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Archimedes

2007 Schools Wikipedia Selection. Related subjects: Mathematicians

                        Classical Greek philosophy
   Ancient philosophy
         Name:       Archimedes of Syracuse (Greek: Ἀρχιμήδης)
        Birth:       c. 287 BC ( Syracuse, Sicily, ancient Magna Graecia)
        Death:       c. 212 BC (Syracuse)
   School/tradition:

   Archimedes ( Greek: Ἀρχιμήδης; c. 287 BC – 212 BC) was a Hellenistic
   mathematician, physicist, engineer, astronomer, and philosopher, born
   on the seaport colony of Syracuse, Magna Graecia, what is now Sicily.
   Many consider him one of the greatest, if not the greatest,
   mathematicians in antiquity. Carl Friedrich Gauss, himself frequently
   called the most influential mathematician of all time, modestly claimed
   that Archimedes was one of the three epoch-making mathematicians (the
   others being Isaac Newton and Ferdinand Eisenstein). Apart from his
   fundamental theoretical contributions to maths, Archimedes also shaped
   the fields of physics and practical engineering, and has been called
   "the greatest scientist ever".

   He was a relative of the Hiero monarchy, which was the ruling family of
   Syracuse, a seaport kingdom. King Hiero II, who was said to be
   Archimedes's uncle, commissioned him to design and fabricate a new
   class of ships for his navy, which were crucial for the preservation of
   the ruling class in Syracuse. Hiero had promised large caches of grain
   to the Romans in the north in return for peace. Faced with war when
   unable to present the promised amount, Hiero commissioned Archimedes to
   develop a large luxury/supply/war barge in order to serve the changing
   requirements of his navy. It is rumored that the Archimedes Screw was
   actually an invention of happenstance, as he needed a tool to remove
   bilge water. The ship, named Syracusia, after its nation, was huge, and
   its construction caused stupor in the Greek world.

   He is credited with many inventions and discoveries, some of which we
   still use today, like his Archimedes screw. He was famous for his
   compound pulley, a system of pulleys used to lift heavy loads such as
   ships. He made several war machines for his patron and friend King
   Hiero II. He did a lot of work in geometry, which included finding the
   surface areas and volumes of solids accurately. The work that has made
   Archimedes famous is his theory of floating bodies. He laid down the
   laws of flotation and developed the famous Archimedes' principle.

Discoveries and inventions

   The Archimedes' screw lifts water to higher levels for irrigation.
   Enlarge
   The Archimedes' screw lifts water to higher levels for irrigation.

   Archimedes became a very popular figure as a result of his involvement
   in the defense of Syracuse against the Roman siege in the Second Punic
   War. He is reputed to have held the Romans at bay with war machines of
   his own design, to have been able to move a full-size ship complete
   with crew and cargo by pulling a single rope , and to have discovered
   the principles of density and buoyancy, also known as Archimedes'
   principle, while taking a bath. The story goes that he then took to the
   streets without any clothing, being so elated with his discovery that
   he forgot to dress, crying " Eureka!" ("I have found it!"). He has also
   been credited with the possible invention of the odometer during the
   First Punic War. One of his inventions used for military defense of
   Syracuse against the invading Romans was the claw of Archimedes.
   A diagram showing how Archimedes may have enabled the defenders of
   Syracuse to aim their mirrors at approaching ships
   Enlarge
   A diagram showing how Archimedes may have enabled the defenders of
   Syracuse to aim their mirrors at approaching ships

   It is said that he prevented one Roman attack on Syracuse by using a
   large array of mirrors (speculated to have been highly polished
   shields) to reflect sunlight onto the attacking ships causing them to
   catch fire. This popular legend, dubbed the "Archimedes's death ray",
   has been tested many times since the Renaissance and often discredited
   as it seemed the ships would have had to have been virtually motionless
   and very close to shore for them to ignite, an unlikely scenario during
   a battle. A group at the MIT have performed their own tests and
   concluded that the mirror weapon was a possibility , although later
   tests of their system showed it to be ineffective in conditions that
   more closely matched the described siege . The television show
   Mythbusters also took on the challenge of recreating the weapon and
   concluded that while it was possible to light a ship on fire, it would
   have to be stationary at a specified distance during the hottest part
   of a very bright, hot day, and would require several hundred troops
   carefully aiming mirrors while under attack. These unlikely conditions
   combined with the availability of other simpler methods, such as
   ballistae with flaming bolts, led the team to believe that the heat ray
   was far too impractical to be used, and probably just a myth.

   It can be argued that even if the reflections didn't induce fire, they
   still could have confused, and temporarily blinded the ship crews,
   making it hard for them to aim and steer. Making them hot and sweaty
   before primary battle may have also tired them faster. The
   effectiveness may have simply been exaggerated.

   Archimedes also has been credited with improving accuracy, range and
   power of the catapult.

