I. Look through the passage and find out the information about:

- the role of Euclid's Elements in university studies

- - who was active in Merton College, Oxford, in the first half of the 14th century

- - where the so-called latitude of forms, began to be discussed

- - what was proved by the Merton school about the quantity of motion

1. Mathematics was studied from a theoretical standpoint in the universities. The universities of Paris and Oxford, which were founded relatively early (c. 1200), were centres for mathematics and philosophy. Of particular importance in these universities were the Arabic-based versions of Euclid, of which there were at least four by the 12th century. Of the numerous redactions and compendia which were made, that of Campanus (c. 1250; first printed in 1482) was easily the most popular, serving as a textbook for many generations. Such redactions of the Elements were made to help students not only to understand Euclid's textbook but also to handle other, particularly philosophical, questions suggested by passages in Aristotle. The ratio theory of the Elements provided a means of expressing the various relations of the quantities associated with moving bodies, relations that now would be expressed by formulas. Also in Euclid were to be found methods of analyzing infinity and continuity (paradoxically, because Euclid always avoided infinity).

2. Studies of such questions led not only to new results but also to a new approach to what is now called physics. Thomas Bradwardine, who was active in Merton College, Oxford, in the first half of the 14th century, was one of the first medieval scholars to ask whether the continuum can be divided infinitely, or whether there are smallest parts (indivisibles). Among other topics, he compared different geometric shapes in terms of the multitude of points that were assumed to compose them, and from such an approach paradoxes were generated that were not to be solved for centuries. Another fertile question stemming from Euclid concerned the angle between a circle and a line tangent to it (called the horn angle): if this angle is not zero, a contradiction quickly ensues, but, if it is zero, then, by definition, there can be no angle. For the relation of force, resistance, and the speed of the body moved by this force, Bradwardine suggested an exponential law. Nicholas Oresme (d. 1382) extended Bradwardine's ideas to fractional exponents.

Figure 2: Uniformly accelerated motion; s = speed, a = acceleration, t = time, and v = velocity. Encyclopædia Britannica, Inc.

3. Another question having to do with the quantification of qualities, the so-called latitude of forms, began to be discussed at about this time in Paris and in Merton College, Oxford. Various Aristotelian qualities (e.g., heat, density, and velocity) were assigned an intensity and extension, which were sometimes represented by the height and bases (respectively) of a geometric figure. The area of the figure was then considered to represent the quantity of the quality. In the important case in which the quality is the motion of a body, the intensity its speed, and the extension its time, the area of the figure was taken to represent the distance covered by the body. Uniformly accelerated motion starting at zero velocity gives rise to a triangular figure.

4. It was proved by the Merton school that the quantity of motion in such a case is equal to the quantity of a uniform motion at the speed achieved halfway through the accelerated motion; in modern formulation: s = ^{1}/_{2}at^{2} (Merton rule). Discussions like this certainly influenced Galileo indirectly and may have influenced the founding of coordinate geometry in the 17th century. Another important development in the scholastic “calculations” was the summation of infinite series.

5. Basing his work on translated Greek sources, the German mathematician and astronomer Regiomontanus wrote the first book in the West on plane and spherical trigonometry independent of astronomy in about 1464 (printed in 1533). He also published tables of sines and tangents that were in constant use for more than two centuries.

II. Match the statements with the paragraphs:

a) Thomas Bradwardine, who was active in Merton College, made some important discoveries in physics

b) Euclid was widely studied at universities in the Middle Ages

c) Mathematical investigations in universities affected later developments in the 17th century.

d) The first book in the West on plane and spherical trigonometry was written in about 1464

e) A question dealing with the quantification of qualities attracted the scholars' attention at that time

III. Read the passage again and single out the mathematical and physical problems discussed and solved at universities in the Middle Ages.

IV. Does the word "compendia" stand in the plural or in the singular? Can we learn about it without looking it up in the dictionary? How? Can we predict the missing form for the pair?

Of the numerous redactions and compendia which were made, that of Campanus (c. 1250; first printed in 1482) was easily the most popular…

In what meaning is the word "easily" used in the sentence? Look it up in the dictionary and then translate the whole sentence.

V. Analyse the meanings of the modal verbs and the forms of the Infinitives they are used with and translate the clauses.

1.…that were not to be solved for centuries.

2.…to ask whether the continuum can be divided infinitely…

3. Discussions like this certainly influenced Galileo indirectly and may have influenced the founding of coordinate geometry in the 17th century.

VI. The passage contains a lot of sentences with the predicate in the Passive Voice. Analyse them and suggest the best way of translating them.

VII. How do you understand the following sentence? What complicates it?

Also in Euclid were to be found methods of analyzing infinity and continuity

VIII. What parts of speech does the word "tangent" represent in the following sentences? Is it used in the same meaning?

1. Another fertile question stemming from Euclid concerned the angle between a circle and a line tangent to it (called the horn angle): if this angle is not zero, a contradiction quickly ensues, but, if it is zero, then, by definition, there can be no angle.

2. He also published tables of sines and tangents that were in constant use for more than two centuries.

Translate the sentences. When doing it, also pay attention to the function and translation of the pronoun "it" and try to find the best way of translating the word "fertile".

IX. Suggest the Russian counterparts for the following words and word combinations:

of particular importance, at least, to handle a question, a moving body, methods of analyzing infinity and continuity, to lead to, an approach to, geometric shapes, in terms of, to solve a paradox, to stem from, the angle between a circle and a line tangent to it, by definition; the relation of force, resistance, and the speed of the body; an exponential law, a fractional exponent, to have to do with, heat, density, and velocity, a geometric figure, an area of the figure, to cover a distance, uniformly accelerated motion, to give rise to, a uniform motion, an accelerated motion, coordinate geometry, the summation of infinite series, plane and spherical trigonometry, tables of sines and tangents, to be in constant use.