The Strangest Mathematician in History: The Life, Genius, and Legacy of Archimedes

Archimedes of Syracuse stands as one of history's most brilliant and eccentric mathematical minds. Living during the Hellenistic period, his pioneering work laid the foundations for hydrostatics, mathematical physics, and approximation calculus. This chapter explores his legendary discoveries, revolutionary mathematical systems, and the dramatic preservation of his lost treatises.

1. The Eureka Moment: Buoyancy and Density

The most famous legend of Archimedes' genius involves Hieron II, the king of Syracuse, who suspected that a local goldsmith had cheated him. The king believed the goldsmith had kept some of the gold intended for his royal crown and substituted it with cheaper, less dense silver. He tasked Archimedes with proving this suspicion without damaging or melting down the crown.

Real-Life Analogy: The Bathtub Insight

Archimedes struggled to find a solution until he stepped into a full bathtub and noticed water spilling over the edge. He realized that the volume of water displaced by his body was exactly equal to the volume of his submerged body.

Shouting "Eureka!" ("I have found it!"), he reportedly ran naked through the streets of Syracuse. To solve the king's challenge, Archimedes compared the crown to a block of pure gold of the exact same weight:

Because silver is less dense than gold, a crown mixed with silver must have a larger volume than a pure gold crown of equal weight. When submerged, the crown displaced more water than the pure gold block, exposing the goldsmith's fraud. This discovery laid the foundation for the law of buoyancy, explaining why massive ships are able to float.

2. Historical Context: Syracuse and Alexandria

Archimedes was born around 287 BCE in Syracuse, a wealthy and powerful Greek city-state on the southeastern coast of Sicily.

• The Era of Peace: He was fortunate to grow up during the stable reign of King Hieron II, a rare period of peace in a region frequently disrupted by foreign invasions and power struggles.
• The Library of Alexandria: To pursue his passion for mathematics, Archimedes traveled to Alexandria, Egypt—the intellectual capital of the Hellenistic world. There, he studied alongside the successors of the legendary mathematician Euclid and immersed himself in the Great Library.
• The Archimedes Screw: While in Egypt, Archimedes invented an ingenious mechanical device known as the Archimedes Screw. This device uses a hollow spiral inside a cylinder to lift water from lower levels (such as rivers or wells) to higher elevations for agricultural irrigation. It remains in use by farmers thousands of years later.

3. The Geometry of Spheres and Cylinders

Despite his practical engineering feats, Archimedes considered his pure mathematical discoveries to be his greatest achievements. Chief among them was uncovering a beautiful geometric relationship between a sphere and a cylinder.

He proved that if a sphere is perfectly enclosed by a cylinder (where the cylinder's height and diameter equal the sphere's diameter):

1. The volume of the sphere is exactly $\frac{2}{3}$ the volume of the cylinder.
2. The surface area of the sphere is exactly $\frac{2}{3}$ the surface area of the cylinder.

Archimedes was so proud of this elegant geometric proof that he requested this specific diagram—a sphere inscribed within a cylinder—be engraved onto his tombstone.

4. Approximating Pi ($\pi$) with Geometric Iteration

Before Archimedes, ancient civilizations like the Babylonians and Egyptians had only rough approximations of $\pi$ (the ratio of a circle's circumference to its diameter). Archimedes sought to calculate its true value using basic geometry.

To do this, he developed a method of exhaustion using inscribed and circumscribed polygons:

Inscribed Polygon (Inside Circle) < Circle < Circumscribed Polygon (Outside Circle)

1. The Starting Point: He began by drawing a triangle snuggly inside a circle (inscribed) and another triangle around the outside of the circle (circumscribed). By calculating the perimeters of these triangles, he established a very wide range for $\pi$ between $2.60$ and $5.19$.
2. Increasing Complexity: He systematically increased the number of sides of the polygons. He calculated the perimeters for squares, hexagons, and progressively complex shapes.
3. The 96-Sided Polygon: By calculating the perimeters of $96$-sided polygons, Archimedes narrowed the value of $\pi$ down to a highly accurate range:

This approximation was so precise that humanity did not calculate the exact value of $\pi$ with absolute certainty until the 18th century.

5. The Cattle Problem of Helios: An Ancient System of Equations

Archimedes enjoyed challenging his contemporary scholars in Alexandria with complex mathematical brain teasers. His most famous puzzle is the Cattle Problem, which asks solvers to calculate the size of the herd of the sun god Helios grazing on the mythical island of Thrinacia.

• The Setup: Archimedes divided the bulls and cows into four colors: white, black, yellow, and dappled.
• The Constraints: He established a series of proportional relationships between the groups, creating a complex system of simultaneous linear equations.
• The Extreme Constraint: The problem became incredibly difficult when he added a second set of conditions, requiring that the sum of certain groups of cattle form a perfect square number, while others form a triangular number.

For centuries, the problem remained unsolved. In 1880, German mathematician A. Amthor partially solved it, proving that the total number of cattle was an astronomical figure. Written out in full, the number would span 206,545 digits. The complete problem was finally solved in 1965 using IBM computers at the University of Waterloo in Canada.

6. The Sand Reckoner: Developing Orders of Magnitude

In his treatise The Sand Reckoner, Archimedes addressed a seemingly impossible question: How many grains of sand would it take to fill the entire known universe?

This exercise was not about literally counting sand, but rather demonstrating that mathematics could represent and calculate unimaginably large numbers.

