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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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