He made a mistake in proving the theorem.
1. The solution of the cubic was, of course, unknown at that time, but Leonardo showed great ability when he obtained a very close approximate solution.
2. The middle years of the seventeenth century constitute the greatest period of mathematical activity, and when we describe the work of this period it must be remembered that mathematicians no longer worked in comparative isolation.
3. The fundamental theorem used when one finds extreme values of functions is the necessary condition that the differential vanishes at an extreme point.
4. When one considers the mathematical work of J. Kepler, one is chiefly impressed by the quality of his imagination.
5. The detailed investigation of this historical question is lengthy, but it is of sufficient interest and importance when we give the main facts.
6. We have already said that Henry IV was much struck by the ability shown by Viet when he was solving a certain problem.
7. Diophantus (Диофант) was satisfied with a single solution of such equations, and his amazing ingenuity (мастерство) was shown when he dealt with each equation as a particular case.
8. The variation plays the same role when one determines extreme values of functionals as the differential does when one finds maxima and minima of functions.
Ex. 6. Answer the following questions using ‘by + Gerund’ structure.
Model: How can you solve this problem? ( to use Pythagoras’ Theorem)
We can solve this problem by using Pythagoras’ Theorem.
1. How will the author introduce new concepts concerning functionals in this section? (to apply some familiar results from the theory of functionals)
2. How does Napier (a Scottish mathematician) obtain the notation of a logarithm? (to compare two motions)
3. How does Wallis (an English mathematician) reach several remarkable results in this work? (often to deduce general propositions from a number of particular cases)
4. How did he verify this result? (to extract the square root of (1 - x²), and also to multiply the above series by itself)
5. How can we find six unknowns? (to solve the system of six equations)
6. How can we obtain the unknown quantities? (to solve the triangle)
7. How did mathematicians find a way out of the dilemma? (to change their conception of what a number is to what we nowadays call the real number)
8. How did he obtain a root of several cubics? (to use conic sections)
9. How does the search begin? (to find the curves that satisfy the fundamental theorem)
Ex. 7. Translate the sentences with the Complex Gerundial Construction.
Вспомните! Герундиальный оборот при переводе на русский язык всегда вводится словами: то, что; тем, что; как; после того, как.
Model. His having failed the exam was a great disappointment.
То, что он не сдал экзамен, было большим разочарованием.
1. Our being invited to take part in the conference is very important for us.
2. We heard of those equations having been solved by the students at the previous seminar.
3. Do you mind my being examined first?
4. Newton’s having discovered the law of gravitation contributed much to the world science.
5. We knew of his having translated the text a week ago.
6. On lecturer’s appearing in the hall, there was loud applause.
7. Their not having done the task was quite a surprise.
8. They know of his presenting a paper at the seminar.
9. His having proved the theorem was very important for him.
Ex. 8. State whether the -ing form is the Gerund, the Participle or the Verbal Noun. Give your reasons.
1. He was not a professional mathematician, being an engineer and an architect, but he came into contact with the greatest mathematicians of his time.
2. We started the reading of English books last year.
3. He repeated Torricelli’s experiments, and showed that barometric readings really did depend on atmospheric pressure by obtaining, at the same moment, readings at different heights on the slope of the hill of the volcano.
4. By considering a number of special cases Cavalieri finally arrived at a theorem.
5. In the last section of the book various theorems are proved relating to the centrifugal force on a body moving in a circle.
6. Even today, despite the simple picture of the real numbers as the points on a line, university students of mathematics always have trouble understanding the formal (and highly abstract) development of the real numbers.
7. Similarly, most people have difficulty in coming to terms with complex numbers – numbers that involve the square root of negative quantities, such as i=√-1.
8. Intuitively speaking, the norm of the difference of two functions should be zero if the functions are identical, small if the functions are “close”, and large if the functions are “far apart”.
9. John Pell was a minor seventeenth-century English mathematician who was mistakenly credited by Euler with having investigated this equation.
Ex. 9. Translate the following word combinations and make up sentences using them:
a) On hearing about the conference, on arriving at the university, on entering the classroom, on learning about the lecture, on hearing the question;
b) The habit of getting up early, the way of doing it, the necessity of organizing it, the hope of meeting them, the chance of winning the prize, the way of travelling, the pleasure of reading this book.
Ex. 10. Translate into English using the following models:
a) I can’t help doing the task. - Я не могу не сделать задание. (Не принять участия в этой конференции; не перевести этот текст с русского на английский, не доказать эти теоремы; не прочитать книгу).
b) It’s no use solving the problem. - Бесполезно решать эту задачу. (Пытаться найти эту книгу; объяснять это правило ему ещё раз; говорить с ними на эту тему; искать решение этого уравнения).
c) We enjoyed reading this book. – Нам понравилось читать эту книгу. (Слушать лекцию этого профессора; участвовать в конференции; делать этот доклад; посещать семинарские занятия).
d) Do you mind our attending your lecture? - Вы не возражаете, если мы посетим вашу лекцию? (Если он представит свою работу первым; если я немного опоздаю на занятие; если она попробует перевести это предложение ещё раз; если они сдадут экзамен раньше).
e) I remember proving the theorem last week. - Помню, что доказывал эту теорему на прошлой неделе. (Читал этот текст на английском, учил эти слова, обсуждал эту книгу с ним, переводил эти статьи).
Pre-Reading Activity
Guess the meaning of the following words:
differential adj. [dIfq'renSql] original adj. [O'rIGqnl]
general adj. ['Genqrql] primitive adj. ['prImItIv]
constant adj. ['kOnstqnt] integration n. [IntI'greISqn]
algebraical adj. [xlGI'breIkql] theoretical adj. [TIq'retIkql]
transcendental adj. [trxnsen'dentl] theorem n. ['TIqrqm]
coefficient n. [kouI'fISqnt] unique adj. [ju:'ni:k]
classify v. ['klxsIfaI] initial adj. [I'nISql]
linear adj. ['lInIq] integrable adj. [In'tIgrqbl]
rationalize v. ['rxSnqlaIz] integral adj. ['IntIgrql]
total adj. [toutl]