Hello everyone!

Mathematics is often presented as if it's a body of knowledge which people have known forever.  But that's not the case; modern math has come about through the work of millions of clever people over thousands of years.  I don't have time to talk about everyone, of course, but I thought it would be interesting to at least discuss a few of the great mathematicians.  There aren't many mathematicians greater than Archimedes, so he seems a natural first choice.

(Portrait by Domenico Fetti (1620).  Image taken from Wikipedia.)

Archimedes of Syracuse (c. 287 - 212 BC) is perhaps most famous for a case of public nudity.  As the story goes, a king suspected his crown was not pure gold, but had instead been made with a mixture of gold and silver.  He asked Archimedes to find a way to check without damaging the crown.  When Archimedes stepped into the bath that evening, he noticed that the water level rose in response.  He realized that he could submerge the crown in water and use the change in water level to find its volume, and thus its density, which would tell him what the crown was made of.  Not bothering to get dressed, he ran through the streets shouting, "Eureka!" ("I have found it").

 

While this story is likely untrue, Archimedes did study the physics of water, particularly relating to the buoyancy of ships.  In fact, most of the work he did in physics and engineering was extremely practical.  For instance, he designed numerous weapons, such as powerful catapults and a giant claw to defend Syracuse from invasion.  He also invented an odometer (to measure the lengths of long paths) and several other useful devices.

 

Like most of the ancient Greeks, his mathematical work was primarily geometric.  His favorite method was to use the law of levers: if two weights sit on a balanced lever, the weight of each times the distance from the fulcrum is equal.He used this law to find ratios of geometric objects.  For instance, he was able to show that for a sphere inscribed in a cylinder, as shown below, both the volume and the surface area were in the ratio .  He was so pleased with this result that he had a sphere and cylinder inscribed into his grave.

 

Archimedes was the first to prove that the area of a circle is .  He also came up with a method for approximating using polygons inscribed in a circle, as below.  He used it to find the bounds .

 

In his works, Archimedes created and used variants of infinite series summation, mechanical curves, and infinitesimals, more than 1800 years before they became common in mathematics.  Truly, he was far ahead of his time.

 

As a final note, Archimedes died during the Second Punic War when Romans captured Syracuse.  A soldier burst into his house while he was studying and killed him.  His last words were supposedly, "Do not disturb my circles."  A mathematician to the very end.

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