Unit 8 Theme:Descartes’s and p.Fermat’s coordinate geometry. Grammar: Conjunctions

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 Descartes’s and P.Fermat’s coordinate geometry.

Students should know the rule of Conjunctions and fulfill grammar exercises.

Students should be better at discussing about Curves of second order.

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 “Descartes’s and P.Fermat’s coordinate geometry.”.Define the important aspects of geometry. Discuss in groups.

Grammar: Introduce and practice conjunctions. Revise the use of conjunctions.

Descartes’s and p.Fermat’s coordinate geometry.

A correspondence is similarly established between the algebraic and analytic properties of the equation f (x, y) = 0, and the geometric properties of the associated curve. The task of proving a theorem in geometry will cleverly be shifted to that of proving a corresponding theorem in algebra and analysis.

There is no unanimity of opinion among historians of mathematics concerning who invented Analytic Geometry, nor even concerning what age should be credited with the invention. Much of this difference of opinion is caused by a lack of agreement regarding just what constitutes Analytic Geometry. There are those who, favouring Antiquity as the era of the invention, point out the well-known fact that the concept of fixing the position of a point by means of suitable coordinates was employed in the ancient world by the Egyptians and the Romans in surveying, and by the Greeks in map-making. And, if Analytic Geometry implies not only the use of coordinates but also the geometric interpretation of relations among coordinates then a particularly strong argument in favour of crediting the Greeks is the fact that Appolonius (c. 225 B. C.) derived the bulk of his geometry of the conic sections from the geometrical equaivalents of certain Cartesian equations of these curves, the idea which originated with Menaechmus about 350 B. C.

Others claim that the invention of Analytic Geometry should be credited to Nicole Oresme, who was born in Normandy about 1323 and died in 1382 after a career that carried him from a mathematics professorship to a bishopric. N. Oresme in one of his mathematical tracts, anticipated another aspect of Analytic Geometry, when he represented certain laws by graphing the dependent variable against the independent one, as the latter variable was permitted to take on small increments. Advocates for N. Oresme as the inventor of Analytic Geometry see in his work such accomplishments as the first, explicit introduction of the equation of a straight line and the extension of some of the notions of the subject from two-dimensional space to three, and even four-dimensional spaces. A century after N. Oresme's tract was written, it enjoyed several printings and in this way it may possibly exert some influence on the succeeding mathematicians.

However, before Analytic geometry could assume its present highly practical form, it had to wait the development of algebraic symbolism, and accordingly it may be more correct to agree with the majority of historians, who regard the decisive contributions made in the seventeenth century by the two French mathematicians, R. Descartes (1596-1650) and P. Fermat (1601-1663), as the essential origin of at least the modern spirit of the subject. After the great impetus given to the subject by these two men, we find Analytic Geometry in a form with which we are familiar today. In the history of mathematics a good deal of space will be devoted to R. Descartes and P. Fermat, for these men left very deep imprints on many subjects. Also, in the history of mathematics, much will be said about the importance of Analytic geometry, not only for the development of Geometry and for the theory of curves and surfaces in particular, but as an indespensable force in the development of the calculus, as the influential power in molding our ideas of such farreaching concepts as those of "function" and "dimension".

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

1. Why was not there unanimity of opinion among scientists about the invention of analytic geometry? Because of … .

a) the position of a point b) a lack of agreement

c) suitable coordinates d) a map-making of the Greeks

2. Who derived the bulk of geometry?

a) Appolonius b) Fermat

c) Menaechmus d) Descartes

3. Who left deep imprints on many subjects?

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

c) Menaechmus c) N. Oresme

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

a. Who invented analytic geometry?

b. Analytic geometry

c. Algebraic symbolism

d. R. Descartes and P. Ferma

Ex.3. Translate the highlighted word correctly.

1. Every student of mathematics meets the remarkable subject called Analytic Geomenry.

a) называемыйb) назывался

c) называетсяd) будет называться

2. This is the task of proving a theorem in geometry.

a) доказыватьb) доказательства

c) доказывающаяd) докажут

3. There is no unanimity of opinion among historians of mathematics concerning who invented analytic geometry.

a) интересующихсяb) интересуются

c) для интересаd) интерес

4. In the history of mathematics much will be said about far-reaching concepts as of “function’ and “dimension”.

a) сказатьb) сказали

c) скажутd) сказали бы

Ex.4.Listening

Follow the link: http://www.youtube.com/watch?v=YxgH5ykGwdQ (Philosophy of Mathematics: Platonism)

Grammar: Conjunctions

Do exercises from Units 113-120 p.226-241 ex.:113.1-120.4 (Raymond Murphy “English Grammar in Use” A self-study reference and practice book for intermediate students of English Third Edition. Cambridge).

БӨЖ тапсырмалар:

Make the presentation «Leading Mathematicians of the Time».

Follow the link and pass the tests for grammar:

http://www.study.ru/test/test.php?id=213

http://www.study.ru/test/test.php?id=214