Euclid was a Greek mathematician from Alexandria known as the "Father of Geometry". His influential work Elements deduced the principles of Euclidean geometry from a small set of axioms. It served as the main textbook for teaching mathematics for over 2000 years. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and rigor. He established an innovative deductive system in geometry based on definitions, axioms, and theorems and used it to prove various geometric results, such as how to construct a regular dodecahedron.

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2. Eclids Work in the field
Eclids Work in the field of
of Mathematics
Mathematics
Euclid was, also known as Euclid of
Alexandria, was a Greek
mathematician, often referred to as the
"Father of Geometry". His Elements is
one of the most influential works in the
history of mathematics, serving as the
main textbook for teaching
mathematics (especially geometry). In
the Elements, Euclid deduced the
principles of what is now called
Euclidean geometry from a small set of
axioms. Euclid also wrote works on
perspective, conic sections, spherical
geometry, number theory and rigor.

3. Euclid's Geometry
● The word 'geometry' is derived from the greek
word 'geo' meaning 'Earth' and 'metron'
meaning 'measuring'. Thus, the word geometry
means 'earth measurement'.
● Euclid was the first mathematician who initiated
a new way of thinking the study of geometry
results by deductive reasoning based upon
previously proved results and some self-evident
specific assumptions called axioms.
AXIOMS : The basic fact which are taken for
granted, without proofs, are called axioms.

4. EUCLID's Definations
In the first Book of Elements, Euclid gave 23
definations and some of them are as follows :-
● A piont is that which has no part. .
● A line is breadthless lenght.
● The ends of a lina are points.
● A staight line is a line whick lies evenly with
the points on itself.
● A surface is that which has lenght and breadth
only.
● The edges of a surface are lines.

5. EUCLID's Axioms and
Postulates
Euclid assumed certain properties, which were
not be proved. These are actually 'obvious
universal truths'. He divided them into two types:
axioms and postulates
Postulates Axioms
● Postulates are universal
truths with out any proofs.
● Postulates are
assumptions used
specifically used for
geometry.
● Axioms are universal
thruths without any
proofs.
● Axioms are assumptions
used throughout
mathematics and not
specifically geometry.

6. Some Axioms and Postulates of
Euclid
Axioms
● Things which are equal
to the same things are
equal to one another.
● If equals are added to
equals, the wholes are
equal.
● The whole is greater
than a part.
● Things which are half of
the same things are
equal
Postulates
● A straight line may be
drawn from any one
point to any other point.
● A terminated line can be
produced indefinitely.
● A circle can be drawn
with any centre and any
radius.
● All right angles are equal
to one another

7. Euclid's Division Lemma
Theorem :If there two positive integers a and b,
there exist unique ineger q and r satisfying
a = bq + r
0 < r < b.
Trough this lemma the formation of the
fundamental theorem of arithematic took place
and Euclid's division algorithm is based on this
lemma.

8. Euclid's division lemma
Euclid's division lemma are used to obtain
the HCF of two positive integer, say c and d,
with c > d, follow the steps below:-
● Step 1: Apply Euclid's division lemma, to c
and d. So we find whole numbers, q and r
such that c = dq + r, 0 < r< d.
● Step 2: If r = 0, dis the HCF of c and d. If r is
not equal to 0, apply the division lemma to d
and r
● Step 3: Continue the process till the
remainder is zero. The divisor at this stage
will be the required HCF.
Euclid's division algorithm is also use full to
find the number of tiles and the dimention to
fill a space as shown in the animation.

9. Euclid's Construction of a
regular dodecahedron

10. Euclid's work on Data
The Data is closely related to the first
four books of the Elements. It opens
with definitions of the different senses
in which things are said to be ``given.''
Thus lines, angles, and ratios may be
given in magnitude, rectilinear figures
may be given in species or given in
form, points and lines may be given in
position,etc.

11. Euclid's work on Catoptrics
Catoptrics, which
concerns the
mathematical theory of
mirrors, particularly the
images formed in plane
and spherical concave
mirrors. The attribution
is held to be
anachronistic however
by J J O'Connor and E
F Robertson who name
Theon of Alexandria as
a more likely author.

12. Euclid's work on Phaenomena
Phaenomena, a treatise on
spherical astronomy,
survives in Greek; it is quite
similar to On the Moving
Sphere by Autolycus of
Pitane, who flourished
around 310 BC.

13. Euclid' work on Optics
Optics is the earliest surviving
Greek treatise on perspective. In
its definitions Euclid follows the
Platonic tradition that vision is
caused by discrete rays which
emanate from the eye. One
important definition is the fourth:
"Things seen under a greater
angle appear greater, and those
under a lesser angle less, while
those under equal angles
appear equal

14. Euclid's Elements
● The Euclid's Elements is a collection of 13
books. Each book contains a sequence of
propositions or theorems, around 10 to 100,
introduced with proper definitions. For
instance in Book I, 23 definitions are followed
by five postulates, after which five common
notions or axioms are included.
● These Elements are dividedinto 13 books in
which
1-6 are of plane geometry
7-9 are of number theory
10 is the theory of irrational numbers
11-13 are of 3-D geometry

15. Made by
Rahul Jaiswal
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