b ~287 BCE Syracuse, d 212 Syracuse
Archimedes is the greatest intellectual figure on this list, and
the most famous mathematician and inventor of classical Greece.. His greatest
discovery can be stated in a sentence: the volume of a sphere is two-thirds
that of the smallest cylinder that contains it. Since the area of the cap of
a cylinder is π and its height is 2r, that can also be written as: V = 4/3 π
.
Cylinders and spheres are about the simplest three-dimensional shapes there are. Visualize a cylinder, and visualize a sphere of the same size. Imagine putting one inside the other. The sphere is smaller, but the empty space around it is of an odd shape. The space is tiny where the sphere touches the cylinder, expands in the corners, and is different on the sides instead of the top. How on earth would you figure out just what that volume is? No one ever did before Archimedes. It looks like such a simple formula, but the proof is difficult without calculus. Archimedes himself had a secret calculus-like technique, which he called the Method of Exhaustion, but was not widely known in classical times. In fact, it wasn't known to moderns until 1906, when a parchment of his appeared that described it. It relied on infinitesimals, on getting closer and closer to a true answer by approximations. This is how he found that pi was about equal to 22/7, a value used until the Middle Ages. That's good to within one part in three thousand, close enough for anything but fine mechanical work.
Yet he was known in classical times less for his mathematics than for his many mechanical inventions:
So that's an engineering record that would guarantee anyone's fame. Yet he is reported to have disdained this merely practical work. He was far more concerned with the pure and spiritual endeavors of mathemetics.
These days we recognize that theory and practice are critical to one another. It was because he understood leverage that he was able to build a catapult. It was probably in trying to make a sphere from a cylinder that he was driven to wonder what the relationship was between their volumes. It's not surprising that the Greeks didn't connect the two - no one else did either until 18th century.
In fact, he asked for a sphere and cylinder to be carved on his tombstone. Over a century later the Roman author Cicero wrote about it in his memoir "On the Good Life":
But from Dionysius's own city of Syracuse I will summon up from the dust—where his measuring rod once traced its lines—an obscure little man who lived many years later, Archimedes. When I was questor in Sicily [in 75 BCE, 137 years after the death of Archimedes] I managed to track down his grave. The Syracusians knew nothing about it, and indeed denied that any such thing existed. But there it was, completely surrounded and hidden by bushes of brambles and thorns. I remembered having heard of some simple lines of verse which had been inscribed on his tomb, referring to a sphere and cylinder modelled in stone on top of the grave. And so I took a good look round all the numerous tombs that stand beside the Agrigentine Gate. Finally I noted a little column just visible above the scrub: it was surmounted by a sphere and a cylinder. I immediately said to the Syracusans, some of whose leading citizens were with me at the time, that I believed this was the very object I had been looking for. Men were sent in with sickles to clear the site, and when a path to the monument had been opened we walked right up to it. And the verses were still visible, though approximately the second half of each line had been worn away.
So one of the most famous cities in the Greek world, and in former days a great centre of learning as well, would have remained in total ignorance of the tomb of the most brilliant citizen it had ever produced, had a man from Arpinum not come and pointed it out!
"Obscure little man", eh?
The Greek world was fading by his day, and being overrun by the monstrous and brutal Roman Empire. The arrogance displayed in the passage above was typical. They were much more impressed with his war machines than they were with a few formulas. This is a not uncommon attitude - serious public funding of science in the US only began with the Manhattan Project. The practical men only understand the value of intellect when it threatens to kill them.
Demise
Remarkably, things are still being learned about Archimedes' works. In 2000, a 10th century manuscript was discovered that contained a treatise of his on floating bodies. It had been erased and written over in the 12th century, but traces of the original text can be seen in ultraviolet. [also describe the Stomachion]
Archimedes (Chris Rorres) - site with complete biographical information
Archimedes on Spheres and Cylinders - a nice discussion of how Archimedes found the area of a sphere. One of Kevin Brown's Math Pages
The Mathematical Achievements and Methodologies of Archimedes - a discussion of his various proofs
(c) John Redford, Apr-2004
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