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05.12.2013, 19:17

Classical Greece. From the point of view of 20 century ancestors of mathematics were Greeks of the classical period (6-4 centuries AD). The mathematics existing during earlier period, was a set of the empirical conclusions. On the contrary, in a deductive reasoning the new statement is deduced from the accepted parcels by the way excluding an opportunity of its aversion.

Insisting of Greeks on the deductive proof was extraordinary step. Any other civilization has not reached idea of reception of the conclusions extremely on the basis of the deductive reasoning which is starting with obviously formulated axioms. The reason is a greek society of the classical period. Mathematics and philosophers (quite often it there were same persons) belonged to the supreme layers of a society where any practical activities were considered as unworthy employment. Mathematics preferred abstract reasoning on numbers and spatial attitudes to the solving of practical problems. The mathematics consisted of a arithmetic - theoretical aspect and logistic - computing aspect. The lowest layers were engaged in logistic.

Deductive character of the Greek mathematics was completely generated by Plato’s and Eratosthenes’ time. Other great Greek, with whose name connect development of mathematics, was Pythagoras. He could meet the Babylon and Egyptian mathematics during the long wanderings. Pythagoras has based movement which blossoming falls at the period around 550-300 AD. Pythagoreans have created pure mathematics in the form of the theory of numbers and geometry. They represented integers as configurations from points or a little stones, classifying these numbers according to the form of arising figures (« figured numbers »). The word "accounting" (counting, calculation) originates from the Greek word meaning "a little stone". Numbers 3, 6, 10, etc. Pythagoreans named triangular as the corresponding number of the stones can be arranged as a triangle, numbers 4, 9, 16, etc. - square as the corresponding number of the stones can be arranged as a square, etc.

From simple geometrical configurations there were some properties of integers. For example, Pythagoreans have found out, that the sum of two consecutive triangular numbers is always equal to some square number. They have opened, that if (in modern designations) n$IMAGE7$                                       - square number, n$IMAGE7$  + 2n +1 = (n + 1) $IMAGE7$ . The number equal to the sum of all own dividers, except for most this number, Pythagoreans named accomplished. As examples of the perfect numbers such integers, as 6, 28 and 496 can serve. Two numbers Pythagoreans named friendly, if each of numbers equally to the sum of dividers of another; for example, 220 and 284 - friendly numbers (here again the number is excluded from own dividers).

For Pythagoreans any number represented something the greater, than quantitative value. For example, number 2 according to their view meant distinction and consequently was identified with opinion. The 4 represented validity, as this first equal to product of two identical multipliers. 

Pythagoreans also have opened, that the sum of some pairs of square numbers is again square number. For example, the sum 9 and 16 is equal 25, and the sum 25 and 144 is equal 169. Such three of numbers as 3, 4 and 5 or 5, 12 and 13, are called "Pythagorean” numbers. They have geometrical interpretation: if two numbers from three to equate to lengths of cathetuses of a rectangular triangle the third will be equal to length of its hypotenuse. Such interpretation, apparently, has led Pythagoreans to  comprehension more common fact known nowadays under the name of a pythagoras’ theorem, according to which the square of length of a hypotenuse is equal the sum of squares of lengths of cathetuses.

Considering a rectangular triangle with cathetuses equaled to 1, Pythagoreans have found out, that the length of its hypotenuse is equal to$IMAGE1$ , and it made them confusion because they tried to present number$IMAGE1$ as the division of two integers that was extremely important for their philosophy. Values, not representable as the division of integers, Pythagoreans have named incommensurable; the modern term - « irrational numbers ». About 300 AD Euclid has proved, that the number$IMAGE1$ is incommensurable. Pythagoreans dealt with irrational numbers, representing all sizes in the geometrical images. If 1 and $IMAGE1$to count lengths of some pieces distinction between rational and irrational numbers smoothes out. Product of numbers $IMAGE2$  also $IMAGE3$is the area of a rectangular with the sides in length $IMAGE2$ and $IMAGE3$ .Today sometimes we speak about number 25 as about a square of 5, and about number 27 - as about a cube of 3.

