How Archimedes Almost Broke Math with Circles

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.

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

•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:

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

Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, by Amir Alexander

Isaac Newton, by James Gleick
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Archimedes' proof from The Measurement of a Circle:

Annotation of the proof from the American Mathematical Society:

More information about Archimedes' inventions and myths:

Circular Reasoning: Who First Proved That C Divided by d Is a Constant? by David Richeson
Discussion of who gets credit for which ideas:

Additional information ("all of Europe knew less in 1500 than Archimedes did on the day he died"):
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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.

[bensyversen]

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.

[calebmcurby]

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.

[billshiff]

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.

[VoightKampf]

The genius of Archimedes is mind boggling and his demise was a monumental loss to science.

[JonathanWhitmore]

Well done! The time that you spend on the visuals really makes the concepts clear. Looking forward to seeing more!

[eeeee]

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

[zetaprnt]

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

[Seershaaaaa]

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 :)

[henrikjørgensendk]

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.

[stormmugger]

This is unironically one of the highest quality math videoes i have seen, and the visuals really help!

[amichayr]

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

[CreateSmarter]

Ben! I'm so happy to see this wonderful video of yours getting the recognition it deserves! Keep it up, man!

[hello-rqkf]

i failed calc 2, but as a classical musician i appreciate how you fit the background music to the time period. that chopin prelude is one of my favs

[Grateful]

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

[se]

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!

[Williamw]

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.

[IanZainea]

1:56 they're lack of pizza held them back

[DisisSid]

I just love seeing great high-level content coming from small channels. Great work! Expecting more videos from you 😀

[eliasschock]

Fantastic video - well edited and well explained!

[kitcutting]

my top 10 maths greats list:
10) Alhazen (optics)
9) Leibniz (binary code)
8) Descartes (coordinates)
7) Ramanujan (fractal proofs)
6) Russell (math philosophy)
5) Euclid (geometry)
4) Newton (calculus)
3) Euler (e^[iπ] + 1 = 0)
2) Archimedes (π)
1) Gauss (non-Euclidean geometry, FFTs, Normal curve, etc.)