History of Calculus: Part 4 - Pre-Calculus Integration

Video Description

This is part 4 of the series: History of Calculus. Where I explore the origins of calculus from ancient times to modern history. In this video, I delve into some of the pre-calculus integration methods of the 17th century and their historical significance. To support this channel: - Buy me a coffee: https://www.buymeacoffee.com/tareksaid - Patreon: https://www.patreon.com/tareksaid - Super Thanks: You can find it by clicking the 3 dots under the video - 0:16 A more accurate term here would have been “one framework” instead of “one method,” as calculus is a collection of interrelated methods. However, I chose “one method” to highlight the transformative power of calculus, particularly when compared to the methods that were commonly used in the 17th century. - 1:00 It’s challenging to pinpoint the exact beginning of modern calculus. While many mathematicians emphasise the publication of this book, others place less importance on it. - 1:22 Kepler wasn’t the first to use infinitesimals during the 2000-year period before calculus. As noted in the last video, medieval thinkers like Oresme employed concepts similar to infinitesimals. However, Kepler used them systematically to calculate areas and volumes, making his approach distinct. - 3:47 Not to be confused with Cavalieri’s Principle, which compares indivisibles of various shapes. The concept discussed here pertains to summing (integrating) indivisibles. - 7:10 While similar, the method described here is not exactly how Cavalieri found the volume of a cone. Cavalieri’s approach was more geometrical in nature. - 8:50 Cavalieri’s method bears a resemblance to Archimedes’ lost method, The Method, which Archimedes used to discover (not prove) areas and volumes of different shapes. This method was rediscovered in the early 20th century, and Ben Syversen has an excellent video on the topic. https://www.youtube.com/watch?v=ZXNIgHov0Nk&t=851s Additionally, the Chinese mathematician Zu Gengzhi employed a similar idea in the 5th century to determine the volume of a sphere. - 11:18 A more generic way to represent the parabola is the equation y=kx^2. where k is a real number. However, I chose k = 1 for simplicity. - 15:00: Fermat extended Cavalieri’s conjecture beyond positive integers, proving it for negative integers, except for n = -1 , as this case results in division by zero. This special case was resolved by Gregoire de St. Vincent, leading to what we now call the natural logarithm. I cover this topic in detail in my other video: The History of the Natural Logarithm. https://www.youtube.com/watch?v=habHK6wLkic&t=836s Later, John Wallis generalised this formula for all real values of n . References: - The Historical Development of the Calculus | C.H. Edwards - The Origins of Infinitesimal Calculus | Margaret E. Baron - The History of the Calculus and its Conceptual Development | Carl B. Boyer - Infinite Powers | Steven H. Strogatz - Calculus Reordered | David M. Bressoud - A History of Mathematics - An Introduction | Victor . Katz - The History of Mathematics - A Brief Course | Roger Cooke - A Source Book in Mathematics,1200 -1800 | Dirk Jan Struik