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Who Was Archimedes?
Archimedes was a Greek philosopher at a time when philosophy included multiple branches of science; he was known for his discoveries in the fields of mathematics, physics, engineering, and astronomy, as well as his inventions. He lived from 287 BC to 212 BC, when he was killed by a Roman soldier during the Siege of Syracuse.
In spite of all his inventions, the best known story in the history of Archimedes is the (probably false) tale of how he discovered the Archimedes Principle, described in the next section.
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The Archimedes Principle
The Archimedes Principle states that any object which is immersed in a fluid is buoyed upwards by a force equal to the weight of the displayed fluid. Because the amount of fluid displaced depends on the volume of the displacing object (that is, the part of the object which is under the water), we can determine whether an object will float by calculating its volume, multiplying to find the weight of the equivalent amount of water (or other fluid), and comparing the result to the weight of the object. If the upwards force is greater than the weight of the object, it will float; otherwise, it will sink.
The story goes that Archimedes was tasked with determining whether the king's new crown was made entirely of gold, or had cheaper silver in it as well. While visiting the public baths, Archimedes observed how his body displaced water and realized that he could use this displacement to determine the volume of the crown; comparing this number with the weight would allow him to determine the density of the crown and compare that to the known density of gold. So excited was he that without stopping to dress, he immediately ran home yelling "Eureka!" (I have it). While the story is of doubtful authenticity, it's a fun bit and is probably what most people think when they think about Archimedes.
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It has been claimed that Archimedes designed machines that could set fire to opposing ships using an array of mirrors, but no evidence has been discovered to support this. However, he did invent a number of other machines, including siege engines and the Archimedes screw pump, which is still used to lift water today.
Archimedes, however, was most proud of his work explaining the principle behind the lever; he was known to say "Give me a place to stand on, and I will move the Earth." Based on this, he designed block and tackle pulley systems that allowed sailors to lift heavy items. He also invented the fodometer.
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While he was, in his day, better known for his inventions, Archimedes made a number of contributions to the field of mathematics. Among other things, he was able to calculate the value of pi as being between 3 1/2 and 3 10/71, and proved the formula A=pi*r*r describing the area of a circle.
His favorite result, however, was the formulas for the volume and surface area of a sphere and cylinder; he was able to show that, given a sphere and cylinder of the same hight and diameter, the sphere's volume and surface area are two-thirds that of the cylinder. At his own request, a sculpted sphere and cylinder were placed on his tomb after his death.
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Most of what we know of Archimedes is from his own writings, or the translations. In particular, much of his writing was discovered in the Archimedes Palimpsest, a goatskin parchment of 174 pages that contains the only surviving copies of several of his works.
Some of Archimedes' work can be found in the book "Greek Mathematical Work: Volume II".