EUCLID’S GEOMETRY

EUCLID

WHAT IS GEOMETRY ???

Geometry is a branch of mathematics which deals with

question related to shape, size, relative position of figures and

the properties of space.

Geometry

is the study of Angles,

Triangles, Length,

Perimeter, Area and

Volume.

A mathematician who works in the field of geometry is

called a geometer.

The word ‘Geometry’ comes from the Greek

words ‘geo’, meaning the ‘Earth’ and ‘Metrein’, meaning ‘To Measure’.

Geometry appears to have originated from the need of measuring land.

Few of the known

Geometers of this

period are Euclid,

Archimedes, Rene

Descartes, Euler and

Gauss.

Geometry was one of the two fields of pre-modern Mathematics. The

other being the study of numbers.

EUCLIDEuclid, also known as ‘Euclid of Alexandria’, was a Greek

mathematician. He is also commonly known as ‘Father of Geometry’ because of his immense contribution in this form of mathematics.

He lived and worked during the period of Ptolemy I.(323B.C. – 283B.C.)

His elements is one of the most influential works in the history of mathematics, serving as the main and only source of imparting

knowledge in mathematics, especially Geometry.

“Euclid” is the anglicized version of the Greek name Εὐκλείδης, meaning “Good Glory”.

It is believed that Euclid would have studied at Plato’s academy at Athen’s.

There are only a few original records of his life has survived through the ages. So much is not known about his life.

Few other works of the famed father of geometry include works on perspective, conic sections, spherical geometry, number theory and

rigor.

Euclidean Geometry is the study of geometry based on definitions, undefined points, which are a point, a line and a plane, and the assumptions of ‘The Father of Geometry’.

The geometry of plane figure is known as ‘Euclid’s Geometry’.

Euclid’s Geometry includes sets of Axioms, and many theorems deduced from them.

Euclid’s text elements was the first systematic discussion of geometry. While many of Euclid’s findings had been previously stated by earlier Greek mathematicians, Euclid is credited with developing the first comprehensive deductive system

Euclid’s approach to geometry consisted of proving all theorems from a finite number of postules and axions.

The concepts of Euclid’s geometry remained unchallenged until the early 19th century. At that time, other forms of geometry started to emerge, called non-Euclidean geometries. It was no longer assumed that Euclid’s geometry could be used to describe all physical space.

Euclid’s Geometry

Basis of Euclid’s GeometryThe elements of Euclid is based on theorems proved by other

mathematical work.

Euclid put together many of Eudoxus’ theorems, many of Theaetetus theorems, and also made the theories, vaguely proved by his predecessors, more relevant.

Most of books I and II were based on Pythagoras,  book III on Hippocrates of Chios, and book V on Eudoxus , while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians

Euclid often replaced misleading proofs with his own.

The use of definitions, postulates, and axioms dated back to Plato.

The Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures.

Books I–IV and VI discuss plane geometry. Many results about plane figures are proved. Pons Asinorum i.e. If a triangle has two equal angles, then the

sides subtended by the angles are equal is proved. The Pythagorean theorem is proved.

Books V and VII–X deal with number theory. It deals with numbers treated geometrically through their

representation as line segments with various lengths. Prime Numbers and Rational and Irrational numbers are

introduced. The infinitude of prime numbers is proved.

Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and

a cylinder with the same height and base.

CONTENTS OF THE BOOKEuclid’s Geometry has 13 books, of which, books I–IV and VI discuss plane geometry; books V and VII–X deal with number theory and books XI–XIII

concern solid geometry.

The Greek mathematicians of Euclid’s time thought of geometry as an abstract model of the world they lived in. The notions of point, line, plain, etc. were derived from what was seen around them. Euclid summarised these notions as definitions.

The Elements begins with a list of definitions. It has been suggested that the definitions were added to

the Elements sometime after Euclid wrote them. Another possibility is

that they are actually from a different work, perhaps older Though Euclid defined a point, a line, and a plane, the

definitions are not accepted by mathematicians. Therefore, these terms are now taken as undefined.

