III. Заполните пропуски подходящими по смыслу словами

III. Заполните пропуски подходящими по смыслу словами.

1. Every age has its own problems that it solves or … by new ones.

a) replaces b) seeks

c) requires d) tests

2. It is by the … of problems that the researcher tests the temper of his steel.

a) question b) solution

c) investigation d) generation

3. The deep … of certain problems for the advance of mathematical science are not to be denied.

a) glance b) advance

c) significance d) importance

4. What new … in the wide and rich field of math – cal thought can the new centuries disclose?

a) methods b) facts

c) solutions d) ideas

Text I

Who of us cannot be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals can there be which the leading mathematical minds of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought can the new centuries disclose?

History teaches the continuity of the development ofscience. We know that every age has its own problems, which the following either solves or casts aside as worthless and replaces by new ones. If we could obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass in our minds and consider the problems which the science of today sets and whose solution we expect from the future. To such a review of present-day problems, raised at the meeting of the centuries, I wish to turn your attention. For the close of a great epoch of the 19th century not only invites us to look back into the past but also directs our thought to the unknown future.

The deep significance of certain problems for the advance of mathematical science, in general and the important role which they play in the work of the individual investigator are not to be denied. As long as a branch of science offers an abundance of problems, so long it is alive, a lack of problems foreshadows extinction or the cessation of independent development. Just as every human undertaking seeks after certain objects, so also mathematical research requires its problems. It is by the solution of problems that the researcher tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon.

IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.

1. What does every age have?

a) its own problems b) its own secrets

c) its own purposes d) its own ideas.

2. How long is the branch of science alive?

a) As long as a branch of science offers an abundance of problems, so long it is alive.

b) The branch of science is alive only when all its problems are solved.

c) The branch of science is alive as long as it is interesting for people.

d) The branch of science is always alive.

3. What foreshadows a lack of problems?

a) extinction or the cessation of independent development.

b) Extinction or the cessation of dependent development.

c) The revival of independent development.

d) The disappearance of independent development.

4. What does mathematical research require?

a) the solution of its problems.

b) it requires new investigations.

c) it requires the disappearance of its problems.

d) it requires new methods and ideas.

V. Выберите заголовок для данного текста, в соответствии с его содержанием.

a. Connection between mathematical problems in past and in future

b. Mathematical problem in past

c. Mathematical problems in future

d. Mathematical problems of our days

VI. Укажите правильный перевод подчеркнутой части предложения.

1. Mathematical minds of coming generations will solve the problems.

a) наступать b) наступающих

c) будет наступать

2. The following conclusion either solves mathematical problems or casts aside by new ones.

a) следующий b) следовал

c) будет следовать

3. Mathematics is the science dealing with many subjects.

a) имеющая дело b) иметь дело

c) будет иметь дело

4. Mathematicians working in any given branch discover new concepts.

a) работающие b) работать

c) работая

III. Заполните пропуски подходящими по смыслу словами.

1. The mathematicians of past centuries knew the … of difficult problems.

a) value b) solution

c) importance d) idea

2. “Problem of quickest …” was proposed by J. Bernoulli.

a) solution b) descent

c) integers d) evidence

3. We study Fermat’s … xⁿ + yⁿ = zⁿ (x, y, z integers).

a) theorem b) axiom

c) assertion d) descent

4. They attempt to prove the impossibility of Fermat’s ….

a) problem b) theory

c) theorem d) example

5. The … of variations owes its origin to the problem of Bernoulli and to similar problems.

a) calculus b) axiom

c) theorem d) assertion

Text II

It is difficult often impossible to judge the value of a problem correctly in advance; for the final award depends upon the gain which science obtains from the problem. Nevertheless, we can ask whether there are general criteria which mark and label a good mathematical problem. An old French mathematician said: "A mathematical theory is not to be considered completed until you made it so clear that you can explain it to the first-man whom you meet in the street". This clearness and ease of understanding, here claimed for a mathematical theory, I should still more demand for a mathematical problem that it ought to be perfect; for what is clear and easily 'understandable attracts, while the complicated repels us. Moreover, a mathematical problem should be difficult in order to appeal to us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths and ultimately a reminder of our pleasure in the successful solution.

