How Archimedes Almost Broke Math with Circles
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Published 2023-11-22
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: • Area of a Circle, the BEST COMPLETE P...
*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." books.google.com/books?id=5cP-81xDWuwC&pg=PA759#v=…
•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: 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 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
amzn.to/3G9o9Pj
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, by Amir Alexander
amzn.to/46s3uR8
Isaac Newton, by James Gleick
amzn.to/49QPNhG
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The Works of Archimedes: archive.org/details/worksofarchimede029517mbp/mode…
Résumé des leçons données à l'École royale polytechnique sur le calcul infinitésimal, par M. Augustin-Louis Cauchy: gallica.bnf.fr/ark:/12148/bpt6k62404287/f1.item
Newton's Fluxions: cudl.lib.cam.ac.uk/view/MS-ADD-03960/27
Archimedes' proof from The Measurement of a Circle:
media.bloomsbury.com/rep/files/primary-source-13-a…
Annotation of the proof from the American Mathematical Society:
www.ams.org/publicoutreach/feature-column/fc-2012-…
More information about Archimedes' inventions and myths:
www.nytimes.com/2013/06/25/science/archimedes-sepa…
Circular Reasoning: Who First Proved That C Divided by d Is a Constant? by David Richeson
Discussion of who gets credit for which ideas:
www.jstor.org/stable/10.4169/college.math.j.46.3.1…
Additional information ("all of Europe knew less in 1500 than Archimedes did on the day he died"):
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.
Note: Amazon links are affiliate links which help support the channel at no additional cost to you.
All Comments (21)
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Thanks for watching! I am looking forward to making more videos like this, so drop a comment if there's anything you'd like to see.
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Centuries later, Cicero visited Syracuse in search of Archimedes' tomb, which had been described having a giant cylinder and a sphere marking it. He finally discovered it hidden among the brambles, and had to tell the Syracusians with him the significance of it, because they didn't know who Archimedes was.
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His last words "μη μου τους κύκλους τάραττε" meaning "do not disturb my circles" are used to this day in Greece when we want to get rid of someone who's annoying us or disrupting our work
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General Marcellus was very upset at Archimedes death. He took only 2 things from Syracuse , 2 machines made by Archimedes which are said to accurately show the positions of all the planets. Until recently it was considered an exaggerated legend but with the discovery of the Antikythera mechanism we now know that it was certainly true.
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The genius of Archimedes is mind boggling and his demise was a monumental loss to science.
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Well done! The time that you spend on the visuals really makes the concepts clear. Looking forward to seeing more!
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8:26 "do not disturn by circles." would be an awesome last words, like he defended something to his death, truly a great mathmatician and inventor
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Math usually scares me, it was always my worst subject. But as someone who is so passionate about astrophysics and other subjects that require math, this video was fun and comprehendable for me. You have a great talent, I wish you luck on your YouTube endeavors :)
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The quality of this video does not match your subscription or view numbers! So under rated!
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i just checked the view count, i thought this would be in the millions. This is very high quality work. both mathematically and storytelling ability
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This is unironically one of the highest quality math videoes i have seen, and the visuals really help!
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Great video, Ben! Love the concept of teaching math through the lens of history, dramatic music, and visuals. I'd like to see more videos like this.
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Ben! I'm so happy to see this wonderful video of yours getting the recognition it deserves! Keep it up, man!
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We always called those proofs "epsilon delta stuff" in first-year calculus. We all dreaded it. I had no idea it wasn't invented until the nineteenth century! No wonder it stood out so much from what was otherwise a fun and relatively easy course.
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I just love seeing great high-level content coming from small channels. Great work! Expecting more videos from you 😀
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Fantastic video - well edited and well explained!
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The creativity, pacing and visual energy of this video were all incredibly excellent! However I did feel underwhelmed by the video’s denouement, wherein Cauchy’s epsilon solution turned out to be the same as Archimedes. Felt like there is a lot more to be explored in the concepts of infinity — hope to see more in future videos!
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Man, where have you been all these years?
The city needs you!
Keep uploading to help me and my generation winning the mathzilla fight.
Please don't ever let anyone delete these videos, these are life saviours -
Congrats.... A well done video. Good luck and success to you and your future content.
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man!! you deserve more, really good video
