Ibrahim Chalhoub. Math Shanyu Ji. October 11 th, Archimedes Apparatus. A paper

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1 Ibrahim Chalhoub Math 4388 Shanyu Ji October 11 th, 2017 Archimedes Apparatus A paper

2 Born in Syracuse, Italy in 287 BC this Greek Mathematician known as an inventor, philosopher, engineer, and astronomer is highly regarded as one of the greatest innovators and minds of years past. Archimedes is known for an astounding number of methods and mathematical principles and laws, but some to live in indistinguishable glory are his Method of Exhaustion Archimedes principle, Archimedes Claw, Archimedes Screw, and Heat Ray. What is exceedingly fascinating about these discoveries were that at no point prior to Archimedes was any significant Mathematical Theory, but only the incorrect Aristotelian mathematical logic. This is what makes Archimedes so gloriously prominent. A reliable assumption that is used throughout this analysis of Archimedes: his widely resourceful mind in a time with little resource to draw from. Take, for example, his Heat Ray. During a time of religious and political strife, many were the attacks on Syracuse by invading naval forces. Asked by the government at the time to help defend his country s coast, Archimedes constructed a series of mirrors strategically placed on the shore to focus beams of light energy at wooden ships and create a combustion reaction, setting their This is while resourceful and relatively ingenious in and around 260 BC, it is, no doubt, impractical and unreliable. There are criticisms of this method:

3 First, attacks would have to be done in the morning for optimal light energy to be used since Syracuse coast faces east. This is especially dependent on the time of year. While the sun always sets in the same relative direction year-round, the angle away from east-west varies, as well as intensity of the Sun s light energy. This was most likely not known at the time as usage of the light energy in a heat ray as a weapon would not have been sufficient. Essentially, every day would harness a differed light energy, at different positioning. Second, the heat ray is dependent on weather conditions such as wind speed and a cloudless sky removing any blockage of the solar energy. It is impossible on a cloudy day to depend on sun that is constantly hiding behind clouds. Third, the targeted ship would need to be stationary for approximately 10 minutes to ignite the extremely thick wood on the vessel. Clearly, usage of the heat ray is extremely dependent on external variables not controlled by defensive forces. At any rate, these experiments were carried out in modern times: in October 2005, MIT traveled to Syracuse and deduced these exact conclusions; it was a viable weapon under perfect conditions. Similarly, the popular show, Mythbusters, deduced its most likely usage was to blind and distract offensive forces attacking Syracuse. They concluded flaming arrows or bolts from a catapult would have been easier to use to light a ship on fire

4 off the coast, thereby proving that the idea of a heat ray was unrealistic and impractical. Archimedes mechanisms were no doubt strange and dependent on uncontrolled forces, but his intentions miraculously succeeded. Another one of Archimedes inventions was the so-called Claw of Archimedes which essentially was a large grasping hand. It was planted on ground and controlled from there which would grab ships by its hull and swing upwards causing the ship to capsize. Unlike the Heat Ray it was tested in 2005 by a popular television documentary called Superweapons of the Ancient Worlds which concluded it was a viable weapon. Finally, maybe the largest chunk of Archimedes engineering work is Archimedes screw which assisted in bilging water efficiently. At the time this was a common use to remove water build-up in the ship s hull. As a helpful visual, the red ball symbolizing water would travel upwards as the helical screw would rotate inside the shaft. His works of 9 Treatises which include On Floating Bodies, Method Concerning Mechanical Theorems, On the Equilibrium of Planes, The Sand-Reckoner, On Conoids and Spheroids, and Measurement of the Circle are some of his most important works and recognizably relevant in the world of Mathematics today. A little bit of analysis on a few of his treaties: First, On Conoids and Spheroids, Archimedes delves into determining volumes of pieces of solids formed by either circles/ellipses/parabolas/hyperbolas relative to an

This Treaty is mostly conjecture, and unauthentic as the basis on the Center of Gravity was contributed by earlier mathematicians who Archimedes was not familiar with.

