V. В математике много думают, рассуждают, доказывают, обосновывают.

There is much thinking, reasoning, proving and justifying in maths. Is there? There is no arguing (not any argument) in this theory. There exists (emerged) a new proof of this theorem. Does there exist…? There does not exist… Did there emerge…? There did not emerge a new proof of this theorem.

VI. Impers. (Пред) полагается (можно, нельзя, вряд ли следует…) It is (pre) supposed that mathematicians prove(d) theorems that way.

One (does not) suppose(s) (can hardly suppose)…

We (you, they) must (not) suppose…

People should (not) suppose…

VII. Emph. Именно (как раз) математики ( в самом деле, ведь) доказывают теоремы.

It is (was) mathematicians who prove(d) theorems. It is (was) deductively that mathematicians prove(d) theorems. Do prove theorems deductively Mathematicians do (did) prove theorems deductively. Whatever (Whichever) Euclid’s proof you take, it is deductive. The earlier you master the procedure of a deductive proof, the sooner you appreciate mathematical rigorous.

VIII. Noun Subst. The proof(s) by deductions is (are) much more rigorous than that of (those of) by induction. Deduction and rigorous are essentials of a mathematical proof. The former and the latter are essentials of a mathematical proof. These proofs are valid but try to establish more rigorous ones.

IX. Verb. Subst. Mathematicians prove theorems inductively rather rarely but physicists do it regularly. Mathematicians prove what they do (=prove) deductively and rigorously.

X. The verb “to do” functions.

1. These students do maths. 2. What do these student do? 3. They do prove theorems. Do prove this theorem deductively? 4. They do not prove theorems but we do. They prove what they do deductively.

CONVERSATIONAL EXERCISES

I.Use the following sentences in your oral description of the ancient practical (empirical, experimental) geometry:

1.The word “geometry” refers to the branch of mathematics. 2. Geometry

arose from practical necessities and appeared several thousand years before our era. 3. The need to bound land led to the notion of simple geometric concepts. 4. The notions of vertical, of parallel, and of perpendicular were suggested by the construction of buildings. 5. Empirical geometry was developed in certain areas of the ancient Orient – Egypt, Mesopotamia, India, China. 7. Geometry appeared as a science to assist surveying, agricultural and engineering pursuits. 8. The river basins (the Nile, the Tigris and Euphrates, the Indus, and Ganges, the Yangtze) cradled advanced forms of society known for their engineering skill. 9. Surveying and engineering project regdired the creation of much practical geometry. 10. Trial-and error methods and empirical procedures were the tools and means of discovery. 11. Geometry was a collection of laboratory results, some correct, others only approximate, concerning areas and volumes. 12. The earliest existing records of man’s geometrical activity were found.