
Grandpa Pencil Archimedes 
Archimedes was a Greek mathematician, astronomer, philosopher, physicist and engineer born in the Greek seaport colony of Syracuse. He is considered by some math historians to be one of history's greatest mathematicians, along with possibly Newton, Gauss, Euler and Gödel. Archimedes became a popular figure as a result of his involvement in the defense of Syracuse against the Roman siege in the First and Second Punic Wars. He is reputed to have held the Romans at bay with war machines of his own design; to have been able to move a fullsize ship complete with crew and cargo by pulling a single rope; to have discovered the principles of density and buoyancy, also known as Archimedes' principle, while taking a bath (thereupon taking to the streets naked calling "eureka"  "I have found it!"); and to have invented the irrigation device known as Archimedes' screw. He has 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. It is said that he prevented one Roman attack on Syracuse by using a large array of mirrors (thought to have been highly polished shields) to reflect sunlight onto the attacking ships causing them to catch fire. Archimedes was killed by a Roman soldier in 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, and told this story to contrast their highmindedness with Roman hamhandedness; 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, he exceeded any other 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 1064. He devised a heuristic method based on statistics to do private calculation that we would classify today as integral calculus, but then presented rigorous geometric proofs for his results. He proved that the ratio of a circle's perimeter 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 π but he gave a procedure to approximate it to arbitrary accuracy and gave an approximation of it as between 3 + 1/7 and 3 + 10/71. Archimedes 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 crown 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 crown, and the decrease in the weight of the crown 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 center of gravity, and he found the centers of gravity of various geometric figures, assuming uniform density in their interiors, including triangles, paraboloids, and hemispheres. Apart from general physics he was an astronomer, and Cicero writes that the Roman consul Marcellus brought two devices back to Rome from the sacked 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). 
