Similarly, there is little detailed information concerning Archimedes life as a child and adolescent. According to history, Heracliedes wrote a biography of Archimedes which is subjected to have had detailed information about various aspects of Archimedes private life. Unfortunately, the biography was destroyed and many aspects of Archimedes personal were lost in the process. It seems that the p of the years has erased the memories of his childhood upbringing. In fact so little is known of his personal life, that there exists no specifics on his coupled life. Whether he had a wife and children remains unknown to the present day.
In comparison, his professional adult life has been studied and retraced century after century, relating of his incredible prowess with Mathematics, and of its unusual genius for technological inventions; some of which are still being used today, two millennia after his death. Historical texts mention his relation to King Hiero II, then the King of Syracuse and presumed uncle of Archimedes (Crystalinks, 2008). The validity of this relation to the Royal Family of Syracuse comes again and again in writings concerning Archimedes, and the few elements of his young adult life seem to confirm his privileged ranking in Syracusian society.
In fact, Archimedes was schooled in Alexandria, Egypt where he traveled to as a teenager to study mathematics (University of St Andrews, 1999). The many counts of his spectacular professional life as a mathematician, scientist, and inventor seem to retrace an origin to that period of his life. Certainly the ability to pursue university study confirms of his family ties to the Aristocratic society of Syracuse. He would later in his life collaborate closely with King Hiero II to come up with inventions to prevent Roman invasion of Syracuse.
Some of those inventions of warfare are reviewed in further detail in the portion of this text dedicated to Archimedes technological inventions and innovations. Archimedes began study in Alexandria at the age of 18. He was then brought to study mathematics along with Conon of Samos, and Eratosthenes (Crystalinks, 2008). As a scholar in Alexandria, he was allowed to study both the theoretical and practical aspects of science and technology, that he often retransmitted back to Greece via letters of correspondence he wrote. It is believed that Archimedes spent five to six years in Alexandria at study.
There are no other accounts during the life of Archimedes where he would have spent a comparable amount of time being educated in the formal sense. Following his study, he returned to Syracuse to become one of the most prolific scientist and inventors known to mankind. History tells that Archimedes invented the Archimedes screw while at study in Alexandria. The famous screw used to carry water from a low lying position to a higher position would found many useful applications and is presently used in modern day sewage plants. An amazing feat indeed.
His ingenuity continued after he returned home to Syracuse, and was fueled by the desire to find adequate solutions in order to protect the city from Roman invasion. In fact, often under the demand of the King, he undertook and completed several inventions targeted at warfare. For so doing, he used mechanisms of destruction and others of dissuasion that proved efficient as they held the roman invader, General Marcus Claudius Marcellus, from entering the city of Syracuse for two consecutive years. Archimedes died in 212BC, while Syracuse was under siege by the Roman invaders.
The story tells that he was killed by a roman soldier during the attack of Syracuse (Crystalinks, 2008). His mathematical Genius Archimedes of Syracuse is particularly known the world over for his stunning ability with mathematics, and in particular with geometry. In this section of the biography, we are to retrace the most important theorems he came up with, and relate of his most impressive scientific discoveries. On the contrary to most mathematicians, Archimedes mathematical inspirations often came from his work on Mechanics, thereby suggesting of an influence he brought to mathematics by making hypothesis based in the practical world.
This is a very interesting practice which is peculiar and certainly differentiates his work from other mathematicians who mostly would come up with a mathematical theorem and then attempt to verify it in the physical world. Archimedes wrote extensively on his work, although most of his work vanished over the years. In particular, he wrote a treatise on mechanics and hydrostatics entitled the “Method Concerning Mechanical Theorems”, which according to history often inspired his work as a mathematician. As he seemed to find his inspiration in the physical mechanical world, Archimedes excelled in the field of Geometry.
One of his famous discoveries was in relation to the comparable volume of a sphere and that of a cylinder. Archimedes was able to prove that the volume of a sphere equaled two-thirds of the volume of a cylinder for which the height equaled the diameter of the sphere (University of St Andrews, 1999). Archimedes was so proud for having found that mathematical reality that he insisted on having it carve on his tomb. Although Archimedes is often thought of as more of an inventor than a mathematician, he participated in several key developments in mathematics.
