What Are Euclid’s Axioms and Postulates in Geometry?
Euclid's Elements of Geometry
Based on various theorems and axioms, the study of geometrical figures and shapes is called Euclidean geometry. Generally, this type of geometry is inaugurated for flat surfaces. This geometry helps in explaining the shapes of geometrical figures better, especially the plane figures. Greek mathematician, popularly known as Euclid, employed Euclidean geometry and even described it in his own written book named 'Elements'. Also, he named this geometry on his name 'Euclid Geometry'. He introduced fundamentals of geometry in his book Elements which include geometric figures and shapes as well as he stated major five postulates and axioms which are universal truths. However, these axioms and postulates are not yet proved.
Euclid's Elements
In Alexandria, Ptolemaic Egypt, an ancient Greek mathematician named Euclid wrote his mathematical and geometrical work which are present in a total of 13 books. It is known as Euclid's Elements. These thirteen books in which Elements was divided were responsible for popularized geometry all over the globe. Euclid Elements as a whole is a compilation of postulates, axioms, definitions, theorems, propositions and constructions as well as the mathematical proofs of the propositions.
Plane Geometry is mainly discussed in book 1 to 4th and also 6th. Five postulates were given by Euclid which he referred to as Euclid's Postulates. This geometry is popularly considered as Euclidean geometry. Due to the work of Euclid, today we have a vast and collective source and reference for learning and practising geometry as it laid the foundation for geometry.
Euclid's Window
Euclid's Window (The story of geometry from parallel lines to hyperspace) is a book written by the author popularly known as Leonard Mlodinow. Leonard Mlodinow was an American theoretical physicist who, with the help of Euclid's Window, had delightfully and brilliantly led us on a journey with the help of 5 revolutions in geometry. As in this revolution of geometry, he explained geometry with the Greek concept of parallel lines to the latest notions of hyperspace.
Euclid's Axioms
Euclid has given seven axioms for geometry which are considered as Euclid axioms. The axioms are listed below:
Things that are equal to identical things are also equal to each other.
In any case, equals of something are added to equals, then as a whole, they are considered as equals.
In any case, equals of something when subtracted from equals, then their remainders come are equal.
Things that are coinciding with each other are equal.
The whole of something is greater as compared to the part.
Things that are double of identical things are equal to each other.
Things that are halves of identical things are equal to each other.
Euclid's Postulates
Euclid has given five postulates for geometry which are considered as Euclid Postulates. These postulates include the following:
From any one point to any other point, a straight line may be drawn.
A terminated line can be produced indefinitely.
By taking any center and also any radius, a circle can be drawn.
All right angles are equal.
When on two straight lines, if a straight line falls, then the interior angles made are on the same side of the line which if taken together is less as compared to the two right angles. Also, then the two straight lines, if produced indefinitely, meet together on the same side on which the sum of angles is less as compared to the right angles.
FAQs on Euclid Geometry: Concepts, Axioms & Elements
1. What is Euclidean Geometry and why is it also called 'plane geometry'?
Euclidean Geometry is a mathematical system developed by the Greek mathematician Euclid, which describes the properties of space using a set of fundamental truths called axioms and postulates. It primarily deals with flat surfaces and two-dimensional shapes like points, lines, angles, and polygons, which is why it is commonly referred to as 'plane geometry'.
2. What are the basic undefined terms in Euclid's geometry according to the NCERT syllabus?
In modern geometry, to avoid circular definitions, some terms from Euclid's work are accepted as 'undefined'. According to the Class 9 syllabus, the three fundamental undefined terms are:
- Point: A specific location that has no dimension (no length, width, or height).
- Line: A straight path that has only one dimension (length) and extends infinitely in both directions.
- Plane: A flat surface that has two dimensions (length and width) and extends infinitely.
3. What is the key difference between an axiom and a postulate in Euclidean geometry?
The key difference between an axiom and a postulate lies in their scope of application. Axioms, or common notions, are self-evident assumptions that are considered true throughout all of mathematics, not just geometry. For example, 'the whole is greater than the part.' In contrast, postulates are assumptions that are specific only to geometry, used to prove geometric theorems.
4. What are Euclid's seven main axioms?
Euclid's seven main axioms, or common notions, are fundamental principles that apply universally in mathematics:
- Things which are equal to the same thing are equal to one another.
- If equals are added to equals, the wholes are equal.
- If equals are subtracted from equals, the remainders are equal.
- Things which coincide with one another are equal to one another.
- 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.
5. What are Euclid's five postulates?
Euclid's five postulates are the foundational assumptions specific to geometry:
- A straight line segment can be drawn between any two points.
- Any straight line segment can be extended indefinitely.
- A circle can be drawn with any center and any radius.
- All right angles are equal to one another.
- If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two straight lines, if extended indefinitely, meet on that side.
6. Why is Euclid's fifth postulate so important in mathematics?
Euclid's fifth postulate is critically important because it is less intuitive than the others, which led mathematicians for centuries to attempt to prove it using the first four postulates. Their failure to do so led to a revolutionary conclusion: the fifth postulate is independent. By creating valid geometries that contradicted it, they developed entirely new systems of non-Euclidean geometry (like spherical and hyperbolic), which are now essential in fields such as physics and cosmology.
7. How does Euclidean geometry differ from non-Euclidean geometries in simple terms?
The primary difference relates to the concept of parallel lines on different types of surfaces:
- Euclidean Geometry (Flat Surface): For any given line and a point not on the line, there is exactly one parallel line that can be drawn through that point.
- Non-Euclidean Geometry (Curved Surface): On a sphere (spherical geometry), there are no parallel lines. On a saddle-shaped surface (hyperbolic geometry), there are infinitely many parallel lines through that same point.
8. Can you provide a real-world example of one of Euclid's axioms?
A practical example of Euclid's first axiom, "Things which are equal to the same thing are equal to one another," can be seen in standardized measurements. If a carpenter cuts a piece of wood (A) to be 1 meter long, and then cuts another piece (B) to be 1 meter long, both pieces A and B are equal to the same thing (1 meter). Therefore, piece A and piece B are equal to each other, allowing for consistent and predictable construction.
