Did Archimedes Write a Problem That Took 2,200 Years to Solve? Ben Syversen 2024-05-24 | A poem from the 3rd century BC, attributed to Archimedes, about counting the number of cattle in a Sicilian herd wasn’t fully solved until the 20th century.Join my Patreon! patreon.com/BenSyversenOr, support me on Ko-Fi with a one-time donation! ko-fi.com/bensyversen**Errata** At 03:13, there is a typo in the second equation from the top (the black cows). It should say 9,717/10 + 143/2800*{sigma}, not 9,717/10 + 143/1,260*{sigma}. The subsequent steps are correct. **Time Stamps:00:00 - Intro00:51 - Part I: The Bulls02:36 - Part II: The Cows04:31 - Part III: The White and the Black Bulls05:38 - Part IV: The Yellow and the Dappled Bulls07:33 - History of Pell Equations09:12 - Path to a Solution10:36 - Archimedes________Some useful resources about the Cattle Problem of Archimedes:Archimedes related web site maintained by Chris Rorres: https://math.nyu.edu/Archimedes/Cattle/Statement.html@numberphile video on the Cattle Problem: youtu.be/dNxyFtqcNss?si=Nj9uMMpSHL5ZlA43Video by @OliverThomas1 explaining methods to solve the problem: youtu.be/qn_mJzm5QII?si=y24GZyl5f0bKqqWc“Archimedes’ Cattle Problem” - Ilan Vardi - https://math.nyu.edu/Archimedes/Cattle/cattle_vardi.pdf“A Simple Solution to Archimedes’ Cattle Problem” - Antti Nygren - https://math.nyu.edu/Archimedes/Cattle/isbn9514259327.pdfPresentation by Alex Walker about the Cattle Problem, which includes the hypothesis about Apollonius of Perga: awwalker.com/wp-content/uploads/2018/02/homslides_7_cattle_problem.pdf_______Music from Epidemic SoundImages of mathematicians are from Wikimedia Commons**A Note about how I use AI generated images in my videos**The emerging ability of artificial intelligence to generate compelling images from text prompts opens new possibilities for compelling storytelling. However, when mixed with real historical imagery, it has the potential to create confusion, or worse, if not handled properly. Additionally, there is important discourse about issues of ethics, intellectual property, etc surrounding AI imagery.I have used Midjourney AI to create “stock” image elements including backgrounds, illustrations, and objects. I do this because it allows much greater specificity than is possible with stock image services (you need a white cow and bull, a yellow cow and bull, a spotted cow and bull, and a black cow and bull, and you want them to be in an "old" painted style? You got it!). Additionally, in this video I used AI generated images of what we popularly imagine Archimedes to look like. Not only are there (obviously) no photographs of Archimedes, there are no surviving images of Archimedes whatsoever. Most of the modern imagery of him is based on a statue, located in the Naples, Italy Archeological Museum. The problem is that scholars eventually figured out that this was actually a statue of ArchiDAMUS, a 4th century king of Sparta, not ArchiMEDES.... All of this being the case, I see no risk of misinforming or creating historically inaccurate images in generating his likeness using AI (after all, it is just as "inaccurate" as any other drawing or painting that exists).
