2. Geometry
Geometry ( geo "earth", metron "measurement") is a
branch of mathematics concerned with questions of
shape, size, relative position of figures, and the
properties of space. A mathematician who works in
the field of geometry is called a geometer.
Geometry was one of the two fields of pre-modern
mathematics, the other being the study of numbers.
3. Euclidean Geometry
Euclidean Geometry is the study of geometry based on
definitions, undefined terms (point, line and plane) and the
assumptions of the mathematician Euclid (330 BC)
Euclidean Geometry is the study of flat space
Euclid's text Elements was the first systematic discussion of
geometry. While many of Euclid's findings had been previously
stated by earlier Greek mathematicians, Euclid is credited with
developing the first comprehensive deductive system. Euclid's
approach to geometry consisted of proving all theorems from a
finite number of postulates and axioms.
The concepts in Euclid's geometry remained unchallenged until
the early 19th century. At that time, other forms of geometry
started to emerge, called non-Euclidean geometries. It was no
longer assumed that Euclid's geometry could be used to describe
all physical space.
4. Euclid’s Definitions
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,plane etc. were derived from what was seen around
them.
Euclid summarised these notions as definitions. A few are given
below:
A point is that of which there is no part.
A line is a widthless length.
A straight line is one which lies evenly with the points on itself.
The extermities of lines are called points.
A surface is that which has only length and breadth.
The edges of surface are lines.
A plane surface is a surface which lies evenly with straight lines
on itself.
5. Euclid’s Axioms
Axioms are assumptions used throughout mathematics and not
specifically linked to geometry.
Here are some of euclid’s axioms:
Axiom 1: Things that are equal to the same thing are also equal to
one another (Transitive property of equality).
Axiom 2: If equals are added to equals, then the wholes are equal.
Axiom 3: If equals are subtracted from equals, then the remainders are
equal.
Axiom 4: Things that coincide with one another equal one another
(Reflexive Property).
Axiom 5: The whole is greater than the part.
Axiom 6: Things which are halves of the same things are equal to one
another
Axiom 7: Things which are double of the same things are equal to one
another
6. Euclid’s Postulates
Postulates are assumptions specific to geometry.
Let the following be postulated:
To draw a straight line from any point to any point.
To produce [extend] a finite straight line continuously in a
straight line.
To describe a circle with any centre and distance [radius].
That all right angles are equal to one another.
The parallel postulate: That, if a straight line falling on two
straight lines make the interior angles on the same side less
than two right angles, the two straight lines, if produced
indefinitely, meet on that side on which are the angles less
than the two right angles.
7. Equivalent Versions of Euclid’s Fifth
Postulate (parallel postulate)
To the ancients, the parallel postulate seemed less obvious than the
others. It seemed as if the parallel line postulate should have been able
to be proven rather than simply accepted as a fact. It is now known that
such a proof is impossible.
Euclid himself seems to have considered it as being qualitatively
different from the others, as evidenced by the organization of
the Elements: the first 28 propositions he presents are those that can be
proved without it.
Many alternative axioms can be formulated that have the same logical
consequences as the parallel postulate. For example Playfair's
axiom states:
For every line l and for every point P lying outside l, there exists a
unique line m passing through P and parallel to l
This result can also be stated in the following way:
Two distinct intersecting lines cannot be parallel to the same line