Archimedes often made use of infinitesimal sums to arrive at proving his hypotheses (Crystalinks, 2008). The method is often compared to modern day integral calculus which is very similar to the methods he employed then. One of his famous mathematical proofs was the approximation of Pi. Archimedes often used his ingenious notion of the mechanical world to arrive at more conclusive mathematical realities. In order to estimate the value of pi more accurately, he designed a circle. He placed a polygon on the outside and on the inside of the circle (University of Utah, 1999).
As he would raise the number of sides of each polygon, he came closer and closer to having a circle; effectively made of a series of small and connecting distances. As he reached 96 sides for the inner and outer polygons, he measured them to obtain a higher and lower boundary limit of the approximation of Pi. Archimedes concluded from the experiment that the value of pi was contained between 3+1/7 and 3+10/71 (Crystalinks, 2008). A remarkable feat leading to an impressive conclusion, which we consider today one of the most important proofs of mathematics.
The formula for the area of a circle is also attributed to Archimedes who came up with the fact that the area was equal to the square root of the radius of the circle multiplied by Pi. His interest for arriving at mathematical truths based on geometrical realities as we can perceived them in a multi-dimensional system, led him to prove more theorems often relating to infinite series or infinite sums. Archimedes is known for determining the equivalency of certain rational numbers by determining their infinite sum.
A rational number differentiates itself from a whole number (an integer for instance), as it has an integer portion and a decimal portion. The infinite sum approximation is often used in mathematics today to estimate areas and volumes in two dimensional and three dimensional spaces primarily. The technique he employed in his infinitesimal related theorems are commonly called method of exhaustion in modern day mathematics (University of St Andrews, 1999). As impressive as his ability for arriving at mathematical reality was, it made even more physical sense when he applied it to the physical world in which we live.
Many of Archimedes theories relating to physics are closely relating to the fields of geometry and physics in general. Often the geometrical mystique of an object would eventually lead to a physical mathematical reality of our world. It is seemingly in such proceeding that Archimedes came up with several theorems of mechanical nature. In fact, Archimedes discovered several theorems on the center of gravity of planes, and solids, and on the mathematical tools and methods to approximate those.
It is interesting to mention that his work, whether in theory or practice was often commanded by the search of the infinite in the mundane reality of the finite. Archimedes is known to have worked on the mathematical theories of spirals, where he helped to determine the mathematical formulation to describe spirals based on polar geometry. The work was compiled in a treatise called the Archimedean Spiral. The treatise describes in mathematical terms the function of a point moving away from a fixed coordinate at a constant speed and with constant angular velocity.
The function described in the treatise corresponds to the geographical representation of a spiral, which in the treatise is the result of moving set of points in a given pattern, that of a spiral (University of St Andrews, 1999). Several of his written theoretical work came as correspondence letters, in particular to a person of the name of Dositheus, who was a student of Conon (Crystalinks, 2008). In some of his letters, Archimedes referred to the calculation of the area enclosed in a parabola and determined by a line secant to the parabolic curve.
In the letters to Dositheus, Archimedes was able to prove that such area would equal to four thirds the area of an isosceles triangle having for base and height the magnitude of the intersecting line in the parabola. He arrived at the result using an infinite summation of the rational number one fourth. This particular mathematical demonstration would later prove invaluable in calculating the areas and volumes of various objects in using integral calculus, a modern form of Archimedes infinite expansion.
One of his most famous scientific discoveries relates to the buoyancy effect of a liquid on a given object: often referred to as Archimedes’ principle. The principle explains that any body immersed in a fluid experiences a force of buoyancy which is equal to the magnitude of the equivalent gravitational force of the liquid displaced during immersion. In other words, Archimedes arrived at the reality that any object plunged in a liquid plentiful enough to maintain such object in equilibrium, would experiment a force in reality equal to the body of water displaced to maintain such equilibrium.
