III. Choose the proper modal verb or equivalent for the Russian words in the brackets.

1. You (нужно) construct a square with the area equal to the area of the given circle. 2. We (предстоит) find a good approximation to the number π value. 3. You (следует) specify the conditions of the experiment. 4. They (разрешат) use a dictionary if necessary. 5. I (в состоянии) solve this difficulty myself. 6. They (вынуждены) check all the calculations again. 7. We (пришлось) consult the reference book to fulfill the task. 8. The students (следует) appreciate the ancient mathematics in a proper way. 9. He (должен был) complete his experiment, that’s why he stayed longer in the laboratory. 10. If you like you (можете) stay here and listen to the professor’s lecture. 11. According to our curriculum we (должны) be through with integrals this term.

IV. Modal compound predicate. Study the rules of translation of modal verbs followed by different Infinitives and translate the sentences below the table into Russian.

He must solve the problem	Он должен решить задачу
He must be solving the problem	Он, должно быть, решает задачу
He must have solved the problem	Он, должно быть, решил задачу

He can (may) do his research			Он может сделать свое исследование
He	can (could) may (might)	be doing his research	Он, возможно, делает исследование
He	can (could) may(might)	have done his research	Он, возможно, сделал исследование

1. Babylonians must have been the first to solve the cubic equations by substitution. 2. Before Archimedes there might have been no systematic way of expressing numbers. 3. Algebraic formulas for finding the volumes of cylinders and spheres may have been used in Ancient Egypt to compute the amount of grain contained in them. 4. The discovery of the theorem of Pythagoras can hardly have been made by Pythagoras himself, but it was certainly made in his school. 5. Regardless of what mystical reasons could have motivated the early Pythagorean investigators, they discovered many curious and fascinating number properties. 6. Newton’s earliest manifestations of the higher mathematical talent may have passed unnoticed. 7. The symbol √ might have been used in the XVI century. 8. Now he is in the library – he must be getting ready for tomorrow’s math classes. 9. R. Descartes’s geometric representation of negative numbers could have helped mathematicians to make negative numbers more acceptable. 10. She may be having classes now.

Listening & Speaking