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Archimedes


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Archimedes was an aristocrat, the son of an astronomer, but little is known about his early life except that he studied for some time in Alexandria, Egypt. Several of his books have been preserved by the Greeks and the Arabs in the Middle Ages, and fortunately, the Roman historian Plutarch describes some episodes of his life. In many areas of mathematics, and in the hydrostatic and static work and achievements that are not exceeded for more than 1500 years!

He approached the area of ​​the media (and the value of ¼) summing the areas of inscribed and circumscribed rectangles, and the general “method of exhaustion”, having smaller and smaller rectangular areas and add them to find the same areas and volumes in several other ways. The expected results of the calculation of Newton and Leibniz for almost 2000 years!

He found the area and tangent to the curve traced by a point moving with uniform velocity along a straight line rotating with uniform angular velocity around a fixed point. This curve, described by r = Atheta in polar coordinates, which is now called the “spiral of Archimedes.” In the calculation, is an easy problem, without calculation, it is very difficult.

The king of Syracuse Archimedes when asked to find a way to determine if some of his money was pure gold without destroying the crown in the process. The crown weighed the exact amount, but it was not a guarantee that it was pure gold. The story is told that Archimedes himself down in a bath, he noticed that some water was expelled from her body and flows over the edge of the tub. It was precisely the insight he needed to realize that the crown should not only evaluate the right amount, but should displace the same volume as the same weight of pure gold. He was so excited at the idea that he would run naked through the streets shouting “Eureka” (“I found it”).

“Give me a fulcrum and I will move the world” was his boast, when he discovered the laws of levers and pulleys. Since it was impossible to question this statement directly, he was asked to move the ship, which had asked for a large number of workers are put in place. Archimedes was so easily made using the pulley system.

During the war between Rome and Carthage, the Roman fleet decided to attack Syracuse, but Archimedes was the design work of a few surprises. There were catapults with adjustable intervals could throw objects that weighed over 500 pounds. The ship survived the catapults were greeted with batteries that have reached the city walls and heavy stone fell on ships. Great grabs attached to levers lifted the ship out of the water and I left. During the second attack failed, it is said that Archimedes had used soldiers Syracuse shields specially shaped and brilliant to focus sunlight on the sails to set it on fire. It was more frightened that the sailors could stand, and the fleet withdrew. Unfortunately, the city started celebrating a little early, and Marcellus captured Syracuse by attacking from the landward side of the celebration.

“Archimedes, then, as fate would have it, willing to work some problems with a diagram, and having fixed his mind and his eyes as when the object of his speculation, he never noticed the introduction of Roman or that. the city was taken in this transport of study and contemplation, a soldier, unexpectedly came upon him, ordered him to follow to Marcellus, he refused to do earlier, he had worked on his problem in a demonstration, the soldier enraged, drew his sword and he crossed. “(Plutarch)

Archimedes requested that his tomb is decorated with a sphere contained in the cylinder as small as possible and with the way the relationship between volume of the cylinder of the ball. Archimedes is considered in relation to this discovery the greatest of all his successes.

The results of Archimedes are more remarkable when you consider the times in which he lived. Archimedes made fundamental discoveries in many areas, and also advanced that their results do not improve for many centuries. Archimedes certainly ranks as one of the greatest minds in history.

Archimedes father was Phidias, an astronomer. We know nothing of Phidias other than this, and we do not know since Archimedes gives us this information in one of his works, The Sandreckoner. A friend called Heraclides Archimedes wrote a biography on him, but unfortunately, this work is lost. How our knowledge of Archimedes would be transformed if this lost work has never been found, or extracts from the same found in the writing of others.

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Archimedes was a native of Syracuse, Sicily. It has been reported by some authors who visited Egypt and there invented a device now known as the Archimedes screw. This is a pump, still used in many parts of the world. It is likely that when he was a young man, Archimedes studied with successors of Euclid in Alexandria. Certainly, he was completely familiar with the mathematics developed there, but what makes this conjecture much more certain, he knew personally the mathematicians working there and sent his results to Alexandria with personal messages. He regarded Conon of Samos, one of the mathematicians at Alexandria, both very high for his abilities as a mathematician and considered him a close friend.

In the preface to the spiral of Archimedes relates an amusing story about his friends in Alexandria. He tells us he was in the habit of sending them information about his last theorem, but without giving evidence. Apparently some of the mathematicians there had claimed that the results of their own to Archimedes said that the last time when he sent them theorems he included two which were false.

Archimedes was able to apply the method of exhaustion, which is the first form of integration, for a number of important results and a few of the descriptions of his works below. Archimedes also gave an accurate approximation of π and showed that it could approach the square roots accurately. He invented a system for expressing large numbers. In mechanics Archimedes discovered fundamental theorems concerning the center of gravity of plane figures and solids. His most famous theorem gives the weight of a body immersed in a liquid, called Archimedes’ principle.

The works of Archimedes, who survived are as follows. On the balance plane (two books), Quadrature of the parabola, the sphere and cylinder (two books), on spirals, On conoids and spheroids, floating bodies (two books), as a circle, and the Sandreckoner. In the summer of 1906, discovered JL Heiberg, professor of classical philology at the University of Copenhagen, a 10th century manuscript which includes the working method of Archimedes. This gives a remarkable insight into how Archimedes discovered many of his results and we will discuss this below once we have given a detailed description of what is in the books survivor.

The order in which Archimedes wrote his works is not known with certainty. We used the chronological order suggested by Heath in the list above this work, except for the method that Heath has placed immediately before On the sphere and cylinder. The paper examines the arguments in favor of a chronological order different from the works of Archimedes.

The Treaty on the plan balances the basic principles of mechanics, using the methods of geometry. Archimedes discovered fundamental theorems concerning the center of gravity of plane figures and presented in this work. In particular, in the book 1, the center of gravity of a parallelogram, a triangle and a trapezoid. Book two is devoted entirely to find the center of gravity of a parabolic segment. In the quadrature of the parabola Archimedes found the area of ​​a parabolic segment cut agreement.

It ‘the first book of the sphere and cylinder Archimedes shows that the surface of a sphere is four times higher than the great circle, is the area of ​​any segment of the dance, shows that the volume of the ball for two-thirds of the limited amount of a cylinder , the surface of a sphere, and two thirds of the surface of the cylinder is limited to the grounds. A discussion of how Archimedes may lead to some of these results using infinitesimals is given. In the second book of the work of Archimedes main result is to show how to cut a ball on the floor so that the relationship between two quantities of the segment is prescribed ratio.

In the spiral of Archimedes defines a spiral, gives fundamental properties connecting the length of the radius vector with the angles through which it has become. He is working on the tangents to the spiral and the determination of the surface portion of the spiral. In the work of conoid and spheroids Archimedes examines paraboloids of revolution, hyperboloids of revolution, and spheroids obtained by rotating an ellipse around its major axis or lower its axis. The main objective of this study was to investigate the volume of the segments of these figures in three dimensions. Some argue that there is a lack of rigor in some of the results of this work, but the discussion interesting attributes it to a modern reconstruction.

The floating body is a work in which Archimedes lays down the basic principles of hydrostatics. His most famous theorem which gives the weight of a body immersed in a liquid, called Archimedes’ principle in this work. He also studied the stability of various floating bodies of different shapes and densities. Able Circle Archimedes shows that the exact value of π is between 310/71 and those of 31 / 7 What is achieved by defining and registering a circle with regular polygons of 96 sides.

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