The Invention That Saved Science Ben Syversen 2025-06-17 | 80,000 Hours want to help you find a fulfilling career that makes a positive difference in the world: https://80000Hours.org/bensyversenHow a numerical table with 5,400 entries became the world's calculator for the next 350 years._____Join my (new) Patreon! patreon.com/BenSyversenSign up for my (new) mailing list! ben-syversen.kit.comOr, support me on Ko-Fi with a one-time donation: ko-fi.com/bensyversen_____Credits:Written, recorded, edited by Ben SyversenIllustrations: Kendall EddyCharacter animation for introduction: Alex ZepherinThanks to Mostafa Hasan for his last minute help with the layout and details of my thumbnails.Thanks to Emily Zhang and David Liu for additional help.Special thanks to Stephen Welch for sharing his Kepler resources with me. _____@WelchLabsVideo's series on Kepler is amazing and does a far more thorough treatment of the math behind the story of Kepler’s discovery of Mars’s orbit than I do here. Take a look (his two videos on Burgi’s red and black numbers and Napier’s logarithms are also excellent):youtube.com/watch?v=Phscjl0u6TI_____More useful videos:youtu.be/habHK6wLkic?si=HfBOQQKNsRpmH4hR youtu.be/3B6FymMv8b4?si=JGrESVyvndEbFFi6_____Recommended Reading:•Tycho & Kepler: The Unlikely Partnership that Forever Changed Our Understanding of the Heavens - Kitty Ferguson•Kepler - Max Caspar•The Daring Invention of Logarithm Tables: How Jost Burgi, John Napier, and Henry Briggs Simplified Arithmetic and Started the Computing Revolution - Klaus Truemper•John Napier: Life, Logarithms, and Legacy - Julian HavilFull list of sources and credits: docs.google.com/document/d/1kYSHvftLfMic09Y_yqu984jzxGuHjRyIqugUg151cwA/edit?usp=sharing_____Chapters:00:00 - Intro00:42 - 1) Kepler and Tycho05:08 - 2) Arithmetica Integra07:12 - 3) Kepler and Tycho cont'd12:10 - 4) Burgi16:51 - 5) Napier22:25 - 6) Kepler Reads Napier24:35 - 7) Briggs25:34 - 8) The Rudolphine Tables26:41 - 9) The Age of the LogarithmImages are public domain via Wikimedia; music from Epidemic SoundThis video was sponsored by 80,000 Hours.Correction:09:56 This is a typo. It should read: *Astrology* is the "foolish little daughter" of respectable astronomy.
The Ancient Mathematician Who Measured the Sun Ben Syversen 2025-02-28 | In the 3rd century BC, Aristarchus of Samos found a method for calculating the relative sizes and distances of the sun, moon, and earth using some simple measurements and...triangles.Join my Patreon! patreon.com/BenSyversenOr, support me on Ko-Fi with a one-time donation! ko-fi.com/bensyversenSpecial thank you to Alex Kontorovich for working with me on this video! This video is loosely based on his Princeton University lecture about Aristarchus: youtu.be/VdeyI8KMqts?si=WSEr15AfL-CUS-z5____Written by Ben Syversen and Alex KontorovichProduced, edited, and animated by Ben SyversenAristarchus and Archimedes Illustrations: Kendall EddyAdditional editing: Sara AdelbergAdditional animation: Pranav KumarCamera: Giacomo BellettiThumbnail design: Mostafa HasanMusic: Epidemic SoundAdditional thank you to Viktor Blåsjö for speaking with me about this topic.____Chapters:00:00 - Intro01:03 - 1) The Half Moon04:41 - 2) The Solar Eclipse07:24 - 3) The Dime Experiment09:00 - 4) The Lunar Eclipse13:24 - Heliocentrism16:10 - Eratosthenes18:19 - Copernicus19:45 - The Universe____Works consulted for this video:•Aristarchus of Samos, The Ancient Copernicus - translated and annotated by Thomas Heath - archive.org/details/aristarchusofsam00heatuoft/page/n5/mode/2up•Aristarchus’s On the Sizes and Distances of the Sun and the Moon: Greek and Arabic Texts - Berggren and Sidoli - http://individual.utoronto.ca/acephalous/Berggren_Sidoli_2007.pdf•Archimedes - The Sand Reckoner - https://web.