The mathematicians of past centuries were accustomed to devoting themselves to the solution of difficult particular problems with passionate zeal. They knew the value of difficult problems. I remind you only of the "problem of quickest descent" proposed by J. Bernoulli, of Fermat's assertion; xn + yn =zn, (x, y, z integers) which is unsolvable except in certain self-evident cases. The calculus of variations owes its origin to this problem of Bernoulli and to similar problems. The attempt to prove the impossibility of Fermat's theorem offers a striking example of the inspiring effect which such a very special and apparently unimportant problem may have upon science. I can remind you as well of the Problem of Three Bodies. The fruitful methods and the far-reaching principles which Poincare brought into celestial mechanics and which are today recognized and applied in practical astronomy are due to the fact that he sought to treat anew that difficult problem and to come nearer to its solution.

IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.

1. Whom do the following words belong to? “A mathematical theory is not to be considered completed until you made it so clear that you can explain it to the first man whom you meet in the street”.

a) to an old French mathematician b) to the Greek mathematician

c) to an Alexandrian mathematician d) to the Swiss mathematician

2. What were the mathematicians of past centuries accustomed to?

a) To devoting themselves to the solution of easy problems without passionate zeal.

b) To devoting themselves to the solution of difficult particular problems with passionate zeal.

c) To devoting themselves to finding the new mathematical problems.

d) To devoting themselves to proving mathematical theorems.

3. What mathematical problem was proposed by J. Bernoulli?

a) “The problem of Three Bodies”.

b) “Problem of quickest descent”.

c) “Problem of the straight line”.

d) “The general problem of boundary values”.

4. Whom does the next assertion (xⁿ + yⁿ = zⁿ) belong to?

a) J. Bernoulli b) Fermat

c) Cantor

5. What is it often difficult to do?

a) to judge the value of a problem correctly in advance.

b) to solve the mathematical problem.

c) to find the correct answer to the mathematical question.

d) to prove any mathematical assertion.

6. What did mathematicians of the past know.

a) they knew the solution of problems.

b) they knew the value of difficult problems.

c) they knew the general criteria which mark and label a good mathematical problem.

V. Выберите заголовок для данного текста, в соответствии с его содержанием.

a. A general criteria which mark and label a good mathematical problem

b. The mathematicians of past centuries

c. The clearness and ease of understanding a mathematical theory

d. Fermat’s assertion xn + yn = zn

VI. Укажите правильный перевод подчеркнутой части предложения.

1. The students know the clearness and ease of understanding a mathematical theory.

a) понимания b) понимая

c) поняв

2. The mathematicians of past were accustomed to devoting themselves to the solution of mathematical problems.

a) посвящению b) посвящая

c) посвятив

3. “Problem of quickest descent”, was proposed by Bernoulli.

a) будет предложена b) была предложена

c) была бы предложена

4. The attempt to prove the impossibility of Fermat's theorem offers a striking example.

a) докажет b) доказав

c) доказать d) доказал бы

III. Заполните пропуски подходящими по смыслу словами.

1. The same special problem finds… in the most diverse and unrelated branches of mathematics.

a) duplication b) calculation

c) equations d) application

2. The students know the theory of … and the theory of equation.

a) curved lines surfaces b) polyhedra, curved lines

c) icosahedron, surfaces d) linear differential equations

3. The rules of … with natural numbers were discovered in this fashion.

a) cube b) squaring of the circle

c) calculation d) duplication

4. The oldest problems in the theory of curves and the differential and... calculus belong to mechanics astronomy and physics.

a) integral b) linear

c) squaring d) cube

Text III

But it often happens also that the same special problem finds application in the most diverse and unrelated branches of mathematics. So for example, the problem of the shortest line plays a chief and historically important part in the foundations of Geometry, in the theory of curved lines and surfaces, in mechanics and in the calculus of variations. And F. Klein convincingly pictured, in his work on the icosahedron, the significance which is attached to the problem of the regular polyhedra in elementary Geometry, in group theory, in the theory of equations and in the theory of linear differential equations.

After referring to the general importance of problems in mathematics, let us return to the question from what sources this science derives its problems. Surely, the first and oldest problems in every field of mathematics spring from experience and are suggested by the world of external phenomena. Even the rules of calculation with natural numbers were discovered in this fashion in a lower stage of human civilization, just as the child of today learns the application of these laws by empirical methods. The same is true of the first unsolved problems of antiquity, such as the duplication of the cube, the squaring of the circle. Also the oldest problems in the theory of the solution of numerical equations, in the theory of curves and the differential and integral calculus, in the calculus of variations, the theory of Fourier series and the theory of potential to say nothing of the abundance of problems properly belonging to mechanics, astronomy and physics.

