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Every school child has learned of Archimedes (287 BC to circa 212 BC). We learn of his inventions and his basic machines that we use in almost every appliance and device in our modern world. He was a scientist and inventor without peer. He was over 1500 years ahead of his time.
We know the famous story about the King's gold crown and how Archimedes had that Eureka moment when he found the law of specific gravity and the nature of the gold in the King's crown.
What was his greatest discovery? Was it some war machine that he invented? Was it his astronomy? Was it his discovery of the nature of the lever, inclined plane, screw or pulley? We seek not what others said, but what he said himself. He was proudest of his work in geometry and especially his finding the area and volumes of the cylinder, sphere and cone.
We must remember he was working in an age without the modern system of numbers. Roman Numerals are not good for computation.
He was able to show with his approximation of π the following relationships:
The volume of a sphere is
while the surface area of the sphere is
What is so interesting about this? Looking closer we see that the surface area is 4 times the area of the great circle of the sphere. Where did that come from? Until Archimedes nobody could guess that result. The great Euclid is totally silent on this subject. In the ancient world the sphere was considered the perfect 3 dimensional shape and the circle the perfect 2 dimensional shape. They were very interested in finding the secrets of these shapes. Not stopping at the volume and area of a sphere Archimedes proved that the volume of a cone with radius r and height r is
Things are more curious still. As you can see the volume of the cone is such that there are 4 cones with height r and circular base r inside a sphere of the same radius. That is, these 4 cones 'fill up' a sphere of the same radius.
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Wednesday, June 15, 2011