Introduction to Euclid's Geometry

1.0History-Euclid and geometry in India

Ancient India

In the Indian subcontinent, the excavations at Harappa and Mohenjo-Daro, etc. show that the Indus Valley Civilization (about 3000 BCE) made extensive use of geometry. The bricks used of constructions were kiln fired and the ratio length : breadth : thickness, of the bricks was found to be $4:2$ : 1 . The geometry of the Vedic period originated with the construction of altars (or vedis) and fireplaces for performing Vedic rites. Square and circular altars were used for household rituals, while altars, whose shapes were combinations of rectangles, triangles and trapeziums, were required for public worship.

Egypt, Babylonia and Greece

Egyptians developed a number of geometric techniques and rules for calculating simple areas and doing simple constructions. Babylonians and Egyptians used geometry mostly for practical purposes and did very little to develop it as a systematic science. The Greeks were interested in establishing the truth of the statements they discovered using deductive reasoning.

• The word 'geometry' comes from the Greek words 'geo' meaning the 'earth' and 'metrein' meaning to 'measure'.

Thales and Pythagoras

A Greek mathematician, Thales is credited with giving the first known proof. This proof was of the statement that a circle is bisected (i.e., cut into two equal parts) by its diameter. One of Thales' most famous pupils was Pythagoras ( 572 BCE ). Pythagoras and his group discovered many geometric properties and developed the theory of geometry to a great extent.

Euclid's elements

Euclid around 300 B.C. also known as father of geometry collected all known work in the field of mathematics and arranged it in his famous treatise called Elements. He divided the 'Elements' into thirteen chapters, each called a book and total number of prohibitions in the elements are 465. Euclid assumed certain properties, which were not to be proved. These assumptions are actually "obvious universal truths". He divided them into two types-Axioms and Postulates.

2.0Euclid's axioms and postulates

The Greek mathematicians of Euclid's time expressed some basic terms in geometry such as point, line, plane, solid, etc. According to what they observed in the world around them. From their observation, a solid is an object in space which has three dimensions called length, breadth and thickness. It has shape, size, position and place. It can be moved from one place to another. The boundaries of a solid are called surfaces. Surface has two dimensions which are length and breadth. It has no thickness. The boundaries of a surface are curves (lines are also curves). Lines and curves has no breadth and no thickness. The ends of a line or a curve are points. A point has no dimension. Euclid presented his work in the form of definitions, axioms, postulates and theorems. Some terms defined by Euclid and other mathematicians of that time may not be fully explained but still the observations were very strong and hence, form the basis for the further development of the subject.

Euclid defined some terms precisely as below

Point : A point is that which has no part. Line: A line is breadthless length. Surface: A surface is that which has length and breadth only. Ends of a line : The ends of a line are points. Straight line : A straight line is a line which lies evenly with the point on itself. Plane surface: A plane surface is a surface which lies evenly with the straight lines on itself. If you carefully study these definitions you find that some of the terms like part, breadth, length, evenly, etc. need to be further explained clearly. For example, consider his definition of a point. In this definition, 'a part' needs to be defined. Suppose if you define 'a part' to be that which occupies 'area', again 'an area' needs to be defined. So, to define one thing, you need to define many other things and you may get a long chain of definition without an end. For such reasons, mathematicians agree to leave some geometric terms undefined.

• Ray: It is a line in which one point has an infinitely extending & one fixed end.
• Line segment: It is a part of line having two endpoints.

3.0The basic undefined terms in geometry are

(i) Point (ii) Line (iii) Plane (i) Point : A point has position only. It has no length, no width and no thickness. (ii) Line : A line has length but no width and no thickness. (iii) Plane : If any two points are taken anywhere on a surface and joined by a straight line, then if each and every point of this line lies in the surface, the surface is called a plane or a plane surface. Though Euclid defined a point, a line and a plane, the definitions are not accepted by today's mathematicians. They take these terms as undefined terms. Because these terms can be represented intuitively or explained with the help of 'Physical Models'. For example : The tip of a fine sharp pencil, or the tip of a needle represent a point. A thread hold tightly by two hands represent a line. The top of a table represents a plane etc.

4.0Undefined properties (axioms and postulates)

On the basis of above definitions and some more observations, Euclid states some obvious universal truths as axioms and postulates. Euclid assumed these universal truths as such, which were not to be proved. In present time, the terms axioms and postulates can be used interchangeably but Euclid made a fine distinction between the two terms. Axioms

The assumptions, which are granted without proof and are used throughout in mathematics which are obvious universal truths, and not specifically linked to geometry are termed as axioms. Some of the Euclid's Axioms are given below :

(i) Things which are equal to the same thing are equal to one another. i.e. if $\mathrm{x}=\mathrm{y}$ and $\mathrm{y}=\mathrm{z}$, then $\mathrm{x}=\mathrm{z}$. E.g. If area of a circle is equal to that of a square and the area of the square is equal to that of a rectangle, then the area of the circle is equal to the area of the rectangle.

(ii) If equals are added to equals, the wholes are equal. i.e. if $a=b$ and $c=d$, then $a+c=b+d$ Also, $\mathrm{a}=\mathrm{b}\Rightarrow \mathrm{a}+\mathrm{c}=\mathrm{b}+\mathrm{c}$ E.g. If $5=5,\Rightarrow 5+2=5+2$ or $7=7$

(iii) If equals are subtracted from equals, the remainders are equal. i.e. if $a=b$ and $c=d$, then $a-c=b-d$. E.g. $3=3$, $\Rightarrow 3-1=3-1$ or $2=2$ Here magnitudes of same kinds can be compared and subtracted. We cannot subtract a line from a triangle. Similarly, we cannot subtract kg from litres.

