2018-02-13
735
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 Greek geometria, 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.
Pronunciation guide
Abu Sahl al-Quhi
Girard Desargues
Hellenistic
Interior
Janos Bolyai
Postulate
Spatial
B Comprehension
Read the text and choose the correct answer.
1 Geometry was first used to solve
A common problems.
B abstract problems.
2 During the Classical Period of Greek civilisation,
A the way problems were solved changed.
B people were only interested in geometry.
3 The Greeks made important advances in geometry
A only during the Classical Period.
B during both the Classical and Hellenistic Periods.
4 After the decline of Greek civilisation,
A mathematicians used geometry to solve other kinds of problems.
B nothing new was discovered.
5 Between the 17th and 19th centuries, European thinkers
A ignored the Greeks' ideas about geometry.
B created new types of geometry.
Before you read
Discuss these questions with your partner.
-» What developments in geometry do you know about that occurred during the Renaissance
(c 15th -17th centuries AD)?
-» What differences and what similarities can you think of between maths and philosophy?
D Vocabulary
Find a synonym in the box for the words or phrases in green in the sentences.
Then check your answers in the text.
n knowledgeable n tutor
n prosperity n synthetic geometry
n goal n advance
n on the continent n analytic geometry
N harsh
1 The country was going through a time of successfulness..............................................
2 The climate was very cold and unpleasant..............................................
3 Her aim in life was to become a philosopher.............................................
4 He lived in Europe, not Britain..............................................
5 She studied a kind of geometry that used theorems and observations to reach conclusions.............................................
6 He taught a kind of geometry involving curves and understanding them..............................................
7She is very well-educated; she always beats me at Trivial Pursuit..............................................
8 Computer technology is expected to develop immensely in the next couple of years.
.............................................
9 She wants to find a Maths teacher to teach her children at home..............................................
Reading
Rene Descartes
Rene Descartes was born in France on 31st March, 1596, at a time of major change in the world. The great wars which had been going on throughout Europe had finally ended, creating an atmosphere of peace and stability which encouraged creative thinking, experimentation and the questioning of old beliefs and ways. After the fall of Constantinople in 1453, Greek and Islamic texts had been rediscovered and read by learned men around Europe. Ideas of the great Renaissance artists and thinkers had quickly spread across the continent. What is more, with the discovery of the New World by Columbus in 1492, a period of exploration, expansion and prosperity had begun.
After completing his education at the Jesuit College and the University of Poitiers, both in France, Descartes began to work on his goal of presenting a new way of looking at philosophy and mathematics. Although his first essays were probably written earlier than 1628, the year he moved to Holland, he was not well known until 1637, when a collection of his essays appeared and attracted the interest of the scientific world. His great work Discourse on the Method was one of the essays included in this collection.
Descartes was knowledgeable about the work of Plato and Aristotle, as well as that of earlier European philosophers like Augustine and Aquinas. Descartes' goal was to reach true knowledge about things by applying mathematical methodology to find answers to philosophical questions. Starting with the principle that the only thing he could be sure of was that he himself existed (Cogito, ergo sum meaning, I think, there/ore I am), he reached his own conclusions about God and the physical world. Because his ideas were very different from traditional ideas of his time, he was often criticised by religious leaders. His work had a great influence on later philosophers, including Benedict de Spinoza, Blaise Pascal, John Locke and Immanuel Kant.
Another of his goals was to advance the field of mathematics, particularly geometry. Until that time, Euclidean geometry was the type most well known. Also known as synthetic geometry, Euclidean geometry uses theorems and observations to reach conclusions.
Building on the work of the ancient Greek, Apollonius of Perga (262-190 BG), Descartes realised that it would be useful and important to be able to measure curved lines in addition to straight ones. This led to his invention of the Cartesian coordinate system, a way of algebraically measuring curves and understanding things about them. This was the start of analytic geometry (also called coordinate geometry and Cartesian geometry) and eventually led to the invention of calculus. In addition to his work in philosophy and geometry, Descartes contributed to algebra, optics and even physiology and psychology.
Descartes became one of the most important figures of his time. Queen Christina of Sweden invited Descartes to tutor her, which he did. However, he became ill in Sweden, possibly because he was not used to the cold, harsh climate, and died on 11th February, 1650. To honour him for his many contributions, people call him the 'Founder of Modern Philosophy' and the 'Father of Modern Mathematics'.
Pronunciation guide
Aquinas
Cartesian
Constantinople
Poitiers
Psychology
Renaissance
E Comprehension
