Analysis incarnate — Leonard Euler.

Though P. Fermat and R. Descartes founded analytic geometry they did not advance the subject far enough and did not elaborate it purely analytically either. A century later L. Euler (1707-1783) a Swiss mathematician who lived the greater part of his life in Russia, engaged in scientific research, lecturing and textbook writing in St. Petersburg Academy, developed the subject matter of both Plane and Solid Analytic Geometry far beyond its elementary stages. Euler's mathematical career opened when Analytic Geometry (made public in 1637) was ninety years old, the calculus about fifty. In each of these fields a vast number of isolated problems were solved, but no systematic unification of the whole of the then mathematics, pure and applied, was made. In particular, the powerful analytic methods of Fermat, Descartes, Newton and Leibnitz were not exploited to the limit of what they were capable, especially in Calculus, Geometry and Mechanics, where Euler proved himself the master.

In the XVIII c. the Universities were not the principal centres of science in Europe. The lead in scientific research was taken by the various royal academies. In Euler's case St. Petersburg and Berlin furnished the sinews of mathematical creation. Both of these foci of creativity owed their inspiration to the restless ambition of Leibnitz. These academies were like some of these today: they were research organizations which paid their leading members to produce scientific research. Euler became famous for his great output of original mathematics and for the wide range of subjects he covered. He contributed new ideas to calculus, geometry, Algerba, Number Theory, Calculus of variations, probability and Topology. He also worked in many areas of applied mathematics, such as Acoustics, Optics, Meachanics, Astronomy, Ballistics, Navigation, Statistics and Finance. His industry was as remarkable as his genius. Euler was the most prolific mathematician in histoty; his scientific heritage is vast, a list of some 850, works of which 550 were published in the lifetime. Euler wrote his great memoirs quite easily and total blindness during the last seventeen years of his life did not regard his unparalleled productivity. He overcame the difficulty of blindness chiefly by means of his remarkable memory. Indeed, if anything, the loss of his eyesight sharpened Euler's perception of the inner world of his imagination.

Ex.1. Choose a,b,c or d.

1. Who founded analytic geometry?

a) Lagrange and Fermat b) P. Fermat and R. Descartes

c) Euler and Fermat d) Lagrange and Euler

2. Where did Euler live the greater part of his life?

a) in Russia b) Alexandria

c) Switzerland d) Greece

3. What did Euler become famous for ….

a) his great out put of original mathematics.

b) his great out put of analitic geometry.

c) his great out put of calculas.

d) his great out put of mechanics.

4. In what parts of mathematics did Euler especially prove himself as a master?

a) in differential equations.

b) in analytic geometry and caleulus.

c) in calculus, geometry and mechanics.

d) in arithmetics and physics.

Ex.2. Choose the title of the text according to summary.

a. New discovery in analytic geometry

b. Euler is innovator of mathematics

c. Euler’s genius and remarkablity of his industry

d. Euler is a swiss scientist

Ex.3. Translate the highlighted word correctly.

1. Euler engaged in scientific research lecturing many subjects.

a) читая лекция b) читающий лекции

c) читать лекции d) читал бы лекции

2. In each of the fields a vast number of isolated problems were solved.

a) решались b) решая

c) будут решаться d) решали бы

3. The powerful analytic methods were not exploited to the limit.

a) не будут разработаны b) не разрабатываются

c) не были разработаны d) не разрабатывались бы

4. These academies were research organizations which paid their leading members to produce scientific research..

a) чтобы проводить научно-исследовательскую работу.

b) проводя научно-исследовательскую работу.

c) проводящим научно-исследовательскую работу.

Ex.4.Listening

Follow the link: http://www.youtube.com/watch?v=NX9YbtYJZ8Y (Elementary Mathematics (K-6) Explained 0: Introduction)

Grammar: Prepositions

Do exercises from Units 121-128 p.242-257 ex.:121.1-128.6 (Raymond Murphy “English Grammar in Use” A self-study reference and practice book for intermediate students of English Third Edition. Cambridge).

БӨЖ тапсырмалар: 9

Speak about what you understood from the text “A modern view of geometry ” or write an essay.

Follow the link and pass the test:

http://www.study.ru/test/test.php?id=215

Unit 10

Theme:The basic and new concepts.

Objectives: By the end of this unit, students should be able to use active vocabulary of this theme in different forms of speech exercises.

Students should be better at discussing about The basic and new concepts.

Students should know the rule of Conditionals and ‘wish’

Methodical instructions: This theme must be worked out during two lessons a week according to timetable.

Lexical material: Introduce and fix new vocabulary on theme “The basic and new concepts”.

Define the important aspects of geometry. Discuss in groups.

Grammar: Conditionals and ‘wish’. Introduce and practice Conditionals. Revise the use of Conditionals.

The basic and new concepts.

The basic concepts of the main branches of mathematics are abstractions from experience, implied by their obvious physical counterparts. But it is noteworthy, that many more concepts are introduced which are, in essence, creations of the human mind with or without any help of experience. Irrational numbers, negative numbers and so forth are not wholly abstracted from the physical practice, for the man's mind must create the notion of entirely new types of numbers to which operations such as addition, multiplication, and the like can be applied. The notion of a variable that represents the quantitative values of some changing physical phenomena, such as temperature and time, is also at least one mental step beyond the mere observation of change. The concept of a function, relationship between variables, is almost totally a mental creation.

The more we study mathematics the more we see that the ideas and conceptions involved become more divorced and remote from experience, and the role played by the mind of the mathematician becomes larger and larger. The gradual introduction of new concepts which more and more depart from forms of experience finds its parallel in geometry and many of the specific geometrical terms are mental creations.

As mathematicians nowadays working in any given branch discover new concepts which are less and less drawn from experience and more and more from human mind the development of concepts is progressive and later concepts are built on earlier notions. These facts have unpleasant consequences. Because the more advanced ideas are purely mental creations rather than abstractions from physical experience and because they are defined in terms, of prior concepts it is more difficult to understand them and illustrate their meanings even for a specialist in some other province of mathematics. Nevertheless, the current introduction of new concepts in any field enables mathematics to grow rapidly. Indeed, the growth of modern mathematics is, in part, due to the introduction of new concepts and new systems of axioms.

Ex.1. Choose a,b,c or d.

1. What are the basic concepts of the main branches of mathematics?

a) abstractions from experience b) experience

c) physical abstractions d) quantitative values

2. What does the notion of a variable represent?

a) the quantitative values of some constant physical phenomena.

b) the quantitative values of some changing physical phenomena.

c) mathematical concepts.

d) functions.

3. Where does the gradual introduction of new mathematical concepts find its parallel in?

a) physics b) geometry

c) experience d) physical practice

4. Where are the new concepts remoted from?

a) experience b) human mind

c) geometry d) mathematics

5. Why are the new concepts more difficult to understand?

a) they are not defined in terms.

b) they are defined in terms of prior concepts.

c) they are defined in terms of physical concepts.

d) they involve many difficult notions.

6. What do new concepts enable mathematics to do?

a)to grow rapidly.

b) to grow slowly.

c) to stop the development.

Ex.2. Choose the title of the text according to summary.

a. The basic and new concepts c. Modern mathematicians

b. The basic concepts d. Irrational numbers