Archimedes

Intro

Archimedes was a Greek polymath: a major innovator in mathematics, geometry, physics, and engineering. He formulated the principle of leverage. He proved that fluids falling toward a central point will eventually form themselves into a sphere, an important contribution to the Ancient Greek hypothesis that the Earth was round (Christopher Columbus was by no means the first to this insight – by the time he came along, it had been understood by every educated person for almost two thousand years). He derived an extraordinarily precise value for the mathematical constant pi (π). And he famously developed a method for measuring the density of irregularly-shaped objects, which would prove extremely useful in spotting forgeries.

Bio

Like a lot of ancient and classical philosophers, Archimedes is a somewhat mysterious figure. Ancient people didn’t spend much time writing biographies, much less autobiographies, so there isn’t much evidence to go on. Only one contemporary biography was written, and no copies survive. We know that Archimedes lived in the 3^rd century BC, and that he spent most of his life in Syracuse, a Greek colony in Sicily near the Italian Peninsula. He appears to have come from a prominent family and may have been a relative of the Syracusan king.

Although we know almost nothing about Archimedes’ life, we do know something about his death. It seems that the great philosopher, who would have been about 75 by this time, was killed by soldiers of the Roman Republic when they laid siege to Syracuse as part of the Second Punic War. In one version of the story, the Roman soldiers had orders to capture Archimedes and bring them to their general. When the soldiers burst in on Archimedes, he was sketching out a geometric proof in the sand of his courtyard. Apparently the siege had not been enough to tear him from his work. Startled, he cried, “Do not disturb my circles!” and tried to prevent the soldiers from trampling over his intricate drawings. Taking this as resistance, the Romans immediately killed him.

Archimedes’ Ideas

Leverage and the Archimedean Point

The idea of a lever is pretty intuitive: if you’re using a bar to pry something open, a longer bar will work better than a short one. If you’re throwing a lacrosse ball, a loner stick will throw it further than a short one. This basic principle underpins not only obvious levers like crowbars and wrenches, but also the wheel and axle, the archer’s bow, the shovel, and even the human arm. Obviously all of these inventions were more or less taken for granted by Archimedes’ time, but he was the first to formulate the underlying principle and to understand how all these different kinds of levers are connected. The principle of “leverage” goes back to his work.

To illustrate his principle, Archimedes famously said, “Give me a place to stand and I will move the world.” He meant that a long enough lever, with its fulcrum in the right place, could move any object, no matter how heavy. The problem, of course, was that Archimedes would have to be standing outside the world in order to move it, and with space travel thousands of years in the future that seemed flatly impossible.

Later philosophers, starting with Descartes, would use this image to describe problems of consciousness. In order for Archimedes to move the Earth, he would have to find a place outside of it to stand, and yet the very concept of standing seemed to require having the Earth underfoot. Descartes argued that there was a similar problem in understanding consciousness: in order to understand the nature of consciousness, we would have to find an “Archimedean point” outside of consciousness, from which we could see our object of inquiry. Yet any inquiry from outside of consciousness is impossible, since consciousness is a necessary precondition for inquiry itself. It’s a problem that dogs philosophers of consciousness to this day.

Quotes

“Eureka!”

The Oxford Dictionary defines “eureka” as “a cry of joy or satisfaction when one finds or discovers something.” Greek for “I’ve found it,” it’s become the mantra of cartoon scientists in moments of epiphany. And that image goes directly back to Archimedes, who is said to have shouted it as he ran naked through the streets of Syracuse. Here’s what happened:

The king of Syracuse, Hiero, had commanded his goldsmith to fashion him a crown of pure gold. But he had no way to check whether the crown was really 100% pure, or whether his goldsmith had cheated him by alloying the gold with silver and pocketing the remainder. Everyone knew that gold was more dense than silver, so if you could compare the crown’s weight to its volume it would be easy to tell whether it was made from pure gold. Archimedes was the world expert on measuring volume, but his formulas only allowed for measuring the volume of regularly-shaped objects: cubes and cylinders and so on. Because the crown was an irregular shape, there was no way to measure its volume without melting it down, and thus no way to check its density and know what it was made from.

Archimedes went home, puzzled by the king’s question. As the night wore on, he got ready to take a bath, possibly to take his mind off the problem. But as he stepped into the bath, he noticed that the water level rose just a little bit when he immersed his body in the water. He realized in a flash that the volume of displaced water was precisely equal to the volume of his submerged body. Fluid volumes were easy to measure, since they could be forced into regular shapes like cylinders. So all he would have to do was put the crown in water, measure the displacement, and compare the resulting ratio to the known density of gold.

In a frenzy, he jumped out of the bath and ran all the way out into the street, shouting “Eureka! Eureka!” He informed the king of his discovery (one assumes he stopped to get dressed first) and the test was immediately performed. Archimedes’ test confirmed the king’s suspicions, proving decisively that the crown had been faked. We are left to guess what happened to the goldsmith.

In Pop Culture

In Disney’s The Sword in the Stone, Merlin’s talking owl is called Archimedes. The owl, however, doesn’t have much in common with the Greek philosopher. In one of his most prominent scenes, Archimedes the owl says to young Arthur: “Now, boy, flying is not merely some crude mechanical process! It’s a delicate art! Purely aesthetic. Poetry of motion.”

To the real Archimedes, that would have made no sense at all. To him, there was nothing more purely aesthetic than the mechanical processes of geometry and physics. There was nothing “crude” about them. In fact, studying these things was so aesthetically compelling to Archimedes that his last act was a desperate attempt to keep from being shaken out of his mathematical reverie.