Euclid’s Definitions, Axioms and Postulates
The Greek mathematicians of Euclid’s time thought of geometry as an abstract model of the world in which they lived. The notions of point, line, plane (or surface) and so on were derived from what was seen around them. From studies of the space and solids in the space around them, an abstract geometrical notion of a solid object was developed.
A solid has shape, size, position, and can be moved from one place to another. Its boundaries are called surfaces. They separate one part of the space from another, and are said to have no thickness. The boundaries of the surfaces are curves or straight lines. These lines end in
Consider the three steps from solids to points (solids-surfaces-lines-points). In each step we lose one extension, also called a dimension. So, a solid has
Euclid summarised these statements as definitions. He began his exposition by listing 23 definitions in Book 1 of the ‘Elements’. A few of them are given below :
A point is that which has no part.
In this , labelled dots indicate interactive points you can move around, while not labelled dots indicate fixed points which you can’t move.
- A line is breadthless 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 and breadth only.
The edges of a surface are lines.
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 Euler's 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 definitions without an end. For such reasons, mathematicians agree to leave some geometric terms undefined.
However, we do have a intuitive feeling for the geometric concept of a point than what the ‘definition’ above gives us. So, we represent a point as a dot, even though a dot has some dimension.
A similar problem arises in Definition 2 above, since it refers to breadth and length, neither of which has been defined. Because of this, a few terms are kept undefined while developing any course of study.
So, in geometry, we take a point, a line and a plane (in Euclid‘s words a plane surface) as
The only thing is that we can represent them intuitively, or explain them with the help of ‘physical models’.
Starting with his definitions, 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.
He used the term ‘postulate’ for the assumptions that were specific to geometry.
Common notions (often called axioms), on the other hand, were assumptions used throughout mathematics and not specifically linked to geometry.
Some of Euclid’s axioms, not in his order, are given below :
(1) Things which are equal to the same thing are equal to one another.
(2) If equals are added to equals, the wholes are
(3) If equals are subtracted from equals, the differences are
(4) Things which coincide with one another are equal to one another.
(5) The whole is
(6) Things which are double of the same things are
(7) Things which are halves of the same things are
These ‘common notions’ refer to magnitudes of some kind. The first common notion could be applied to plane figures.
For example: if an area of a triangle equals the area of a rectangle and the area of the rectangle equals that of a square, then the area of the triangle also equals the area of the square.
Magnitudes of the same kind can be compared and added, but magnitudes of different kinds cannot be compared. For example: a line cannot be compared to a rectangle, nor can an angle be compared to a pentagon.
The 4th axiom given above seems to say that if two things are identical (that is, they are the same), then they are
In other words, everything equals itself. It is the justification of the principle of superposition.
Axiom (5) gives us the definition of ‘greater than’. For example, if a quantity B is a part of another quantity A, then A can be written as the sum of B and some third quantity C. Symbolically, A > B means that there is some C such that A =
Now let us discuss Euclid’s five postulates. They are :
Postulate 1 : A straight line may be drawn from any one point to any other point. Note that this postulate tells us that at least one straight line passes through two distinct points, but it does not say that there cannot be more than one such line.
However, in his work, Euclid has frequently assumed, without mentioning, that there is a unique line joining two distinct points. We state this result in the form of an axiom as follows:
Axiom 5.1 : Given two distinct points, there is a unique line that passes through them.
How many lines passing through P also pass through Q ? Only
How many lines passing through Q also pass through P? Only
Thus, the statement above is self-evident, and so is taken as an axiom.
Postulate 2 : A terminated line can be produced infinitely long line. Note that what we call a line segment now-a-days is what Euclid called a terminated line.
So, according to the present day terms, the second postulate says that a line segment can be extended on either side to form a
Postulate 3 : A circle can be drawn with any centre and any radius. Given a point P and a distance r, you can draw a circle with centre P and radius r.
Postulate 4 : All right angles are equal to one another. Any two right angles are
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.
For example, the line PQ in the above figure, falls on lines AB and CD such that the sum of the interior angles 1 and 2 is less than 180° on the left side of PQ. Therefore, the lines AB and CD will eventually intersect on the left side of PQ.
A brief look at the five postulates brings to your notice that Postulate 5 is far more complex than any other postulate.
On the other hand, Postulates 1 through 4 are so simple and obvious that these are taken as ‘self-evident truths’. However, it is not possible to prove them.
So, these statements are accepted without any proof. Because of its complexity, the fifth postulate will be given more attention in the next section.
Now-a-days, ‘postulates’ and ‘axioms’ are terms that are used interchangeably and in the same sense. ‘Postulate’ is actually a verb. When we say “let us postulate”, we mean, “let us make some statement based on the observed phenomenon in the Universe".
Its truth/validity is checked afterwards. If it is true, then it is accepted as a ‘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.
So, when any system of axioms is given, it needs to be ensured that the system is
After Euclid stated his postulates and axioms, he used them to prove other results. Then using these results, he proved some more results by applying deductive reasoning. 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. In the next few chapters on geometry, you will be using these axioms to prove some theorems.
Now, let us see in the following examples how Euclid used his axioms and postulates for proving some of the results:
Example 1 : If A, B and C are three points on a line, and B lies between A and C, then prove that AB + BC = AC.
Solution : In the figure given above, AC coincides with
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.
Example 2 : Prove that an equilateral triangle can be constructed on any given line segment.
Solution : In the statement above, a line segment of any length is given, say AB
Draw a triangle that has sides of lengths 6cm each.
In the box of the length, draw the base side of the triangle, which is 6cm. Now we already have two of the three vertices of the triangle – the challenge is to find the last one.
Next, draw a circle of radius 6cm around one of the vertices say A, and a circle of radius 6cm around the other one say B.
The third vertex of the triangle say C is the
So, you have to prove that this triangle is
Now, AB = AC and Similarly, AB = BC since they are the
From these two facts, and Euclid’s axiom that things which are equal to the same thing are equal to one another, you can conclude that AB =
So, ∆ ABC is an equilateral triangle.
Note that here Euclid has assumed, without mentioning anywhere, that the two circles drawn with centres A and B will meet each other at a point.
Now we prove a theorem, which is frequently used in different results.
Theorem 5.1 : Two distinct lines cannot have more than one point in common.
Proof : Here we are given two lines l and m. We need to prove that they have only one point in common.
For the time being, let us suppose that the two lines intersect in two distinct points, say P and Q. So, you have two lines passing through two distinct points P and Q.
But this assumption clashes with the axiom that only one line can pass through two distinct points. So, the assumption that we started with, that two lines can pass through two distinct points is wrong.
From this, what can we conclude? We are forced to conclude that two distinct lines cannot have more than one point in common.