1. As a consequence of the exponential growth of science most mathematics has developed since the 15th century AD, and it is a historical fact that from the 15th century to the late 20th century new developments in mathematics were largely concentrated in Europe and North America. For these reasons the bulk of this article is devoted to European developments since 1500.

2. This does not mean, however, that developments elsewhere have been unimportant. Indeed, to understand the history of mathematics in Europe it is necessary to know its history at least in Mesopotamia and Egypt, in ancient Greece, and in Islāmic civilization from the 9th to the 15th centuries. The ways in which these civilizations influenced one another, and the important direct contributions Greece and Islām made to later developments, are discussed in the first parts of this article.

3. India's contributions to the development of contemporary mathematics were made through the considerable influence of Indian achievements on Islāmic mathematics during its formative years. In order to provide a portrait of the mathematical achievements of one major Asian civilization, the article contains an overview of some of the principal periods and achievements of mathematics in China.

4. It is important to be aware of the character of the sources for the study of the history of mathematics. The history of Mesopotamian and Egyptian mathematics is based on the many extant original documents written by scribes. Although in the case of Egypt these documents are few, they are all of a type and leave little doubt that Egyptian mathematics was, on the whole, elementary and profoundly practical in its orientation. For Mesopotamian mathematics, on the other hand, there are a large number of clay tablets, which reveal mathematical achievements of a much higher order than those of the Egyptians. The tablets indicate that the Mesopotamians had a great deal of remarkable mathematical knowledge, although they offer no evidence that this knowledge was organized into a deductive system. Future research may reveal more about the early development of mathematics in Mesopotamia or about its influence on Greek mathematics, but it seems likely that this picture of Mesopotamian mathematics will stand.

5. From the period before Alexander the Great no Greek mathematical documents have been preserved except for fragmentary paraphrases, and even for the subsequent period it is well to remember that the oldest copies of Euclid's Elements are in Byzantine manuscripts dating from the 10th century AD. This stands in complete contrast to the situation described above for Egyptian and Babylonian documents. Although in general outline the present account of Greek mathematics is secure, in such important matters as the origin of the axiomatic method, the pre-Euclidean theory of ratios, and the discovery of the conic sections, historians have given competing accounts based on fragmentary texts, quotations of early writings culled from nonmathematical sources, and a considerable amount of conjecture.

6. Many important treatises from the early period of Islāmic mathematics have not survived or have survived only in Latin translations, so that there are still many unanswered questions about the relationship between early Islāmic mathematics and the mathematics of Greece and India. In addition, the amount of surviving material from later centuries is so large in comparison with that which has been studied that it is not yet possible to offer any sure judgment of what medieval Islāmic mathematics did not contain, and this means that it is not yet possible to evaluate with any assurance what was original in European mathematics from the 11th to the 15th century.

7. In modern times the invention of printing has largely solved the problem of obtaining secure texts and has allowed historians of mathematics to concentrate their editorial efforts on the correspondence or the unpublished works of mathematicians. However, the exponential growth of mathematics means that, for the period from the 19th century on, historians are able to treat only the major figures in any detail. In addition there is, as the period gets nearer the present, the problem of perspective. Mathematics, like any other human activity, has its fashions, and the nearer one is to a given period, the more likely these fashions are to look like the wave of the future. For this reason, the present article makes no attempt to assess the most recent developments in the subject.

I. Read the passage and match the ideas with the right paragraph:

The achievements in mathematics in other ares of the world are also of great importance

The character of the sources for the study of the history of mathematics

The relationship between early Islāmic mathematics and the mathematics of Greece and India

Reason why the article is devoted to European developments

Reasons why from the 19th century on historians are able to treat only the major figures in any detail.

The mathematical achievements in Islāmic civilization

Greek mathematical documents have been preserved in fragmentary texts

II. Answer the questions:

1. Where were new developments in mathematics concentrated from the 15th century to the late 20th century?

2. What is necessary to understand the history of mathematics in Europe?

3. Did Islam make a great contribution to the development of mathematics?

4. Why is it important to be aware of the character of the sources for the study of the history of mathematics?

5. What do Egiptian mathematical documents show?

6. What was characteristic of Mesopotamisn mathematics?

7. What is the history of Greek mathematics based on? What mathematical spheres has it influenced?

8. How have important treatises from the early period of Islāmic mathematics survived?

9. Why is it impossible to offer any sure judgment of what medieval Islāmic mathematics did not contain?

10. How has the invention of printing affected historians of mathematics?

11. Why are historians able to treat only the major figures in detail for the period from the 19th century on?

12. What does the problem of perspective mean?

III. Do you agree that mathematics has fashions? Provide your arguments for or against.

IV. In mathematics the prase "exponential growth" means "ýêñïîíåíöèàëüíûé ðîñò ". Is it used in the same meaning in the following phrases from the passage:

the exponential growth of science

the exponential growth of mathematics

Translate the phrases into Russian.

V. Some nouns change their meanings when used in the plural. In what meaning are the italicized words used in the passage? How will you translate the sentences into Russian? Study the dictionary entries for these words.

…to the late 20th century new developments in mathematics have been largely concentrated in Europe and North America.

India's contributions to the developmentof contemporary mathematics…

VI. The word "evidence" is sometimes difficult to translate. Study the dictionaty entry and decide which Russian variants are applicable to mathematical texts:

How will you translate the following clause?

…although they offer no evidence that this knowledge was organized into a deductive system

Mathematics in ancient Mesopotamia

Until the 1920s it was commonly supposed that mathematics had its birth among the ancient Greeks. What was known of earlier traditions, such as the Egyptian as represented by the Rhind 's Papyrus (itself edited for the first time only in 1877), offered at best a meagre precedent. This impression gave way to a very different view as Orientalists succeeded in deciphering and interpreting the technical materials from ancient Mesopotamia.

Owing to the durability of the Mesopotamian scribes' clay tablets, the surviving evidence of this culture is substantial. Existing specimens of mathematics represent all the major eras—the Sumerian kingdoms of the 3rd millennium BC, the Akkadian and Babylonian regimes (2nd millennium), and the empires of the Assyrians (early 1st millennium), Persians (6th through 4th centuries BC), and Greeks (3rd century BC to 1st century AD). The level of competence was already high as early as the Old Babylonian dynasty, the time of the lawgiver king Hammurabi (c. 18th century BC), but after that there were few notable advances. The application of mathematics to astronomy, however, flourished during the Persian and Seleucid (Greek) periods.