MATHEMATICS THE HANDMADEN OF SCIENCE BY ARCHIMEDES

Archimedes of Syracuse (287- 212 B.C.)Archimedes was one of the three greatest mathematicians of all the time the other two being Newton and Gauss. The son of an astronomer. Archimedes had an appreciation for both mathematics and science and made major contributions to the both. He gave the accurate estimation to pie, developed much of solid geometry, and anticipated the theory of integration well before it was discovered by others many years later.

He was also a brilliant and valued engineer who developed many weapons of war which were used to defend his home city of Syracuse while under siege by the Romans.Rumors have it that Archimedes also developed a parabolic mirror which could direct sunlight to Roman ships to set them on fire! This is most likely true but it sure makes a good story!

His writings are considered works of art. T.L. health, a scholar and translator of Archimedes work stated The treatises are, without exception, monuments of mathematical exposition, the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern elimination of everything not immediately relevant to the purpose, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in mind of the reader.

ARCHIMEDESA Greek mathematician, physicist and inventor, who made profoundly original contributions in mathematics, founded all by himself the fields of statistics, hydrostatics and mathematical devices useful in war and peace. The achievements of Archimedes probably make him the foremost scientist before Newton.

LIFE AND MAJOR DISCOVERIES

He was born about 287 B.C. in Syracuse, Sicily, a Greek colony. His father, Pheidias, was an astronomer.He studied at Alexandria, then the center of the scientific world, as a student of followers of Euclid. During his stay in Alexandria he invented a screw for raising water from the Nile to irrigate fields.

The design of the screw is based on two geometrical forms, the helix and the cylinder. After his return to Syracuse, he devoted himself mainly to science.

Plutarch, the Roman historian, wrote that often his intense concentration on mathematics made him forget his food and neglect his person to that degree that, when he was carried by absolute violence to bathe or have his body anointed, he used to trace geometrical diagrams in the ashes of the fire, and diagrams in the oil on his body,

being in a state of entire preoccupation, and, in the truest sense, divine possession with his love and delight in science.

In his treatise Floating Bodies, Archimedes stated that a body immersed in a fluid is buoyed up a force equal to the weight of the displaced fluid. By means of this principle, known as Archimedes principle, he determined that a crown was not pure gold.Hieron II, king of Syracuse, a friend and perhaps kinsman of Archimedes, ordered a crown of pure gold but suspected that the artist had fraudulently added alloys to the crown.

Archimedes discovered the solution to this problem while lying in a bath, and it caused him so much joy that he hastened home from the bath undressed, crying Eureka! Eureka! 9 I have found it! I have found it!) The ratio of the weight of the crown to the weight of the water it displaced when completely immersed gave him what is now called the specific gravity of the 2 material.

Archimedes only had to take a quality of pure gold and find its specific gravity by the same process. If the two specific gravities did not agree, the crown was not pure gold. By using his law of hydrostatics, Archimedes would have calculated the precise qualities of gold and silver. What happened to the corrupt goldsmith is not recorded.

It was Archimedes, as the fist true mathematical modelers, who stated clearly the law of levers in his book On the equilibrium of planes. To dramatize its application he claimed that any weight might be moved, and, if there were another earth, by going into it he could move this one. King Hieron was told of this, and challenged Archimedes to demonstrate it, which he did; however he chose to use pulleys rather than levers. As Plutarch described it:

He fixed accordingly upon a ship of burden out of the Kings arsenal which could not be drawn out of the dock without great labour and many men; and, loading her with many passengers and a full freight, sitting himself the while afar off, with no great endeavor, but only holding the head of the pulley in his hand and drawing the cords by degrees, he drew the ship in a straight line, as smoothly and evenly as if she been in the sea.

In his treatise The Sand Reckoner, Archimedes described a system for expressing numbers and gave an answer to his own question how many 8 grains of sand does the universe old? Using Aristarchus estimate of the size of the universe, Archimedes calculated that it would contain approximately 10 raise to 63 grains of sand. Archimedes could express numbers as large as 10 raised to80,000,000,000,000,000, which is followed by 80,000 million zeros.

In his Measurement of the circle, Archimedes gave an approximation for the value of pie, namely [220/70] >pie> [220/71]. He arrived at the upper and lower bounds on the value of pie by starting that circumscribed a circle and another hexagon inscribed in the circle. He doubled the number of sides of each hexagon until both were 96- sided figures, and so trapped the circumference of the circle between their two perimeters.

In Spirals, Archimedes defined what is known as the spiral of Archimedes ( r= a teta, where r and teta are polar coordinates). In Conoids and Spheroids he gave the sum for the first n integers. He summed other finite series and also infinite series like } n= 0 4-n. in addition, he could solved certain types of cubic equations.

In about 2174 BC, Archimedess native city of Syracuse was besieged by the Romans under their general Marcus Claudius Marcellus. To the Romans great discomfort, Syracuse was very effectively defended by various military machines designed by Archimedes. These included catapults, missile throwers, and grappling hooks that could capsize the Roman boats and so set them alight! When the Romans finally gained possession of the city in 212 BC, Plutarch tells us:

As fate would have it, he was intent on working out some problem with a diagram, and, having fixed his mind and his eyes alike on his investigation, he never noticed the incursion of the Romans nor the capture of the city, and, when a soldier came up to him suddenly and bade him follow to Marcellus, he refused to do so until he had worked out his problem to a demonstration.

As Marcellus had given strict orders that the famous mathematics was to be kept alive, the soldier was probably executed. Archimedes had requested that his gravestone be engraved with the diagram of one of his favorite results the volume of a sphere inscribed in a cylinder is 2/3 of the volume of the cylinder.

METHODS OF DISCOVERYWork that have come down to us, besides those already mentioned are Sphere and Cylinder, Quadrature of the Parabola, Stomachion ( a geometric puzzle), Book of Lemmas, and Cattle Problem. Lost works include an investigation of polyhedra, a book dealing with the naming of numbers, books on balances and levers and on centers of gravity,

Sphere- making, and Catoptrica. All his works is characterized by rigour, imagination and power. But Archimedes was different from earlier Greek mathematicians like Euclid in that he was as concerned about how to discover new results and how to apply them, as about how to prove them rigorously. So how did Archimedes come to actually discover such formulas as that volume of a sphere, in the first place?

In 1899 a previously unknown text by Archimedes was found in Jerusalem, on a 10th century parchment, covered over by some religious writing. This work studied by the scholar Johan Heiberg, and published amidst great excitement in 1906. the text is called The method, and is Archimedes description of how he came up with many of his results, before proving them rigorously. He imagined his plane figures sliced up into linear sections

And his solids sliced up into planar cross- sections; then he applied his law of levers to these slices, and summed. This method involved considering an infinite number of infinitely thin slices, so would be considered very dubious by the rigorous standards of the Greek mathematics. Archimedes was practical enough to use it without scruple, but he was also pure mathematician enough to see the imperative for geometric proof as well.

His methods of discovery and his rigorous proofs are the forerunners of what we today call integration. No- one in the history of mathematics better illustrates the perfect combination of applied resourcefulness and pure rigour than Archimedes.