, we know that a person can indeed go and touch a wall. So this paradox was just a foreshadowing of our modern understanding that we can do something infinitely many times and get a finite result, like some of the infinite series we looked at that actually have finite sums or calculate the area under a curve, such as the case of the infinite rectangles.

Leibniz was a good man. Unfortunately, like every good man, such as Georg Cantor, he encountered many ugly accusations. When Leibniz published his work about calculus in 1684 for the first time, he didn’t know that his life was going to become a nightmare. Obsessive Newton hadn’t published anything on calculus yet; therefore, he didn’t want to share the credit for the discovery of calculus and started to blame Leibniz for stealing his ideas! When Leibniz traveled to London in 1676, he didn’t know that Newton was going to use that as a weapon against him. With the order of the president of the Royal Society, Isaac Newton, mathematicians accused the German of having glimpsed Newton’s unpublished notes. They started the dispute of who invented calculus first.

Scientists and philosophers held weekly discussions and debated Newton and Leibniz’s case unfairly by favoring Newton from the start. Newton was the president of the Royal Society and never gave Leibniz a chance to defend himself. Today, only dictators such as Recep Tayyip Erdogan, don’t give people a chance to defend themselves! Eventually, Newton was credited as a first discoverer. Until his death, Leibniz fought to prove that he had invented calculus without consulting Newton’s notes. He never truly got the credit he deserved, and therefore there is still no complete edition of Leibniz’s writings in English!

I wish Newton, who had already found the law of gravitation, would have exhibited kindness and left calculus to Leibniz, but he didn’t!

However, Mathematics

Life is good for only two things, discovering mathematics…


Written by


Math Teacher. Content Curator. Soccer player. Maradona fan. Mostly write about the lectures I love to learn better.

However, Mathematics

Life is good for only two things, discovering mathematics and teaching mathematics.


Written by


Math Teacher. Content Curator. Soccer player. Maradona fan. Mostly write about the lectures I love to learn better.

However, Mathematics

Life is good for only two things, discovering mathematics and teaching mathematics.

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