There is a famous anecdote on how Archimedes came up with the physical theorem. Legend has it that it was during a bath that he came up with the concept for the buoyancy theorem. According to history, he came up with the answer to the buoyancy theorem in wanting to help his uncle, King Hiero II, to solve the Golden Crown Mystery. In fact, the story relates that the King, Hiero II, sent a certain amount of gold to his goldsmith to be made into a crown. When the crown returned from the goldsmith, the King apparently noticed that it was lighter than the presumed amount of gold that was given to the goldsmith.
King Hiero II presented the dilemma to his nephew Archimedes of Syracuse, who supposedly came up with an answer to the problem that very night. The legend states that Archimedes came up with the buoyancy theorem by filling his bathtub to the top. When he entered the bath, a certain amount of water poured out of the bath. He later on realized that the mass of the amount of water dispersed from the bathtub was equivalent to the mass of his own body. From arriving at this discovery, the story claims that Archimedes ran the streets of Syracuse naked and screaming “Eureka”, which means “I have found it”.
The next day he reiterated the experiment with the Golden Crown and the same amount of gold that was initially given to the goldsmith, when he was able to confirm King Hiero’s assumption that not all the gold given to the goldsmith was used in making the Golden Crown (Andrews University, 1998). This amazingly simple proof carries one of the most important theoretical truths of physics. The principle of buoyancy is better known today as the Law of Hydrostatics, and is directly attributed to Archimedes of Syracuse.
The above anecdote is a classic example of Archimedes’ability to confront complex theoretical problems by transcribing them into practical life. A considerable number of his experiments and scientific theorems were similarly found through empirical and methodical practical proceedings. Archimedes Inventions As a keen mathematician, Archimedes was particularly talented in determining physical solutions to various problems encountered in his life. Often, the mechanical tools that he devised were a direct projection of a theorem he wanted to prove or vice versa. One of his most famous inventions was the Archimedes screw.
Sometimes referred to as Archimedes water pump, the device was created by the Greek mathematician during his study in Alexandria. Archimedes screw is a machine made to pump water from a lower level to a higher level. In short, an ingenious method for carrying water over distances thereby apparently defeating the law of gravity. The screw is made of a cylindrical pipe angled at fourty five degrees and containing a helix. When the bottom end of the device is plunged into water and set to rotate, the helix’s rotation carries water from the bottom end of the cylinder to the top end (Crystalinks, 2008).
Archimedes according to historians, devised another form of the screw in a comparable yet dissimilar shape. In our day, the system is being used primarily in waste-water treatment plants to pump sewage waters. There is little account however on the applications for which the Archimedes screw may have served during Archimedes life, other than its use for irrigation of the Hanging Gardens of Babylon, and for removing water in the hull of ships. In fact, most counts of using the technique point to its modern day utilization.
Other inventions brought by Archimedes received a considerable amount of attention, and found direct applications during his lifetime. From his close relationship with the King Hiero II, Archimedes was requested to build machines to keep the Roman assailant at bay. Archimedes successfully created several weapons of war that held the Roman invader several years. Archimedes is in fact known for inventing the catapult for that purpose. The catapult is a device based on the principle of the lever, which is capable of carrying an object several times its weight.
When the catapult is fired, the object “flies” in describing a parabolic curve, prior to hitting its target. The catapult was often used during warfare as a defensive method to protect a territory from invaders. He would later on be used as on offensive weapon for attacking protected areas or castles. The catapult can be assimilated as the early form of a canon, which solely relied on mechanical means to operate. The device served Syracuse of Sicily well during the Punic wars of Rome vs. Carthage. Archimedes, at the King’s request, created several weapons to defend the city (Biography Shelf, 2008).
Among such weapons were the catapult, the crossbow, and the claw; which could be used to cover several ranges. These various methods of defense allowed Archimedes to keep the Roman assailants at shore for two long years, according to historical reports. Archimedes also came up with the Archimedes ray, a device which was created to set invading ships on fire at a large distance. The device is made up of several mirrors forming a parabolic shape where the rays are reflected to subsequently interfere at a point which can be considered the focus of the parabolic shape.
