Archimedes was born in 287 BC in Syracuse, Sicily. The only time that Archimedes would come out of his house would be to go to his school at Euclid's school in Alexandria. Archimedes father was an Astronomer named Phidias. Archimedes was a genius when it came to the subject of mathematics and most of his time contemplating new problems to solve. Sometimes he got so involved in his work that he would forget to eat. He never gave up the opportunity to think about his work. So sometimes after bathing and putting olive oil on himself he would trace figures in the oil on his own skin.
Much of Archimedes comes from his relationship with Hiero, the king of Syracuse and Gelon, Hiero's son. Archimedes had many inventions including Archimedes' watering screw and a miniature planetarium. His approximation of pi between 3-1/2 and 3-10/71 was the most accurate of his time and showed that he could approximate square roots accurately too.
Archimedes was also unhappy with the Greek number system, so he decided to devise his own number system that could accommodate larger numbers more easily. He invented the entire field of hydrostatics with the discovery of Archimedes' Principle. However, his greatest invention was integral calculus. To find out the area of sections bounded by geometric figures such as parabolas and ellipses, Archimedes broke the sections into an infinite number of rectangles and then added the areas together. This is known as integration.
At one time, the king ordered a gold crown and gave the goldsmith the exact amount of gold needed. When Hiero received it, the crown had the correct weight but the monarch suspected that some silver had been used instead of gold. Since he couldn't prove it, he asked Archimedes to solve the problem. One day while pondering about the question, Archimedes dipped into his bathtub and noticed that the amount of water overflowing the tub was proportional to the amount of his body that was being submerged. This gave him the idea for solving the problem of the crown, and he was so thrilled that he ran naked through the streets of Syracuse shouting "Eureka! Eureka!" which means "I have found it!" There are a few ways Archimedes could determine the proportion of silver in the crown. One method has to do with a proposition that Archimedes later wrote in a piece called "On Floating Bodies", which is now called Archimedes' Principle: a body immersed in a fluid is buoyed up by a force equal to the weight of fluid displaced by the body. Using this method, he would take two equal weights when immersed in water. Next he would have compared the weight of the crown and an equal weight of pure silver in water the same way. The difference between these two comparisons would show that the crown was not pure gold.
Another time, Archimedes told Hiero that with a given force, he could move any given weight. So he investigated the properties of the lever and pulley, and stated, "Give me a place to stand and I will move the earth." King Hiero, who was amazed at that, asked him to prove it. In the harbor was a ship that had proved to be impossible to launch by the combined strength of all the men in Syracuse. Archimedes used a mechanical device that allowed him to move the ship from a distance. Hiero saw that Archimedes had a very creative and inventive mind for constructing mechanical aids. Hiero persuaded Archimedes to construct machines for use in combat, both defensive and offensive. Archimedes had many works including, "On the Sphere and Cylinder", which shows that the surface of a sphere is four times that of a great circle. In Archimedes second book, shows how to cut a given sphere by a plane so that the ratio of the volumes of the two segments of these three-dimensional figures.
The end of Archimedes life was anything but boring. In 212 B.C., Marcellus, a Roman general, decided to conquer Syracuse. While Archimedes was drawing figures in the dust, a Roman soldier stepped on his drawings and demanded him to come with the soldier. Archimedes then responded, "Don't disturb my circles!" The soldier was so angry that he pulled out his sword and slays the great geometer. When Archimedes was buried, they placed on his tombstone the figure of a sphere inscribed inside a cylinder and the 2:3 ratios of the volumes between them, the solution to the problem he considered his greatest achievement.
The achievements of Archimedes are outstanding. He is considered by most historians of mathematics as one of the greatest mathematicians of all time.