5. Перепишите предложения. Подчеркните Participle I и Participle II, определите их функцию в предложении. Переведите предложения.

1. The applied mathematician takes the pure mathematician’s findings and applies them to the varied concrete situations.

2. Geometry was constructed on the basis of a number of axioms and theorems which were derived from axioms.

3. Having finished his talk the speaker smiled and waited for comments.

4. He couldn’t catch up with them though working very hard.

5. Trigonometry is the branch of mathematics dealing with the relations between the sides and angles of triangles.

6. Прочитайте текст, перепишите и письменно переведите 2, 3, 4 и 6 абзацы.

A MODERN VIEW ON GEOMETRY

For a long time geometry was intimately tied to physical space, actually beginning as a gradual accumulation of subconscious notions about physical space and about forms, content, and spatial relations of specific objects in that space. We call this very early geometry “subconscious geometry”. Later human intelligence evolved to this point where it became possible to consolidate some of the yearly geometrical notions into a collection of somewhat general laws or rules. We call this laboratory phase in the development of geometry “scientific geometry”. About 600 B. C. the Greeks began to inject deduction into geometry giving rise to what we call “demonstrative geometry”.

In time demonstrative geometry becomes a material-axiomatic study of idealized physical space and of the shapes, sizes, and relations of idealized physical objects in that space.

With the elaboration of analytic geometry in the first half of the seventeenth century, space was regarded as a collection of points; and with the invention, about two hundred years later, of the classical non-Euclidean geometries; mathematicians accepted the situation that there is more than one conceivable space and hence more than one geometry.

At the end of the nineteenth century, Hilbert and others formulated the concept of formal axiomatic, and there developed the idea of branch of maths as an abstract body of theorems deduced from a set of postulates. Each geometry became, from this point of view, a particular branch of maths. Postulates sets for a large variety of geometries were studied.

In the twentieth century the study of abstract spaces was inaugurated and some very general studies came into being. A space became merely a set of objects together with a set of relations in which the objects are involved, and geometry became the theory of such a space. It must be confessed that this latter notion of geometry is so embracive that the boundary lines between geometry and other areas of maths became very blurred, if not entirely obliterated. It is essentially only the terminology and the mode of thinking involved that makes the subject “geometric”.

There are many areas of maths where the introduction of geometric terminology and procedure greatly simplifies both the understanding, and the presentation of some concept or development. This becomes increasingly evident in so much of maths that some mathematicians of the second half of the twentieth century feel that perhaps the best way to describe geometry today is not as some separate and prescribed body of knowledge but as a point of view – a particular way of looking at a subject.