Archimedes (~287 – 212 BCE) Universal Genius

b ~287 BCE Syracuse, d 212 Syracuse

Renaissance mosaic in Stadelsches Kunstinstitut, Frankfurt

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 r² and its height is 2r, that can also be written as: V = 4/3 π r³ .

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:

• The Archimedes Screw – a helix inside a cylinder that draws fluids upwards when turned. Its tradtional use has been in irrigation, but it is now also common in sewage treatment plants. It has the same advantage in both cases – a resistance to clogging by solid objects. Since it has no valves and no pistons sliding against walls, there is little for a particle to jam in to. He originally came up with it as a way to pump water out of ships.
• The compound pulley – Like the lever, this works by transforming a small force over a great distance into a large force over a short distance. With the compound pulley, the motion of the one rope that is being is pulled on is transformed into the motion of the many ropes between the pulleys. Archimedes demonstrated it by single-handed pulling a ship into dock.
• Ship grapplers (Archimedes Claw) – a hook mounted on a swinging crane that could grab a ship by the bow and upend it. A lead weight was released at one end of the crane to draw up the other end. Another example of leverage.
• Catapults – were known in the ancient world, but Archimedes developed several for the defense of Syracuse when it was being attacked by the Romans under Marcellus.
• Burning Mirrors – an array of mirrors mounted on a parabola, and used to ignite the sails of Roman ships. From his geometric studies he would have known about the focusing properties of the parabola. Still, this would have needed a lot of small mirrors with good surfaces and accurate placement to work. No one else to my knowledge has ever tried this, so it’s probably myth..
• Star globes – He was said to be bringing a map of the stars on a globe to the Roman general Marcellus when he was killed.
• Orrery – A model of the solar system with planets swinging around the sun. This would have been an impressively complex mechanism for the classical world, but it was lost. The modern name comes from the fourth Earl of Orrery, who commissioned one from George Graham in the early 18th century.

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 mathematics.

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 theory and practice – no one else did either until the 18th century.

Archimedes was so proud of this formula that 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 modeled 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

The only account of his death was written by Plutarch about 300 years later.   He writes in “Parallel Lives” of the capture of Syracuse by Marcellus:

But nothing afflicted Marcellus so much as the death of Archimedes, who was then, as fate would have it, intent upon working out some problem by a diagram, and having fixed his mind alike and his eyes upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming up to him, commanded him to follow to Marcellus; which he declining to do before he had worked out his problem to a demonstration, the soldier, enraged, drew his sword and ran him through. Others write that a Roman soldier, running upon him with a drawn sword, offered to kill him; and that Archimedes, looking back, earnestly besought him to hold his hand a little while, that he might not leave what he was then at work upon inconclusive and imperfect; but the soldier, nothing moved by his entreaty, instantly killed him. Others again relate that, as Archimedes was carrying to Marcellus mathematical instruments, dials, spheres, and angles, by which the magnitude of the sun might be measured to the sight, some soldiers seeing him, and thinking that he carried gold in a vessel, slew him. Certain it is that his death was very afflicting to Marcellus; and that Marcellus ever after regarded him that killed him as a murderer; and that he sought for his kindred and honoured them with signal favours.

If he had survived, one shudders to think how much more harm the Romans could have done with his technologies.  It’s likely, though, that even if they could have gotten Archimedes to work for them, they would have disregarded him as wooly-headed.  Who needs these incomprehensible cranes and screws when you have a good sharp sword?  Why waste time building all this complicated stuff when you can just kill everyone?  That worked pretty well for them for a long time.   Marcellus knew that he had lost a great treasure with Archimedes’ death, but I doubt the Romans appreciated just how far he had gotten.

Links

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

Apr-2004

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