V. Make the sentences negative:

Model: The early Greeks dealt with numeration systems and counting. (practically)

The early Greeks did not practically deal with numeration systems and counting.

1.The Pythagoreans inherited more superior Babylonian positional numeration system. (In spite of infinite contact) 2. Their minds were inclined toward the elementary mechanical aspect of mathematics. (apparently) 3. The Greeks’ system of counting was simple (obviously) 4. It was positional system based on 10 (non-positional without place value or symbol zero) 5. They used special symbols for numbers. ( letters of their alphabet for numbers) 6. The Pythagoreans investigated and discovered many operations on real numbers, (properties of natural numbers) 7. They represented numbers as algebraic structures (geometric patterns) 8. They developed algebra (buried algebra in geometry) 9. They created a grand structure of number-systems similar to that for Geometry. (since their general outlook was geometric rather than arith-metical) 10. The principal Greek contributions were number systems. (since they were lascinated by the properties and not the operations on numbers) 11. There still exists only the Greeks’ classical number theory. (modern number theory) 12. Modern approach is definitely oriented toward the structural properties of numeration systems. (number systems, i.e., the properties of operations on numbers).

VI. Make the sentences impersonal using the noun-substitutes and modal verbs:

Model: Mathematicians claim that mathematics is man’s greatest intellectual

achievement (accomplishment)

One can claim that mathematics is man’s greatest intellectual achievement.

(accomplishment).

1. Historians assert that Geometry- the study of forms- was the special

concern of the classical Greeks. 2. The scientists failed to trace the exact date of the emergence of abstract notions of number and geometric figure. 3. Tradition credits the Pythagoreans with this greatest contribution. 4. It is necessary to emphasize that the appreciation of number as an abstract idea was one of the major advances in the history of thought. 5. It is true to say that pure abstract geometric form is the common property of all solids. 6. Scientists do not perform any experiments with pure forms. 7. Mathematicians deal exclusively with abstract pure forms in mathematics. 8. The Greeks introduced and applied extensively the pure deductive reasoning with abstract forms. 9. By means of deductive reasoning geometers reveal the essential properties and the relationships of geometric forms. 10. By abstracting the concept of a geometric figure the Greeks achieved the greatest level of generalization then known. 11. Nowadays mathematicians gain a new understanding of the abstract foundations of mathematics.

Model: Objects have both essential and accidental properties, the former – According to the Greeks only the former should enter, the latter into definition of an object.

1. The classical Greeks neglected experimentation and practical applications,

According to the their principle … was mechanical, and … was vulgar. 2. Current mathematics is a method of inquiry known as postulation thinking and a field for creative endeavor… constitutes responces to purely and purely intellectual challenges. 3. The Greeks had only one space and one geometry; these were absolute concepts. Today … is the set of objects together with a set of relations in which the objects are involved, and… is the theory of such a space. 4. The modern concept of Geometry is so embracive that the boundary lines between… and the other arcas of mathematics became very obliterated. 5. Some mathematicians claim that geometry is not a separate mathematical discipline but a point of view…. implies a particular way f at a subject. 6. The concept of axiomatic method originated in in Ancient Greece in the fifth century B.C. The modern form of axiomatic method was in a stage of evolution for more than of the XIX century. The application by Hilbert’s geometry. 7.Without doubt the most outstanding contribution of the early Greek was the formulation of the pattern of axiomatic. Unlike Hilbert’s pattern of “formal axiomatic”… is usually referred to as “material axiomatic”. 8. The modern form of axiomatic method and a very high level of abstraction characterize today’s mathematics,… is the unifying principle for all the branches of mathematics,… implies a higher order of abstraction compared to Euclid’s; as the objects, relations and operations are already themselves abstractions. 9.From the axiomatic point of view mathematics is a storehouse of abstract forms and mathematical structures. Most of… had originally a very definite intuitive content. 10. The interdependence in mathematics and the internationalism of its appeal are displayed by simultaneous discoveries in mathematics. The evidence of … is the discovery of non-Euclidean geometry by a German, a Russian and a Hungarian, who had no connections with or knowledge of each other.