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Rajat AgrawalFolgen

- 2. • Geometry, branch of mathematics that deals with shapes and sizes. • Geometry may be thought of as the science of space. Just as arithmetic deals with experiences that involve counting, so geometry describes and relates experiences that involve space. • Basic geometry allows us to determine properties such as the areas and perimeters of two-dimensional shapes and the surface areas and volumes of three-dimensional shapes. • People use formulas derived from geometry in everyday life for tasks such as figuring how much paint they will need to cover the walls of a house or calculating the amount of water a fish tank holds.
- 3. • Euclid (lived circa 300 BC), Greek mathematician, whose chief work, Elements, is a comprehensive treatise on mathematics in 13 volumes on subjects such as plane geometry, proportion in general, the properties of numbers, incommensurable magnitudes, and solid geometry. • He probably was educated at Athens by pupils of Plato. He taught geometry in Alexandria and founded a school of mathematics there. • The Data, a collection of geometrical theorems; the Phenomena, a description of the heavens; the Optics; the Division of the Scale, a mathematical discussion of music; and several other books have long been attributed to Euclid; most historians believe, however, that some or all of these works (other than the Elements) have been spuriously credited to him. • Historians disagree as to the originality of some of his other contributions. Probably the geometrical sections of the Elements were primarily a rearrangement of the works of previous mathematicians such as those of Eudoxus, but Euclid himself is thought to have made several original discoveries in the theory of numbers (see Number Theory).
- 4. • Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. • Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Euclid was the first to show how these propositions could FIT into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It only works for geometry of three dimensions . Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.
- 5. Euclid's axioms: In his dissertation to Trinity College, Cambridge, Bertrand Russell summarized the changing role of Euclid's geometry in the minds of philosophers up to that time. It was a conflict between certain knowledge, independent of experiment, and empiricism, requiring experimental input. This issue became clear as it was discovered that the parallel postulate was not necessarily valid and its applicability was an empirical matter, deciding whether the applicable geometry was Euclidean or non-Euclidean.
- 6. Euclid gives five axioms for plane geometry, stated in terms of constructions. It was further translated by Thomas Heath. These are- • Things that are equal to the same thing are also equal to one another (Transitive property of equality). • If equals are added to equals, then the wholes are equal (Addition property of equality). • If equals are subtracted from equals, then the remainders are equal (Subtraction property of equality). • Things that coincide with one another are equal to one another (Reflexive Property). • The whole is greater than the part.
- 7. Euclid, who lived about 300 bc, realized that only a small number of postulates underlay the various geometric theorems known at the time. He determined that these theorems could be deduced from just five postulates. 1. A straight line may be drawn through any two given points. 2. A straight line may be drawn infinitely or be limited at any point. 3. A circle may be drawn using any given point as the center, and with any given radius (the distance from the center to any point on the circle). 4. All right angles are congruent. (A right angle is an angle that measures 90°. Two geometric figures are congruent if they can be moved or rotated so that they exactly overlap.) 5. Given a straight line and a point that does not lie on the line, one and only one straight line may be drawn that is parallel to the first line and passes through the point. These five postulates can be used in combination with various defined terms to prove the properties of two- and three-dimensional figures, such as areas and circumferences. These properties can in turn be used to prove more complex geometric theorems.
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