But, in the further development of the special domain of mathematics, the human mind, encouraged by the success of its solutions become convinced of: its independence. It evolves from itself alone, often without appreciable influence from outside by means of logical combination, generalization, specialization, by separating and collecting ideas in elegant ways, by new and fruitful problems and the mind appears then as the real questioner and the source of the new problems. Thus arose the problem of prime numbers and the other unsolved problems of number theory, Galois' theory of equations, the theory of algebraic invariants, the theory of abelian and automorphic functions; indeed, almost all the nicer problems of modern arithmetic and function theory arose in this way.

IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.

1. In what branches of mathematics can the same special problem find application?

a) in the related branches.

b) in the most diverse and unrelated branches of mathematics.

c) in the connected branches of mathematics.

d) in the different branches of mathematics.

2. What problem plays a chief and historically important part in the foundations of Geometry?

a) the problem of the shortest line.

b) the problem of the longest line.

c) the problem of Three Bodies.

d) “the problem of quickest descent”.

3. Where do the first and oldest problems in every field of mathematics spring from?

a) the nature b) experience

c) the human mind d) the external phenomena

4. What happens with the human mind, encouraged by the success of its solution?

a) it becomes unconvinced of its independence.

b) It becomes convinced of its independence.

c) It becomes sure in itself.

d) It becomes clear.

V. Выберите заголовок для данного текста, в соответствии с его содержанием.

a. The sources of mathematical problems

d. The rules of calculation

c. The application of mathematical problems

d. Galois’ theory of equations

VI. Выберите правильный перевод подчеркнутой части предложения.

1. After referring to the general importance of problem, we were able to find its solution.

a) ссылаясь b) при ссылке на

c) сославшись

2. We study the problems properly belonging to mechanics.

a) принадлежащие b) принадлежали бы

c) принадлежали

3. Mathematical thought involves special kind of thinking.

a) мышление b) мыслит

c) будет мыслить

4. The human mind encouraged by the success of its solutions became convinced of its independence.

a) поддерживать b) поддержанный

c) поддерживал

III. Заполните пропуски подходящими по смыслу словами.

1. New questions open up new... of mathematics.

a) solutions b) divisions

c) problems d) experiences

2. It shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of ….

a) hypotheses b) solutions

c) deductions d) requirements

3. The demand for logical … by means of finite number of processes is simply the requirement of rigour in reasoning.

a) reason b) deduction

c) division d) solution

4. The theory of algebraic … experienced a considerable simplification.

a) curves b) deduction

c) methods d) solution

Text IV

In the meantime, while the creative power of pure reason is at work, the outer world again comes into play, forces upon us new questions from actual experience, opens up new divisions of mathematics and whilewe seek to conquer these new fields of knowledge for the realm of pure thought, we often find the answers to old unsolved problems and thus simultaneously advance most successfully the old theories, thanks' to this ever-recurring interplay between pure thought and experience.

It remains to discuss briefly what general requirements may be proposed and laid down for the solution of a mathematical problem. I want first of all say this: that it shall be possible to establish the correctness of the solution by means of.a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This demand for logical deduction by means of a finite number of processes is simply the requirement of rigour in reasoning. Indeed, this requirement of rigour, which became proverbial in mathematics, corresponds to a universal philosophical necessity of our understanding; and on the other hand, only by satisfying this claim do the problems attain their full effect.

Besides it is an error to believe that rigour in the proofis the enemy of simplicity. On the contrary, we find it proved by numerous examples that the rigorous method is at the same time the simpler and worthy in the long run and easier to understand. The very effort for rigour helps us come across a simpler method of proof. It also frequently leads the way to methods which are more capable of development than the old methods of less rigour. Thus, the theory of algebraic curves experienced a considerable simplification and attained greater unity by means of a more rigorous function-theoretical methods and the introduction of transcendental curves.

IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.

1. What play does the outer world come into?

a) It forces upon us new questions from actual experience.

b) It opens up new divisions of mathematics.

c) It gives us answers to the old questions.

d) It does not come into play.

2. What does the outer world open up?

a) new questions from actual experience.

b) new divisions of mathematics.

c) new field of knowledge.

d) old unsolved problems.

3. What does the very effort for rigour help us to do?

a) to solve mathematical problems.

b) to come across a simpler method of proof.

c) to find new problems.

d) to make an exact formulation of the problem.

4. What is the main demand for the solution of a mathematical problem?

a) it must be exactly formulated.

b) it must be solved.

c) it must not be understandable.

d) it must be based on experience.