By aligning the mirrors adequately, it was then feasible to set ships on fire by focusing light reflected from the mirrors directly onto the ships. However, not all of Archimedes inventions were meant for warfare. The Greek mathematician and inventor came up with several devices to assist sailors to carry large objects from the water. Most of those devices operated based on the principle of the lever that was also used in the conception of the catapult. Off all of his work both in theory and in practice, only his writing remained to this day.
In fact several of his correspondence letters were compiled into a repository of treatise commonly called the Archimedean Palimpsest (Cryslalinks 2008). According to ancient history, a palimpsest is a literal compilation of writings that were transcribed onto parchments and contained several layers of text on a given page. It seems evident to modern day historians and archeologists that the multiple writings on a single page indicated that parchment were expensive and hard to come by, and thus demanded that the author writes several times on the same page in order to conserve the precious parchment.
The Archimedean palimpsest was made of the following treatises: 1- On the Equilibrium of Planes The treatise was focused on the principle of the lever and its various applications. The document describes how the principle of the lever can be applied to the calculation of the center of gravity of various bodies including parabola, hemispheres, and triangles. 2- On spirals The treatise “On Spirals” describes the mathematical function of point moving in a curvilinear direction in a three dimensional setting. The work is better known under the appellation of the Archimedean Spiral. 3- On the Sphere and the Cylinder
The treatise describes the mathematical derivation on the relationship between a given sphere and a cylinder having for height the diameter of the sphere. Archimedes was able to mathematically prove that in that very context, the volume of the sphere equaled two thirds to that of the cylinder. 4- On Conoids and Spheroids In this treatise, Archimedes demonstrates how to calculate the areas and volumes of conical sections, spherical sections, and parabolic sections. 5- On Floating Bodies Probably one of the most famous works of Archimedes, the “On Floating Bodies” treatise describes the theorem of equilibrium of fluidic materials.
In this document, Archimedes proved that a body of water would take a spherical form around a given center of gravity. In the second volume of the treatise, he describes the equilibrium states of parabolic sections partially immersed in a body of water. 6- The Quadrature of the Parabola This treatise corresponds to the mathematical derivation that the area made by the intersection of a line with a parabola equals four thirds of a triangle having a base and height both equal to the segment of the line intersecting the parabola. 7- Stomachion
The treatise describes a problem in which Archimedes attempted to estimate the number of strips of paper of various shapes and quantities that would be necessary to reconstruct a square. The method is said to be an early version of the field of combinatorics. The Archimedean palimpsest was conserved over two millennia, and his now conserved at the Walters Art Museum in Baltimore, MD. It is the only remaining work directly written by Archimedes himself to be existing today. Unlike the mathematician’s writings which were compiled in the Archimedean palimpsest, there are apparently no remains or originals of his practical inventions.
The Archimedean palimpsest is direct testimony of Archimedes’ inclination for mathematical prowess, in particular when it concerned geometry. Most of his work that was conserved in the redaction of the palimpsest describes some very important rules and theorems for Mathematics. Among those, the approximation of pi and the use of the method of exhaustion to estimate areas, volumes, and surface areas of solids of varying forms were key elements leading to modern day mathematical practice.
Some of his inventions are in use today, yet his name is most often associated in our era with the buoyancy theorem also known as Archimede’s Principle. References Andrews University, 1998, Biographies of Mathematicians – Archimedes, website available at http://www. andrews. edu/~calkins/math/biograph/bioarch. htm Biography Shelf, 2008, Short Biography of Archimedes, website available at http://www. biographyshelf. com/archimedes_biography. html Crystalinks, 2008, Archimedes, Biographical Sketch of the Mathematician, available at http://www.
crystalinks. com/archimedes. html Trebuchetstore, 2008, Archimedes: A biography, website available at http://www. redstoneprojects. com/trebuchetstore/archimedes_1. html University of St Andrews, 1999, Archimedes of Syracuse, School of Mathematics and Statistics, available at http://www-groups. dcs. st-and. ac. uk/~history/Biographies/Archimedes. html University of Utah, 1999, Archimedes and the Computation of Pi, website available at http://www. math. utah. edu/~alfeld/Archimedes/Archimedes. html