How Archimedes Almost Broke Math with Circles
Channel: Ben Syversen
Date: 2023-11-22

Archimedes proved the area formula for a circle by dividing the shape into infinitesimally small pieces. The concept was behind some of his greatest mathematical achievements, but rested on a paradox that wasn't addressed rigorously for thousands of years. Even Isaac Newton left some key questions unresolved when he wrestled with this paradox 1800 years later.

I skipped the detailed proof in this video - you can find an excellent full explanation of Archimedes' proof of the area formula of the circle by @DanielRubin1 here: https://youtu.be/R1HUtt2oo7A?si=r-FQ-OfaES7XVmUM

**A few clarifications and comments, based on helpful viewer feedback**

•While Archimedes is often credited with the invention of the screw style water pump, and it bears his name, evidence shows that this invention existed in Egypt before Archimedes, and he was likely the first to "demonstrate and fully explain its properties." https://books.google.com/books?id=5cP-81xDWuwC&pg=PA759#v=onepage&q&f=false

•Another source, cited in Wikipedia, says that the screw may have been an improvement on a similar pump that existed in the hanging gardens of Babylon: https://muse.jhu.edu/issue/2473 (I have not independently vetted this article since it comprises such a small detail in my video)

•Some have pointed out that Leibniz was given short shrift in my video. Had I known that this video would go viral, I would have held it back for a week or two to fix that problem! As it is, I hope to cover the Newton/Leibniz controversy in a future video.

•It's worth clarifying that parts of the story of Archimedes' death are likely apocryphal, in particular his last words. In the video, I referred to it as a "legend" because there's no question that it's a good story. Some 300 years after Archimedes, Plutarch reported multiple versions of Archimedes' death and his last words. There is probably a version of the truth in there somewhere, but storytelling and myth making surely played a role as well.

Time Stamps:
00:00 - Introduction
00:51 - The challenge of curves
01:44 - The area of a circle
03:27 - The paradox of infinitesimals
04:42 - History after Archimedes
06:45 - Calculus in the modern world
07:11 - Archimedes' life and death
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The inspiration for this video comes from:

Infinite Powers: How Calculus Reveals the Secrets of the Universe, by Steven Strogatz https://amzn.to/3RagI0v

I highly recommend this book to anybody interested in reading more. It's a great read, even if you're not a "math person."
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Additional Resources:

Calculus Reordered: A History of the Big Ideas, by David M. Bressoud 
https://amzn.to/3G9o9Pj

Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, by Amir Alexander
https://amzn.to/46s3uR8

Isaac Newton, by James Gleick
https://amzn.to/49QPNhG
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The Works of Archimedes: https://archive.org/details/worksofarchimede029517mbp/mode/2up

Résumé des leçons données à l'École royale polytechnique sur le calcul infinitésimal, par M. Augustin-Louis Cauchy: https://gallica.bnf.fr/ark:/12148/bpt6k62404287/f1.item

Newton's Fluxions: https://cudl.lib.cam.ac.uk/view/MS-ADD-03960/27

Archimedes' proof from The Measurement of a Circle:
http://media.bloomsbury.com/rep/files/primary-source-13-archimedes-measurement-of-a-circle.pdf

Annotation of the proof from the American Mathematical Society:
https://www.ams.org/publicoutreach/feature-column/fc-2012-02

More information about Archimedes' inventions and myths:
https://www.nytimes.com/2013/06/25/science/archimedes-separating-myth-from-science.html

Circular Reasoning: Who First Proved That C Divided by d Is a Constant? by David Richeson
Discussion of who gets credit for which ideas:
https://www.jstor.org/stable/10.4169/college.math.j.46.3.162

Additional information ("all of Europe knew less in 1500 than Archimedes did on the day he died"):
https://inters.org/Whitehead-Western-Development-Science
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Thank you to @JonathanWhitmore for feedback on the script and providing the Manim animation at 00:12. Thank you to @CreateSmarter for technical help and valuable editing feedback on this video.


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