GEOMETRYGEOMETRYFrom: From:

http://encarta.msn.com/encyclopedia_761569706_6/ Gehttp://encarta.msn.com/encyclopedia_761569706_6/ Geometry.htmlometry.html

http://math.youngzones.org/geom_history.htmlhttp://math.youngzones.org/geom_history.html

HISTORYHISTORY

Geometry—from the Greek words geô, “earth,” Geometry—from the Greek words geô, “earth,” and metrein, “to measure”and metrein, “to measure”– an accurate description of the works of the an accurate description of the works of the

earliest geometers earliest geometers – concerned with problems such as measuring concerned with problems such as measuring

the size of fields and laying out accurate right the size of fields and laying out accurate right angles for the corners of buildings.angles for the corners of buildings.

Geometry: Timeline Geometry: Timeline EgyptiansEgyptians c. 2000 - 500 B.C. - Ancient Egyptians c. 2000 - 500 B.C. - Ancient Egyptians

demonstrated a practical knowledge of demonstrated a practical knowledge of geometry through surveying and construction geometry through surveying and construction projects. projects. – The Nile River overflowed its banks every The Nile River overflowed its banks every

year, and the river banks would have to be year, and the river banks would have to be re-surveyed. In the Rhind Papyrus, pi is re-surveyed. In the Rhind Papyrus, pi is approximated. approximated.

Babylonians Babylonians c. 2000 - 500 B.C. - Ancient clay tablets reveal c. 2000 - 500 B.C. - Ancient clay tablets reveal

that the Babylonians knew the Pythagorean that the Babylonians knew the Pythagorean relationships. One clay tablet reads, “4 is the relationships. One clay tablet reads, “4 is the length and 5 the diagonal. What is the breadth? length and 5 the diagonal. What is the breadth? Its size is not known. 4 times 4 is 16. 5 times 5 Its size is not known. 4 times 4 is 16. 5 times 5 is 25. You take 16 from 25 and there remains is 25. You take 16 from 25 and there remains 9. What times what shall I take in order to get 9. What times what shall I take in order to get 9? 3 times 3 is 9. 3 is the breadth”. 9? 3 times 3 is 9. 3 is the breadth”.

Greeks Greeks c. 750-250 B.C. - Ancient Greeks practiced c. 750-250 B.C. - Ancient Greeks practiced

centuries of experimental geometry like Egypt centuries of experimental geometry like Egypt and Babylonia had, and they absorbed the and Babylonia had, and they absorbed the experimental geometry of both of those experimental geometry of both of those cultures. cultures.

Created the first formal mathematics of any kind Created the first formal mathematics of any kind by organizing geometry with rules of logic. by organizing geometry with rules of logic. Euclid's (400BC) important geometry book The Euclid's (400BC) important geometry book The Elements formed the basis for most of the Elements formed the basis for most of the geometry studied in schools ever since.geometry studied in schools ever since.

After the fall of the Greek and Roman After the fall of the Greek and Roman civilizations, Europe entered the Dark Ages. civilizations, Europe entered the Dark Ages.

Advances in geometry were made largely by Advances in geometry were made largely by Muslims in the Middle East and North Africa Muslims in the Middle East and North Africa and Hindus in India. and Hindus in India.

Most of the works of Greek mathematics were Most of the works of Greek mathematics were scattered or lost. Some of these, including scattered or lost. Some of these, including Elements, were translated and studied by the Elements, were translated and studied by the Muslims and Hindus. Muslims and Hindus.

During the 12th and 13th centuries Elements was During the 12th and 13th centuries Elements was translated from Greek and Arabic into Latin and the translated from Greek and Arabic into Latin and the modern European languages, and geometry was modern European languages, and geometry was added to the curriculum of monastery schools.added to the curriculum of monastery schools.

The next great stride in geometry was taken by The next great stride in geometry was taken by French philosopher and mathematician René French philosopher and mathematician René Descartes, whose influential treatise Discourse on Descartes, whose influential treatise Discourse on Method was published in 1637. He introduced a Method was published in 1637. He introduced a method of representing geometric figures within a method of representing geometric figures within a coordinate system. His work forged a link between coordinate system. His work forged a link between geometry and algebra by showing how to apply the geometry and algebra by showing how to apply the methods of one discipline to the other. This link is methods of one discipline to the other. This link is the basis of analytic geometry, a subject that the basis of analytic geometry, a subject that underlies much modern work in geometry.underlies much modern work in geometry.

In the 18th century, Gaspard Monge, a French In the 18th century, Gaspard Monge, a French professor of mathematics, developed still another professor of mathematics, developed still another branch of geometry, called descriptive geometry. branch of geometry, called descriptive geometry. – Descriptive geometry is the science of making Descriptive geometry is the science of making

accurate, two-dimensional drawings, or accurate, two-dimensional drawings, or representations, of three-dimensional representations, of three-dimensional geometrical forms and of graphically solving geometrical forms and of graphically solving problems relating to the size and position in problems relating to the size and position in space of such forms. Descriptive geometry is space of such forms. Descriptive geometry is the basis of much of engineering and the basis of much of engineering and architectural drafting.architectural drafting.

