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VI. Укажите правильный перевод подчеркнутой части предложения.

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

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