(iv) The things which coincide with one another are equal to one another. In figure, two-line segment AB and CD coincide with each other. $\therefore \mathrm{AB}=\mathrm{CD}=2\mathrm{cm}$

(v) The whole is greater than the part. i.e. if $\mathrm{a}>\mathrm{b}$, then there exists c such that $\mathrm{a}=\mathrm{b}+\mathrm{c}$. Here, $b$ is a part of $a$ and therefore, $a$ is greater than $b$. In figure, $AB=AC+BC$ $=1+2$ $=3\mathrm{cm}$ $\therefore AB>AC$ and $AB>BC$

(vi) Things which are double of the same things are equal to one another. In Figure, CD and EF are double of AB $\therefore \mathrm{CD}=\mathrm{EF}$

(vii) Things which are halves of the same things are equal to one another. In Figure, CD and EF are halves of $AB$ $\therefore \mathrm{CD}=\mathrm{EF}$

Postulates

The assumptions, which are specifically linked to geometry and are obvious universal truths, are termed as postulates.

Euclid gave five postulates as stated below :

(i) A straight line may be drawn from any one point to any other point. Let $A$ be a given point and $B$ be some other point. If we draw several lines passing through the point A , we see that only one of these lines passes through the point B also. Similarly, if we draw several lines passing through the point B we see that only one of these lines passes through the point A also. So, we can say a unique line passes through the points A and B. See figure.

(ii) A terminated line (i.e., a line segment) can be produced indefinitely on either side. A terminated line is called line segment these days.

(iii) A circle can be drawn with any centre and any radius.

(iv) All right angles are equal to one another.

(v) 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. (a) The figure below shows a line $PQ$ falling on lines $AB$ and $CD$ such that the sum of the interior angles $\angle 1$ and $\angle 2$ is less than $180^{\circ }$ or 2 right angles. Therefore, lines $AB$ and CD if produced on left side will meet somewhere.

5.0A statement as a theorem

Propositions or theorems are the statements which are proved, using definition, axioms, previously proved statements and by deductive reasoning. E.g., (i) "The sum of all angles of a triangle is equal to $180^{\circ }$ " is a theorem. (ii) "The sum of all angles of a quadrilateral is $360^{\circ }$ " is a theorem.

Euclid deduced 465 propositions (theorem) in a logical chain using his axioms, postulates, definitions and already proved theorems. When we prove a theorem, then it becomes a general statement for other theorem. A theorem and assumption (postulate) are both statements. The difference is that an assumption (postulate) is accepted to be true without any proof while a theorem is accepted to be true only when it has been proved. 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. So, when any system of axioms is given, it needs to be ensured that the system is consistent.

• Collinear points : Three or more points are said to be collinear, if there is a line which contains all of them.
• Concurrent lines : Three or more lines are said to be concurrent, if there is a point which lies on all of them.

6.0Some axioms of points and lines

Here, we shall assume some properties about lines and points without any proof but these properties are obvious universal truths. These properties are taken as axioms. (i) A line contains infinitely many points. (See figure below) (ii) Through a given point, infinitely many lines can be drawn.

In figure, we observe that, out of all lines passing through the point $P$ there is exactly one line ' $\ell$ ' which also passes through Q. Similarly, out of all lines passing through the point $Q$ there is exactly one line ' $\ell$ ' which also passes through P. Hence, we find exactly one line ' $\ell$ ' which can be drawn through two points $P$ and Q .

• Q. Ram and Ravi have the same weight. If they each gain weight by 2 kg , how will their new weights be compared? Explanation: Let x kg be the weight each of Ram and Ravi. On gaining 2 kg , weight of Ram and Ravi will be ( $\mathrm{x}+2$ ) each. According to Euclid's second axiom, when equals are added to equals, the wholes are equal. So, weight of Ram and Ravi are again equal.
• Q. Solve the equation a-15 = 25 and state which axiom do you use here. Explanation: a $-15=25$ On adding 15 to both sides, we have a $-15+15=25+15=40$ (using Euclid's second axiom) or $\mathrm{a}=40$.
• Q. In the given figure, if $\angle 1=\angle 3,\angle 2=\angle 4$ and $\angle 3=\angle 4$, write the relation between $\angle 1$ and $\angle 2$, using an Euclid's axiom.
  
  Explanation: Here, $\angle 3=\angle 4$ and $\angle 1=\angle 3$ and $\angle 2=\angle 4$. Euclid's first axiom says, the things which are equal to the same thing are equal to one another. So, $\angle 1=\angle 2$.

Theorem 1:

Theorem 2:

Theorem 3:

Equivalent versions of Euclid's fifth postulate

There are several equivalent versions of the fifth postulate of Euclid. One such version is stated as 'Playfair's Axiom' which is given below :

7.0Playfair's axiom (axiom for parallel lines)

For every line $\ell$ and for every point P not lying on $\ell$, there exists a unique line m passing through $P$ and parallel to $\ell$.

Let us observe it in figure.

Another version of the above axiom is as stated below : Two distinct intersecting lines cannot be parallel to the same line. In figure there are infinitely many straight lines passing through $P$ but there is exactly one-line $m$ which is parallel to $\ell$. Thus, two intersecting lines cannot be parallel to the same line.