Euclid deduced a total of 131 definitions. There were 23 in Book I, 2 in Book II, 11 in Book III, 7 in Book IV, 18 in Book V, 4 in Book VI, 22 in Book VII, 16 in Book X and 28 in Book XI.

EUCLID’S DEFINITIONS

Some of the definitions are:1. A point is that which has no part. 2. A line is breathless length. 3. Equal circles are those whose diameters are equal, or whose

radii are equal. 4. Circles are said to touch one another which meet one another

but do not cut one another. 5. A straight line is said to be fitted into a circle when its ends

are on the circumference of the circle. 6. A ratio is a sort of relation in respect of size between two

magnitudes of the same kind. 7. Magnitudes which have the same ratio be called proportional. 8. The height of any figure is the perpendicular drawn from the

vertex to the base. 9. An even number is that which is divisible into two equal parts.10.A solid is that which has length, breadth, and depth.11.Parallel planes are those which do not meet.

EUCLID’S DEFINITIONS

EUCLID’S AXIOMS & POSTULATES

Euclid assumed certain properties, which were not to be proved. These assumptions are actually ‘obvious universal truths (Axioms)’. He divided them into two types:

A. POSTULATES – He used the term ‘postulate’ for the assumptions that were specific to geometry.

B. COMMON NOTIONS – Common notions, on the other hand, were assumptions used throughout mathematics and not specifically linked to geometry.

Euclidean geometry is an axiomatic system, in which all theorems ("true statements") are derived from a small number of

axioms.These are ‘self-evident truths’ which we take to be true without proof.Axioms have been chosen based on our intuition and what appears to

be self-evident. Therefore, we expect them to be true.

• Things which equal the same thing also equal one another.– If a=b and b=c, then a=c

• If equals are added to equals, then the wholes are equal.– If a=b, then a+c = b+c

• If equals are subtracted from equals, then the remainders are equal.– If a=b, then a-c=b-c

• The whole is greater than the part.– 1 > ½

• Things which are double of the same things are equal to one another.– If a=2b and c=2b, then a=b

• Things which are halves of the same things are equal to one another.– If a= ½ b and c= ½ b, then a=c

EUCLID’S AXIOMS

The 5 postules which Euclid gave for plane geometry

1. A straight line may be drawn from any one point to any other point.

The first postulate says that given any two points such as A and B, there is a line AB which has them as

endpoints.2. A terminated line can be produced indefinitely.

The second postulate says that a line segment ( terminated line) can be extended on either side to

form a line.3. A circle can be drawn with any centre and any radius.

4. All right angles are equal to one another. This postulate says that an angle at the foot of one perpendicular, equals an angle at the foot of any other

perpendicular.5. If a straight line falling on two straight lines makes the

interior angles on the same side of it taken together less than two right angles, then the two straight lines, if

produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

EUCLID’S POSTULATES

After Euclid stated his postulates and axioms, he used them to prove other results. Then using these results, he proved some more results by applying deductive reasoning. The statements that were proved are called propositions or theorems.

Euclid deduced 465 propositions using his axioms, postulates, definitions and theorems proved earlier in the chain.

There were 48 propositions in Book I, 14 in Book II, 37 in Book III, 16 in Book IV, 25 in Book V, 33 in Book VI, 39 in Book VII, 27 in Book VIII, 36 in Book IX, 115 in Book X, 39 in Book XI, 18 in Book XII and 18 in Book XIII.

EUCLID’S PROPOSITIONS

EUCLID’S PROPOSITIONS1. In isosceles triangles the angles at the base equal one another,

and, if the equal straight lines are produced further, then the angles under the base equal one another.

2. If a straight line touches a circle, and a straight line is joined from the center to the point of contact, the straight line so joined will be perpendicular to the tangent.

3. If magnitudes are proportional taken separately, then they are also proportional taken jointly.

4. If two triangles have their sides proportional, then the triangles are equiangular with the equal angles opposite the corresponding sides.

5. If two numbers are relatively prime to any number, then their product is also relatively prime to the same.

6. If two similar plane numbers multiplied by one another make some number, then the product is square.

7. Any cone is a third part of the cylinder with the same base and equal height.

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