2. INTODUCTION TO EUCLID’S
• THE WORD ‘GEOMETRY’ COMES FROM GREEK WORD ‘GEO’
MEANING THE ‘EARTH’ AND ‘METRENI’ MEANING TO
• GEOMETRY APPEARS TO HAVE ORIGINATED FROM THE NEED
FOR MEASURING LAND.
• NEARLY 5000 YEARS AGO GEOMETRY ORIGINATED IN EGYPT
AS AN ART OF EARTH MEASUREMENT.
• EGYPTIAN GEOMETRY WAS THE STATEMENTS OF RESULTS.
• EUCLID WAS THE FIRST GREEK MATHEMATICAN WHO
INITIATED A NEW WAY OF THINKING THE STUDY OF
• HE INTRODUCED THE METHOD OF PROVING A GEOMETRICAL
RESULTS BY DEDUCTIVE REASONING BASED UPON PREVIOUSLY
PROVED RESULT AND SOME SELF EVIDENT SPECIFIC
ASSUMPTIONS CALLED AXIOMS.
• THE GEOMETRY OF PLANE FIGURE IS KNOWN AS ‘EUCLIDEAN
4. EUCLID’S DEFINITIONS
• A POINT IS THAT WHICH HAS NO PART.
• A LINE IS BREADTHLESS LENGTH.
• THE ENDS OF THE LINE ARE POINT’S.
• A STRAIGHT LINE IS A LINE WHICH LIES EVENLY WITH
THE POINT ON ITSELF.
• A SURFACE IS THAT WHICH HAS LENGTH AND BREADTH
5. EUCLID’S DEFINITIONS
• THE EDGES OF A SURFACE ARE LINES.
• A PLANE SURFACE IS A SURFACE WHICH LIES
EVENLY WITH THE STRAIGHT LINES ON IT SELF.
6. EUCLID’S AXIOMS
• THING’S WHICH ARE EQUAL TO THE SAME THINGS ARE
EQUAL TO ONE ANOTHER.
• IF EQUALS ARE ADDEDTO EQUALS,THE WHOLES ARE
• IF EQUALS ARE SUBTRACTED FROM EQUALS,THE
REMAINDERS ARE EQUAL.
• THINGS WHICH COINCIDE WITH ONE ANOTHER ARE
EQUAL TO ONE ANOTHER
7. EUCLID’S AXIOMS
• THE WHOLE IS GREATER THAN THE PART.
• THINGS WHICH ARE DOUBLE OF THE SAME
THINGS ARE EQUAL TO ONE ANOTHER.
• THINGS WHICH ARE HALVES OF THE SAME
THINGS ARE EQUAL TO ONE ANOTHER.
8. EUCLID’S FIVE POSTULATE
• POSTULATE 1 : A STRAIGHT LINE MAY BE DRAWN
FROM ANY ONE POINT TO ANY OTHER POINT.
• POSTULATE 2: A TERMINATED LINE CAN BE
• POSTULATE 3: A CIRCLE CAN BE DRAWN WITH ANY
CENTRE AND ANY RADIUS.
• POSTULATE 4: ALL RIGHT ANGLES ARE EQUAL TO
9. EUCLID’S FIVE POSTULATE
• 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
10. PROBLEM ON EUCLID’S GEOMRTRY
• If A, B and C are three points on a line, and B lies between A and C (see Fig.
5.7), then prove that AB + BC = AC.
11. PROBLEMS ON EUCLID’S GEOMRTRY
• SOLUTION: In the figure given above, AC coincides with AB + BC.
• Also, Euclid’s Axiom (4) says that things which coincide with one another are
equal to one another. So, it can be deduced that AB + BC = AC
• Note that in this solution, it has been assumed that there is a unique line
passing through two points.
12. Equivalent Versions of Euclid’s Fifth
• Euclid’s fifth postulate is very significant in the history of mathematics.
Recall it again from Section 5.2. We see that by implication, no intersection
of lines will take place when the sum of the measures of the interior angles
on the same side of the falling lines exactly 180°. There are several
equivalent versions of this postulate. One of them is‘Play fair’s Axiom’ (given
by a Scottish mathematician John Play fair 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 and parallel to l’.
• From Fig. 5.11, you can see that of all the lines passing through the point P,
only line m is parallel to line l.