5. Use of Geometry in Indus Valley Civilization
Excavations at Harappa and Mohenjodaro.
Cities , roads, drainage systems, rooms of
different types(mensuration and practical
arithmetic), bricks (4:2:1)
10. Euclid’s Definitions
1. A point is that which has no part.
2. A line is breadthless length.
3. The ends of a line are points.
4. A straight line is a line which lies evenly with the
points on itself.
5. A surface is that which has length and breadth
6. The edges of a surface are lines.
7. A plane surface is a surface which lies evenly with
the straight lines on itself.
11. The definitions of a point, a line, and a plane, are not accepted
by mathematicians. Therefore, these terms are taken as
Euclid’s Definitions problem
13. Euclid's Postulates:
1. A straight line can be drawn from any point to any point.
2. A terminated line can be produced indefinitely.
3. It is possible to describe a circle with any centre any
4. All right angles are equal to one another.
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.
14. Two equivalent versions of the Fifth
(i). ‘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’.
(ii). Two distinct intersecting lines cannot be
parallel to the same line.
15. Euclid's Axioms:
1. Things which are equal to the same things are also equal to one
E.g. If A=B & C=B. that is both A & C are equal to B, then A & C will be equal.
2. If equals are added to equals, then the wholes are equal.
E.g. Two glasses A & A’ has same volume of water. Now we add equal quantity of
water B, to both glass A & A’, then the final volume of water in the jar will be
same. A+ B will be equal to A’ + B.
3. If equals are subtracted from equals, then the remainders are
E.g. Two glasses A & A’ has same volume of water. Now we remove equal quantity of
water B, from both glass A & A’, then the final volume of water in the jar will be
same. A- B will be equal to A’ - B.
4. Things which coincide with one another are equal to one
E.g. if two triangles coincide with each other then they are equal.
16. Euclid's Axioms:
5. The whole is greater than the part.
This statement is true in physics, chemistry, mathematics, geometry, biology,
6. Things which are double of the same things are equal to one
If A= A’ then 2A= 2A’.
7. Things which are halves of the same things are equal to one
If A= A’ then ½ A= ½ A’.
17. Theorems or Prepositions:
After stating Postulates & Axioms, Euclid used
these to prove other results by applying
E.g.: “Diameter divides circle in 2 parts” is a
theorem. Euclid deduced 465 Theorems.
“Two distinct lines cannot have more than one
point in common” is a theorem.
18. Non Euclidean Geometry/Spherical geometry
Lines are not straight.
They are parts of great circles (i.e., circles obtained by the
intersection of a sphere and planes passing through the
centre of the sphere).
5th postulate of Euclid-
Lines AN & BN should not meet
but they meet at point N.
It has been proved that Euclidean geometry is valid only
for the figures in the plane. On the curved surfaces, it fails.