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Archimedes/SandReckoner/Ch.1/Ch1.html•Viktor Blåsjö - podcast - Opinionated History of Mathematics - Heliocentrism in Antiquity - open.spotify.com/episode/6q80OeT8x1apeZr6JC9EKi?si=239fa7115d944f3b____To learn more about Aristarchus:•3Blue1Brown: Terence Tao on how we measure the cosmos | The Distance Ladder Part 1 - youtu.be/YdOXS_9_P4U?si=T4FpcsQVtoViKilj•terrytao.wordpress.com/2025/02/13/cosmic-distance-ladder-video-with-grant-sanderson-3blue1brown-commentary-and-corrections____Images:Images are public domain via Wikimedia and Metropolitan Museum of ArtAdditional images licensed via Envato ElementsImage of the “well” in Syene created in MidjourneyStellar parallax diagram - ESA/ATG medialab - https://www.esa.int/Science_Exploration/Space_Science/Gaia/Parallax
The $2 Million Lost Book of Archimedes Ben Syversen 2024-10-31 | A prayer book holding hidden text reveals long-buried secrets from the greatest mathematician of the ancient world. Join my Patreon! patreon.com/BenSyversenOr, support me on Ko-Fi with a one-time donation! ko-fi.com/bensyversenTo try everything Brilliant has to offer—free—for a full 30 days, visit brilliant.org/BenSyversen . You’ll also get 20% off an annual premium subscription.___Special thank you to Steven Strogatz and Amir Alexander for lending their time and expertise! I highly recommend both of their books that cover ideas discussed here:Infinite Powers by Steven Strogatz tinyurl.com/4bpnfnusInfinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World by Amir Alexander tinyurl.com/yuu4urwwAlso check out Amir Alexander's latest book: Liberty's Grid: A Founding Father, a Mathematical Dreamland, and the Shaping of America tinyurl.com/ysfy4afv(these are NOT affiliate links. Support local bookstores!)___Much of the story told here, as well as quite a bit of additional information not covered in this video, can be found in this excellent book:The Archimedes Codex by William Noel and Reviel Netz tinyurl.com/3v3s7ubcThis is a touching and thoughtful remembrance and celebration of William Noel: boltonhillmd.org/bulletin/mourning-and-celebrating-will-noel___Credits:Written and Produced by Ben SyversenIllustrations: Kendall EddyEditing: Ben Syversen, Pranav KumarAnimation: Ben Syversen, Alex ZepherinCamera and sound for A-roll footage: Anthony Forma and Mike KarasThumbnail design: Mostafa Hasan and Ben SyversenMusic: Epidemic SoundAlso thank you to Gary Rubinstein and Viktor Blåsjö, who both offered helpful feedback while I was writing this script.___Chapters:00:00 - Intro01:35 - The Method (The Basics)08:18 - Archimedes10:33 - The Dark Ages12:48 - Brilliant14:02 - The Method (Parabola)20:37 - The Scribe21:31 - Renaissance26:35 - Discovered29:33 - Lost31:51 - Found35:46 - New Discoveries___A few helpful Archimedes resources:•William Noel's TED Talk about the Palimpsest: tinyurl.com/2rcnuhkd•Archimedes at NYU Math: tinyurl.com/5ax5n9zj•Gary Rubinstein's explanation of The Method for the parabola: tinyurl.com/yzbssjr3•An excellent explanation of Archimedes' Method for the volume of a sphere: tinyurl.com/yz5mvu7w___Full list of sources: tinyurl.com/5b2rsxzkPictures and video from http://archimedespalimpsest.orgThe digital Palimpsest from http://archimedespalimpsest.net. Palimpsest images are in the Public Domain, courtesy of University of Pennsylvania Libraries and the owner of the PalimpsestOther books shown on screen: Torricelli - Opera Geometrica - Courtesy of The Linda Hall Library of Science, Engineering & Technology - Creative Commons License CC by 4.0. The Works of Archimedes (English translation from Heiberg’s German editions) - Edited by T.L. Heath, originally published 1897 by Cambridge University PressMaps: Public domain, courtesy of the University of Texas Libraries, The University of Texas at Austin. https://maps.lib.utexas.edu/maps/historical/history_europe.htmlWilliam Noel Google Talks Archive: youtube.com/watch?v=Xe9uQVGkz9kImages: public domain via Wikimedia and Metropolitan Museum of Art Other images via Wikimedia Commons:Image of the Death of Archimedes - via https://math.nyu.edu/Archimedes/Death/DeathIllus.htmlIsidore