V. Выберите заголовок для данного текста, в соответствии с его содержанием.

a. The outer world again comes into play

b. The solution of mathematical problem

c. The very effort for rigour helps us come across a simpler method of proof

d. The theory of algebraic curves

VI. Укажите правильный перевод подчеркнутой части предложения.

1. Modern methods of carrying out mathematic operations become sophisticated through modern computers.

a) выполнения b) выполнить

c) выполняя

2. There is a universal philosophical necessity of our understanding.

a) понимания b) понимающий

c) понимавший

3. Only by satisfying this claim the problems attain their full effect.

a) удовлетворяя b) удовлетворив

c) удовлетворяющий

4. The rigorous methods are easier to understand.

a) понять b) понял бы

c) понятный

II. Заполните пропуски подходящими по смыслу словами.

1.Here is the double … a>b>c.

a) equality b) inequality

c) triangle d)rectangle

2. Who can do without the figure of the …, the circle with its centre, or with the cross of three perpendicular axes?

a) triangle b) rectangle

c) axes d) circle

3. On the diagram there is the cross of three perpendicular ….

a) axes b) rectangles

c) triangles d) inequalities

4. Geometric … are graphic formulas and no mathematician can do without them.

a) figures b) segments

c) circles d) axes

5. Students draw segments and … closed in one another.

a) points b) rectangles

c) formulas d) symbols

Text V

To the new concepts correspond, necessarily, new signs. These we choose in such a way that they remind us of the phenomena of the external world. Likewise the geometric figures are signs or symbols of space intrusion and are used as such by all mathematicians. Who does not always use along with the double inequality a>b>c the picture or drawing of three points following one another on a straight-line as the geometrical idea of "betweenness"? Who does not make use of drawings of segments and rectangles closed in one another, when it is required to prove with perfect rigour a difficult theorem on the continuity of functions or the existence of points of condensation? Who can do without the figure of the triangle, the circle with its centre, or with the cross of three perpendicular axes? The arithmetical symbols are written diagrams and the geometric figures are graphic formulas and no mathematician can do without them or avoid them.

Some remarks upon the difficulties which mathematical problems may offer and the means of overcoming and coping with them may be worth discussing. If we do not manage andare not able to solve a mathematical problem the reason often consists in our failure to recoenize the more general standpoint from which the problem under study appears only as a single link in a chain of related problems. After finding this standpoint, the problem becomes more accessible to our investigations and we possess then a method which is applicable also to related problems. This way for finding general methods is certainly the most fruitful and the most certain; for who seeks for methods without having: a definite problem in mind seeks for the most part in vain.

In dealing with mathematical problems, specialization plays, to my mind a still more important part than. generalization. Perhaps in mostcases where we seek in vain the answer to a question, the cause of the failure lies in the fact that problems simpler and easier than the one at issue were either not at all or incompletely solved. All depends, then, on finding out these easier problems, and on solving them by means of devices as perfect as possible and of concepts capable of generalization. This rule is one of the most important levers for overcoming mathematical difficulties and I think, that it is used wherever it is possible, though sometimes unconsciously.

IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.

1. What do the new concepts correspond?

a) new signs b) new symbols

c) new solutions d) new problems

2. What does the formula a>b>c mean?

a) It is a double inequality b) It is a double equality

c) It is an equation d) It is triangle

3. What are the arithmetical symbols?

a) graphic formulas b) written diagrams

c) double inequality d) algebraic figures

4. When does the problem become more accessible to our investigations?

a) after finding proper standpoint.

b) after finding the solution of concepts.

c) after making drawing.

5. What does the solution of the problem depend on?

a) on it’s formulation.

b) on it’s source.

c) on finding out easier problems and on their solving.

d) on it’s difficulties.

