Exercises to Non-Euclidian geometries

Упражнение 5. Выпишите из текста предложения, содержащие страдательный залог.

… the existence of the special quadrilaterals.. is based

This postulate is…stated

... the conclusions which may be drawn are consequences of the assumption which have been made.

It should be borne in mind…

 

Упражнение 6. Выпишите из текста предложения, в которых it не переводится.

It is interesting to note… that the existence of the special…– интересно заметить

it should be borne in mind… - необходимо помнить

 

Упражнение 7. Определите, к какой части речи относятся подчеркнутые в тексте слова.

interesting – adjective (прил.)

assuming – gerund

stated – participle II (причастие II)

deductive – adjective (прил.)

considering – participle I (причастие I)

through – preposition

independently – adverb (наречие)

development – noun (сущ.)

internally – adverb (наречие)

valid – adjective (прил.)

undefined – participle II (причастие II)

terms – noun (сущ.)

reasoning – noun (сущ.)

straight – adjective (прил.)

 

Упражнение 8. Назовите слова, от которых образованы следующие производные.

1. important (важный), importance (важность), unimportant (неважный);
2. deduce (выводить), deduction (вывод), deductive (дедуктивный);
3. consequence (последствие), consequent (последовательный), consequently (последовательно);
4. essence (суть), essential (основной), essentially (по существу);
5. valid (действенный), validity (вескость, действенность), invalid (недейственный)
6. intuitive (интуитивный), intuitively (интуитивно), intuition (интуиция);
7. imagine (представить), image (образ), imaginable (вообразимый), imaginary (воображаемый), imagination (воображение);
8. complete (полный), completely (полностью), completion (завершение), incomplete (неполный);
9. correspond (соответствовать), correspondence (соответствие), correspondingly (соответственно);
10. according to (в соответствии с), accordingly (соответственно), to accord (согласовываться).

 

Упражнение 9. Найдите подлежащее или группу подлежащего в следующих предложениях.

1. It is interesting to note that the existence of the special quadrilateral discussed above is based upon the so called parallel postulate of Euclidean geometry.
2. Through a point not on line L there is no more than one line parallel to L.
3. An important aspect of geometry as a deductive system is that the conclusions which may be drawn are consequences of the assumption which have been made.
4. However, the conclusions drawn in non-Euclidean geometries are often completely inconsistent with Euclidean conclusions.
5. For example, according to Euclidean geometry parallelograms and rectangles (in the sense of a parallelogram with four 90-degree angles) exist.
6. It should be borne in mind that the conclusions of non-Euclidean geometry are just as valid as those of Euclidean geometry, even though the conclusions of non-Euclidean geometry contradict those of Euclidean geometry.
7. The reason for forming these intuitive images is only to help our reasoning within a certain deductive system.

 

 

Упражнение 10. Переведите предложения на русский язык.

1. The result of the work depends on our finding a proper solution for this problem.

Результат работы зависит от того, найдем ли мы верное решение этой задачи.

 

2. The earth’s moving around the sun is common knowledge nowadays.

То, что земля вращается вокруг солнца, сегодня известно всем.

3. Because of its having no dimensions you cannot see a point, feel a point, or move a point.

Т.к. точка не имеет размеров, вы не можете видеть, чувствовать или перемещать ее.

 

4. We discussed their having presented the paper.

Мы обсудили то, что они представили этот документ.

 

5. A segment being extended indefinitely is easy to imagine.

Легко представить отрезок, продлеваемый в бесконечность.

 

6. Computers being able to control things make them very helpful.

То, что компьютеры могут контролировать явления действительности, делает их очень полезными.

 

 

Упражнение 11. Составьте план текста.

 

Упражнение 12. Измените данные предложения по образцу.

 

А.

I enjoyed this book (to read). -  I enjoyed reading this book.

1. We enjoyed the concert (to listen to). – We enjoyed listening to the concert.

2. Everybody enjoyed the lecture (listen to). – Everybody enjoyed listening to the lecture.

I enjoyed my stay in the country (to stay). – I enjoyed my staying in the country.

They enjoyed the performance (to watch). – They enjoyed watching the performance.

I enjoyed that game (to play). – I enjoyed playing the game.

 

B.

I am going to do it right away. – I thing of doing it right away.

1. I am going to buy a new TV set. – I think of buying a new TV set.

2. I am going to spend my holidays in the Crimea. – I think of spending my holidays in the Crimea.

3. We are going to set off tomorrow. – We think of setting off tomorrow.

4. He is going to pay for the books right away. – He thinks of paying for the books right away.

5. I am going to cook this food later today. – I think of cooking this food later today.

 

C.

I am sorry to ask you such questions. Excuse me for asking you such questions.

Sorry to come here so late (to make so much noise, to enter the room without a knock; to be late; to refuse to help them; to interrupt, not to come on time).

 

Excuse me for coming he so late

Excuse me for making so much noise.

Excuse me for entering the room without a knock.

Excuse me for being late.

Excuse me for refusing to help them.

Excuse me for interrupting.

Excuse me for not coming on time.

 

D. Задайте вопросы по образцу.

Do you mind my asking you a few questions?

(to leave you for a moment, shut the door, stay here for a while, to play the piano, to air the room, to put down your address and telephone number, to cook that delicious food for us again).  

 

Do you mind my leaving you for a moment?

Do you mind my shutting the door?

Do you mind my staying here for a while?

Do you mind my playing the piano?

Do you mind my airing the room?

Do you mind my putting down your address and telephone number?

Do you mind my cooking that delicious food for us again?

