阿基米德對數學的貢獻[用英文](10分)

Archimedes

Mathematics
Although he is often regarded as a designer of mechanical devices, Archimedes also made important contributions to the field of mathematics. Plutarch wrote: “He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life.”



圖片參考：http://upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Archimedes_pi.svg/300px-Archimedes_pi.svg.png



圖片參考：http://en.wikipedia.org/skins-1.5/common/images/magnify-clip.png
Archimedes used the method of exhaustion to approximate the value of π
Some of his mathematical proofs involve the use of infinitesimals in a way that is similar to modern integral calculus. By assuming a proposition to be true and showing that this would lead to a contradiction, Archimedes was able to give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the method of exhaustion, and he employed it to approximate the value of π (Pi). He did this by drawing a larger polygon outside a circle, and a smaller polygon inside the circle. When the polygons had 96 sides each, he calculated the lengths of their sides and showed that the value of π lay between 3 + 1/7 (approximately 3.1429) and 3 + 10/71 (approximately 3.1408). This was a remarkable achievement, since the ancient Greek number system was unwieldy and used letters rather than the symbols used today. He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle.

He used the method of exhaustion to show that the value of the square root of 3 lay between 265/153 (approximately 1.732) and 1351/780 (approximately 1.7320512). The modern value is ~1.7320508076, making this a very accurate estimate.

Another noted mathematical work by Archimedes is The Sand Reckoner. In this work he set out to calculate the number of grains of sand that the universe could contain. In doing so, he challenged the notion that the number of grains of sand was too large to be counted. He wrote: "There are some, King Gelon (Gelon II, son of Hieron II), who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited." To solve the problem, Archimedes devised a system of counting based around the myriad. This was the ancient Greek word for infinity, based on the Greek word for uncountable, murious. The word myriad was also used to denote the number 10,000. He proposed a number system using powers of myriad myriads (100 million) and concluded that the number of grains of sand required to fill the universe would be 8×1063 in modern notation.[21]


圖片參考：http://upload.wikimedia.org/wikipedia/commons/3/32/Parabola-and-inscribed_triangle.png



In the field of geometry, Archimedes proved that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height (see illustration on right).

He expressed the solution to the problem as a geometric progression that summed to infinity with the ratio 1/4:



圖片參考：http://upload.wikimedia.org/math/e/9/5/e9536b14f759bdb301399dc41be5c168.png

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, and so on. This proof is a variation of the infinite series 1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1/3.

It has been suggested that Heron's formula for calculating the area of a triangle from the length of its sides was known to Archimedes. However, the first reliable reference to this formula occurs in the work of Heron of Alexandria in the 1st century AD. [22]