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1. Read and translate this text
Topic 2
N.L. Lobachevsky
N.I. Lobachevsky was born in 1792 in Nizhny Novgorod. After
his father death in 1797 the family moved to Kazan where Lo-
bachevsky graduated from the University. He stayed in Kazan all his
life occupying the position of dean of the faculty of Physics and
Mathematics and rector of the University. He lectured on mathematics,
physics and astronomy.
Lobachevsky is the creator of non-Euclidean geometry. His first
book appeared in 1829. Few people took notice of it. Non-Euclidean
geometry remained for several decades an obscure field of science.
Most mathematicians ignored it. The first leading scientist who realized
its full importance was Riemann.
There is one axiom of Euclidean geometry. This is the famous
postulate of the unique parallel, which states that through any point not
on a given line one and only one line can be drawn parallel to the given
line. It goes without saying that there are many lines through a point,
which do not intersect a given line within any distance, however large.
So, this axiom can never be verified by experiment. All the other axi-
oms of Euclidean geometry have a finite character since they deal with
finite portions of lines and with plane figures of finite extent. The fact
that the parallel axiom is not experimentally verifiable raises the ques-
tion of whether or not it is independent of the other axioms. If it were a
necessary logical consequence of the other, then "it would be possible
not to regard it as an axiom and to give a proof of it in terms of the
other Euclidean axioms. For centuries mathematicians have tried to
find such a proof and in the long run were appeared that the parallel
postulate is really independent of the others.
What does the independence of the parallel postulate mean?
Simply that it is possible to construct a consistent system of geometri-
cal statements dealing with points and lines, by deduction from a set of
axioms in which the parallel postulate is replaced by a contrary postu-
late. Such a system is called non-Euclidean geometry.
Lobachevsky settled the question by constructing in all detail a
geometry in which the parallel postulate does not hold.
14
Non-Euclidean geometry has developed into an extremely useful
instrument for application in the physical world.
After 1840 Lobachevsky published a number of papers on con-
vergence of infinite series and the solution of defined integrals. In
modern books on defined integrals about 200 integrals was solved by
Lobachevsky.
2. Retell the text
Additional text (Topic 2)
Read and translate this text
Sofia Kovalevskaya
January 15, 1850 – February 10, 1891
Kovalevskaya Stamps issued in 1951 and 1996.
Written by Becky Wilson, Class of 1997 (Agnes Scott College)
An extraordinary woman, Sofia Kovalevskaya was not only a
great mathematician, but also a writer and advocate of women's rights
in the 19
th
century. It was her struggle to obtain the best education
available which began to open doors at universities to women. In addi-
tion, her ground-breaking work in mathematics made her male coun-
terparts reconsider their archaic notions of women's inferiority to men
in such scientific arenas.
Sofia Kovalevskaya was born in 1850. As the child of a Russian
family of minor nobility, Sofia was raised in plush surroundings. She
was not a typically happy child, though. She felt very neglected as the
middle child in the family of a well admired, first-born daughter, Anya,
and of the younger male heir, Fedya. For much of her childhood she
was also under the care of a very strict governess who made it her per-
sonal duty to turn Sofia into a young lady. As a result, Sofia became
fairly nervous and withdrawn – traits which were evident throughout
her lifetime.
Sofia's exposure to mathematics began at a very young age. She
claims to have studied her father's old calculus notes that were papered
on her nursery wall in replacement for a shortage of wallpaper. Sofia
1. Read and translate this text Non-Euclidean geometry has developed into an extremely useful
Topic 2 instrument for application in the physical world.
After 1840 Lobachevsky published a number of papers on con-
N.L. Lobachevsky
vergence of infinite series and the solution of defined integrals. In
N.I. Lobachevsky was born in 1792 in Nizhny Novgorod. After modern books on defined integrals about 200 integrals was solved by
his father death in 1797 the family moved to Kazan where Lo- Lobachevsky.
bachevsky graduated from the University. He stayed in Kazan all his
life occupying the position of dean of the faculty of Physics and 2. Retell the text
Mathematics and rector of the University. He lectured on mathematics,
physics and astronomy.
Lobachevsky is the creator of non-Euclidean geometry. His first Additional text (Topic 2)
book appeared in 1829. Few people took notice of it. Non-Euclidean Read and translate this text
geometry remained for several decades an obscure field of science.
Most mathematicians ignored it. The first leading scientist who realized Sofia Kovalevskaya
its full importance was Riemann. January 15, 1850 – February 10, 1891
There is one axiom of Euclidean geometry. This is the famous
Kovalevskaya Stamps issued in 1951 and 1996.
postulate of the unique parallel, which states that through any point not
Written by Becky Wilson, Class of 1997 (Agnes Scott College)
on a given line one and only one line can be drawn parallel to the given
line. It goes without saying that there are many lines through a point, An extraordinary woman, Sofia Kovalevskaya was not only a
which do not intersect a given line within any distance, however large. great mathematician, but also a writer and advocate of women's rights
So, this axiom can never be verified by experiment. All the other axi- in the 19th century. It was her struggle to obtain the best education
oms of Euclidean geometry have a finite character since they deal with available which began to open doors at universities to women. In addi-
finite portions of lines and with plane figures of finite extent. The fact tion, her ground-breaking work in mathematics made her male coun-
that the parallel axiom is not experimentally verifiable raises the ques- terparts reconsider their archaic notions of women's inferiority to men
tion of whether or not it is independent of the other axioms. If it were a in such scientific arenas.
necessary logical consequence of the other, then "it would be possible Sofia Kovalevskaya was born in 1850. As the child of a Russian
not to regard it as an axiom and to give a proof of it in terms of the family of minor nobility, Sofia was raised in plush surroundings. She
other Euclidean axioms. For centuries mathematicians have tried to was not a typically happy child, though. She felt very neglected as the
find such a proof and in the long run were appeared that the parallel middle child in the family of a well admired, first-born daughter, Anya,
postulate is really independent of the others. and of the younger male heir, Fedya. For much of her childhood she
What does the independence of the parallel postulate mean? was also under the care of a very strict governess who made it her per-
Simply that it is possible to construct a consistent system of geometri- sonal duty to turn Sofia into a young lady. As a result, Sofia became
cal statements dealing with points and lines, by deduction from a set of fairly nervous and withdrawn – traits which were evident throughout
axioms in which the parallel postulate is replaced by a contrary postu- her lifetime.
late. Such a system is called non-Euclidean geometry. Sofia's exposure to mathematics began at a very young age. She
Lobachevsky settled the question by constructing in all detail a claims to have studied her father's old calculus notes that were papered
geometry in which the parallel postulate does not hold. on her nursery wall in replacement for a shortage of wallpaper. Sofia
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