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Match these words with their definitions.

1 spatial A a way of drawing things so that they look real

2 roots B relating to physical space

3 perspective C beginning

4 concentric D with a common centre

5 intersect E limited

6 deduction F a conclusion

7 finite G to cut across

8 radius H a free two-dimensional shape

9 plane I a straight line from the centre to the edge of a circle

 

Reading

Geometry

Geometry (from the Greekgeometria, the Earth's measure} has its roots in the ancient world, where people used basic techniques to solve everyday problems involving measurement and spatial relationships. The Indus Valley Civilisation, for example, had an advanced level of geometrical knowledge - they had weights in definite geometrical shapes and they made carvings with concentric and intersecting circles and triangles. Gradually, over the centuries, geometrical concepts became more generalised and people began to use geometry to solve more difficult, abstract problems.

However, even though people in those times knew that certain relationships existed between things, they did not have a scientific means of proving how or why. That changed during the Classical Period of the ancient Greek civilisation (490 BC-323 BC). Because the ancient Greeks were interested in philosophy and wanted to understand the world around them, they developed a system of logical thinking (or deduction) to help them discover the truth. This methodology resulted in the discovery of many important geometrical theorems and principles and in the proving of other geometrical principles that had been known by earlier civilisations. For example, the Greek mathematician Pythagoras was the first person that we know of to have proved the theorem

a2 + b2 = c2.

Some of the most significant Greek contributions occurred later, during the Hellenistic Period (323BC-31 BC). Euclid, a Greek living in Egypt, wrote Elements, in which, among other things, he defined basic geometrical terms and stated five basic axioms which could be deduced by logical reasoning. These axioms or postulates, were: 1. Two points determine a straight line. 2. A line segment extended infinitely in both directions produces a straight line. 3. A circle is determined by a centre and distance. 4. All right angles are equal to one another. 5. If a straight line intersecting two straight lines forms interior angles on the same side and those angles combined are less than 180 degrees, the two straight lines if continued, will intersect each other on that side. This is also referred to as the parallel postulate. The type of geometry based on his ideas is called Euclidean geometry, a type that we still know, use and study today.

With the decline of Greek civilisation, there was little interest in geometry until the 7th century AD, when Islamic mathematicians were active in the field. Ibrahim ibn Sinan and Abu Sahl al-Quhi continued the work of the Greeks, while others used geometry to solve problems in other fields, such as optics, astronomy, timekeeping and map-making. Omar Khayyam's comments on problems in Euclid's work eventually led to the development of non-Euclidean geometry in the 19th century.



During the 17th and IS1*1 centuries, Europeans once again began to take an interest in geometry. They studied Greek and Islamic texts which had been forgotten about, and this led to important developments. Rene Descartes and Pierre de Fermat, each working alone, created analytic geometry, which made it possible to measure curved lines. Girard Desargues created projective geometry, a system used by artists to plan the perspective of a painting. In the 19th century, Carl Friedrich Gauss, Janos Bolyai and Nikolai Ivanovich Lobachevsky, each working alone, created non-Euclidean geometry. Their work influenced later researchers, including Albert Einstein.


Date: 2016-01-14; view: 1040


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