10.28M

St-Lobachevsky_04-05-26

1.

Lobachev
sky
Nikolai
Ivanovi
ch
For the 1st-year
students,
1792-1856
doing
the
2nd
semester
in
Mathematics 1
Mechanics Faculty.
&

Task C4
Give a talk on
Lobachevsky N.I.
You must include the issues given below.
Introduce the topic
1.
Speak
about
Lobachevsky as an
eminent person.
2. Describe Lobachevskian
geometry.
Conclude the talk
You have up to 4 minutes to give your talk.
2

3.

4.

5.

Part 1:
https://quizlet.com/ru/1028798314/lobach
evsky-ni-part-1-flashcards/?i=2spbc6&x=1jqtrds/?i=2spbc6&x
=1jqt
Part 2:
https://quizlet.com/ru/1028804442/lobac
hevsky-ni-part-2-flashcards/?i=2spbc6&x=1jqt
5

6.

Men of science consider Lobachevsky to be a great
mathematician. The whole world knows Lobachevsky to
have strictly demonstrated and explained the principles of
the theory of parallel lines. We consider him to be a great
organizer of popular education, and we know him to have
written much on the problems of education.
p27
8
Nikolai Ivanovich Lobachevsky was born on December 1, 1792
in Nizhny Novgorod. His father died when he was only a child,
leaving the family in extreme poverty. After his father’s death in
1797 the family moved to Kazan where Lobachevsky was admitted
to the gymnasium. We know his progress to have been extremely
rapid in mathematics and classics. At the age of 14 he entered the
University of Kazan, where he is known to have spent 40 years as a
student, assistant professor, and finally rector. He stayed in Kazan
all his life. He started as a lecturer. In 1816 he was promoted to
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associate professor, and in 1822, at the age of 30, he became a full

7.

He served in many administrative positions and
became the rector of Kazan University in 1827. Under his
direction great improvements were made at the
University. We know an observatory to have been
founded and equipped and a mechanical workshop to
have been established.
For 2200 years all the mankind believed Euclid to have
discovered an absolute truth. Lobachevsky proved Euclid’s axiom
on parallel lines not to be true. He built a new geometrical theory
quite different from that of Euclid. Lobachevsky is acknowledged to
be the creator of a non-Euclidean geometry. His first book appeared
in 1829. Few people took notice of it. Non-Euclidean geometry (as a
matter of fact the name is due to Gauss) remained for several
decades an obscure field of science. Most leading mathematicians
ignored it. It was Riemann who realized its full importance. In
mathematics there is such a notion as “Euclid’s fifth’. It is a rule in
Euclidean geometry which states that for any given line and point
not on the line, there is one parallel line through the point not
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8.

The non-Euclidian geometry that Lobachevsky developed is
referred to as hyperbolic geometry. Lobachevsky replaced Playfair’s
axiom with the statement that for any given point there exists more than one
line that can be extended through that point and run parallel to another line
of which that point is not a part. He developed the angle of parallelism
which depends on the distance the point is off the given line. In hyperbolic
geometry the sum of angles in a hyperbolic triangle must be less than 180
degrees. Lobachevsky settled the question by constructing in all detail a
geometry in which the parallel postulate does not hold. Non-Euclidean
geometry stimulated the development of differential geometry which has
many applications. Hyperbolic geometry is frequently referred to as
“Lobachevsky geometry”.
Another of Lobachevsky’s achievements was developing a method for
the approximation of the roots of algebraic equations. This method is now
known as the Dandelin-Graffe method, named after two other
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mathematicians who discovered it independently. In Russia, it is called the

9.

In conclusion, we must admit the revolutionary character of
Lobachevsky’s work. We know his ideas to have greatly influenced
not only geometry, but mechanics, physics, astronomy as well. The
boldness of his challenge and its successful outcome have inspired
mathematicians to challenge other “axioms” or “accepted truths”. It
is my firm belief that like Galileo, Copernicus and Newton,
Lobachevsky is one of those who laid the foundation of science.
9

10.

Write a
sentence
summary
the text
3of
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11.

Write
an
abstract
of the text.
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12.

Liste
n
Write
&
Talk
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13.

• Part
#1
• Part
#2
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14.

