EUCLID’SGEOMETRY

Snehal Bhargava

WHAT IS GEOMETRY???

Geometry (geo-earth, metron-measurement) is a branch of mathematics which deals with questions related to shape, size, relative position of figures,

and the properties of space.Geometry is the study of angles and triangles,

length, perimeter, area and volume.

A mathematician who works in the field of geometry is called a geometer.

Few of the known geometers are Euclid, Archimedes, René Descartes,

Euler and Gauss.Euclid

WHAT IS THE HISTORY OF GEOMETRY???

Early geometry was a collection of principles concerning lengths, angles, areas, and volumes, which

were developed to meet some practical need in surveying, construction, astronomy, and various

crafts. In the 7th century BC, a Greek mathematician used to

solve problems such as calculating the height of pyramids and the distance of ships from the shore. The oldest surviving Latin translation of

the Elements Around 300 BC, geometry was revolutionized by

Euclid, whose Elements, widely considered the most successful and influential textbook of all time,

introduced the axiomatic method and is the earliest example of the format still used in mathematics

today, that of definition, axiom, theorem, and proof. A European and an Arab practicing geometry in the

15th century.

WHO WAS EUCLID???

Euclid was a Greek mathematician, who is known as the "Father of Geometry".

His Elements is one of the most influential works in the history of mathematics, In the Elements, Euclid

deduced the principles of what is now called Euclidean geometry from a small set of axioms.

Euclid’s life is not known. Nothing is known about his birth or death. Even description of Euclid's physical

appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art is the

product of the artist's imagination. It is believed that Euclid may have studied at Plato's Academy in Athens.

Euclid in Raphael's School of Athens

WHAT IS THE EUCLID’S GEOMETRY???

It includes sets of axioms, and many theorems deduced from them.

Euclid was the first to show how these theorems could fit into a comprehensive

deductive and logical system.It has 13 books, of which, books 1–4 and 6 discuss plane geometry; books 5 and 7–10 deal with number theory and books 11–13

concern solid geometry.

Fragment of Euclid's Elements

WHAT IS THE BASIS OF EUCLID’S GEOMETRY???

• The Elements is based on theorems proved by other mathematics supplemented by some original work.

• Euclid put together many of Eudoxus' theorems, many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors.

• Most of books 1 and 2 were based on Pythagoras,  book 3 on Hippocrates of Chios, and book 5 on Eudoxus , while books 4, 6, 11 and 12 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.

In Triangle ABC, Angle B = Angle C

Therefore, by Pons Asinorum, AB=AC.

WHAT IS THE CONTENT IN THE BOOK???

Books 1–4 and 6 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.A

B C

Books 5 and 7–10 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.

2, 3, 5, 7, 9 etc. are Prime Nos..-1/2, 5/7, 11/16 etc. are Rational Nos.

√2, 1.74653875…… , √71 etc. are Irrational Nos.

Books 11–13 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.

EUCLID’S DEFINITIONS• The Elements begins with a list of definitions.• Euclid deduced a total of 131 definitions.• 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.

• Some of the Definitions by Euclid are :-1. A point is that which has no part.2. A line is breathless length.3. An even number is that which is divisible into two

equal parts. 4. A solid is that which has length, breadth, and depth.5. Parallel planes are those which do not meet.

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EUCLID’S AXIOMS• 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.

• Some of the Axioms stated by Euclid are :-1. Things which equal the same thing also equal one another.

– If A=B and C=B, then A=C

2. If equals are added to equals, then the wholes are equal.– If A=B, then A+C=B+C

3. If equals are subtracted from equals, then the remainders are equal.– If A=B, then A-C=B-C

4. The whole is greater than the part.– 1 > 1/2

5. Things which are halves or double of the same things are equal to one another.– If A= ½ B and C= ½ B or A=2B and A=2C, then A=C

EUCLID’S POSTULATES• Each postulate is an axiom—which means a statement which is

accepted without proof— specific to the subject matter. Most of them are constructions. 

• Some of the Postulates by Euclid are :-1. A straight line may be drawn from any one point to any other

point. 2. A terminated line can be produced indefinitely.3. A circle can be drawn with any centre and any radius.4. All right angles are equal to one another.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 PROPOSITIONS• 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.

• Some of the Propositions stated by Euclid are :-1. 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 magnitudes are proportional taken separately, then they are also proportional taken jointly.

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

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

“I tell you that I accept God simply. But you must note this: If God exists and if He really did create the world, then, as we all

know, He created it according to the geometry of Euclid.” - Ivan, in The

Brothers Karamazov, by Fyodor Dostoyevsky (1821-1881)

THANK YOU