of Meletus: © Hubertl / Wikimedia Commons / CC BY-SA 4.0Hagia Sophia: via Wikimedia CommonsConica of Apollonius of Perga fol. 162b and 164a.jpg - via Wikimedia CommonsMar Saba (Photo by Jean & Nathalie, 2011).jpg - via Wikimedia CommonsAerial view of Mar Saba 03.jpg - Andrew Shiva / Wikipedia / CC BY-SA 4.0National Library of Greece: © Thomas Wolf, www.foto-tw.de (CC BY-SA 3.0 DE) - via Wikimedia CommonsBundesarchiv, Bild 146-1994-036-09A / CC-BY-SA - via Wikimedia CommonsBundesarchiv, Bild 183-1985-1216-509 / CC-BY-SA 3.0- via Wikimedia CommonsBundesarchiv, Bild 183-2008-0710-500 / CC-BY-SA 3.0 - via Wikimedia CommonsBundesarchiv, Bild 183-L05487 / CC-BY-SA 3.0 - via Wikimedia CommonsBundesarchiv, Bild 183-N0827-318 / CC-BY-SA 3.0 - via Wikimedia CommonsŞükrü Naili Paşa Atatürk ile birlikte.jpg - Ozdemirfth - CC-BY_SA 4.0 - via Wikimedia CommonsThis video was sponsored by BrilliantCorrections: 31:51 - Pronounced "Noll," like the first name
The Childhood Proof That Led Einstein to Reshape the Universe Ben Syversen 2024-03-30 | Einstein turned the world on its head in November of 1919, when data collected during a solar eclipse matched the predictions of his Theory of General Relativity. But Einstein’s path to discovering his theory traces back much further, to when he was 12 years old and he first learned about an ancient mathematical method…Join my Patreon! patreon.com/BenSyversenOr, support me on Ko-Fi with a one-time donation! ko-fi.com/bensyversenSpecial thank you to Professor @AlexKontorovichMath of Rutgers University and Museum of Mathematics (MoMath) for his help and participation.Additional credits:Giacomo Belletti - cameraBrandt Adams - newscaster voiceKolja Gjoni - drum rollValentin Cazako - help with creating the “Pringle chip” model and 3D animationsMusic from Epidemic Sound and Envato Elements.Thank you to Joel Simser (@JoelSimser) for valuable feedback on edits, and everybody else who gave me feedback and advice during this process._____________This video was originally inspired by a 2015 article by Steven Strogatz in The New Yorker Magazine about Einstein’s proof of the Pythagorean Theorem: newyorker.com/tech/annals-of-technology/einsteins-first-proof-pythagorean-theorem“Einstein, His Life and Universe” by Walter Isaacson served as a primary source for the biographical details: amzn.to/3TxtJRS_____________To learn more about Special Relativity and Minkowski space-time:@Mahesh_Shenoy - The Triplet Paradox - youtu.be/3V00tAfcHCI'We all move at speed of light through spacetime'.. What does it really mean? - youtu.be/TJmgKdc7H34@MinutePhysics - SpaceTime Intervals: Not EVERYTHING is Relative | Special Relativity Ch. 7 - youtube.com/watch?v=WFAEHKAR5hUTo learn more about General Relativity: @veritasium - Why Gravity is NOT a Force - youtu.be/XRr1kaXKBsU_____________**A Note about how I use AI generated images in my videos**The emerging ability of artificial intelligence to generate compelling images from text prompts opens new possibilities for compelling storytelling. However, when mixed with real historical imagery, as is in my video, it has the potential to create confusion, or worse, if not handled properly. I have set a few guidelines for my use of AI generated images in this video so that a viewer can easily understand which images are real photographs and which are synthetically generated:ALL images that have been placed in a “frame” (eg a border that resembles an old photo print, etc) are REAL historical images.ALL images that include Einstein's full face, as well as all World War I related images, are REAL historical images.I have used Midjourney AI to create “stock” image elements including backgrounds, illustrations, and objects.I have used Midjourney AI to create some images that are implied to be of Einstein. In these, Einstein’s face is FULLY OR PARTIALLY OBSCURED. Please send me a message or drop a comment and I'll be happy to clarify any specific images._____________Items