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Archimedes of Syracuse (circa 287 BC - 212 BC), was a Greek mathematician, astronomer, philosopher, physicist and engineer. He was killed by a Roman soldier during the sack of the city, despite orders from the Roman general, Marcellus, that he was not to be harmed.

1 Discoveries and inventions

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 WarHistory Military history War The Second Punic War was fought between Carthage and Rome from 218 to 204 BC. It was the second of three major wars fought between the Phoenician colony of Carthage, and the Roman Republic, then still confined to the Italian Ps. He is reputed to have held the Romans at bay with warFor other uses of War, see War (disambiguation). War is conflict, between relatively large groups of people, which involves physical force inflicted by the use of weapons. Other terms for war include armed conflict hostilities and police action''. See Lim engineAn engine is something that produces some effect from a given input. The origin of engineering was the working of engines. There is an overlap in English between two meanings of the word "engineer": 'those who operate engines' and 'those who design and cos of his designWhen applied to fine and applied arts, engineering, and other such creative efforts, design is both a noun and a verb. The verb is the process of originating and developing a plan for an artistic or functional object, which may require countless hours of; to have been able to move a full-size shipA ship like a boat, is a vehicle designed for passage or transportation by water. A ship usually has sufficient size to carry its own boats, such as lifeboats, dinghies, or runabouts. A rule of thumb saying (though it doesn't always apply) goes: "a boat c complete with crew and cargo by pulling a single rope; to have discoverA discovery is a novel observation, usually of a natural phenomenon. Contrast with invention, technique, theory Historical discoveries The discovery that the Earth was not flat Age of Exploration The term has also been used for the beginning of contact beed the principles of densityFor other meanings of density, see density (disambiguation Density (symbol: rho Greek: rho) is a measure of mass per unit of volume. The higher an object's density, the higher its mass per volume. The average density of an object equals its total mass div 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 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.

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 statics to do private calculation 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 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. He was the first, and possibly the only, 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. (This proposition needs to be understood consistently with 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.)

In the process, he calculated the oldest known example of a geometric series with the ratio 1/4:

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. Essentially, this paragraph summarizes the proof. Archimedes also gave a quite different proof of nearly the same proposition by a method using infinitesimals; that different proof is found here.

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 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. 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 an astronomer, and Cicero writes that in the year 212 BC when Syracuse, Italy was raided by Roman troops, the Roman consul Marcellus brought a device which mapped the sky on a sphere and another device that predicted the motions of the sun and the moon and the planets (i.e. a planetarium) to Rome. 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' works were not very influential, 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' work were few and far between. Many of his works were lost when the library of Alexandria was destroyed 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 and 17th centuries.