   Archimedes was killed by a Roman soldier during the sack of Syracuse
   during the Second Punic War, despite orders from the Roman general
   Marcellus that he was not to be harmed. The Greeks said that he was
   killed while drawing an equation in the sand; engrossed in his diagram
   and impatient with being interrupted, he is said to have muttered his
   famous last words before being slain by an enraged Roman soldier: Μη
   μου τους κύκλους τάραττε ("Don't disturb my circles"). The phrase is
   often given in Latin as "Noli turbare circulos meos" but there is no
   direct evidence that Archimedes ever uttered these words. This story
   was sometimes told to contrast the Greek high-mindedness with Roman
   ham-handedness; however, it should be noted that Archimedes designed
   the siege engines that devastated a substantial Roman invasion force,
   so his death may have been out of retribution .

   In creativity and insight, Archimedes exceeded any other European
   mathematician prior to the European Renaissance. In a civilization with
   an awkward numeral system and a language in which "a myriad" (literally
   "ten thousand") meant "infinity", he invented a positional numeral
   system and used it to write numbers up to 10^64. He devised a heuristic
   method based on statistics to do private calculations that we would
   classify today as integral calculus, but then presented rigorous
   geometric proofs for his results. To what extent he actually had a
   correct version of integral calculus is debatable. He proved that the
   ratio of a circle's circumference to its diameter is the same as the
   ratio of the circle's area to the square of the radius. He did not call
   this ratio p but he gave a procedure to approximate it to arbitrary
   accuracy and gave an approximation of it as between 3 + 10/71
   (approximately 3.1408) and 3 + 1/7 (approximately 3.1429). He was the
   first Greek mathematician to introduce mechanical curves (those traced
   by a moving point) as legitimate objects of study. He proved that the
   area enclosed by a parabola and a straight line is 4/3 the area of a
   triangle with equal base and height. (See the illustration below. The
   "base" is any secant line, not necessarily orthogonal to the parabola's
   axis; "the same base" means the same "horizontal" component of the
   length of the base; "horizontal" means orthogonal to the axis. "Height"
   means the length of the segment parallel to the axis from the vertex to
   the base. The vertex must be so placed that the two horizontal
   distances mentioned in the illustration are equal.)
   Image:Parabola-and-inscribed_triangle.png

   In the process, he calculated the earliest known example of a geometric
   progression summed to infinity with the ratio 1/4:

          \sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots
          = {4\over 3} \; .

   If the first term in this series is the area of the triangle in the
   illustration then the second is the sum of the areas of two triangles
   whose bases are the two smaller secant lines in the illustration, and
   so on. Archimedes also gave a quite different proof of nearly the same
   proposition by a method using infinitesimals (see " Archimedes' use of
   infinitesimals").

   He proved that the area and volume of the sphere are in the same ratio
   to the area and volume of a circumscribed straight cylinder, a result
   he was so proud of that he made it his epitaph.

   Archimedes is probably also the first mathematical physicist on record,
   and the best before Galileo and Newton. He invented the field of
   statics, enunciated the law of the lever, the law of equilibrium of
   fluids, and the law of buoyancy. He famously discovered the latter when
   he was asked to determine whether a wreath (often incorrectly reported
   as a crown -- a crown could have been melted down to determine its
   composition, whereas a wreath was sacred) had been made of pure gold,
   or gold adulterated with silver; he realized that the rise in the water
   level when it was immersed would be equal to the volume of the wreath,
   and the decrease in the weight of the wreath would be in proportion; he
   could then compare those with the values of an equal weight of pure
   gold. He was the first to identify the concept of centre of gravity,
   and he found the centers of gravity of various geometric figures,
   assuming uniform density in their interiors, including triangles,
   paraboloids, and hemispheres. Using only ancient Greek geometry, he
   also gave the equilibrium positions of floating sections of paraboloids
   as a function of their height, a feat that would be taxing to a modern
   physicist using calculus.

   Apart from general physics, he was also an astronomer, and Cicero
   writes that the Roman consul Marcellus brought two devices back to Rome
   from the ransacked city of Syracuse. One device mapped the sky on a
   sphere and the other predicted the motions of the sun and the moon and
   the planets (i.e., an orrery). He credits Thales and Eudoxus for
   constructing these devices. For some time this was assumed to be a
   legend of doubtful nature, but the discovery of the Antikythera
   mechanism has changed the view of this issue, and it is indeed probable
   that Archimedes possessed and constructed such devices. Pappus of
   Alexandria writes that Archimedes had written a practical book on the
   construction of such spheres entitled On Sphere-Making.

   Archimedes's works were not widely recognized, even in antiquity. He
   and his contemporaries probably constitute the peak of Greek
   mathematical rigour. During the Middle Ages the mathematicians who
   could understand Archimedes's work were few and far between. Many of
   his works were lost when the library of Alexandria was burnt (twice)
   and survived only in Latin or Arabic translations. As a result, his
   mechanical method was lost until around 1900, after the arithmetization
   of analysis had been carried out successfully. We can only speculate
   about the effect that the "method" would have had on the development of
   calculus had it been known in the 16th to the 17th centuries.

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