VII. Make the sentences more emphatic. Tarn active into Passive, if necessary.

Model: The only numbers accepted by the Greeks

is …. that were the natural numbers.

It {

was … who It was the natural numbers that were the only numbers

accepted by the Greeks.

1.The Greeks first appreciated the power of mathematical reasoning. 2. The

Greeks gave mathematics a major place in their civilization. 3. The Greeks initiated patterns of thought that are still bask in our civilization. 4. The Greeks converted mathematics into abstract, deductive and axiomatic system of thought. 5. In constructing method of proof, mathematicians employ a high order of intuition, imagination and ingenuity. 6. Though Thales of M, proved some geometric theorems deductively, the Pythagoreans applied this process of exclusively and developed it further. 7. Mathematical theory emerged and evolved first in the mathematics of the early Greeks. 8. Euclid’s “Elements” were the first successful attempt to build all Geometry based on postulation thinking. 9. Euclid based his development of Geometry on a logical system. 10. Despite some shortcomings the “Elements” are a work of genius. 11. The Ancient Greece created the intellectual miracle of a logical system. 12. Euclid’s masterpiece was the first magnificent and epoch-making application of the axiomatic method. 13. Mathematicians study Euclid’s “Elements” to master the art of rigorous geometric reasoning. 14. The subject of Geometry was once almost synonymous with the name of Euclid. 15. Euclidean Geometry was the only Geometry for more than two thousand years. It is not any more. 16. “Elements” are no longer all Geometry, but this masterwork is the logical ideal of all science. 17. “Elements” is the most durable and influential textbook in the history of mathematics.

Model. Modern mathematics ignores the distinction between postulates and

axioms.

Modern mathematics does ignore this distinction.

1. Gauss conceived the idea of non-Euclidean Geometry long before other

creators of the subject. The term is due to him. 2. Gauss claimed that there exists a closed and consistent Geometry in which the Euclidean 5^th postulate does not hold. 3. Euclid himself realized the impossibility of deriving the parallel postulate from the rest axioms and postulates. 4. Many mathematicians after Euclid attempted to prove the parallel postulate by an indirect method (i.e., reduction ad absurdum). 5. Mathematicians tried to construct a geometry in which the negation (the converse) of the parallel postulate holds. 6. The futile and fruitless efforts to produce a proof of the parallel postulate led to the idea of a geometry with more than one parallel. 7. The discoverers of non- Euclidean Geometry wanted to show that the 5^th postulate is, in fact, deducible. 8. In this they failed. But they succeeded in finding a new word, a new consistent geometry with infinitely many parallel lines. 9. N. Lobachevsky published the first account dealing with non-Euclidean geometry and created the subject concerned. 10. Priority arguments are very important in science and that is why we honour N. Lobachevsky as the creator of non- Euclidean geometry. 11. Gauss and Bolyai independently drew the same conclusions from the impossibility of proving the parallel postulate. 12. Their contemporaries paid almost no attention at first to their novel and grand ideas. 13. New non-Euclidean geometry lacked intuitive appeal and so was almost ignored. 14. Euclidean Geometry rooted so firmly in thought that their contemporaries hardly recognized and appreciated the latter. 15. A few decades passed before mathematicians took notice of this grand discovery. 16. The new geometry gained intuitive appeal as a result of the “models” for non-Euclidean Geometry constructed by F. Klein, Poincare and Hilbert.