1. В евклидовой геометрии есть одна аксиома, «истинность» которой, то
есть соответствие эмпирическим данным о растяжении нитей или лучей света,
отнюдь не очевидна. Это знаменитый постулат о единственной параллели,
который гласит, что через любую точку, не лежащую на данной прямой,
можно провести одну и только одну прямую, параллельную данной прямой.
Замечательная особенность этой аксиомы состоит в том, что в ней содержится
утверждение, касающееся всего протяжения прямой, представляемой как
неограниченно простирающаяся в любом направлении, поскольку сказать, что
две прямые параллельны, значит сказать, что они никогда не пересекутся, как
бы далеко они ни были проведены. Само собой разумеется, что через точку
проходит множество прямых, которые не пересекают данную прямую ни на
каком фиксированном расстоянии, каким бы большим оно ни было. Поскольку
максимально возможная длина реальной линейки, нити или даже луча света,
видимого в телескоп, конечно, конечна, а внутри любого конечного круга
существует бесконечно много прямых линий, проходящих через данную точку
и не пересекающихся с данной линией внутри круга, из этого следует, что эта
аксиома никогда не может быть проверена экспериментально. Все
остальные аксиомы евклидовой геометрии имеют конечный характер,
поскольку имеют дело с конечными отрезками прямых и с плоскими
фигурами конечной протяженности. Тот факт, что аксиома параллельных не
поддается экспериментальной проверке, ставит вопрос о том, является ли
14 она
независимой от других аксиом. Если бы она была необходимым логическим

15.

2. В то время любая геометрическая система, не находящаяся в
абсолютном согласии с системой Евклида, считалась5 очевидной
бессмыслицей. Кант, самый выдающийся философ того времени,
сформулировал это отношение в своем утверждении, что аксиомы
Евклида присущи человеческому разуму, а потому не имеют
объективной обоснованности для реального пространства. Но в конце
концов6 возникло убеждение, что бесконечные неудачи в поисках
доказательства постулата о параллельных объясняются не
отсутствием изобретательности, а тем, что постулат о параллельных
действительно независим от других.
Что означает независимость постулата о параллельных? Просто
то,
что
можно
построить
непротиворечивую
систему
«геометрических» утверждений, касающихся точек, прямых и т. д.,
путем дедукции из набора аксиом, в которой постулат параллели
заменен противоположным постулатом. Такая система называется
неевклидовой геометрией. Нужно было обладать интеллектуальным
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мужеством Лобачевского, чтобы понять, что такая геометрия,

16.

Write
an
abstract
of the text.
16

17.

Make up 10 direct questions of
different types based on the content
of the text. These questions encourage
exploration of the key concepts presented in the text
while varying in structure and focus.
17

18.

Make up 10 indirect questions
based on the content of the text.
These indirect questions prompt further
reflection and analysis on the key points
covered in the text.
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19.

Ex.2: Analyze all the ing-forms used
in theThe
text.
Great
Russian
p21
8
Mathematician
N. I. Lobachevsky was born in 1792. His father having died very
early, the family was hard up and there was no hope of the boy’s getting a
good education. While studying at school, however, N. Lobachevsky
being extremely mathematically-minded showed extraordinary abilities
and rapid progress which couldn’t help drawing the attention of the
famous mathematicians of the time.
He entered the University of Kazan at the age of 14 and due to his
brilliant work he succeeded in taking a Master’s degree four years later.
His duties as an assistant professor consisting of delivering lectures on
physics and astronomy, of being responsible for building up the
University library and founding the University Museum did not prevent
him from carrying out his math research. N. Lobachevsky’s having been
appointed rector of Kazan University in 1827 resulted in many great
improvements there. His being overburden with teaching and
administration did not keep N. Lobachevsky from creating one of the
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20.

In spite of all the services being rendered to his country, N.
Lobachevsky was relieved by the tsarist government in 1846 of
his rectorship without being given any explanation or reasons. It
was no good arguing the issue and broken by this injustice N.
Lobachevsky soon fell ill and lost his eyesight.
Everyone heard of Pangeometry, the result of all his scientific
work having been dictated by N. Lobachevsky when he was
completely blind. It is worth mentioning that his ideas
influenced greatly not only the development of maths but other
sciences as well: mechanics, physics, astronomy. Few
discoveries in the world of science can equal those of N.
Lobachevsky. That is why we think of N. Lobachevsky, like
Archimedes, Galileo, Copernicus and Newton, belonging to the
20
galaxy of scientists who laid the foundations of science.

21.

22.

Nikolai Ivanovich Lobachevsky, a distinguished Russian mathematician, was
born on December 1, 1792, near Nizhny Novgorod in the Russian Empire. Of
Russian and Polish heritage, he was one of three children. Lobachevsky's
revolutionary work in geometry, specifically Lobachevskian or hyperbolic
geometry, earned him the epithet "the Copernicus of Geometry" by William
Kingdon Clifford. This title underscores the profound and transformative nature of
his contributions, which reshaped the landscape of geometrical understanding.
Lobachevsky's most significant contribution to mathematics was a series of
five papers titled "O nachalakh geometrii" (On the Principles of Geometry),
published in 1829 and 1830. These papers presented a bold challenge to Euclid's
fifth postulate, also known as the parallel postulate. This postulate, a cornerstone
of Euclidean geometry, asserts that through a point not on a given line, only one
line can be drawn parallel to the given line. Lobachevsky dared to question this
fundamental assumption, proposing an alternative postulate that allowed for the
existence of multiple lines parallel to a given line through a single point.
This radical departure from Euclidean principles led to the development of
hyperbolic geometry, a non-Euclidean geometry that describes a curved, saddleshaped space. In this geometry, the angles of a triangle sum to less than 22180

23.

