SDS 486: The History of Calculus

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In episodes 460 and 468 of the SuperDataScience show, I provided a history of algebra and a history of data respectively. Those episodes ended up being episodes that generated a lot of buzz, so I’m back today with a history of calculus.

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The earliest known records of calculus calculations or things related to calculus comes from about 4,000 years ago. Around 1800 before the common era, we have an Egyptian papyrus from that era with area calculations on it. That’s related to the idea of integral calculus that allows us to find the area of shapes. Later, much later, something called the method of exhaustion, which is a specific technique for finding the area of curved shapes, so circles could be notoriously difficult to find the area of without something like the method of exhaustion. This method of exhaustion technique was developed by the Greeks about 2,000 years after Egyptians had started working with area calculations. The Greeks’ Eudoxus around 400 before the common era and Archimedes around 250 before the common era developed this method of exhaustion technique, which you can still use today to find the area of curved shapes.

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Independently, a Chinese scholar named Louie Hugh also discovered this method of exhaustion around the same time as Archimedes. Going now another millennium into the future to now the 11th century of the common era, the Arab, Al-Haytham began using integrals to calculate volumes. Up until this point in history, all of the calculus related calculations that we have record of, involve things like finding the area of a shape, so this is all related to integral calculus, but it wasn’t until just 1,000 years ago in the 11th century of the common era that this Arab, Al-Haytham, developed the kinds of integral techniques that we use still today.

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Alongside integral calculus, the other main branch of calculus is differential calculus. While integrals allow us to go from, to calculate the area of shapes, of curves, the area under curves, say, differential calculus allows us to go in the opposite direction. Differential calculus allows us to go from a curve to calculate the slope of that curve. Differential calculus can let us go from distance to speed to acceleration, while integral calculus goes the other way. It allows us to go from acceleration to speed to distance, to give you an example in the real world.

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It wasn’t until the 14th century of the common era that we had any kind of differentiation-like methods for finding the slope of a curve. It wasn’t until very recently, the 17th century, that the German mathematician, Gottfried Leibniz, and in parallel, the English mathematician, Isaac Newton, both separately developed our modern calculus, our modern differential calculus that we still use today. Both Leibniz and Newton even came up with a higher order differentiation and integration that we still use today. This includes second order, third order derivatives and integrals. This is the kind of thing that allows us to go not only from distance to its first derivative speed, but also to its second derivative of acceleration. Again, another big achievement of Leibniz and Newton very recently.

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They also came up with the rules that we can use today to look at an equation and in a lot of cases calculate the derivatives and integrals by hand. These rules are things like the product rule and the chain rule. Leibniz, the German guy from the 17th century, he named calculus and he devised the notation that is still most popular today. Isaac Newton, on the other hand, he was the first person to apply calculus to physics. For example, he used it to describe the laws of motion as well as gravity. Remember that Newton is the person who really characterized gravity. He was able to define it, and calculus allowed him to do that.

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All right, if you’re into learning more about calculus, particularly its applications to data science and machine learning, you can check out my Calculus for Machine Learning YouTube playlist. We talk more about the history of calculus, the method of exhaustion, for example, in particular, as well as tons and tons of differential calculus rules, integral calculus rules, and how we apply them to allow machine learning algorithms to learn from relatively simple models like regression models to the most complicated deep learning models we have today. That’s my Calculus for Machine Learning YouTube playlist, and if you’d happen to like exclusive, fully-worked solutions to all of the exercises I provide in my calculus curriculum, then you have the option of purchasing, typically for very cheap, my mathematical foundations of machine learning Udemy course, which I published in partnership with SuperDataScience.

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All right, that’s it for today’s episode. Thanks for listening and I’m looking forward to another round of SuperDataScience with you very soon.