Mathematical Discoveries - The Quadrature of A Parabolic Segment

The area of a parabolic segment:

 

 

     As proven by Archimedes, the area of the parabolic segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure.
 

     In The Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is ⁴⁄₃ times the area of a corresponding inscribed triangle as shown in the figure at right. He expressed the solution to the problem as an infinite geometric series with the common ratio ¹⁄₄:

 

     If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines, and so on. This proof uses a variation of the series 1/4 + 1/16 + 1/64 + 1/256 +……. which sums to ¹⁄₃.

Mathematical Discoveries - Claw of Archimedes

The discovery of the Claw of Archimedes:

      The Claw (Greek: Ἁρπάγη, harpágē, "snatcher") of Archimedes (also known as the "iron hand") was an ancient weapon devised by Archimedes to defend the seaward portion of Syracuse's city wall against amphibious assault. The accounts of ancient historians seem to describe it as a sort of crane equipped with a grappling hook that was able to lift attacking ships partly out of the water, then either cause the ship to capsize or suddenly drop it.

     The Claw of Archimedes is a weapon that he is said to have designed in order to defend the city of Syracuse. Also known as "the ship shaker," the claw consisted of a crane-like arm from which a large metal grappling hook was suspended. When the claw was dropped onto an attacking ship the arm would swing upwards, lifting the ship out of the water and possibly sinking it.

     These machines featured prominently during the Second Punic War in 214 BC, when the Roman Republic attacked Syracuse with a fleet of at least 120 Quinqueremes under Marcus Claudius Marcellus. When the Roman fleet approached the city walls under cover of darkness, the machines were deployed, sinking many ships and throwing the attack into confusion. Historians such as Polybius and Livy attributed heavy Roman losses to these machines, together with catapults also devised by Archimedes.

     The plausibility of this invention was tested in 1999 in the BBC series Secrets of the Ancients and again in early 2005 in the Discovery Channel series Superweapons of the Ancient World. The producers of Superweapons brought together a group of engineers tasked with conceiving and implementing a design that was realistic, given what we know about Archimedes. Within seven days they were able to test their creation, and they did succeed in tipping over a model of a Roman ship so that it would sink. While this does not prove the existence of the Claw, it suggests that it would have been possible.

Mathematical Discoveries - Archimedes Screw

The discovery of the Archimedes Screw:

    The Archimedes screw can raise water efficiently.

     A large part of Archimedes' work in engineering arose from fulfilling the needs of his home city of Syracuse. The Greek writer Athenaeus of Naucratis described how King Hiero II commissioned Archimedes to design a huge ship, the Syracusia, which could be used for luxury travel, carrying supplies, and as a naval warship. The Syracusia is said to have been the largest ship built in classical antiquity. According to Athenaeus, it was capable of carrying 600 people and included garden decorations, a gymnasium and a temple dedicated to the goddess Aphrodite among its facilities.

    

     Since a ship of this size would leak a considerable amount of water through the hull, the Archimedes screw was purportedly developed in order to remove the bilge water. Archimedes' machine was a device with a revolving screw-shaped blade inside a cylinder. It was turned by hand, and could also be used to transfer water from a low-lying body of water into irrigation canals. The Archimedes screw is still in use today for pumping liquids and granulated solids such as coal and grain. The Archimedes screw described in Roman times by Vitruvius may have been an improvement on a screw pump that was used to irrigate the Hanging Gardens of Babylon. The world's first seagoing steamship with a screw propeller was the SS Archimedes, which was launched in 1839 and named in honour of Archimedes and his work on the screw.

     Archimedes' screw consists of a screw (a helical surface surrounding a central cylindrical shaft) inside a hollow pipe. The screw is turned usually by a windmill or by manual labour. As the shaft turns, the bottom end scoops up a volume of water. This water will slide up in the spiral tube, until it finally pours out from the top of the tube and feeds the irrigation systems. The screw was used mostly for draining water out of mines or other areas of low lying water.

     The contact surface between the screw and the pipe does not need to be perfectly watertight, as long as the amount of water being scooped at each turn is large compared to the amount of water leaking out of each section of the screw per turn. Water that leaks from one section leaks into the next lower one, so that a sort of mechanical equilibrium is achieved in use.

Mathematical Discoveries - The Value of Pi

The discovery of Pi using the exhaustion method:

While Archimedes is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics.

 

Archimedes used the method of exhaustion to approximate the value of pi.
 

While Archimedes is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. Archimedes used the method of exhaustion to approximate the value of pi.

Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus. Through proof by contradiction, he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answers lay. This technique is known as the method of exhaustion, and he employed it to approximate the value of π.

He did this by drawing a larger regular hexagon outside a circle and a smaller regular hexagon inside the circle, and progressively doubling the number of sides of each regular polygon, calculating the length of a side of each polygon at each step.

As the number of sides increases, it becomes a more accurate approximation of a circle. After four such steps, when the polygons had 96 sides each, he was able to determine that the value of π lay between 3¹⁄₇(approximately 3.1429) and 3¹⁰⁄₇₁ (approximately 3.1408), consistent with its actual value of approximately 3.1416. He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle (πr²).

Archimedes' Life - Archimedes' Death

Archimedes’ death:

 

Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Many people described visiting the tomb of Archimedes, which was surmounted by a sphere inscribed within a cylinder. .

For two years, Archimedes repelled the Romans, enabling the city of Syracuse to survive the lengthy siege. Nevertheless, in 212 BC the forces of Marcellus prevailed and took the city. Marcellus had great respect for Archimedes, and immediately dispatched soldiers to retrieve his foe.

When the Roman soldier demanded Archimedes accompany him to the quarters of Marcellus he simply refused, and continued his ruminations. The enraged soldier flew upon Archimedes, striking the 75 year-old eccentric dead.

Archimedes' Life - Archimedes' Education

Archimedes’ education:

In the third century BC, Syracuse was a hub of commerce, art and science. As a youth in Syracuse, Archimedes developed his natural curiosity and penchant for problem solving.

When he had learned as much as he could from his teachers, Archimedes traveled to Egypt in order to study in Alexandria. Founded by Alexander the Great in 331 BC, Alexandria had, by Archimedes' time, earned a reputation for great learning and scholarship.

After his studies in Alexandria, Archimedes returned to Syracuse and pursued a life of thought and invention. Many apocryphal legends record how Archimedes endeared himself to King Hiero II, discovering solutions to many problems that vexed the king.

Archimedes' Life - Archimedes of Syracuse

Archimedes of Syracuse:

Archimedes was born in the city of Syracuse on the island of Sicily in 287 BC. He was the son of an astronomer and mathematician named Phidias. His family was related to that of Hiero II, King of Syracuse.

Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. He is regarded as one of the leading scientists in classical antiquity.

Among his advances in physics are the foundations of hydrostatics, statics and an explanation of the principle of the lever. He is credited with designing innovative machines, including siege engines and the screw pump.