V. Выберите заголовок для данного текста, в соответствии с его содержанием.

a. The phenomena of the external world

b. The double inequality a>b>c

c. Mathematical symbols

d. Mathematical symbols and the rule for overcoming mathematical difficulties

VI. Укажите правильный перевод подчеркнутой части предложения.

1. Draw three points following one another on a straight line.

a) следующие b) следование

c) следует

2. This way for finding general methods is certainly the most fruitful one.

a) нахождения b) находить

c) находящий

3. We seek for methods without having a definite problem.

a) не имея b) не иметь

c) не имевший

4. After finding this standpoint, the problem becomes more accessible to our investigations.

a) нашел b) находивший                         c) находя

III. Заполните пропуски подходящими по смыслу словами.

1. The ratio of the … to the side of an isosceles right triangle is irrational.

a) hypotheses b) hypotenuse

c) parallel d) ratio

2. Our task is to ….

a) to square the circle b) seek

c) investigate d) show

3. The proof of the axiom of … is very important.

a) hypotenuses b) triangles

c) rectangles d) parallelepiped

4. The problem of “… motion” is the most important one to science.

a) inexhaustible b) hypotenuse

c) perpetual d) triangle

Text VI

Occasionally it happensthat we seek the solution under insufficienthypotheses or in an incorrect sense and for that reason do not surmount the difficulty. The problem then arises to show the impossibility of the solution under the conditions specified. Such proofs of impossibility were effected by the ancients, for instance, when they showed that the ratio of the hypotenuse to the side of an isosceles right triangle is irrational. In later mathematics, the question of the. impossibility of certain solutions plays a great part and we realize in this way that old and difficult problems, such as the proof of tile axiom of parallels, the squaring the circle, of the solution of equations of the fifth degree by radicals found fully satisfactory and rigorous solutions, although in a different sense than that originally intended. It is probably this important fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares but which as yet no one supported by a proof or refuted) that every definite mathematical problem must necessarily be settled, either in the form of a direct answer to the question posed, or by the proof of the impossibility of its solution and hence the necessary failure of all attempts.

Is this axiom of the solvability of every problem a peculiar characteristic of mathematical thought alone, or is it possibly a general law inherent in the nature of the mind, that all questions which it asks must be answerable? For in other sciences there exist also old problems which were handled in a manner most satisfactory and most useful to science by the proof of their impossibility. For example, the problem of perpetual motion. The efforts to construct a perpetual motion machine were not futile as the investigations led to the discovery of the law of the conservation of energy, which, in turn, explained theimpossibility of the perpetual motion in the sense originally presupposed.

This conviction of the solvability of every mathematical problemisa powerful stimulus and impetus to the researcher. We hear within us the perpetual call. There is the problem. Seek its solution. You can find it for in mathematics there is no futile search even if the problem defies solution. The number of problems in mathematics is inexhaustible and as soon as one problem is solved others come forth in its place. Permit me in the following to dwell on particular and definite problems, drawn from various departments of mathematics, whose discussion and possible solution may result in the advancement and progress of science.

IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.

1. Where do we seek the solution of mathematical problems?

a) In the experience.

b) In the outer world.

c) In the hypotheses.

2. What mathematical figure has hypotenuse?

a) circle.

b) right triangle.

c) rectangle.

3. What is the number of mathematical problems?

a) many.

b) exhaustible.

c) inexhaustible.

d) definite.

4. What is the conviction of the solvability of ever mathematical problem for the researcher?

a) powerful stimulus and impetus.

b) problem.

c) difficulty.

d) stimulus.

5. What happens as soon as one problem is solved?

a) others come forth in its place.

b) all other problems are also solved.

c) we can solve the next problems.

d) all other problems disappear.

V. Выберите заголовок для данного текста, в соответствии с его содержанием.

1. The hypotenuse of an isosceles right triangle

2. The problem of perpetual motion

3. The number of problems in mathematics is inexhaustible

4. Unsolved problems

VI. Укажите правильный перевод подчеркнутой части предложения.

1. As soon as one problem is solved the other comes on it’s place.

a) решена b) решили

c) будет решена d) решили бы

2. There are linear differential equations, having a prescribed monogramic group.

a) имеющие b) имевшие

c) иметь d) имели

3. The question is raised whether mathematics can like other scinces.

a) поднять b) подняв

c) будет поднят d) поднят

4. The efforts to construct a perpetual motion machine were not futile.

a) создав b) создающие

c) создал d) создавать

III. Заполните пропуски подходящими по смыслу словами.

1. Pythagoras founded a community which embraced both mystical and rational.

a) members b) axioms

c) doctrines d) theorems

2. They used their own non-positional … system.

a) coordinate b) arithmetic

c) numeration d) geometric

3. Standard Greek alphabet was supplemented by special ….

a) quantities b) theorems

c) symbols d) words

4. The Pythagorean school was the most influential in determining both the nature and … of Greak mathematics.

a) content b) system

c) power d) diminish

5. The symbols represented a … instead of a word.

a) property b) number

c) axiom d) theorem

Text I

Great minds of Greece such as Thales, Pythagoras, Euclid, Archimede, Appolonius, Eudoxus, etc. produced an amazing amount of first class mathematics. The fame of these mathematicians spread to all corners of the Mediterranean world and attracted numerous pupils. Masters and pupils gathered in schools which though theyhad few, buildings and no campus were truly centres of learning. The teaching of these schools dominated the entire life of the Greeks.