Overcoming the Greek Numeral System

The ancient Greek numeral system used letters of the alphabet to represent numbers, making calculations beyond $10,000$ (a "myriad") highly impractical and complex.

To bypass this limitation, Archimedes invented a revolutionary system of orders of magnitude. He grouped numbers into "periods" based on powers of $10^8$ (a myriad of myriads). Using this exponential system, Archimedes calculated that the number of sand grains required to fill the cosmos was:

While modern astrophysics shows that this vastly underestimates the actual size of our universe, Archimedes' work was a monumental leap forward in scientific notation and mathematical reasoning.

7. Peer Review, Plagiarism, and the Lost Palimpsest

Archimedes shared his mathematical breakthroughs by writing letters to fellow scholars.

The "Movie Trailer" Method: Author Alan Hirshfeld noted that Archimedes' letters functioned like modern Hollywood movie trailers. He would tease his mathematical breakthroughs and challenge his peers to solve them themselves. To catch plagiarists who claimed his work as their own, Archimedes intentionally included fake mathematical propositions in his letters.

The Journey of the Archimedes Palimpsest

Because Archimedes wrote on fragile papyrus scrolls, his original works deteriorated over time. The survival of his ideas is a miracle of historical preservation:

In the 13th century, parchment was so scarce that a scribe scraped away Archimedes' mathematical text to reuse the pages for a Christian prayer book (creating a palimpsest).

In 1906, Danish scholar Johan Ludvig Heiberg discovered this 174-page parchment in a monastery library in Constantinople (now Istanbul). Using a magnifying glass, he identified the faint, underlying horizontal Greek text as the lost works of Archimedes. In 1998, an anonymous American collector bought the book at auction for over $2 million and loaned it to the Walters Art Museum in Baltimore, where modern digital imaging finally made the scraped-away text fully legible.

8. The Law of the Lever and the Defense of Syracuse

One of the most profound physical principles preserved in the Palimpsest is the Law of the Lever: a small force applied at a great distance from a fulcrum can lift a much heavier weight.

If one side of a lever carries twice the weight of the other, the lever will balance perfectly if the heavier weight is placed at half the distance from the fulcrum. Archimedes famously summarized this principle of leverage:

"Give me a lever long enough and a fulcrum on which to place it, and I shall move the world."

The Siege of Syracuse (213 BCE)

Archimedes put his understanding of levers and mechanics into action when the Roman Republic besieged Syracuse.

• The Claw of Archimedes: He designed a massive, crane-like claw weapon. Using levers and pulleys, the claw reached over the city walls, hooked onto Roman warships, lifted them out of the water, and shook or capsized them. The Roman soldiers were so terrified that they believed they were fighting against gods.
• The Mirror Myth: Legend claims Archimedes used giant mirrors to focus the sun's rays and burn Roman ships. Historians believe it is far more likely that the defenders used highly polished metal plates to reflect blinding sunlight into the eyes of Roman sailors to disorient them.

9. The Death of a Genius

Despite Archimedes' defensive inventions, Syracuse fell to the Romans in 212 BCE after a prolonged siege.

Archimedes was so engrossed in his mathematical work that he did not realize the Roman army had breached the city. He was hunched over a geometric diagram drawn in the dirt when a Roman soldier approached and ordered him to follow him. Archimedes refused to leave until he finished his proof, famously saying:

"Do not disturb my circles!"

Enraged by his defiance, the soldier drew his sword and killed the 75-year-old mathematician on the spot.

The Roman General Marcellus was devastated by the news. He had issued strict orders to capture Archimedes alive, recognizing his immense scientific value to Rome. Marcellus arranged a grand funeral and honored Archimedes' final wish by building a tomb decorated with a sphere inscribed in a cylinder, commemorating his favorite mathematical ratio of $2:3$.

10. Summary of Key Achievements

Summary and Key Takeaways

• The Birth of Hydrostatics: Archimedes discovered that a submerged object displaces a volume of water equal to its own, establishing the scientific principles of buoyancy and density.
• The Power of Limits: By using inscribed and circumscribed polygons up to 96 sides, Archimedes pioneered the "method of exhaustion," approximating $\pi$ to an incredibly narrow range ($3.1408$ to $3.1429$).
• Inventing Large-Scale Notation: To calculate the grains of sand needed to fill the universe ($10^{63}$), Archimedes bypassed the limitations of the Greek alphabet by inventing a base-$10^8$ system of orders of magnitude.
• The $2:3$ Geometric Ratio: Archimedes proved that a sphere inscribed in a cylinder occupies exactly two-thirds of its volume and surface area, a discovery he cherished so much he had it carved onto his tombstone.
• The Archimedes Screw: His practical engineering genius led to the invention of a spiral water-raising screw that revolutionized agricultural irrigation.
• The Law of the Lever: He mathematically defined how distance from a fulcrum amplifies force, which he used to build devastating defensive weapons like "The Claw" to protect Syracuse from Roman warships.
• The Palimpsest Miracle: Much of Archimedes' work survived only because a 10th-century copy was scraped off and overwritten as a prayer book, preserving faint traces of his original text that were recovered using 20th-century imaging.
• A Dedicated Mind: Archimedes' obsession with mathematics ultimately cost him his life; he was killed by a Roman soldier during the fall of Syracuse because he refused to stop working on a geometric drawing.