Archimedes

Archimedes was born in Syracuse, Sicily, in 287 B.C. His father was Philas, an aristocratic astronomer. He was educated in Alexandria, Egypt; where he met the Alexandrian scholars Conon of Samos and Eratosthenes of Cyrene. For much of his life, Archimedes kept a correspondence with these two scholars, updating them on his most recent discoveries and proofs. Archimedes spent the major part of his life in Sicily, in and around Syracuse. He did not hold any public office but devoted his entire lifetime to research and experiment. Archimedes is credited with the invention of the compound pulley, the hydraulic screw, the burning mirror, and vast improvements made on the catapult. He calculated the exact value of pi, proved that the volume of a sphere is 2/3 that of the circumscribed cylinder, and defined the law of the lever. Perhaps one of Archimedes’ most famous discoveries is the discovery of the hydrostatic principle now called the Archimedes principle. There are three different accounts of Archimedes’ death in 212 B.C. One of the most popular is that a Roman soldier came upon Archimedes while he was drawing diagrams in sand during the Roman siege of Syracuse during the Second Punic War. As legend has it, Archimedes, so involved in his calculations, had not noticed the commotion around him; he offended he intruder by saying, “Do not disturb my diagrams.” The soldier stabbed Archimedes through the chest, killing what historians call one the Three Greatest Mathematicians.

Archimedes wrote many books containing his propositions and proofs before his death, but none were so famous as The Method Treating of Mechanical Problems, or more simply known as The Method. This work is also widely known as the Archimedes Palimpsest. In this work, Archimedes describes in detail not only the problem and his solutions, but also all of his thoughts and trials along the way. The book was relocated to Jerusalem, where a monk copied it and stored it in the library. Unfortunately, during the Dark Ages, where the pursuit of science and mathematics gave way to blind faith in religion, a monk ran out of paper. He happened across The Method, and decided to purge the parchment of its original text in order to make room for a religious text. He cut the book along its binding, and washed the pages until the text was barely visible. He then wrote the new text over that of Archimedes at a perpendicular angle. The Palimpsest was kept safely hidden until 1906, when it was rediscovered—completely by chance—by Danish philologist Johan Ludvig Heiberg, who happened to be looking through the religious texts in the library. Forbidden to remove the Palimpsest from the library, he photographed each page and attempted to decipher it. In the 1920s, when Heiberg tried to go back to reference the book, he discovered it missing. It wasn’t until the 1990s that the Palimpsest was rediscovered. It sold in private auction for two million dollars. Researchers asked for permission to attempt to decipher the Archimedean text underneath the religious text. Permission was granted. Researchers have undertaken the huge project, and have combined the use of ultraviolet light, infrared rays, and digital processing to decipher the coveted text.

In the Palimpsest, Archimedes, used infinitesimals to solve problems similar to those approached in integral calculus. Among these, he calculated the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and that of the area of a region bounded by a parabola and one of its secant lines. An infinitesimal is a number whose absolute value is greater than zero but smaller than any real number. A number x в‰  0 is an infinitesimal if and only if every sum |x| + ... + |x| of finitely many terms is less than 1, no matter how large the finite number of terms. In that case, 1/x is larger than any positive real number. There is no infinitesimal real number, it exists only in notion. Archimedes did not believe in infinitesmals, ironically enough, and states that all of his works in The Method are not cmplete mathematical proofs.

Archimedes used geometric figures in both mechanics and mathematics. In mathematics, inscribed an equilateral triangle on a circle, and calculated the area of the triangle. He then drew two equal sides for each one side, and calculated the area again. He continued this pattern of drawing new sides and recalculating the area until he had drawn a 96-sided polygon. He then did the same thing, only circumscribing the polygon to the circle. Using this method, he calculated the exact value of pi to be somewhere between 3 1/7 and 3 10/71. Archimedes also discovered the thirteen geometric solids made up of different regular polygons, such as: the truncated tetrahedron (see figure 4.1), made up of four triangles and four octagons; the rhombicuboctahedron (figure 4.2), made