Text 2 The history of geometry

The earliest recorded beginnings of geometry can be come down to ancient Mesopotamia, Egypt, and the Indus Valley from around 3000 BCE. Early geometry was a collection of empirically discovered principles. These concerned lengths, angles, areas, and volumes. The principles were developed to meet some practical need in surveying, construction, astronomy, and various crafts. The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets and the Indian Shulba Sutras, while the Chinese had the work of Mozi, Zhang Heng, and the Nine Chapters on the Mathematical Art, edited by Liu Hui. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks.

Until relatively recently (i.e. the last 200 years), the teaching and development of geometry in Europe and the Islamic world was based on Greek geometry. Euclid’s Elements (c. 300 BCE) was one of the most important early texts on geometry, in which he presented geometry in an ideal axiomatic form, which came to be known as Euclidean geometry. Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but the others fell into disuse and were lost.

In the Middle Ages, mathematicians in medieval Islam contributed to the development of geometry, especially algebraic geometry and geometric algebra. Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin) (836–901) dealt with arithmetical operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry. Omar Khayyám (1048–1131) found geometric solutions to cubic equations, contributed to the development of non-Euclidian geometry.

In the early 17th century, there were two important developments in geometry. The first, and most important, was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry is the study of geometry without measurement, just the study of how points align with each other.

Two developments in geometry in the 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Lobachevysky, Bolyai and Gauss and of the formulation of symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann, working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems.

3. Найдите английские эквиваленты следующих слов и словосочетаний:

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