Background Information on Archimedes and his Findings:
Archimedes (287-212 BC), born in Syracuse Sicily, is one of the worlds most renown mathematician and engineers of his time. Archimedes ability to anticipate the discoveries of sciences, and mathematics made him unique and ahead of any mathematician of his time. Being a close friend and military advisor of King Heiro of Sicily, many of Archimedes’ scientific discoveries were donated to Syracuse to help defend the country from Rome. Archimedes often wrote of these inventions and findings, but during the Punic wars, most of these books were destroyed. Archimedes key accomplishments in his lifetime in general were his contributions to calculus, Archimedes Principal, designs of machines for Syracuse, lever and pulley designs, Mirror systems for defense of Syracuse, and the Archimedes screw. All of these findings prove Archimedes to be ahead of his time by many centuries.
Archimedes is known as the “father of geometry” because of his many contributions to mathematics which has generally helped the progression of mathematics to a state of calculus. His contributions are said to have been the creation of calculus. Many say that Archimedes would have been the creator of calculus had he been better at notating himself. Although his notation was bad, he still did accredited to discoveries and findings such as limiting PI, form Archimedes Principal, form the law of Hydrostatics, created the volume of a sphere to the volume of a cylinder ratio. These contributions are all a part of Archimedes love for creation and love for mathematics.
Archimedes in general is the reason calculus today exists. Archimedes was many centuries ahead of his time thinking of integral calculus. Archimedes limited PI to be closer then 3 1/7 which at the time was the closest number achieved. This being one of his more complex findings, he limited PI without having state of the art tools which now are present. Archimedes took a circle, and bisected a hexagon, until he couldn’t bisect anymore, and inscribed a circle inside the hexagon. He then noticed if he increased the number of sides he increased the accuracy of PI. His development of this resulted in showing PI to be in between the value 3.1408 and 3.1428. (Hutchings 447-451)
Archimedes also is thought to be the creator of integral calculus. His findings were found by his interest in the area underneath the curve of a parabola. He took a rectangular segment under it, and found the separate areas of the rectangles and added them together. Today this is what we call an Integral. This finding was the first step which ultimately led to the formation of Integral Calculus. (Tank: webpage)
Another Accomplishment of
Archimedes was the discovery of the Law of Buoyancy (Archimedes’ Principal).
The story behind this discovery revolves around fraud. King Heiro was concerned
about a goldsmith stealing gold that was supposed to be formed into a new
crown. The king’s crown was returned to King Heiro and he then approached his
friend Archimedes to see if the goldsmith was stealing parts of the gold from
the crown because the crown appeared to be light. The same night Archimedes is
said to have found, in his bath tub, the law of buoyancy. Archimedes took
notice when he got into the bath tub that night that the amount of water that
rose out of the bathtub when he immersed himself was equivalent to the weight of
his body. It is rumored at that point that he ran through the streets of
Syracuse yelling “Eureka Eureka”, meaning
”I found it, I found it”. The next morning he went to the king and took the
crown and put it into water and weighed the amount of water that came out of the
water-tub. Then Archimedes compared to the weight of the gold given to the
king, and therefore found that the gold weight did not equal the weight of the
crown which ultimately led to the beheading of the goldsmith. The formal
writing of the law of hydrostatics is,
FB=ρfluid g VD
This equation is accredited to Archimedes, and is formally known as the Archimedes Principal. (Fox & McDonald p.77)
Archimedes claims that his greatest finding is the relationship of a sphere and cylinder’s volume. Archimedes through diligence and derivation found that a cylinder can hold a sphere and the volume of the largest sphere will always be 2:3 of the cylinder in which it’s contained. Archimedes felt so strongly about this finding, he had his last wish fulfilled on his death in which case he had them scribe a sphere with the ratio of 2:3 scribed on his tomb. (Hutchings 403-430)
On top of being one of the most widely respected mathematicians Archimedes was also a large contributor to the defense of Syracuse from Rome. Archimedes is said to have designed most of the defensive mechanisms that Syracuse used to defend off the Roman Empire during the Punic wars. War machines designed by Archimedes were said to be the creation of many different strategic strategies to countries for many years to follow. Some of these creations are a fulcrum with a lever, pulley systems, catapults, forklifts, and Mirroring systems. (Rorres: webpage)
Archimedes’ finding the benefit of a lever with a fulcrum ultimately led him to be able to help with the creation of the defensive strategy for the defense of Syracuse. He noticed that with using lever on a fulcrum he could move an incredible amount of weight. This is where he made his renowned statement “Give me a lever long enough and I can move the world”. He said this prior to having respect to King Heiro in which case the king made him prove his worthiness. The king set up a large “Immovable” ship and allowed Archimedes to provide proof of such a statement, and Archimedes moved this “Immovable ship” in front of the king, and many spectators, to then gain respect from the king. (Rorres: Webpage)
Archimedes realized the importance of a fulcrum and lever which helped the defense from the Roman Empire. Archimedes, King Heiro’s military advisor of many years, designed “forklift” style arms to extend out and pick ships up and throw them into rocks ultimately dismantling the Roman fleet. This held off the Roman Empire for many years because they could not hold the people of Syracuse from getting supplies.
A defensive creation that Archimedes used to help defend off the fleets as well as the forklifts is his mirror systems. It is said that an old man [Archimedes] directed soldiers to hold hexagonal mirrors which directed the direct noon light onto the Roman fleet and set them on fire. This drastically scared the Roman fleet as they were unsure the cause.
Another defensive creation that Archimedes is accredited to is the creation of a catapult. This was a very complex creation for the time. These machines slung up to 500 pound stones and crushed platoons of Roman soldiers. With the depletion of these forces the Roman Empire retreated from battle leaving Syracuse to remain under control.
In the past there have been many contributors to mathematics, however, Archimedes was ahead of his time and was brilliant with numbers and his ability to discover ideas and progress them. His bold statement to King Heiro, “Give me a lever long enough, and a place to stand and I could move the earth”, shows his confidence in his findings. Ultimately these findings would lead him to his death. While the Roman Empire finally penetrated Syracuse in the second Punic war, Archimedes lay in his den on the ground drawing circle like patterns on his way to a new derivation, a Roman soldier entered his residence and Archimedes only words to this soldier was “Don’t disturb my circles”. Despite the soldiers orders to not harm Archimedes (due to his potential of helping the Romans) the soldier thought Archimedes was crazy, and stabbed him with neglect to his orders. This shows Archimedes love for mathematics, and diligence for working these numbers. His confidence is clearly why we today have Integral Calculus, an understanding of buoyancy, levers and pulleys, and many other scientific creations that are said to lead off of Archimedes findings.