Ancient Greeks solved the equations with unknown values by means of geometrical constructions. Special constructions for performance of addition, subtraction, multiplication and division of pieces, extraction of square roots from lengths of pieces have been developed; nowadays this method is called as geometrical algebra.

Reduction of problems to a geometrical kind had a number of the important consequences. In particular, numbers began to be considered separately from geometry because to work with incommensurable divisions it was possible only with the help of geometrical methods. The geometry became a basis almost all strict mathematics at least to 1600 AD. And even in 18 $IMAGE4$ century when the algebra and the mathematical analysis have already been advanced enough, the strict mathematics was treated as geometry, and the word "geometer" was equivalent to a word "mathematician".

One of the most outstanding Pythagoreans was Plato. Plato has been convinced, that the physical world is conceivable only by means of mathematics. It is considered, that exactly to him belongs a merit of the invention of an analytical method of the proof. (the Analytical method begins with the statement which it is required to prove, and then from it consequences, which are consistently deduced until any known fact will be achieved; the proof turns out with the help of return procedure.) It is considered to be, that Plato’s followers have invented the method of the proof which have received the name "rule of contraries". The appreciable place in a history of mathematics is occupied by Aristotle; he was the    Plato’s learner. Aristotle has put in pawn bases of a science of logic and has stated a number of ideas concerning definitions, axioms, infinity and opportunities of geometrical constructions.

About 300 AD results of many Greek mathematicians have been shown in the one work by Euclid, who had written a mathematical masterpiece "the Beginning”. From few selected axioms Euclid has deduced about 500 theorems which have captured all most important results of the classical period. Euclid’s Composition was begun from definition of such terms, as a straight line, with a corner and a circle. Then he has formulated ten axiomatic trues, such, as « the integer more than any of parts ». And from these ten axioms Euclid managed to deduce all theorems.

Apollonius lived during the Alexandria period, but his basic work  is sustained in spirit of classical traditions. The analysis of conic sections suggested by him - circles, an ellipse, a parabola and a hyperbole - was the culmination of development of the Greek geometry. Apollonius also became the founder of quantitative mathematical astronomy.

The Alexandria period. During this period which began about 300 AD, the character of a Greek mathematics has changed. The Alexandria mathematics has arisen as a result of merge of classical Greek mathematics to mathematics of Babylonia and Egypt. Generally the mathematics of the Alexandria period were more inclined to the solving technical problems, than to philosophy. Great Alexandria mathematics - Eratosthenes, Archimedes and Ptolemaist - have shown force of the Greek genius in theoretical abstraction, but also willingly applied the talent for the solving of practical problems and only quantitative problems.

Eratosthenes has found a simple method of exact calculation of length of a circle of the Earth, he possesses a calendar in which each fourth year has for one day more, than others. The astronomer the Aristarch has written the composition "About the sizes and distances of the Sun and the Moon”, containing one of the first attempts of definition of these sizes and distances; the character of the Aristarch’s job was geometrical.

The greatest mathematician of an antiquity was Archimedes. He possesses formulations of many theorems of the areas and volumes of complex figures and the bodies. Archimedes always aspired to receive exact decisions and found the top and bottom estimations for irrational numbers. For example, working with a correct 96-square, he has irreproachably proved, that exact value of number$IMAGE4$  is between 3 $IMAGE5$ and 3$IMAGE6$ Архимед has proved also some theorems, containing new results of geometrical algebra.

Archimedes also was the greatest mathematical physicist of an antiquity. For the proof of theorems of mechanics he used geometrical reasons. His composition "About floating bodies” has put in pawn bases of a hydrostatics.

Decline of Greece. After a gain of Egypt Romans in 31 AD great Greek Alexandria civilization has come to decline. Cicerones with pride approved, that as against Greeks Romans not dreamers that is why put the mathematical knowledge into practice, taking from them real advantage. However in development of the mathematics the contribution of roman was insignificant.
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