5 axis it is placed on, which, if Calculus 3 is in your repertoire, are all problems in integration, which were not known at the time. Next, On the Equilibrium of Planes Archimedes is concerned with center of gravity, which is a huge topic in Physics. This Treaty is mostly conjecture, and unauthentic as the basis on the Center of Gravity was contributed by earlier mathematicians who Archimedes was not familiar with. What he contributed was rather an extension than discovery. Moreover, The Sand-Reckoner demonstrates a numerical notation system on how to express an exceedingly large number. Archimedes uses the number of grains of sand in the world to notate this number with a base of 100,000,000. Archimedes found the current number system tedious and insufficient for counting certain numbers and distances so he essentially created a new counting method. Furthermore, Archimedes proved an interesting property of a parabola in which a triangle is inscribed inside the parabola and a straight line in The Quadrature of the Parabola which is among Archimedes greatest writings in 3rd century BC. Here the area of the parabola enclosed by the straight line is 4 3 the area of the triangle which can be described by an infinite geometric series n=0 4 -n = = 4 3. It is an extension of the method of exhaustion as the remaining slits of the parabola consist of an infinite number of triangles, which, as the number of triangles approach infinity, the actual area of the triangle is approached. It was only until the notion of calculus that this method was surpassed.

6 Last but certainly not least, The Law of Hydrostatics (Archimedes principle), is in my opinion, the most interesting of Archimedes work. The Archimedes Principle states that The upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces and acts in the upward direction at the center of mass of the displaced fluid. This is an essential law of physics in Fluid Mechanics that is still used today. The explicit formula is thus: density of object density of fluid = weight weight apparent immersed weight where the apparent immersed weight thus: a pparent immersed weight = weight of object w eight of displaced fluid Any floating object displaces its own weight of fluid was stated by Archimedes of Syracuse - This was especially important for King Hiero II of Syracuse, who was gifted a crown. To verify was pure gold, he asked Archimedes to prove it. Struggling day and night, he finally realized while he was taking a bath that the water level changed as he entered in the water. With the insurmountable intelligence Archimedes was gifted with, he knew, or at the very least his logic would have it, that a gold crown would displace the same amount of water that a piece of gold would with the same volume and thus the density would differ if the crown the king was given was faux gold. Archimedes was so astonished by this, he jumped out of the bath and shouted Eureka! and ran into the street naked in sheer excitement. What is most significant is that Archimedes determined the volume of an irregular shape which at the time was completely unheard of since

7 Mathematical Theory was unheard of at this time. Geometry and their areas and volumes were only beginning to formulate in the greatest intellectual minds of yesteryear and Archimedes set this path in motion. Archimedes was simply one of the greatest mathematicians of our time - without question, the proverbial platter he served the minds of future mathematicians like Descartes, Leibniz and Newton to birth the calculus and coordinate plane we know today is priceless. Clearly, these inventions of Archimedes are medieval and simply impractical to the eyes of the 21st century. We have weaponized missiles and eyes in the sky to help us prepare for invading and threatening forces. However, to the mind of Archimedes, who was asked to engineer seemingly impossible mechanisms with hardly any tools is astonishing. This is why he is credited with being one of most profound minds in the field of Mathematics. This is why he is sometimes regarded as the Father of Mathematics. An interesting quote by Archimedes that inspired me: Give me a place to stand, and I shall move the earth. This is in regards to the usage of the simple machine: the lever. This quote originates from Archimedes where in response to this bold statement, Archimedes was asked to prove such a statement by King Hiero. Archimedes then proceeded to execute this experiment with a massive ship in the harbor - with enough men they were successful in proving that it indeed could move. This was a tremendous step in the world of simple

8 machinery - and in extension, physics - and mathematics. The analogy he uses in conjunction with the logic does truly make sense. Analyzing this, we can think about it in a broader sense: with a fixed point in which someone can stand and a long enough lever to support tremendous weight, the heaviest of objects can be moved. Many are the myths surrounding Archimedes death - some say he was killed by a soldier of an invading army while he attempted to finish a Math problem. Legend has it before he was slain, he yelled, Do not disturb my circles!, at the soldier. A tragic end to such a profound life. How would mathematics have survived without his intelligence? Where would we be? The world will never know.

A Presentation By: Charlotte Lenz

A Presentation By: Charlotte Lenz http://www.math.nyu.edu/~crorres/archimedes/pictures/ ArchimedesPictures.html Born: 287 BC to an Astronomer named Phidias Grew up in Syracuse, Sicily Only left Sicily