2. EUCLID’S GEOMETRY
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.
3. 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
4. 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.
5. Euclid’s Axioms
Things which are equal to the same thing are equal to
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
Things which coinside with one another are equal to
6. The whole is greater than the part.
i.e. If A > B then there exists C such that A = B +
Here B is a part of A & therefore A is greater
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.
7. 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
Postulate 4: All right angles are equal to one another.
8. 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.
9. 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.
10. Equivalent version of Euclid’s fifth
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.
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-
Example, Spherical geometry, the geometry of the
universe we live in.