Archimedes

Archimedes was an ancient Greek mathematician, philosopher, and inventor who produced important works on geometry, arithmetic, and mechanics. Born in Syracuse, Sicily (now Italy) in the early third century BC, Archimedes was the greatest mathematician of his age, whose contributions to geometry revolutionized the field and anticipated integral calculus. Key achievements include discovering the relationship between the surface and volume of a sphere and its circumscribing cylinder, the formulation of a hydrostatic principle (known as Archimedes’ principle), and a device for raising water known as the Archimedes screw that is still in use today.

Archimedes was born in Syracuse, Sicily in 287 BC, the son of Phidias an astronomer and mathematician. Little is known about his early life or family. Heracleides, a friend of Archimedes, wrote a biography of him; however, this work is lost. It is reported by authors that he visited Egypt and it is there that he invented the Archimedes' screw. It is highly likely Archimedes studied with the successors of Euclid in Alexandria when he was a young man. He was completely familiar with the mathematics developed there and personally knew mathematicians working there, sending his results to Alexandria with personal messages. He regarded Conon of Somos in Alexandria as a close friend and admired his abilities as a mathematician.

He spent most of the rest of his life devoting his time to research and experimentation across many fields. Many apocryphal legends tell of how Archimedes endeared himself to King Hiero II of Syracuse by discovering solutions to problems that vexed the king.

There are nine known treatises of Archimedes. The last was discovered in a 10th-century manuscript in 1906.

Sphere inscribed inside a cylinder.

• On the Sphere and Cylinder—defines the surface area of a sphere and shows that the volume of a sphere is two-thirds that of a cylinder in which it is inscribed.
• On the Measurement of the Circle—found π (ratio between circle's circumference and diameter is between 3 & 10/70 - 3 & 10/71).
• On Conoids and Spheroids—finds the volume of solids formed by the revolutions of a conic section about its axis (circle, ellipse, parabola, hyperbola).

Archimedean Spiral.

• On Spirals—develops properties related to the tangents of the Archimedean Spiral.
• Centers of Gravity—found the center of gravity for various plane figures and conics, also establishes the law of the lever.
• Quadrature of the Parabola—through both mechanical and rigorous geometry the area of any segment of a parabola.

• The Sand Reckoner—an attempt to remedy the inadequacies of the Greek numerical notation system.
• On Floating Bodies—finds the positions that various solids will assume when floating in a fluid establishing the Archimedes' principle.
• On the Method of Mechanical Theorems—describes the process of discovery in mathematics.

Demonstration of the Archimedes' principle.

This was said to have been discovered when he was tasked to determine the purity of King Hiero II's crown. The Archimedes' principle states when a body is totally or partially immersed in a fluid the upward (buoyant) force is equal to the weight of the fluid it displaces. Therefore, the net upward force is the difference between the buoyant force and its weight. When positive the object rises and when negative the object sinks. Archimedes' principle is a law fundamental to fluid mechanics.

Archimedes proved the law of the lever using geometric reasoning. He showed that if the distance from the hinge to where the input force is applied is greater than the distance to the output force, the lever amplifies the input force and that the reverse is also true.

Archimedes made extensive use of the method of exhaustion to make discoveries. The method of exhaustion is a technique of finding the area of a shape by inscribing it inside a sequence of polygons whose areas converge to the area of the containing shape. He also performed the first known use of indivisibles a method similar to Cavalieri's principle to determine whether the volume of two solids is equal. Both of these techniques are regarded as forerunners to modern calculus.

Archimedes applied the method of indivisibles to a sphere inscribed inside a cylinder to derive a formula for the surface area (4πr²) and volume of a sphere (4/3πr³), proving the result through the method of exhaustion. Given the surface area (6πr²) and volume (2πr³) of a cylinder, Archimedes showed that both the volume and surface area of a sphere inscribed inside a cylinder is two-thirds that of the cylinder.

Allegedly invented for removing water from the hold of a large ship, Archimedes invented a device known as the Archimedes screw—a form of positive-displacement pump. Made up of a hollow cylinder and a spiral part, the device traps fluid from a source, forcing it to a discharge location. One end is placed in a low-lying fluid source, with the other end tilted up to a higher discharge area. Water is transported up the screw by rotating the spiral. It was effective and went on to be used to transport water from low-lying areas up into irrigation ditches. The design is still in use today. For example, to lift wastewater in water treatment plants.

Illustration of the Archimedes screw.

In 214 BC, pro-Carthaginian factions within Syracuse chose to side with Carthage against Rome. This caused legions of the Roman army led by General Marcellus to lay siege to Syracuse. Before his death in 216 BC, King Heiro II of Syracuse had set Archimedes to work strengthening the walls of Syracuse and modifying the Euryelos fortress. Archimedes also constructed war machines to defend Syracuse. For two years Archimedes's fortifications and war machines helped repel the Romans, until 212 BC when forces took the city. In his writings about Marcellus, Plutarch described how Archimedes' engines of war were used in the Roman siege of 212 BC:

when Archimedes began to ply his engines, he at once shot against the land forces all sorts of missile weapons, and immense masses of stone that came down with incredible noise and violence; against which no man could stand; for they knocked down those upon whom they fell in heaps, breaking all their ranks and files. In the meantime huge poles thrust out from the walls over the ships and sunk some by great weights which they let down from on high upon them; others they lifted up into the air by an iron hand or beak like a crane's beak and, when they had drawn them up by the prow, and set them on end upon the poop, they plunged them to the bottom of the sea; or else the ships, drawn by engines within, and whirled about, were dashed against steep rocks that stood jutting out under the walls, with great destruction of the soldiers that were aboard them. A ship was frequently lifted up to a great height in the air (a dreadful thing to behold), and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall.

General Marcellus had great respect for Archimedes and dispatched soldiers to retrieve him. A soldier demanded Archimedes accompany him to the quarters of Marcellus. When Archimedes refused, the soldier struck Archimedes dead. Marcellus was greatly distressed and ordered he be buried with honors. As he had wished, Archimedes's tombstone was engraved with an image of a sphere within a cylinder.