Modern GeometryModern Geometry Analytic, projective, and descriptive geometry came Analytic, projective, and descriptive geometry came

into being within the framework of Euclidean into being within the framework of Euclidean geometry. geometry.

For many centuries mathematicians believed that For many centuries mathematicians believed that Euclid’s fifth postulate of the unique parallel could Euclid’s fifth postulate of the unique parallel could be proved on the basis of Euclid’s first four be proved on the basis of Euclid’s first four postulates, but all efforts to discover such a proof postulates, but all efforts to discover such a proof were fruitless. In the 19th century, however, were fruitless. In the 19th century, however, geometries were developed in which Euclid's fifth geometries were developed in which Euclid's fifth postulate was replaced by alternative statements. postulate was replaced by alternative statements. The leaders in developing these non-Euclidean The leaders in developing these non-Euclidean geometries were Carl Friedrich Gauss, János Bolyai, geometries were Carl Friedrich Gauss, János Bolyai, Nikolay I. Lobachevsky, and George Friedrich Nikolay I. Lobachevsky, and George Friedrich Bernhard Riemann.Bernhard Riemann.

In 1872 German mathematician Felix Klein In 1872 German mathematician Felix Klein used a relatively new branch of mathematics used a relatively new branch of mathematics called group theory to unify and classify all the called group theory to unify and classify all the geometries of his time. In 1899 David Hilbert, geometries of his time. In 1899 David Hilbert, another German mathematician, published his another German mathematician, published his Foundations of Geometry, which provided a Foundations of Geometry, which provided a rigorous system of axioms for Euclidean rigorous system of axioms for Euclidean geometry and exerted great influence on other geometry and exerted great influence on other branches of mathematics.branches of mathematics.

In 1916 the theory of relativity showed that In 1916 the theory of relativity showed that many physical phenomena could be deduced many physical phenomena could be deduced from geometric principles. The success of the from geometric principles. The success of the theory gave impetus to studies in differential theory gave impetus to studies in differential geometry and in topology.geometry and in topology.

The Five Postulates of EuclidThe Five Postulates of Euclid

1.1. It is possible to draw a straight It is possible to draw a straight lineline from any from any pointpoint to another to another pointpoint. .

2.2. It is possible to produce a finite straight It is possible to produce a finite straight lineline continuously in a straight continuously in a straight lineline. .

3.3. It is possible to describe a It is possible to describe a circlecircle with any with any centercenter and and radiusradius. .

The Five Postulates of EuclidThe Five Postulates of Euclid

1.1. All All right anglesright angles are equal to one another. are equal to one another.

2.2. If a straight If a straight lineline falling on two straight falling on two straight lineslines makes the interior makes the interior anglesangles on the same side less on the same side less than two than two right anglesright angles, the straight , the straight lineslines (if (if extended indefinitely) meet on the side on extended indefinitely) meet on the side on which the which the anglesangles which are less than two which are less than two right anglesright angles lie. lie.

PointPoint

PointPoint – defined by (x,y) – defined by (x,y)

x = abscissa or x – x = abscissa or x – coordinatecoordinate

y = ordinate or y- y = ordinate or y- coordinatecoordinate

P1

P2d

y

x

LineLine LineLine - shortest distance between two points - shortest distance between two points

d

AngleAngle The union of two raysThe union of two rays

angle

PlanePlane

The union of three noncollinear pointsThe union of three noncollinear points

P

Two lines are parallel if they are on the same Two lines are parallel if they are on the same

plane and will never intersect(alsohave the plane and will never intersect(alsohave the

same slope).same slope).

Two lines are perpendicular if they intersect Two lines are perpendicular if they intersect

and the intersection form a right angle(90 and the intersection form a right angle(90

degrees).degrees).

If two lines do not intersect, then they are either If two lines do not intersect, then they are either

parallel lines or they are skew lines.parallel lines or they are skew lines.

Two Parallel linesTwo Parallel lines

Two skew lines( the two lines do not lie on the Two skew lines( the two lines do not lie on the

same plane and which do not intersect)same plane and which do not intersect)

Triangle PropertiesTriangle Properties

In a triangle, the sum of any two side must be In a triangle, the sum of any two side must be

greater than the third side.greater than the third side.

In a right triangle, the sum of the squares of the In a right triangle, the sum of the squares of the

two legs is equal to the square of the two legs is equal to the square of the

hypotenuse.hypotenuse.

CongruenceCongruence

Three well-known criteria for the congruence of Three well-known criteria for the congruence of

trianglestriangles

a) sasa) sas

b) asab) asa

c) sssc) sss