that Appear in the Video:•“Notes for an Autobiography” by Albert Einstein - originally published in the Saturday Review of Literature, November 26, 1949•Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise - Manfred Schroeder - amzn.to/3viq1n6•Documents from Einstein’s Studies at the Zurich Polytechnical Institute - tinyurl.com/33ds27xa•On the Electrodynamics of Moving Bodies - Albert Einstein - tinyurl.com/42hmyftj•Space and Time: Minkowski’s Papers on Relativity - Hermann Minkowski - tinyurl.com/5n79k9ef•An Introduction to Riemannian Geometry - tinyurl.com/yc5x2ac7•The Collected Papers of Albert Einstein - https://einsteinpapers.press.princeton.edu•A Peek Into Einstein’s Zurich Notebook - John D. Norton - tinyurl.com/4w2df5xt•Hilbert Paper - tinyurl.com/3mdcercw•Solar Eclipse Maps from 1911 - 1920 - tinyurl.com/3p8pmx8b_____________Additional Sources:•Relativity: The Special and General Theory - Albert Einstein - amzn.to/43xh4Tx•Richard Feynman’s lecture on the Special Theory of Relativity - https://www.feynmanlectures.caltech.edu/I_15.html•Hermann Minkowski’s Spacetime: The Theory That Einstein Overlooked - David D. Nolte - tinyurl.com/27aa9nw5•How Einstein Lost His Bearings, and with them, General Relativity - Kevin Hartnett - tinyurl.com/3zv5awwk•The 1919 Eclipse Results that verified General Relativity and their later detractors: a story retold - Gerard Gilmore and Gudrun Tausch-Pebody - tinyurl.com/54p2e4kt•Marcel Grossman and his contribution to the General Theory of Relativity - T. Sauer - tinyurl.com/5ate8m6r•Einstein’s Pathway to General Relativity - John D. Norton - tinyurl.com/3mjajz4cPhotos: •Wikimedia Commons - commons.wikimedia.org/wiki/Main_Page•Leo Baeck Institute - tinyurl.com/f6vt5uhk
How Archimedes Almost Broke Math with Circles Ben Syversen 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. Join my Patreon! patreon.com/BenSyversenOr, support me on Ko-Fi with a one-time donation! ko-fi.com/bensyversenI 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: 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." 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 - Introduction00:51 - The challenge of curves01:44 - The area of a circle03:27 - The paradox of infinitesimals04:42 - History after Archimedes06:45 - Calculus in the modern world07:11 - Archimedes' life and death_____The inspiration for this video comes from:Infinite Powers: How Calculus Reveals the Secrets of the Universe, by Steven Strogatz amzn.to/3RagI0vI highly recommend this book to anybody interested in reading more. It's a great read, even if you're not a "math person."______Additional Resources:Calculus Reordered: A History of the Big Ideas, by David M. Bressoud amzn.to/3G9o9PjInfinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, by Amir Alexanderamzn.to/46s3uR8Isaac Newton, by James Gleickamzn.to/49QPNhG_____The Works of Archimedes: archive.org/details/worksofarchimede029517mbp/mode/2upRésumé 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.itemNewton's Fluxions: cudl.lib.cam.ac.uk/view/MS-ADD-03960/27Archimedes' proof from The Measurement of a Circle:http://media.bloomsbury.com/rep/files/primary-source-13-archimedes-measurement-of-a-circle.pdfAnnotation of the proof from the American Mathematical Society:ams.org/publicoutreach/feature-column/fc-2012-02More information about Archimedes' inventions and myths:nytimes.com/2013/06/25/science/archimedes-separating-myth-from-science.htmlCircular Reasoning: Who First Proved That C Divided by d Is a Constant? by David RichesonDiscussion of who gets credit for which ideas:jstor.org/stable/10.4169/college.math.j.46.3.162Additional information ("all of Europe knew less in 1500 than Archimedes did on the day he died"):inters.org/Whitehead-Western-Development-Science_____Thank you to @JonathanWhitmore for feedback on the script and providing the Manim animation at 00:12. Thank you to @JoelSimser 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.
