best seen with animations. the texts and picture might not be clear without animations.

1. EUCLID‘S
GEOMETRY

3. 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.
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.
The oldest surviving Latin translation of the Elements

5. 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 I–IV and VI
discuss plane geometry; books V and VII–X deal
with number theory and books XI–XIII concern
solid geometry.
Fragment of Euclid's Elements

6. 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 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

7. CONTENTS OF THE BOOK
• 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.

8. Some of the definitions are:
1. A point is that which has no part. (D1, B1)
• The Elements begins with a list of definitions.
2. A line is breathless length. (D2, B1)
• It has been suggested that the definitions were added to the
3. Equal circles are those whose diameters are equal, or whose radii are
Elements sometime after Euclid wrote them. Another
equal. (D1, B3)
possibility is that they are actually from a different work,
4. Circles are said to touch one another which meet one another but do not
perhaps older.
cut one another. (D3, B3)
5. • A Though line is said to be fitted into a circle when its ends are on the
straight Euclid defined a point, a line, and a plane, the
definitions of the circle. (D7, by
circumference are not acceptedB4) mathematicians. Therefore,
6. A theseis a sortare now taken as undefined.
ratio terms of relation in respect of size between two magnitudes of
• the same kind. (D3, B5)
Euclid deduced a total of 131 definitions. There were 23 in
7. Magnitudes which have the in Book III, 7 in Book IV, 18 in BookB5)4
Book I, 2 in Book II, 11 same ratio be called proportional. (D6, V,
in Book of any in Book VII, 16 in Book X and 28 in the vertex
8. The height VI, 22 figure is the perpendicular drawn fromBook XI. to
the base. (D4, B6)
9. An even number is that which is divisible into two equal parts. (D6, B7)
10.A solid is that which has length, breadth, and depth.(D1, B11)
11.Parallel planes are those which do not meet. (D8, B11)

9. • Euclid assumed certain properties, which were not to be
proved. These assumptions are actually ‗obvious universal
truths (Axioms)‘. He divided them into two types:
1. POSTULATES – He used the term ‗postulate‘ for the
assumptions that were specific to geometry.
2. COMMON NOTIONS– Common notions, on the other hand,
were assumptions used throughout mathematics and not
specifically linked to geometry.

10. Some of the common notions are:
• Things which equal the same thing also equal one another.
• Euclidean geometry is an axiomatic system, in which
– If a=b and b=c, then a=c
all theorems ("true statements") are derived from a small
• If equals of axioms. to equals, then the wholes are equal.
number are added
– If a=b, ‗self-evident truths‘ which we take to be true
• These arethen a+c = b+c
• If equals are subtracted from equals, then the remainders are
without proof.
• equal. have been chosen based on our intuition and what
Axioms
– If a=b, then a-c=b-c
appears to be self-evident. Therefore, we expect them to be
• The whole is greater than the part.
true.
– 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=b

11. Euclid gave 5 postulates for plane geometry:
1. A straight line may be drawn from any one point to any other
point.
• Each postulate is an axiom—which means a statement
 The first postulate says that given any two to the subject
which is accepted without proof— specific points such
as A and B, there is a line AB which has
matter. Most of them are constructions. them as endpoints.
A terminated actually a verb. When we say ―let us
•2. ‗Postulate‘ is line can be produced indefinitely.
 The second postulate says that a line segment ( terminated
postulate‖, we mean, ―let us make some statement based on
theline) can bephenomenon in the Universe‖. line.
observed extended on either side to form a Its
3. truth/validity is checkedany centre and anyis true, then it is
A circle can be drawn with afterwards. If it radius.
4. acceptedangles are equal to one another.
All right as a ‗Postulate‘.
 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.

12. Few Euclid‘s Propositions are:
1. In isosceles triangles the angles at the base equal one another,
and, if the equal straight lines are produced further, then the
• After Euclidthe basehis postulates and axioms, he used them
angles under stated equal one another. (P1.5)
to prove other results. Then using these results, he proved
2. If a straight line touches a circle, and a straight line is joined from
some more results by of contact, deductive reasoning. The
the center to the point applying the straight line so joined will
statements that were proved are called propositions or
be perpendicular to the tangent. (P3.18)
3. theorems.
If magnitudes are proportional taken separately, then they are
also proportional taken jointly. (P5.18)
• Euclid deduced 465 propositions using his axioms,
4. postulates, definitions and theorems proved earlier in the
If two triangles have their sides proportional, then the triangles
are equiangular with the equal angles opposite the corresponding
chain.
• sides. (P6.5) 48 propositions in Book I, 14 in Book II, 37 in
There were
5. If two III, 16 in are relativelyin Book V, 33 in Book VI, their
Book numbers Book IV, 25 prime to any number, then 39 in
product is also relatively prime to the same. (P7.24)
Book VII, 27 in Book VIII, 36 in Book IX, 115 in Book X, 39
6. If two similar plane numbers and 18 in Book XIII.
in Book XI, 18 in Book XII multiplied by one another make
some number, then the product is square. (P9.1)
7. Any cone is a third part of the cylinder with the same base and
equal height. (P12. 10)
Px. y – x = Book No. ; y = Proposition No.