What are Definitions?
Definitions explain the precise meaning of geometric terms. Euclid started his "Elements" with 23 definitions.
Common Definitions by Euclid:
- A point is that which has no part
- A line is breadthless length
- The ends of a line are points
- A straight line lies evenly with points on itself
- A circle is a plane figure bounded by one line such that all lines from a point inside (center) to the boundary are equal
What are Axioms?
Axioms (or "common notions") are general statements accepted as true without proof. They are not specific to geometry — they apply to all of mathematics.
Euclid's Axioms (Common Notions):
| # | Axiom | Meaning |
|---|---|---|
| 2 | If equals are added to equals, the wholes are equal | Addition property |
| 3 | If equals are subtracted from equals, the remainders are equal | Subtraction property |
| 4 | Things which coincide with one another are equal to one another | Superposition |
| 5 | The whole is greater than the part | Part-whole relationship |
What are Postulates?
Postulates are assumptions specific to geometry that are accepted without proof.
Euclid's Five Postulates:
- A straight line may be drawn from any point to any other point
- A finite straight line can be extended continuously in a straight line
- A circle may be described with any center and any radius
- All right angles are equal to one another
- If a straight line falling on two straight lines makes interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side