Text. 10. Basic Geometric Concepts

The practical value of geometry lies in the fact that we can abstract and illustrate physical objects by drawings and models. For example, a drawing of a circle is not a circle, it suggests the idea of a circle. In our study of geometry we separate all geometric figures into two groups: plane figures whose points lie in one plane and space figures or solids. A point is a primary and starting concept in geometry. Line segments, rays, triangles and circles are definite sets of points. A simple closed curve with line segments as its boundaries is a polygon. The line segments are sides of the polygon and the end points of the segments are vertices of the polygon. A polygon with four sides is a quadrilateral. We can name some important quadrilaterals. Remember, that in each case we name a specific set of points. A trapezoid is a quadrilateral with one pair of parallel sides. A rectangle is a parallelogram with four right angles. A square is a rectangle with all sides of the same length. The regular polyhedra are a part of geometric study chiefly in antiquity. They have a symmetrical beauty that fascinates men of all ages. The first question in connection with regular polyhedra is: How many different types are there? Thanks to the ancient Greeks we know that there are exactly five types of polyhedra. All objects in their view are composed of four basic elements: earth, air, fire and water. They believe that the fundamental particles of fire have the shape of tetrahedron, the air particles have the shape of octahedron, of water - the icosahedron, and the earth - the cube. The fifth shape, the dodecahedron, they reserve for the shape of the universe itself. Plane geometry is the science of the fundamental properties of the sizes and shapes of objects and treats geometric properties of figures. The first question is, under what conditions two objects are equal or congruent in size and shape. Next, if figures are not equal, what significant relationship may they possess to each other and what geometric properties can they have in common? The basic relationship is shape. Figures of unequal size but of the same shape, that is, similar figures have many geometric properties in common. If figures have neither shape nor size in common, they may have the same area, or, in geometric terms, they may be equivalent, or may have endless other possible relationships. Geometry is the science of the properties, measurement and construction of lines, planes, surfaces and different geometric figures. What do we call “constructions” in our study of geometry? Ruler-compass constructions are simply the drawings which we can make when we use only a straightedge and a compass. A compass is a misleading word. It is not only “компас” in the maths, it is usually “циркуль”. We call such misleading words “ложные друзья переводчика”. For a ruler you ought to use an unmarked straightedge because measurement has no role in ruler-compass constructions. Of course, you can use a marked straightedge if you don’t permit yourself to use these marks for measurement. Later you ought to do some measurement to “check” your constructions. We measure segments in terms of other segments and angles in terms of other angles. It seems only natural that we find areas indirectly as well.

Vocabulary

value	величина, значение
drawing	чертеж
suggest	предполагать
plane	плоскость
solid	геометрическое тело
ray	луч
set	набор
polygon	многоугольник
vertices (pl) от vertex	вершина
quadrilateral	четырехсторонний
rectangle	прямоугольник
chiefly	главным образом
to fascinate	очаровывать
to be composed of	состоять из
to possess = to have

1. Answer the following questions:

1) What is the practical value of geometry?

2) How many types of polyhedral are there?

3) What is the shape of the universe?

4) Under what conditions are two objects equal or congruent in size and shape?

5) What figures have many geometric properties in common?

6) What misleading words for geometry can you find in the text?

7) How can segments be measured?

2. Use the opening phrases to agree or disagree with the following statements.

That’s right.	Not quite so, I am afraid.
Exactly. Certainly.	I don’t think this is just the case.
This is the case.	I doubt it. Far from that.
I accept it fully.	Just the other way round.
	Not at all. Quite the reverse.

1) Geometry is the science of geometric figures.

2) If figures are not equal they can have similar properties.

3) A square is a rectangle with all sides of different length.

4) The line segments are sides of the polygon and the end points of the segments are vertices of the polygon.

5) A trapezoid is a quadrilateral with two pairs of parallel sides.

Text. 11.

Non-Euclidean Geometry

There is evidence that a logical development of the theory of parallels gave the early Greeks a lot of trouble. Euclid met the difficulties by defining parallel lines as coplanar straight lines that do not meet one another however far they may be produced in either direction, and by adopting as an initial assumption his now famous parallel postulate: "If straight line intersect two straight lines so as to make the interior angles on one side of it together less than two right angles, the two straight lines will intersect if indefinitely produced, on the side on which are the angles which are together less than two right angles". Actually, the postulate is the converse of Proposition 17 of Euclid's Book II and it seemed more like a proposition than a postulate. It was natural to ask if the postulate was really needed at all, or perhaps it could be derived as a theorem, or, at least, it could be replaced by a more acceptable equivalent. The attempts to devise substitutes and to derive it as a theorem from the rest of Euclid's postulates occupied geometers for over two thousand years and culminated in the most far-reaching development of modem maths — non-Euclidean geometry.

Topology started as a branch of geometry, but during the second quarter of the twentieth century it underwent such generalization and became involved with so many other branches of maths that it is now more properly considered, along with geometry, algebra, and analysis, a fundamental division of maths. Today topology may roughly be defined as the math study of continuity, though it still reflects its geometric origin. Topology is the study of those properties of geometric figures which remain invariant under so-called topological transformations, that is, under single-valued continuous mapping possessing single-valued continuous inverses.

Vocabulary

evidence	основание, доказательство
trouble	тревога, заботы
coplanar	копланарный
direction	направление
adopt	принимать
assumption	предположение
initial	первоначальный
converse	обратно (-ая теорема)
proposition	теорема, утверждение
acceptable	приемлемый
to devise	придумывать, изобретать
substitute	заменять
to undergo	переносить, испытывать
continuity	непрерывность
that is	то есть

1. Give Russian equivalents to the following English word combinations:

More like a proposition, a more acceptable equivalent, to devise substitutes, the most far-reaching development, underwent generalization, became involved with, along with geometry, single-valued continuous inverses, indefinitely produced, however far produced, initial assumption, it was natural to ask, culminated in the development of modern math, properly considered, a fundamental division of maths.

2. Answer the following questions:

1) What theory caused a lot of troubles to early Greeks?

2) What did the attempts to devise substitutes culminate in?

3) What is topology?

4) What branches of maths did it become involved with?

5) How can you define topology?

6) What definition did Euclid give to parallel lines?

7) What ideas occupied geometers for over two thousand year?

8) How did they culminate?

3. Find in the text nouns with the following suffixes.