Notes of chapter: Introduction to Euclid’s Geometry are presented below. Indepth notes along with worksheets and NCERT Solutions for Class 9.
(1) The word geometry comes from the Greek word ‘geo’ meaning the ‘earth’ and ‘metrein’ means ‘to measure’.
(2)The people of ancient civilisations like Egypt, Babylonia, China, India and Greece etc. used geometry to solve many daily life problems.
(i) To measure land.
(ii) To construct pyramid based on a geometry shape.
(iii) In town planning.
(3) The ‘Sulbasutras’ were the guide of geometrical constructions for the ancient Indians. They used these sutras for constructing altars (or vedis) and fireplaces for performing Vedic rites. Square and circular altars were used for household rituals, while combinations of rectangles, triangles and trapeziums altars were required for public worship.
(4) Geometry developed as a science-
(i) Geometry is developed as a systematic science by Greeks. Thales, a Greek mathematician gave first proof of statement that a circle can be divided by its diameter in two equal parts.
(ii) Pythagoras (572 BC), a pupil of Thales discovered many geometrical properties and contribute in development of geometry.
(iii)Euclid, a teacher of mathematics at Alexandria in Egypt, collected all known work and arranged it in his famous treatise “Elements”. The Elements has thirteen chapters. Each chapter is called a book. These thirteen books became base for understanding of geometry.
(5)Euclid’s definitions, Axioms and Postulates
Euclid began his exposition by listing 23 definitions in Book 1 of the “Elements”. A few of them are given below:
(i) A Point is that which has no parts.
(ii) A line is breadthless length.
(iii)The ends of a line are points.
(iv) A straight line is a line which lies evenly with the points on itself.
(v) A surface is that which has length and breadth only.
(vi) The edges of a surface are lines.
(vii) A plane surface is a surface which lies evenly with straight lines on itself.
There are some terms of the definitions are not defined by Euclid or mathematician of his time. For example, part, length, breadth evenly etc.
Postulates and axioms
The assumptions of Euclid which are universal truth and specific to only geometry are known as postulates.
The assumptions of Euclid which are universal truth and used throughout in mathematics and not specific to only geometry are known as axioms.
Some of Euclid’s axioms which are not in his order are given below-
(i) Things which are equal to the same thing are equal to one another.
(ii) If equals are added to equals, the wholes are equals.
(iii) If equals are subtracted from equals, the remainders are equals.
(iv) Things which coincide with one another are equal to one another.
(v) The whole is greater than the part.
(vi) Things which are double of the same things are equal to one another.
(vii) Things which are halves of the same things are equal to one another.
These “common notions” refer to magnitudes of some kind.
The first common notion could be applied to plane figures.
An area of a triangle = Area of a rectangle
Area of rectangle = Area of square
Area of triangle = Area of square
According to second notion, magnitudes of the same kind can be compared and added, but magnitudes of different kinds can not be compared. For example, a line can not be added to a rectangle, nor can an angle be compared to a pentagon.
Five postulates of Euclid
Postulate 1- A straight line can be drawn from any one point to any other point.
We state this result in the form of axiom as follows:-
Axiom 5.1- Given two distinct points, there is a unique line that passes through them.
Only one line passes through from point P to point Q. There is only one line which is passes through point Q to point P is PQ. Thus, the above statement is self – evident and so is taken as an axiom.
Postulate 2- A terminated line can be produced indefinitely.
Euclid’s terminated line is same as line segment of today. Therefore, a line segment can be extended on either side to form a line.
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.
Eg:- The line PQ in fig given below falls on lines AB and CD such that the sum of the interior angles 1 and 2 is less than 1800 on the left side of PQ. Therefore, the lines AB and CD will eventually interest on the left side of PQ.
Today, ‘postulates’ and ‘axioms’ are term in the same sense. “Postulate” means “let us make some statement based on the observed phenomenon in the Universe”. It means, first postulate makes and then check if it is truth or not. If it is truth or valid then accepted as “postulate”
A system of axioms is called consistent, if it is impossible to deduce from these axioms a statement that contradicts any axiom or previously proved statement.
The statements that were proved are called propositions or theorems. Euclid deduced 465 propositions in a logical; chain using his axioms, postulates, definitions and theorems proved earlier in the chain.
Equivalent versions of Euclid’s fifth postulate
Fifth postulate is very significant in mathematics. There are several equivalent versions of this postulate. One of them is given by Scottish mathematician John Playfair known as ‘Playfair Axiom”, stated below:
‘For every line l and for every point P and lying on l, there exists a unique line m passing through P and parallel to l.
This result can also be stated I the following form:
‘Two distinct intersecting lines cannot be parallel to the same line.’