The Strangest Mathematician in History

Archimedes was so thrilled by his discovery that he ran through the streets naked, shouting at the top of his lungs, Eureka! I have found it! The story goes that the king of Syracuse, a powerful ancient Greek city-state, suspected that a goldsmith had cheated him by mixing cheaper silver into his crown. He turned to Archimedes to figure out if the crown was pure gold without damaging it. Archimedes solved the problem when, one day, he stepped into a full bathtub and noticed water spilling over the edge. The more he immersed himself, the more water overflowed. That's when he realized, the amount of water displaced must be equal to the volume of the object submerged. He then leaped from his bath and straight into the pages of history. Armed with this newfound realization, he solved the king's challenge by comparing the crown to a gold bar of equal weight. When Archimedes submerged the crown, it displaced more water than pure gold of the same weight. This revealed it had a larger volume and wasn't pure gold. It was mixed with silver. His discovery became the foundation for understanding buoyancy, the reason ships float. The Roman statesman and philosopher Cicero claimed that Archimedes possessed a genius greater than one would imagine it possible for a human being. The Eureka moment was just a glimpse of his brilliance. Archimedes was born around 287 BC in Syracuse, a prosperous Greek city-state on the southeastern shore of Sicily. He was fortunate to be born during the reign of Heron II, a rare period of peace in a region otherwise marked by power struggles and invasions. His passion for math led him to Alexandria, the intellectual capital of the Hellenistic world, where he studied with the successors of Euclid and soaked in knowledge at the Great Library. It was in Egypt that the young Archimedes invented his famous screw, an ingenious device that uses a hollow spiral to lift water from rivers and wells for irrigation. For thousands of years, farmers relied on this simple yet effective tool, rooted in mathematical precision. His passion for math would field a lifetime of discoveries. He considered his greatest achievement to be uncovering a fascinating mathematical relationship between spheres and cylinders. He found that the volume of a sphere is two-thirds the volume of a cylinder that Archimedes was so proud of this discovery that he requested it to be engraved on his tombstone. That passion for math led him to tackle another timeless challenge, calculating the true value of pi using only basic geometry. Pi is the ratio of a circle's circumference to its diameter. It had already been approximated earlier by the Babylonians and Egyptians. Archimedes began by inscribing a triangle snugly inside the circle and another around it. By calculating their perimeters, he estimated that pi was between 2.60 and 5.19. Then he used shapes that had more and more sides, more closely resembling a circle, squares, hexagons, and even a 96-sided polygon. Each step brought him closer to the value of pi. He eventually figured out it was between 3.1408 and 3.1429, which was very close to their real value of pi, about 3.14159, which wasn't determined with certainty until the 18th century. But Archimedes wasn't just a serious mathematician. He also had a playful side, as seen in the brain teasers he posed to scholars in Alexandria. His cattle problem involved calculating the number of cattle in the herd of the sun god Helios grazing on the mythical island of Parnassia. Archimedes divided the herd of bulls and cows into four groups, white, black, yellow, and dappled. He then laid out a set of proportional relationships and constraints that had to be satisfied, resulting in a complex set of equations. The problem became extraordinarily difficult when a second set of restrictions was added, such as requiring the total number of cattle to be a triangular number. For centuries, the problem remained unsolved. It wasn't until 1880 that German mathematician A.M. Torr partially solved it, revealing that the total number of cattle was an alarmingly large number. Written out in full, the number would stretch to 206,545 digits. In 1965, the full problem was finally solved using IBM computers at the University of Waterloo in Canada. Another seemingly impossible question Archimedes tackled was, This wasn't about literally counting grains of sand, but about demonstrating how mathematics could represent and reason about unimaginably large numbers. The ancient Greek numeral system used letters of the alphabet to represent numbers, but this system became impractical for larger numbers. Representing numbers beyond 10,000 required increasingly complex symbols, which made calculations difficult. To overcome this challenge, Archimedes introduced a revolutionary system grouping numbers so that each order was 10 to the power of 8 times larger than the one before. He could represent numbers as large as 10 raised to the power of 8 times 10 to the power of 8. With his innovation, Archimedes estimated that it would take While modern science shows this vastly underestimates the actual size of the universe, it was a groundbreaking demonstration of mathematical reasoning at the time. Archimedes shared