 

 

Упражнение 13. Объедините два предложения в одно по образцу

I told him about it. I remember that very well.

I remember telling him about it.

 

1. I shut the door. I remember that very well.

I remember shutting the door.

 

2. I have already examined that patient. I remember that very well.

I remember examining that patient.

 

3. I paid him for that service. I remember that very well.

I remember my paying him for that service.

 

4. I mentioned that figure in my report. I remember that very well.

I remember my mentioning that figure in my report.

 

 

Упражнение 14. Ответьте на следующие вопросы о треугольниках и четырехугольниках, изображенных ниже.

 

1. Will you name the diagonal [daI'xgqnl] in this parallelogram?

These are AC and DB.

 

2. Are opposite sides of a parallelogram congruent?

Yes, they are congruent.

 

3. Can we say the same about opposite angles?

Yes, opposite angles of the parallelogram are congruent.

 

4. Do diagonals AC and DB intersect at T or at some other point?

Yes, they intersect at T.

 

5. What kind of an angle is angle A, B? Why?

Angle A is acute, and angle B is obtuse because angle A is less than the right angle, and angle B is more than the right angle.

 

6. Where do the altitudes of the triangles A, B, C drawn below lie?

The altitudes of the triangles A, B, C lie on the lines perpendicular ["pE:pqn'dIkjVlq] to their bases.

 

7. How many altitudes has a triangle?

A triangle has three altitudes.

 

8. How can you find the area of a right triangle?

We should multiple its legs and divide the result by two.

 

9. Can you deduce what the area of a parallelogram is?

You should multiply the altitude by the base.

 

10. What can you say about the figure below? Name all the elements in it.

It is a trapezium [trq'pi:z|Iqm]. BC and AD are its parallel sides, they are called bases. BD is a diagonal. BP, CQ and KD are altitudes, PBCQ is a rectangle.

 

 

Упражнение 15. Ответьте на следующие вопросы по тексту.

 

1. How is the so called parallel postulate usually stated now?

The parallel postulate is stated as follows: through a point not on line L there is no more than one line parallel to L.

 

2. What is our definition of the special quadrilateral having pairs of parallel sides based upon? It is a parallelogram or a rectangle.

 

3. What are the names of those mathematicians who developed non Euclidean geometries? They are Lobachevski, Bolyai and Riemann.

 

4. What was Euclid’s assumption concerning a given point and a given line?

Through a point not on line L there is no more than one line parallel to L.

 

5. What did Lobachevski assume?

Through a given point not on a given line there is more than one line parallel to the given line.

 

6. What was Riemann’s idea of the relation between a given point and a given line?

Riemann assumed that through a given point not on a given line there is no line parallel to the given line.

 

7. What did those variations of the parallel postulate lead to?

The variations of the parallel postulate have led to the development of non-Euclidean geometries.

 

8. What is the difference between Lobachevski and Riemann’s ideas on parallelograms and rectangles?

According to the geometries of Lobachevski and Bolyai parallelograms exist but rectangles do not; according to the geometry of Riemann neither parallelograms nor rectangles exist.

 

9. Can we say that the conclusions of non-Euclidean geometry are just as valid as those of Euclidean geometry? Why?

Yes, we can say that the conclusions of non-Euclidean geometry are just as valid as those of Euclidean geometry because they are true.

 

10. What are the intuitive images corresponding to the undefined terms straight line and plane in Euclidean and non-Euclidean geometries?

The plane of Euclid is a flat surface; the plane of Lobachevski is a saddle-shaped or pseudospherical surface; the plane of Riemann is an ellipsoidal or spherical surface.

 

 

Упражнение 16. Перескажите кратко текст на английском языке.

 

Упражнение 17. Прочтите текст и скажите, какова его основная идея.

Geometry was developed originally to solve practical problems.

Изначально геометрия была предназначена для решения практических проблем.

 

The ancient Egyptians knew many geometric facts: for example, they knew that a triangle whose sides have lengths of 3, 4, and 5 – units is a right triangle (a triangle having a 90 degree angle).

Древние египтяне знали многие геометрические законы: например, они знали, что треугольник со сторонами, кратными 3, 4, 5, является прямоугольным (т.е. имеющим угол в 90 градусов).

 

But it was the ancient Greeks who created a magnificent, logically organized body of geometric knowledge.

Однако только древние греки сумели создать прекрасную, логически организованную структуру геометрических знаний.

 

The abstract, theoretical system of the Creeks unexpectedly yielded an enormous variety of practical applications.

Абстрактная, теоретическая система греков неожиданно нашла широкое практическое применение.

 

That is one reason why modern mathematical systems, including the geometric system developed in this book, are fashioned in the Greek manner.

Эта одна из причин, почему современные математические теории, в том числе и геометрическая теория, представленная в данной книге, изложены на греческий манер.

 

In our geometric system there are three undefined terms: point, line and plane.

В нашей геометрической теории три исходных (неопределенных) понятия: точка, линия и плоскость.

 

Other geometric terms will be defined on the basis of undefined terms.

Другие геометрические понятия будут определены на основе исходных понятий.

 

Certain basic properties of points, lines, and planes will be stated in the postulates (assumed statements).

Некоторые базовые свойства точек, линий, плоскостей будут определены в постулатах (предположениях, не требующих доказательств).

 

From postulates and definitions, we shall deduce other important statements called theorems.

На основе постулатов и определений мы выведем прочие важные утверждения, называемые теоремами.

 

 

The main idea of the text is that Greeks developed a magnificent, logically organized body of geometric knowledge which yielded an enormous variety of practical applications.