Despite the unquestioned influence of Egypt and BabyloniaonGreek mathematicians, the mathematics produced by the Greeks differed fundamentally from that which preceded it. It were the Greeks who founded mathematics as a scientific discipline. The Pythagoreanschool was the most influential in determining both the nature and content Greek mathematics. Its leader Pythagoras foundedacommunity which embraced both mystical and rational doctrines.

The original Pythagorean brotherhood (c. 550—300 B. C.) was a secret aristocratic society whose members preferred to operate from behind the scenes and, from there, to rule social and intellectual affairs with an iron hand. Their noble born initiates were taught entirely by word of mouth. Written documentation was not permitted, since anything written might give away the secrets largely responsible for their power. Among these early Pythagoreans were men who knew more about mathematics then available than most other people of their time. They recognized that vastly superior in design and manageability Babylonian base-ten positional numeration system might make computational skills available to people in all walks of life and rapidly democratize mathematics and diminish their power over the masses. They used their own non possitional numeration system (standard Greek alphabet supplemented by special symbols). Although there was no difficulty in determining when the symbols represented a number instead of a word, for computation the people of the lower classes had to consult an exclusive group of experts or to use complicated tables and both of these sources of help were controlled by the brotherhood.

IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.

1. Who produced an amazing amount of first-class mathematicians?

a) Great minds of Greece.

b) Great minds of Mediterranian world.

c) Great minds of Babylonia.

2. Who influenced Greek mathematics?

a) Egypt and Fromce b) Egypt

c) Egypt and Babylonia d) Babylonia

3. What did the Pythagoreans represent instead of a word?

a) number b) hieroglyph

c) symbol d) figure

4. What did the people of the lower classes use for their communication?

a) coordinate system b) complicated tables

c) book d)"Pythagorean" theorem

5. What did Pythagoras community embrace?

a) property b) numeration system

c) mystical and rational doctrines d) real number

V. Выберите заголовок для данного текста, в соответствии с его содержанием.

a. Great minds of Greece

b. Numeration system

c. The Pythagorean school

VI. Укажите правильный перевод подчеркнутой части предложения.

1.They were taught entirely by word of mouth.

a) обучаются b) обучили бы

c) обучались

2. The sources of help were controlled by the brotherhood.

a) контролируются b) контролировали бы

c) контролировались

3. The symbols represented a number instead of a word.

а) представляли b) представили бы

с) были представлены

4. Written documentation was not permitted.

a) не разрешала бы b) не была разрешена

c) не будет разрешена

III. Заполните пропуски подходящими по смыслу словами.

1. In geometry they (the Pythagoreans) developed the properties of ….

a) number b) parallel line

c) similar figures d) equation

2. The Pythagoreans used to prove that “the sum of the angles of any triangle is equal to ….

a) 180 degrees b) right angles

c) two right angles d) three right angles

3. The Pythagoreans were aware of the existence of at least three of the regular ….

a) triangles b) squares

c) polyhedral solids d) circles

4. The Pythagoreans discovered the … of a side and a diagonal of a square.

a) angle of a square b) sides of a triangle

c) angle of a triangle d) incommensurability

5. The Pythagoreans developed a fairly complete theory of ….

a) fractions b) proportions

c)real numbers d) numbers

Text II

For Pythagoras and his followers the fundamental studies were geometry, arithmetic, music, and astronomy. The basic element of all these studies was number not in its practical computational aspect, but as the very essence of their being; they meant that the nature of numbers should be conceived with the mind only. In spite of themystical nature of much of the Pythagorean study the members of community contributed during the two hundred or so years following the founding of their organization, a good deal of sound mathematics. Thus, in geometry they developed the properties of parallel lines and used them to prove that “the sum of the angles of any triangle is equal to two right angles”. They contributed in a noteworthy manner to Greek geometrical algebra, and they developed a fairly complete theory of proportional though it was limited to commensurable magnitudes, and used it to deduce properties of similar figures. They were aware of the existence of at least three of the regular polyhedral solids, and they discovered the incommensurability of a side and a diagonal of a square.

Details concerning the discovery of the existence of incommensurable quantities is lacking, but it is apparent that the Pythagoreans found it as difficult to accept incommensurable quantities as to discover them. Two segments are commensurable if there is a segment that “measures” each of them – that is, it contains exactly a whole number of times in each of the segments.

IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.

