Euclid's Geometry: The Fundamentals Explained
Anurag Vijay
EUCLID’S
GEOMETRY
GEOMETRY
GEO
METREIN
Earth
To
measure
Geometry
originated
in
Egypt
as an art
of
Earth
measurement
Euclid
(325
BCE-265
BCE):
The
Father
of
Geometry
The
first
Egyptian
mathematician
who
initiated
a
new
way
of
thinking the
study of
geometry.
Introduced
the
method of
proving
a
geometrical
result by
deductive
reasoning
based upon
previously
proved
result
&
some self
evident
specific assumptions called
axioms.
Euclid’s
Definitions
Euclid
has
listed
23
definitions
in Book-1 of
the
‘Elements’; a
few
are
:
A
point
is
that
which has no
part.
A
line
is
breadth
less
length.
The
ends of a line
are
points.
A
straight
line
is a line which lies
evenly
with the
points
on
itself.
A
surface
is
that
which has
length
&
breadth
only.
The edges
of a
surface are
lines.
A
plane
surface
is a
surface
which lies
evenly
with the
straight
lines
on
itself.
Euclid’s
Axioms &
Postulates
Euclid
assumed
certain
properties
which
are
actually
obvious
universal
truths &
divided
them
into two
types:
Postulates
– the assumptions
specific
to
geometry.
Axioms
- the assumptions
used throughout
mathematics
&
not specifically
linked to
geometry
.
Euclid’s
Axioms
Things
which
are
equal
to
the same thing
are
equal
to
one
another.
i.e. If A = C & B =
C,
then A =
B.
Here
A,
B & C
are
same kind of
things.
If
equals
are
added
to
equals, the whole
are
equal
i.e.
If
A = B & C =
D,
then A + C = B +
D
Also
A = B then this
implies that
A + C = B +
C.
If
equals
are
subtracted
from
equals, the
remainders
are
equal.
Things
which
coinside
with
one
another
are
equal
to
one
another.
The whole is
greater
than the
part.
i.e. If A > B then
there
exists
C
such that
A = B +
C.
Here
B is a part of A &
therefore
A is
greater
than
B.
Things
which
are
double of the
same
things
are
equal
to
one
another.
The things
are
halves
of the same things
are
equal
to
one
another.
Euclid’s
Five
Postulates
Postulate
1
:
A
straight
line
may
be
drawn
from
any
one
point to
any
other
point.
Axiom:
given two
distinct points, there
is a
unique
line
that
passes
through
them.
Postulate
2
:
A
terminated
line
can
be
produced
indefinitely.
Postulate
3
:
A
circle
can
be
drawn
with
any
centre
and
any
radius.
Postulate
4
:
All
right
angles
are
equal
to
one
another.
Postulate
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
Theorems
Theorems
-
proved
statements,
465
theorems
in a
logical
chain
using his
axioms,
postulates
&
definitions.
Theorem
1:
Two
distinct
lines
cannot
have
more
than
one
point
in
common.
Equivalent version
of
Euclid’s
fifth
postulate
There are
several
equivalent versions
of
fifth
postulate;
one of
them is
‘
Playfair’s
axiom
’(a
Scottish
mathematician
in 1729) as
stated
below:
For
every
line
l
and
for
every point
p
not
lying
on
l
,
there
exists
a
unique
line
m
passing through
p
&
parallel
to
l
.
Conclusion
Euclid did not
require
his fifth
postulate
to
prove
his
first
28
theorems but
he
himself including
many
mathematicians
were convinced
that
the
fifth
postulate
is actually a
theorem that can
be
proved
using
just
the
four postulates
and other
axioms
.
However
all
attempts to
prove
the
fifth
postulate
as a
theorem
have
failed
& this led
to
a
great
achievement
– the
creation
of
several
other
geometries
,
quite
different
from
Euclidean geometries called
as
non-
Euclidean
geometries.
Example,
Spherical
geometry
,
the
geometry
of the
universe
we live
in.
Thank
You
Euclid, known as the Father of Geometry, introduced the principles of geometry in Egypt. His work included definitions, axioms, and postulates that laid the foundation for geometric reasoning. Euclid's Five Postulates are crucial in understanding the basic concepts of geometry. This article provides insights into Euclid's contributions to the field of mathematics.
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Presentation Transcript
EUCLIDS GEOMETRY GEOMETRY GEO METREIN Earth Tomeasure Geometry originated in Egypt as an art of Earthmeasurement Euclid (325 BCE-265 BCE): The Father of Geometry The first Egyptian mathematician who initiated a new way of thinking the study of geometry. Introduced the method of proving a geometrical result by deductive reasoning based upon previously proved result & some self evident specific assumptions called axioms.
Euclids Definitions Euclid has listed 23 definitions in Book-1 of the Elements ; a few are : A point is that which has no part. A line is breadth less length. The ends of a line are points. A straight line is a line which lies evenly with the points on itself. A surface is that which has length & breadth only. The edges of a surface are lines. A plane surface is a surface which lies evenly with the straight lines on itself.
Euclids Axioms & Postulates Euclid assumed certain properties which are actually obvious universal truths & divided them into twotypes: Postulates the assumptions specific togeometry. Axioms - the assumptions used throughoutmathematics & not specifically linked to geometry.
Euclids Axioms Things which are equal to the same thing are equal to one another. i.e. If A = C & B = C, then A = B. Here A, B & C are same kind of things. If equals are added to equals, the whole are equal i.e. If A = B & C = D, then A + C = B + D Also A = B then this implies that A + C = B + C. If equals are subtracted from equals, the remainders are equal. Things which coinside with one another are equal to one another.
The whole is greater than the part. i.e. If A > B then there exists C such that A = B + C. Here B is a part of A & therefore A is greater than B. Things which are double of the same things are equal to one another. The things are halves of the same things are equal to one another.
Euclids Five Postulates Postulate 1: A straight line may be drawn from any one point to any other point. Axiom: given two distinct points, there is a unique line that passes throughthem. Postulate 2: A terminated line can be producedindefinitely. Postulate 3: A circle can be drawn with any centre and any radius. Postulate 4: All right angles are equal to one another.
Postulate 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 rightangles.
EuclidsTheorems Theorems- provedstatements, 465 theorems in a logical chain using his axioms, postulates & definitions. Theorem 1: Two distinct lines cannot have more than one point in common.
Equivalent version of Euclids fifth postulate There are several equivalent versions of fifth postulate; one of them is Playfair s axiom (a Scottish mathematician in 1729) as statedbelow: For every line l and for every point p not lying on l, there exists a unique line m passing through p & parallel to l.
Conclusion Euclid did not require his fifth postulate to prove his first 28 theorems but he himself including many mathematicians were convinced that the fifth postulate is actually a theorem that can be proved using just the four postulates and other axioms. However all attempts to prove the fifth postulate as a theorem have failed & this led to a great achievement the creation of several other geometries , quite different from Euclidean geometries called as non- Euclidean geometries. Example, Spherical geometry, the geometry of the universe we live in.