his discoveries in unconventional letters to fellow mathematicians. Alan Hirschfeld, author of Eureka Man, the Life and Legacy of Archimedes, likened them to a Hollywood movie trailer, teasing the breakthrough and challenging his peers to solve it. He even included fake propositions to catch plagiarists. As he explained, We may never have fully grasped Archimedes' genius because one of his greatest works was nearly lost forever. Archimedes recorded his ideas on papyrus scrolls, the most common writing material in the ancient Greek world. But these fragile scrolls deteriorated over time. In the 10th century, a Byzantine scribe copied seven of his treatises onto parchment to save them from being lost forever. But by the 13th century, parchment had become so scarce that another scribe scraped off Archimedes' text, unaware of the mathematical treasure he was holding, and reused the pages to create a Christian prayer book. Remarkably, beneath the prayers, faint traces of Archimedes' work could be seen running from left to right. In 1906, the Danish scholar Johan Ludwig Heiberg discovered the 174-page parchment, called the Palimpsest, in a monastery library in Constantinople, now Istanbul. Using only a magnifying glass, he uncovered the faint traces of an underlining text and identified it as the work of Archimedes. In 1998, an anonymous private American collector purchased the medieval prayer book at a Christie's auction for over $2 million. The collector then loaned it to the Walters Art Museum in Baltimore, Maryland, where it was carefully conserved, digitally imaged, and made legible. One of the most famous ideas preserved in the manuscript is the law of the lever, a principle that is simple yet profound. A small force applied at the right distance can lift a much heavier weight. For example, even if one side of the lever carries twice the weight of the other, the lever will balance if the heavier side is half as far from the fulcrum. Archimedes captured the power of this principle when he famously said, Give me a lever long enough, in a fulcrum on which to place it, and I shall move the world. His understanding of balance and force applied to real-world scenarios when he famously defended Syracuse during the siege of the Roman Republic in 213 BC. At the time, Syracuse was aligned with Carthage to the frustration of the Romans. Archimedes created a crane-like mechanism resembling a claw that grabbed onto Roman ships, tipping them over or lifting them out of the water and shaking them violently. The Greek philosopher and historian Plutarch describes how the Romans began to think they were fighting with the gods. The ingenuity of his war machines even gave rise to myths, including one about Archimedes using mirrors to focus the sun's rays and burn Roman ships. What's more likely is that the defenders may have used polished metal plates to reflect sunlight into the eyes of Roman sailors, disorienting them as they approached. Despite Archimedes' efforts, the city ultimately fell to the vastly more experienced and better equipped Romans in 212 BC. Archimedes was so focused on his work that he didn't even realize that the Romans had stormed Legend has it that he was hunched over a mathematical diagram, deep in thought, when a Roman soldier approached him and ordered him to leave. Archimedes refused to abandon his calculations, reportedly saying, Do not disturb my circles, a reference to his mathematical work. Enraged, the soldier killed him on the spot. As author Alan Hirschfeld declared, he is said to have taken his dying breath poised between the point of a sword and a point he was pondering in mathematics. Archimedes was around 75 years old. The Roman general Marcellus was devastated to learn of Archimedes' death. He had specifically ordered his soldiers to spare the life of the great mathematician, understanding the immense value he could bring to Rome. To honor him, Marcellus arranged for a grand funeral and ordered a tomb inscribed with Archimedes' favorite discovery, the sphere and cylinder, in a surface area ratio of 2 to 3. Archimedes' discoveries laid the groundwork for so much of what we understand in math and science today. If you'd like to sharpen your own math skills, Brilliant is the perfect place to start. Brilliant is an interactive learning platform that helps you build real-world skills in math, science, programming, and AI through hands-on problem solving. If you want to boost your understanding of geometry, Brilliant's courses let you visualize and solve problems in a fun and engaging way. Or explore algebra and calculus to dive deeper into the formulas that shape the world around us. I personally enjoy spending 10 to 15 minutes a day with Brilliant. Right now, I'm exploring their How Large Language Models work course, which breaks down the principles behind AI systems like ChatGPT in an easy-to-understand way. It's a powerful habit for anyone who loves learning. You can try Brilliant for free for 30 days by visiting brilliant.org slash newsthink. That's brilliant.org slash newsthink for a free trial. My link is in the description. Or simply scan the QR code on your screen. Plus, you'll get 20% off an annual premium subscription. Thanks for watching. For Newsthink, I'm Cindy Pom.