1. What study was the fundamental for Pythagoras?

a) mathematics b) philosophy

c) arithmetic, geometry, music and astronomy

d) algebra

2. What was the basic element of all studies?

a) word b) number

c) symbol d) point

3. What property did the Pythagoreans develop in geometry of?

a) angles b) diagonal

c) parallel lines d) sides

4. What did the Pythagoreans discover?

a) a diagonal of a square b) a right angle

c) a computational aspect d) a parallel line

V. Выберите заголовок для данного текста, в соответствии с его содержанием.

a. Non-positional numeration system

b. Number as the basic element of all studies

c. Incommensurable quantities

d. The properties of geometry

VI. Укажите правильный перевод подчеркнутой части предложения.

1. In geometry were developed the properties of parallel lines.

a) были развиты b) развили бы

c) будут развиваться d) развиватья

2. The theory was limited to commensurable magnitudes.

a) была ограничена b) ограничила

c) ограничивала d) ограничить

3. To discover the incommensurability of a side and a diagonal of a square is our UNIT.

a) были открыты b) открылись

c) открыли d) открыть

4. To contribute in a noteworthy manner to Greek geometrical algebra was very important.

a) внесли вклад b) вносили бы вклад

c) был внесен вклад d) вносить

III. Заполните пропуски подходящими по смыслу словами.

1. The same geometric procedure can be adapted to the side and ….

a) diagonal of a square b) diagonal numbers

c) regular pentagon d) star pentagon

2. A number can’t be expressed as the ….

a) rational approximation b) irrational approximation

c) ratio of associated pairs d) ratio of two integers

3. The first pair of segments to be incommensurable is the side and diagonal of a ….

a) square b) right triangle

c) number d) regular pentagon

4. The ration of associated pairs of the numbers give closer and closer ….

a) rational number b) fraction

c) rational approximation d) irrational number

Text III

The fact that there revealed pairs of segments for which such a measure does not exist provides the incommensurable case. It is possible that the first pair of segments found to be incommensurable is the side and diagonal of a regular pentagon, the favourite-figure of the Pythagoreans because its diagonals form the star pentagon, the distinctive, mark of their so­ciety. This same geometric procedure can also be adapted to the side and diagonal of a square. Here there exists an association with the so-called Pythagoreans side and diagonal numbers. The ratio of associated pairs of these numbers gives successively closer and closer rational approximations to √2; in fact, they are the approximations obtained by computing successive convergents of the continued fraction form of √2. This is reflected in modern mathematics in the concept of irrational number, a number that cannot be expressed as the ratio of two integers, e. g., п, e, 1/2. This devastating discovery was ascribed to Pythagoras himself, but more probably it was made by some Pythagorean. Since the philosophy of the Pythagorean school was that whole numbers or whole numbers in ratio are the essence of all existing things, the followers of that school regarded the emergence of irrationals as a "logical scandal". As the revelation of geomet­rical magnitudes whose ratio cannot be represented by pairs of integers led to the "crisis" in the foundations of their mathematics, the Pythagoreans tryed to conceal their greatest discovery. A Pythagorean Hippasus (c. 400 B. C.) who first brought out the irrationals from concealment into the open supposedly perished in a shipwreck at sea. But great discoveries could not be suppressed. The discovery of incommensurables was a turning-point, a landmark in the history of mathematics, and its significance can hardly be over appreciated. It resulted in a need to establish a new theory of proportions indepen­dent of commensubarility. This was accomplished byEudoxus (c. 370 B. C.). The details of the gradual transition from a theory of proportions which includedincommensurable quantities to a clear realization of.the concept of an irrational number covered a wide range of sophisti­cated mathematical topics and this concept was fully clarified only in the nineteenth century by R. Dedekind and G. Cantor. In mathematics of today the irrationals form an important subset of real numbers the basis of both algebra and analysis.

IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.

