Euclidean geometry, founded by Euclid, is based on definitions, axioms, and theorems from which a systematic study of geometry emerged, notably in his text 'Elements'. The work includes 13 books covering plane geometry, number theory, and solid geometry, providing proofs for various propositions and theorems, including the Pythagorean theorem. Euclid's approach remained influential until the rise of non-Euclidean geometries in the 19th century, challenging previous assumptions about geometry's universality in describing physical space.

2. WHAT IS GEOMETRY ???

4. EuclideanGeometryis the study of geometrybasedon definitions,undefinedpoints, whichare a point,a lineand a
plane,and the assumptionsof ‘TheFatherof Geometry’.
The geometry of plane figure is known as ‘Euclid’sGeometry’.
Euclid’sGeometryincludessets of Axioms,and many theorems deducedfromthem.
Euclid’stext elements was the first systematicdiscussionof geometry.While many of Euclid’sfindingshad been
previouslystatedby earlierGreek mathematicians, Euclidis creditedwith developingthe first comprehensive deductive
system
Euclid’sapproachto geometryconsistedof proving all theoremsfroma finite number of postulesand axions.
The concepts of Euclid’sgeometry remainedunchallengeduntilthe early 19th century. At that time, otherformsof
geometry startedto emerge, callednon-Euclidean geometries.It was no longerassumedthatEuclid’sgeometry could be
used to describeall physical space.
Euclid’s Geometry

5. Basis of Euclid’s Geometry
The 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.

6. 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.
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.
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 BOOK
Euclid’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.

7. 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.

10. • 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

12. 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.

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