Grammar: Complex Modals

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

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 “Geometry as a science”. Define the basic peculiarities of its functions and its role in our life. Grammar: Introduce and practice the Complex Modals.

Ex.1 Read and translate the text

Geometry as a science

Geometry (Ancient Greek: geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of a formal mathematical science emerging in the West as early as Thales (6th Century BC). By the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.^[1] Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. Both geometry and astronomy were considered in the classical world to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.

In Euclid's time there was no clear distinction between physical space and geometrical space. Since the 19th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation, and the question arose: which geometrical space best fits physical space? With the rise of formal mathematics in the 20th century, also 'space' (and 'point', 'line', 'plane') lost its intuitive contents, so today we have to distinguish between physical space, geometrical spaces (in which 'space', 'point' etc. still have their intuitive meaning) and abstract spaces. Contemporary geometry considers manifolds, spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales. These spaces may be endowed with additional structure, allowing one to speak about length. Modern geometry has multiple strong bonds with physics, exemplified by the ties between pseudo-Riemannian geometry and general relativity. One of the youngest physical theories, string theory, is also very geometric in flavour.

Gottfried Wilhelm von Leibniz (1646-1716) Germany

Leibniz was one of the most brilliant and prolific intellectuals ever; and his influence in mathematics (especially his co-invention of the infinitesimal calculus) was immense. His childhood IQ has been estimated as second-highest in all of history, behind only Goethe. Descriptions which have been applied to Leibniz include "one of the two greatest universal geniuses" (da Vinci was the other); "the most important logician between Aristotle and Boole;" and the "Father of Applied Science." Leibniz described himself as "the most teachable of mortals."

Mathematics was just a self-taught sideline for Leibniz, who was a philosopher, lawyer, historian, diplomat and renowned inventor. Because he "wasted his youth" before learning mathematics, he probably ranked behind the Bernoullis as well as Newton in pure mathematical talent, and thus he may be the only mathematician among the Top Ten who was never the greatest living algorist or theorem prover. We won't try to summarize Leibniz' contributions to philosophy and diverse other fields including biology; as just three examples: he predicted the Earth's molten core, introduced the notion of subconscious mind, and built the first calculator that could do multiplication. Leibniz also had

political influence: he consulted to both the Holy Roman and Russian Emperors; another of his patrons was Sophia Wittelsbach, who was only distantly in line for the British throne, but was made Heir Presumptive. (Sophia died before Queen Anne, but her son was crowned King George I of England.)

Ex.2 Mark true and false sentences:

1 Geometry (Ancient Greek: geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures

2 Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes

3 By the 2nd century BC geometry was put into an axiomatic form by Euclid

4 Newton developed ingenious techniques for calculating areas and volumes

5 Since the 17th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation

6 Both geometry and astronomy were considered in the classical world to be part of the Quadrivium

7 So today we have to distinguish between physical space, geometrical spaces (in which 'space', 'point' etc. still have their intuitive meaning

8 Modern geometry has not multiple strong bonds with physics, exemplified by the ties between pseudo-Riemannian geometry and general relativity

Ex 3 Answer the questions:

1 What is the meaning of the word “Geometry”?

2 When was geometry put into an axiomatic form by Euclid?

3 What do you know about Wilhelm von Leibniz?

Grammar: Complex Modals.