1. What was the distinctive mark of the Pythagoreans society?

a) segment b) side and diagonal

c) polyhedral d) star pentagon

2. What gives the ratio of pairs of numbers?

a) real number b) approximation to 0

c) rational approximation to y2 d) irrational number

3. Why cannot a number be expressed as the ratio of two integers? Because it was ….

a) irrational number b) rational number

c) real number d) fractional number

4. Who first brought out the irrationals from concealment?

a) Cantor b) Eudoxus

c) Hippasus d) Descart

5. What new theory did Eudoxus establish?

a) a theory of numbers b) a theory of proportions

c) a theory of equations d) the“Pythagorean” theory

V. Выберите заголовок для данного текста, в соответствии с его содержанием.

a. The concept of irrational number

b. Geometrical magnitudes

c. The discovery of incommensurables

VI. Укажите правильный перевод подчеркнутой части предложения.

1. The ratio of associated pairs of side and diagonal numbers give rational approximations to y2.

a) соединять b) соединенные

c) соединение d) соединены

2. The discovery of incommensurables and its significance can hardly beoverappreciated.

a) могут быть переоценены b) могли бы переоценить

c) смогут быть переоценены d) могли бы быть переоценены

3. The discovery was made by some Pythagoreans.

a) сделало b) будет сделано

c) сделало бы d) было сделано

4. This number can’t be expressed as the ratio of two integers.

a) не может быть выражена b) не могло быть выражено

c) не сможет быть выражена d)не смогло бы быть выражено

III. Заполните пропуски подходящими по смыслу словами.

1. Under Plato’s influence they emphasized … to the extent of ignoring all practical applications.

a) geometry b) axioms

c) pure mathematics d) new theory

2. They proved the fundamental theorems of a plane and ….

a) curve b) plane geometry

c) symbol d) solid geometry

3. The Egyptians were able to prove the … property.

a) right angle b) theorem

c) information d) number

4. The ancient’s knew the … of some theorems.

a) subjects b) proofs

c) groups d) columns

Text IV

The "Pythagorean" theorem isone of the most important propositions in the entire realm of geometry. There is no doubt, how­ever, that the "Pythagorean property": c² =a² + b² was known priortothe time of Pythagoras; there existed clay tablet texts which con­tained columns of figures related to Pythagorean triples. The frequent text­book reference to Egyptian "rope-stretchers" and their knotted surveying ropes as proof that these ancients knew the theorem was erro­neous. While it isknown that the Egyptians realized as early as 2000 B. C. that 4²+ 3²=5², there is no evidence that the Egyptians knew or were able to prove the right angle property of the figure involved. Pythagoras is credited with the proof of this property which is true for all right trianges, and for all natural numbers. Although much of this information was known to the ancients of earlier times, the deductive aspect of geometry was exploited and advanced considerably in the work of the Pythagoreans.

The mysticism of this celebrated school aroused the suspicion and dislike of the people who finally drove the Pythagoreans out Crotona. A Greek seaport in Southern Italy and burnt their buildings. Pythagoras was murdered but his followers were scattened to other Greek centres and continued his teachings. The Pythagoreans were credited with giving the subject of mathematics special and independent status. They were the first group to treat mathematical concepts as abstractions and they distinguished mathematical theory from practices or calculations. They proved the fundamental theorems of plane and solid geometry and of “arithmetica”- the theory of numbers.

More widely known than the Pythagoreans was the Academy of Plato which had Aristotle as its most distinguished students. The latter then founded his own school at the time of Plato’s death pupils were the most famous philosophers, mathematicians and astronomers of their age. Under Plato’s influence they emphasized pure mathematics to the extent of ignoring all practical applications and they added immensely to the range of mathematics.

IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.

1. What is one of the most important proportions in the entire realm of geometry?

a) theory of proportions b) theory of numbers

c) the “Pythagorean” theorem d) numeration system

2. What was the “Pythagorean property?”

a) rational approximation b) ratio of two integers

c) sum of the angles is equal to two right angles d) c2 =a2 + b2

3. What was the “Pythagorean property” true for?

a) diagonals b) right angles

c) right triangle and all natural numbers d) irrational numbers

4. What was considerably advanced in the work of the Pythagoreans?

a) written documentation b) numeration system

c) incommensurable quantities d) deductive aspect of

geometry

5. What did they distinguish mathematical theory from?

a) deductive aspect b) theory of number

c) practice or calculation d)geometrical

magnitudes

6. Who was the most distinguished student in the Academy of Plato?

a) Plato b) Aristotle

c) Pythagoras d) Cantor

V. Выберите заголовок для данного текста в соответствии с его содержанием.

a. The “Pythagorean property”

b. The Pythagorean theorem

c. The academy of Plato

d. Aristotle – the founder of his own school

VI. Укажите правильный перевод подчеркнутой части предложения.

1. Much of this information was known to the ancients of earlier times.

a) известна b) была известна

c) была бы известна d) будет известна

2. The deductive aspect of geometry was exploited in the works of the Pythagoreans.

a) разработал b) разработался бы

c) был разработан d) будет разработан

3. They proved that fundamental theorems of plane and solid geometry were known to them.

a) доказали b) были доказаны

c) доказывались d) доказанные

4. The mysticism of the celebrated school aroused suspicion of the people.

a) вызывает b) вызвало

c) вызовет d) вызвало бы