### Ideas

– Convergence methods, in which a curvilinear figure is bounded inside and out by similar rectilinear figures, can approximate the area of the curved figure to any degree of accuracy: Using convergence methods, the value of pi is found to be greater than 3 10/71 and less than 3 1/7.

– Objects have a center of gravity and for mathematical purposes can be treated as if they wre composed of parallel lines (plane figures) or parallel planes (solid figures).

– Mechanical methods can suggest the truth of mathematical propositions, which must then be rigorously proven by mathematical methods.

– Numbers of any size can be expressed using a suitable system of notation.

– Magnitudes cannot be infinitely small.

### Biography

Greek mathematician who flourished in Sicily. He is generally considered to be the greatest mathematician of ancient times. Most of the facts about his life come from a biography about the Roman soldier Marcellus written by the Roman biographer Plutarch.

Archimedes performed numerous geometric proofs using the rigid geometric formalism outlined by Euclid, excelling especially at computing areas and volumes using the method of exhaustion. He was especially proud of his discovery for finding the volume of a sphere, showing that it is two thirds the volume of the smallest cylinder that can contain it. At his request, the figure of a sphere and cylinder was engraved on his tombstone. In fact, it is often said that Archimedes would have invented calculus if the Greeks had only possessed a more tractable mathematical notation.

Some of Archimedes’s geometric proofs were actually motivated by mechanical arguments which led him to the correct answer. During the Roman siege of Syracuse, he is said to have single-handedly defended the city by constructing lenses to focus the Sun’s light on Roman ships and huge cranes to turn them upside down. When the Romans finally broke the siege, Archimedes was killed by a Roman soldier.

### Major Works of Archimedes

– On the Sphere and the Cylinder

– On the Measurement of the Circle

– On the Equilibrium of Planes

– On Floating Bodies

– On the Method of Mechanical Theorems

– The Sandreckoner

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