Archimedes
"Give me a place to stand and I will move the earth."
The Man
Archimedes was born around 287 B.C. in Syracuse, which was a Greek settlement on the
southeastern coast of Sicily (and at the time the largest city in the Hellenistic world).
His father was an astronomer named Phidias. Archimedes came from the same royal family as
King Hieron II, the ruler of the city. Because of this heritage, he was able to spend most
of his productive years under the protection of the king devoting himself to wholehearted
study and experimentation, although he also spent some time studying in Alexandria.
The Inventor
Some of Archimedes' fame comes from his inventions. One of his most famous inventions, which
is still used in some parts of the world, is the Archimedean screw, which pumps water from lower
elevation to higher elevation. Archimedes invented this screw while on a visit to Egypt for the
purpose of raising canal water over levees into fields that needed irrigation. Later on it was also
used to pump water out of mines and ships' holds. A Youtube video demonstrating how it works can be found
here .
The Engineer
The quote above was spoken by Archimedes after he discovered the law of levers and pulleys. Upon the
request of his king and patron, Hieron, Archimedes designed a system of pulleys and levers that allowed
him to launch a large ship on his own. Hieron's men were having difficulty getting the ship launched
because it was much heavier than usual.
The Scientist
Perhaps the most famous story of Archimedes is that of his discover of the law of buoyancy. His friend
and patron, King Hieron, asked him to find a way to determine if the golden crown he had ordered to be
made had in fact been made entirely of gold as he had asked. The king suspected that an equal weight of
silver had been added with the gold. He needed a way to prove his suspicion, since the crown's weight
matched the weight of gold assigned to the goldsmith. The answer to the problem came to Archimedes as he
entering the bathtub one day. He noticed that as he lowered his body farther into the water, more water
was displaced and flowed over the top of the tub. He realized that even though the weight was the same,
if the crown was indeed made of gold and not a combination of other metals, it should displace the same
amount of water as the same weight of pure gold. As he came to this realization, he is reported to have
jumped out of the tub and rushed home naked crying "Eureka, eureka!" This translates to "I have found it,
I have found it!"
The Warrior
Archimedes also used his vast knowledge to help defend Syracuse from Roman invasion. He directed the
fortifying of the city with a series of powerful catapults and crossbos to send missiles at specified
ranges.In this way he ensured that no matter how close the Roman soldier came to the city, they would
always be under fire. He also engineered devices that could drop huge stones or masses of lead from the
city walls. Large cranes caught the remaining ships with grapling hooks, lifted them out of the water,
and dropped them. It is also legend that during this siege Archimedes used mirrors to set the enemy ships
on fire, although this part of the story is likely myth. (The Mythbusters busted this myth. A video clip
can be seen here.)
The Mathematician
Archimedes made many great contributions to the world of mathematics. The one I would like to focus on
here is his approximation of pi. At this time in the ancient world, people had an idea of what pi was,
and they used it frequently, especially in their buildings. The Pyramids at Giza and Solomon's Temple
are two examples of this. Even though they had a concept of what it was, nobody had found a way to
calculate it. Archimedes used the method of exhaustion to find an approximation. He used cicumscribed
and inscribed polygons and gradually increased the number of sides until he found the following
approximation of pi using a 96-gon: 3 1/7 < pi < 3 10/71 . (This calculation by Archimedes is where the
estimate of 22/7 for pi originated. 22/7 was the lower bound found by Archimedes.) This is particularly
amazing because Calculus had not yet been invented. He did all the calculations by hand, which involved
a lot of work and tedious calculations. Below are two different applets that demonstrate his process,
although they don't show the amount of calculations involved.
Approximating Pi
NOVA Applet
I found
this website very helpful for explaining the process that Archimedes went through. Basically he
began with a circle with diameter of one and added a circumscribed and inscribed hexagon. He calculated
the perimeter of each of the hexagons, and concluded that pi had to be somewhere in between the perimeter
of the inscribed hexagon and the circumscribed hexagon. To obtain a better approximation he then repeated
the process with inscribed and circumscribed dodecagons, then with 24-gons, then 48-gons, and finally
96-gons. Each time he increased the number of sides, he got closer to the perimeter (or circumference)
of the circle, which in this case would be pi since the diameter is 1. This follows the idea of limits
and integrals, although Calculus wouldn't be invented for several hundred years.
Archimedes died in 212 B.C. following the famous siege mentioned above. The Romans retreated after being
defeated by sea with Archimedes' clever contraptions. They temporarily withdrew their forces, and the
people of Syracuse relaxed in their vigilance. The Romans then invaded the city by land, and although
Marcellus, the Roman general, gave explicit orders that the Archimedes' house and life should be spared,
Archimedes was killed by a Roman soldier. Marcellus erected an elaborate monument in Archimedes' honor.
The Romans complied with his wish that a figure of a sphere inscribed in a right cylinder be put on his
tombstone (Archimedes was very proud of his discovery of the relation between a sphere and a right
cylinder: the volume of the sphere is equal to two-thirds the volume of a circumscribing cylinder).
"Don't spoil my circles!"