Initially, Lobachevsky's groundbreaking work was met with
skepticism and indifference from the mathematical community. His
ideas were so far removed from conventional thinking that they
struggled to gain acceptance. However, his genius was eventually
recognized in the 20th century when Albert Einstein's theory of general
relativity provided a physical context for non-Euclidean geometries.
Einstein's theory demonstrated that space itself could be curved by
gravity, and Lobachevsky's hyperbolic geometry became an essential
Lobachevsky's academic journey began at Kazan University, where he excelled
tool for understanding the relativistic universe.
and earned his Master of Science degree in 1811. His dedication and talent led him
to become a professor at the university, and he later served as its rector for an
impressive 19 years. Beyond his work in non-Euclidean geometry, Lobachevsky
made significant contributions to other areas of mathematics, including his
fundamental research on Dirichlet integrals, which resulted in the formulation of
the Lobachevsky integral formula. His diverse mathematical pursuits highlight his
comprehensive understanding and mastery of the field.
Lobachevsky's legacy is one of intellectual courage and unwavering dedication
to the pursuit of truth. He famously wrote to his father, "From nothing I have
created a new different world," capturing his excitement and sense of
accomplishment in developing his revolutionary ideas about parallels. His
commitment to his research is further exemplified by his statement, "I thought I
would sacrifice myself for the sake of the truth. I was ready to become a martyr
23
who would remove the flaw from geometry and return it purified to mankind."

24.

Write a
sentence
summary
the text
3of
24

25.

https://docs.google.com/forms/d/e/1FAIpQLSf8nhXjtj2b
39sh93Cw2Xd2pez1aiQwHMNDMlhUGoZcL8UZjQ/vie
wform?usp=preview
25

26.

Short
Answer
Questions
1.
2.
3.
4.
5.
6.
7.
8.
9.
What was the name of the series of papers that Lobachevsky published in
1829 and 1830, and what was their significance?
How did the mathematical community initially react to Lobachevsky's
work, and what eventually led to its acceptance?
Beyond his work in non-Euclidean geometry, what other significant
contribution did Lobachevsky make to mathematics?
What was Lobachevsky's academic background, and how did it contribute
to his later success?
What is the connection between Lobachevsky's work on Dirichlet integrals
and his broader contributions to mathematics?
How does the text portray Lobachevsky's character and his approach to his
work?
Explain how Lobachevsky's work on hyperbolic geometry challenged the
prevailing understanding of space and geometry.
Based on the text, what can we infer about the significance of
Lobachevsky's academic background in shaping his contributions to
mathematics?
26
How does the text illustrate the concept of 'scientific progress' as a process

27.

10. Explain how Lobachevsky's work on hyperbolic geometry challenged the
prevailing understanding of space and geometry, and how this challenge
ultimately contributed to the development of modern physics.
11. Analyze the significance of Lobachevsky's statement, "From nothing I have
created a new different world," in relation to his contributions to
mathematics and the impact of his work on subsequent generations of
mathematicians and scientists.
12. Based on the text, how did Lobachevsky's academic journey at Kazan
University contribute to his intellectual development and his ability to
challenge established ideas in mathematics?
13. Compare and contrast Lobachevsky's revolutionary work in geometry to the
scientific revolution that occurred in the 16th and 17th centuries. How did
both movements challenge established paradigms and lead to new
understandings of the world?
14. Drawing parallels between Lobachevsky's work and the development of
quantum mechanics, explain how both areas of scientific inquiry challenged
classical assumptions and led to a deeper understanding of the universe.
15. Relate Lobachevsky's dedication to his research, as exemplified by his
statement, "I thought I would sacrifice myself for the sake of the truth,"27to
the concept of 'artistic integrity' as seen in the work of artists like Vincent

28.

Open-ended
Prompts
1. Lobachevsky's work was initially met with skepticism and indifference. How
have you experienced similar situations where your ideas or perspectives were
not immediately accepted? How did you handle those situations, and what did
you learn from them?
2.
Lobachevsky's dedication to his research is evident in his statement, 'I thought
I would sacrifice myself for the sake of the truth.' What are you passionate
about, and how far are you willing to go to pursue your passions? What does it
mean to you to 'sacrifice yourself for the sake of the truth’?
3.
The text describes Lobachevsky as a 'visionary thinker who dared to challenge
the established order.' What are some examples of people in your own life or
in history who have challenged the status quo? What impact did their actions
have, and what lessons can we learn from their experiences?
4.
Lobachevsky's work in non-Euclidean geometry was initially considered
radical and even absurd. How do you think new ideas and concepts are often
perceived, and what can we do to foster an environment that is more open to
innovation and challenging existing paradigms?
28

29.

