Learning Objectives
- Understand the historical development of geometry
- Learn Euclid's definitions, axioms, and postulates
- Distinguish between axioms and postulates
- Understand Euclid's five postulates
- Appreciate the concept of equivalent versions of Euclid's fifth postulate
Key Concepts
Euclid's Definitions
Euclid defined basic geometric terms in his famous work Elements:
- A point is that which has no part.
- A line is breadthless length.
- The ends of a line are points.
- A surface is that which has length and breadth only.
- The edges of a surface are lines.
Axioms and Postulates
Axioms are assumptions used throughout mathematics and not specifically linked to geometry.
Postulates are assumptions specific to geometry.
Euclid's Axioms
- 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.
Euclid's Five Postulates
- A straight line may be drawn from any one point to any other point.
- A terminated line can be produced indefinitely.
- A circle can be drawn with any centre 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 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.
Equivalent of the Fifth Postulate
Playfair's Axiom: For every line l and for every point P not lying on l, there exists a unique line through P which is parallel to l. This is equivalent to Euclid's fifth postulate.
Summary
Euclid's geometry, presented in his book Elements, is built upon a system of definitions, axioms, and postulates. Axioms are general assumptions while postulates are specific to geometry. His five postulates form the foundation of plane geometry. The fifth postulate about parallel lines has been extensively studied and has equivalent formulations like Playfair's Axiom.
Important Terms
- Axiom: A statement accepted as true without proof, used across mathematics
- Postulate: A statement accepted as true without proof, specific to geometry
- Theorem: A statement that is proved using axioms, postulates, and previously proved statements
- Corollary: A result that follows directly from a theorem
Quick Revision
- Euclid wrote Elements — the most influential geometry text in history
- Axioms are universal truths; postulates are geometric truths
- There are 7 axioms and 5 postulates
- Fifth postulate deals with parallel lines
- Playfair's Axiom: Through a point outside a line, exactly one parallel line exists