Exercises
I. Word-study.
Read and memorize the following words and word combinations:
kudos – слава
indignity – пренебрежение, унижение
steadfastness – стойкость
stature – рост, фигура
to stem – происходить
to defend one’s thesis – защитить диссертацию
atomic and nuclear weapon – атомное и ядерное оружие
to smb’s credit – благодаря кому-либо
to deprive of – лишать чего-либо
to ban – запрещать
title - титул
II. Comprehension check-up.
1) Answer the following questions.
When and where was A. Sakharov born?
A. D. Sakharov took an active part in developing the hydrogen bomb, didn't he?
When was he awarded the Nobel Peace prize?
Why did he live in Gorky?
Who helped A. D. Sakharov to return to Moscow?
2) Choose the statement you think to be correct.
1. Sakharov was born
a) in Samara
b)
c)
2.
a)
b)
c)
3.
a)
b)
c)
III. Render the contents of the text in 10 sentences. Nikolai ivanovich lobachevsky
(1792-1856)
B
orn
in Nizhni Novgorod, Russia, Lobachevsky possessed an early aptitude
for mathematics,
and after studies at Kazan University (1807-11), became a professor
of mathematics
at that institution. Upon becoming rector in 1827, he expanded the
scientific mission of
the university and began publishing a Scientific
Proceedings. His
first communication of his discovery
of a Non-Euclidean or Hyperbolic Geometry came in a lecture of 1826,
later published
as "On the Principles of Geometry" in the Kazan
Herald (1829-30),
but his ideas were not
understood or well received at the time. Mathematics was experiencing
rapid change at this
time and the nineteenth century has been called the "Golden Age
in mathematics" (Boyer
496)—even such sciences as biology and psychology grew more
quantitative and statistically-based
by the end of the century, and new mathematical concepts such as
non-Euclidean
geometries, noncommutative algebra, and n-dimensional spaces were
developed.
The story of Lobachevsky's new geometry begins with Euclid's work the Elements (c. 300 B.C.E.) which outlined a geometrical system of definitions (primitive terms), axioms, and theorems logically deduced from the axioms. Lobachevskian geometry in fact accepts all of the Euclidean propositions, except for the fifth parallel-postulate (here expressed in simplified terms): only one straight line can be drawn parallel to a given straight line through a point not on that line (in the same plane). Fruitless attempts were made by Proclus and Posidonius in classical times and by eighteenth-century mathematicians Girolamo Saccheri and Johann Lambert to prove the validity of this assumption as a theorem instead of accepting it as a self-evident axiom. Lobachevsky in his attempt to prove the parallel-postulate (by demonstrating that denial of Euclid's parallel-postulate leads to contradiction) found that a new set of theorems could be formulated describing a consistent, non-contradictory and logical system which he called "imaginary geometry."
He began with the axiom "through a point not lying on a given line one can draw in the plane determined by this point and line at least two lines which do not have a point of intersection with the given line". Lobachevsky, similarly to Gauss and Wachter, further demonstrated that on the imaginary figure the horosphere (a sphere of infinite radius, comprised of an infinite number of horo-cycles or circles of infinite radius), both Euclidean and normal plane geometry were valid, suggesting that these geometries were subsets of Lobachevky's more powerful and more generalized system. Lobachevsky published three book-length explanations of his new geometrical system: New Foundations of Geometry (1835-38); Geometrical Researches on the Theory оf Parallels (1840), of which the first section is reprinted here below; and Pangeometry (1855), the final expression of his complete system.
The possibility of a non-Euclidean geometry had also been envisioned by Carl Friedrich Gauss (1777-1855), the most influential mathematician of the nineteenth century, but Gauss, perhaps fearing ridicule, did not publish or systematize his insights in his lifetime. The Hungarian mathematician Jands Bolyai (1802-1860), at the same time as Gauss and Lobachevsky, also developed a non-Euclidean "Absolute Science of Space," published as an appendix to one of his father's mathematical treatises in 1832, although Gauss surprisingly withheld public support for Bolyai's views. Obviously involving a case of simultaneous discovery among researchers interested in the same questions, credit for the full elaboration of non-Euclidean geometry, however, is most often awarded to Lobachevsky.
Lobachevsky's work wrought a profound change in mathematical thinking which had relied for over a thousand years on Euclid's system as an exact description of space and everyday geometrical reality. The realization of alternate geometries helped to dislodge the notion "that Euclidean geometry is 'innate,' 'unique,' 'natural,' or 'god-given'" (Yaglom vi). Work on non-Euclidean geometries, however, was not fully appreciated before G.F.B. Rie-mann's Uber die Hypothesen welche der Geometric zu Grunde liegen (1867), in which he proposed an even more general approach to geometry, thus subsuming both Euclidean and Lobachevskian geometries as special cases of a more universalized geometry. Riemann by theorizing curved metric spaces paved the way for Einstein's relativity theory in the twentieth century.