Before you read: Are analysis and mechanics connected with each other or do they exist separately? Do you think the article will treat them in contrast or in close relationship?

1. The scientific revolution had bequeathed to mathematics a major program of research in analysis and mechanics. The period from 1700 to 1800, “the century of analysis,” witnessed the consolidation of the calculus and its extensive application to mechanics. With expansion came specialization, as different parts of the subject acquired their own identity: ordinary and partial differential equations, calculus of variations, infinite series, and differential geometry. The applications of analysis were also varied, including the theory of the vibrating string, particle dynamics, the theory of rigid bodies, the mechanics of flexible and elastic media, and the theory of compressible and incompressible fluids. Analysis and mechanics developed in close association, with problems in one giving rise to concepts and techniques in the other, and all the leading mathematicians of the period made important contributions to mechanics.

2. The close relationship between mathematics and mechanics in the 18th century had roots extending deep into Enlightenment thought. In the organizational chart of knowledge at the beginning of the preliminary discourse to the Encyclopédie, d'Alembert distinguished between “pure” mathematics (geometry, arithmetic, algebra, calculus) and “mixed” mathematics (mechanics, geometric astronomy, optics, art of conjecturing). Mathematics generally was classified as a “science of nature” and separated from logic, a “science of man.” The modern disciplinary division between physics and mathematics and the association of the latter with logic had not yet developed.

3. Mathematical mechanics itself as it was practiced in the 18th century differed in important respects from later physics. The goal of modern physics is to explore the ultimate particulate structure of matter and to arrive at fundamental laws of nature to explain physical phenomena. The character of applied investigation in the 18th century was rather different. The material parts of a given system and their interrelationship were idealized for the purposes of analysis. A material object could be treated as a point-mass (a mathematical point at which it is assumed all the mass of the object is concentrated), as a rigid body, as a continuously deformable medium, and so on. The intent was to obtain a mathematical description of the macroscopic behaviour of the system rather than to ascertain the ultimate physical basis of the phenomena. In this respect the 18th-century viewpoint is closer to modern mathematical engineering than it is to physics.

4. Mathematical research in the 18th century was coordinated by the Paris, Berlin, and St. Petersburg academies, as well as by several smaller provincial scientific academies and societies. Although England and Scotland were important centres early in the century, with Maclaurin's death in 1746 the British flame was all but extinguished

5. During the period 1600–1800 significant advances occurred in the theory of equations, foundations of Euclidean geometry, number theory, projective geometry, and probability theory. These subjects, which became mature branches of mathematics only in the 19th century, never rivaled analysis and mechanics as programs of research.

I. Look up the words and expressions in the English-Russian dictonary. They will help you to understand the passage better.

to acquire identity, a partial differential equation, an ordinary differential equation, calculus of variations, infinite series, differential geometry, the theory of the vibrating string, particle dynamics, the theory of rigid bodies, the mechanics of flexible and elastic media, the theory of compressible and incompressible fluids, to develop in close association, to give rise to, a preliminary discourse , fundamental laws of nature, a point-mass, a continuously deformable medium, mathematical engineering.

II. The study of meaning. Look up the meanings of the following words in the English –Russian dictionary to understand how they are used in the passage. Translate the corresponding sentences into Russian.

Thought, identity, develop, respect

III. Suggest the best translation of the Absolute Participial Construction:

Analysis and mechanics developed in close association, with problems in one giving rise to concepts and techniques in the other…

IV. Fill in the missing form:

Singular

Plural

medium

phenomena

basis

analysis

calculi, calculuses

V. Look through the article and match the statements with paragraphs.

a) the classification of mathematics

b) centres coordinating mathematical research

c) the applications of calculus and analysis

d) the place of analysis and mechanics in mathematical research in the 18^{th} century

e) the difference between mathematical mechanics in the 18th century and later physics

VI. Read the article in details and answer the questions:

1. What did the period from 1700 to 1800 see?

2. What kind of specialisation did it lead to?

3. What did applications of analysis include?

4. How did analysis and mechanics develop?

5. Where did the roots of mathematics and mechanics extend?

6. How was mathematics classified and what was the place of mechanics in this classification?

7. What was the character of applied investigation in mathematical mechanics in the 18th century?

8. Does it differ from modern approach to physics?

9. How was mathematical research in the 18th century coordinated?

VII. Speak about:

- the consolidation of the calculus and its extensive application to mechanics

- the classification of mathematics in the 18th century

- how mathematical mechanics was practiced in the 18th century and at present