6.
Lobachevsky's work was eventually recognized and validated
by Einstein's theory of relativity. How can we learn to
appreciate the value of seemingly abstract or theoretical ideas,
even if their practical applications are not immediately
apparent?
7.
8.
Lobachevsky's contributions to mathematics were diverse,
encompassing non-Euclidean geometry, Dirichlet integrals, and
other areas. How does exploring different fields or disciplines
broaden our understanding and perspectives? What are the
benefits
of havingLobachevsky's
a multi-faceted approach
to learning
and
The text
describes
work as
'transformative'
and
problem-solving?
'revolutionary.'
What are some examples of transformative ideas or
innovations that have shaped our world? How do these innovations challenge
us to think differently and adapt to change?
9.
The text mentions that Lobachevsky's work was initially met with skepticism
and indifference. How do you think this experience might have shaped his
character and his approach to research? What are some of the potential
benefits and drawbacks of facing resistance and criticism in the pursuit of
knowledge?
10. Lobachevsky's work in non-Euclidean geometry challenged the prevailing
understanding of space and geometry. How do you think our understanding
29
of the world is shaped by the assumptions and frameworks we use to

30.

Provide information
schemes.
on
the
30

31.

Read - Learn - Act out
Exploring
Lobachevsky’s
Contributions to Mathematics
-
-
-
Transformative
Characters:
Alex: A mathematics major with a keen interest in the history
of math.
Alex: Hey
Sarah, have you ever come across Nikolai Lobachevsky? I was delving
A history
major
about scientific
into hisSarah:
contributions
to geometry
and passionate
found them absolutely
fascinating!
andHe’s
theiroften
impacts.
Sarah:revolutions
Yes, I have!
called the "Copernicus of Geometry," isn’t he?
That really emphasizes how his work revolutionized mathematical thought in
much the same way that Copernicus changed our understanding of astronomy.
Alex: Exactly! Lobachevsky courageously challenged Euclid’s fifth postulate,
which posits that for any point not on a given line, there can be only one line
drawn parallel to that line. Lobachevsky proposed a remarkable alternative,
suggesting that more than one parallel line could pass through that point.
Sarah: That’s such a radical departure from the traditional framework! What does
this actually mean in practical terms, though?
Alex: Well, it led to the development of what we now refer to as hyperbolic
geometry. In this type of geometry, for example, the angles of a triangle can sum
to less than 180 degrees! Imagine trying to draw a triangle on the surface of a
saddle; you’d see that the angles behave quite differently than we expect in
31 flat,
Euclidean space.

32.

Read - Learn - Act out
- Alex: They certainly were! His theories were so unconventional that
many mathematicians dismissed them outright. It took a long time for his
work to gain traction. Surprisingly, it wasn’t until the 20th century, when
Einstein introduced his theory of general relativity, that Lobachevsky’s
non-Euclidean geometries were recognized for their significance in
understanding a curved space-time.
- Sarah: That's really interesting. It’s a testament to how innovative
thinking can take time to be appreciated. Did Lobachevsky contribute to
any other fields beyond geometry?
- Alex: Absolutely! In addition to his pioneering work in hyperbolic
geometry, he made notable contributions to integral calculus, particularly
in his studies on Dirichlet integrals. He formulated what is now known
as the Lobachevsky integral formula, which showcases the depth of his
mathematical expertise.
- Sarah: I really admire his dedication to mathematics. I read that he
famously wrote to his father about how he had created “a new different
world,” capturing the essence of what he achieved.
32
- Alex: Yes! His commitment to exploring truth, despite facing

33.

Read - Learn - Act out
- Sarah: Definitely! It’s amazing to see how his ideas not only
transformed geometry but also paved the way for advancements in
physics, influencing concepts related to the curvature of space itself.
Lobachevsky's legacy is indeed remarkable.
- Alex: For sure! It’s crucial to highlight figures like him in discussions
about the evolution of science, as they dared to think outside the box.
Their innovative ideas often lead to groundbreaking discoveries that
shape our understanding of the world.
- Sarah: I couldn’t agree more! Perhaps we should consider doing a
joint project that explores the impact of Lobachevsky's work on
modern mathematics and its implications in fields like physics. It
would be fascinating to investigate!
- Alex: That sounds like a brilliant idea! Let’s dive deeper into his
theories and examine how they reshaped our understanding of space,
geometry, and even the universe itself. I think it would yield some
compelling insights!
- Sarah: Absolutely! I’m looking forward to our exploration. This33 is

St-Lobachevsky_04-05-26

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