Учебное пособие по английскому языку для студентов 2 курса факультета математики и информационных технологий Уфа риц башгу 2020



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пособие для математиков II Мотина О П , Кулыева А А -4

UNIT 8. GEOMETRY




Before you start
Read the key words:
ingenious – оригинальный, гениальный
anticipate – предвосхищать
celestial sphere – небесная сфера
concurrent – одновременный, параллельный
plane curve – плоская кривая, кривая в плоскости
intrinsic – зд. внутренний
manifold – множество
approximately – приблизительно
resemble – напоминать
endow – наделять
polygon – многоугольник
conic section – коническая секция/сечение
assumption – предположение
pervasive – распространенный, повсеместный
Fourier Transform – преобразование Фурье
surveying – геодезия, геологическое изыскание
geometric entity – геометрический элемент/объект


Before you read
Answer the following questions:
1) What does geometry study?
2) When did geometry emerge? What was its application in the ancient world?
3) What branches of geometry can you name? What do they study?
4) What geometrical shapes can you name?
5) What professions require good knowledge of geometry?


Reading
Geometry


Geometry (Ancient Greek: geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of a formal mathematical science emerging in the West as early as Thales (6th Century BC). By the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow. Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia.
The introduction of coordinates by René Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures, such as plane curves, could now be represented analytically, i.e., with functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. Furthermore, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry. The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry.
In Euclid's time there was no clear distinction between physical space and geometrical space. Since the 19th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation, and the question arose: which geometrical space best fits physical space? With the rise of formal mathematics in the 20th century, also 'space' (and 'point', 'line', 'plane') lost its intuitive contents, so today we have to distinguish between physical space, geometrical spaces (in which 'space', 'point' etc. still have their intuitive meaning) and abstract spaces. Contemporary geometry considers manifolds, spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales. These spaces may be endowed with additional structure, allowing one to speak about length.




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