King Archimedes Research Paper

Archimedes, considered on of the greatest minds of the ancient world was born on the island of Sicily in the Greek city of Syracuse in the year 287 B.C.. Syracuse at the time was an independent Greek city-state with a 500-year history. He was the son of Phidias who was a Greek Astronomer and Mathematician. All that we know about Archimedes comes from his existing manuscripts, and from ancient historians such as Plutarch and Cicero among others centuries after his death. Considering the length of time between Archimedes death and the historians' accounts, along with the nonuniformity of their writings, some details of his life have to be subject to question. For example, Plutarch has been stated saying that Archimedes was related to King
Hieron II, but Cicero had claimed that he was of a low birth. It was also reported that he would become so engrossed in his thoughts that he would forget to eat or bathe making his grooming habits more to be desired.
It is believed that his early schooling came from Syracuse, then traveled to study with the Egyptian mathematician and astronomer Conon in the city of Alexandria. Archimedes had become close friends with Conon and also Eratosthenes, the custodian of the Alexandrian library. Long after completing his studies and returning to Syracuse he continued his correspondence with both of them about his different mathematical and scientific discoveries. The contributions of Archimedes to mathematical knowledge are abundant setting force principles of plane and solid geometry. Only three of his treatises on plane geometry have survived, these are; Measurement of a Circle, Quadrature of a Parabola, and On the Sphere and Cylinder. In his book Measurement of a Circle Archimedes shows that the ratio between the circumference of a circle and its diameter can be calculated as he calls it the method for calculating Pi. He correctly determined that the value of Pi is somewhere between 3.1408 and 3.1428. Along the same plane he determined that a circle's area could be determined by using the formula PI( r^2 ).
In the book On the Sphere and Cylinder Archimedes dealt with the topic of solid geometry contains several famous discoveries, including his theory that the volume of a sphere is equal to 4/3Pi( r^3). He also showed in his work that as long as the cylinder's height and width is equal to the sphere's diameter, and the volume of the sphere will be 2/3 the volume of the cylinder surrounding it. He was so proud of this discovery that he had the symbol etched into his tombstone.
Quadrature of a Parabola discusses a refined version of the method of exhaustion which was originally developed by the Egyptians. Archimedes developed the technique for determining the area under a curve, which is elaborated about in a later volume, The Method. In the book The Method it shows that Archimedes has theorized the development of integral calculus by two thousand years. Of the several essays that Archimedes has written The Sand Reckoning is the only one that remains. The essay proposed the problem of determining the number of grains of sand in the universe, containing a special notation for estimating and expressing very large numbers. The numerical system he uses is based on units of 10,000, because in ancient Greece 10,000 was the largest number they thought they needed to work with. The ancient Greeks use the word "myriad" to express the biggest number word, where the words million, billion, etc. didn't even exist. Archimedes also invented the mathematical puzzle called the Cattle Problem, where one had to determine the number of bulls and cows of different colors, given that the color of the cattle was in a particular ratio to the others. Along with the infinite number of possible solutions to the problem this one was especially challenging to the Greek who had no knowledge of algebra.
Archimedes established fundamental rules of mechanics and many different devices used in everyday life as well as machines to aid in the war against Rome.
For one of his discoveries Archimedes was called upon by King Hieron to find out if the crown the king just had made was of pure gold. The king had a suspicion the goldsmith had substituted some other metal for some of the gold. As Archimedes was taking a bath one day he noticed that the more he moved into the tub the more the water was displaced and running over the edge. This led him to believe that items of different mass would put a different amount of water out of the tub. Excited about his discovery it is said that he ran down the streets of Syracuse naked screaming "Eureka, Eureka!". At the castle Archimedes presented to the King the proof by taking a piece of gold with the same mass as the crown and submerged it into the water, when the crown itself was dropped into the water they noticed that more water came out with the crown showing that there was some other metal in the crown besides gold proving the guilt and making the goldsmith a very unhappy
person.

Along with the work that Archimedes did with levers and compound pulleys which he is noted in saying " Give me a long enough lever and I could move the Earth." A device that is called the Archimedes Screw was said to be invented to empty water out of one of the kings ships.
A sketch of the screw is on a separate page at the back of the report. The object consisted of a screw, swaddled by a cylinder, that was used with a hand crank. As the crank was turned, the screw spiraled in an upward motion carrying water with it as it went. Even to this day a similar device is in use at the Nile Delta of Egypt to carry water to crops. The King challenged Archimedes to put one of his ships that had run an aground back into the water. To meet the challenge Archimedes contrived a device of compound pulleys that an effortless pull of a rope he was able to guide the ship back into the water. With his interest in astronomy Archimedes also built a model planetarium showing how the planets and moons revolved around the
sun. For his contribution to the war effort Archimedes encountered the first two of the three Punic Wars between the Romans and the Carthaginians. Since Syracuse sided with the Carthage, when the Roman General Marcellus began his siege of the city in 214 B.C. the King called upon Archimedes to help in the defense of the city. Some of the war weapons that Archimedes conceived were some catapults that would throw large timbers at the enemy putting large holes in the ships, a large mirror like a device that would reflect the suns rays and catch the enemy ships on fire. He also contrived a large grappling device (which is also illustrated in the back of the report) which he could use the hook at the end of a rope entangled throughout an intricate set of pulleys to grab and either pick up or capsize and sink the enemy ship. The reputation of Archimedes was known far and wide so when the city of Syracuse was finally sacked in 212 B.C. General Marcellus gave orders that Archimedes would be taken alive. While Archimedes was etching in the sand, some mathematical problem, a roman soldier stumbled upon him. As Archimedes yelled and ordered at the Roman soldier not to disturb his writings the enraged soldier removed his sword and impaled Archimedes thus ending the life of on of the greatest mathematicians of all times. On a personal note, It seems to me once the General found out that Archimedes was dead by one of his soldiers, that soldier suffered the same fate as did the goldsmith earlier.