Lecture · 15 min · intermediate
Euclid's postulates
What we take as given, and what we must prove
Euclid, Elements · Book I, Postulates and Common Notions
Euclid does not begin with theorems. He begins with a contract: here are the moves you are allowed to make with a straightedge and compass. Every later proof is a choreography of those moves. If a step is not licensed by a postulate, a common notion, or a prior proposition, it is smuggling.
On the board today
- —State the five postulates in modern language.
- —Separate postulates (geometry) from common notions (equality).
- —See why the parallel postulate is the odd one out.
Equivalent to Euclid's fifth, and the seed of non-Euclidean geometry.
Postulates 1–3 are construction licenses: draw the unique segment between two points; extend a segment; draw a circle with given centre and radius. Postulate 4 says all right angles are equal — the plane does not have a preferred corner. Postulate 5 is a claim about how lines behave at infinity. Drop it, and you still have a coherent geometry (spherical or hyperbolic); you just lose “angle sum .”
Common notions
Things equal to the same thing are equal to each other; equals added to equals are equal; the whole is greater than the part.
These are not geometric. They are the arithmetic of equality, used in every book of the Elements.
Worked example · A licensed construction
Given two points , construct an equilateral triangle on . (Elements I.1)
- 1.Draw circle centre through (Postulate 3).
- 2.Draw circle centre through .
- 3.The circles meet at a point ; join and (Postulate 1).
- 4. as radii, so is equilateral.
The first proposition of Euclid uses only postulates 1 and 3 and the definition of a circle.
Pause the lecture
Why is “two distinct lines meet in at most one point” not listed as a postulate in Book I?
Common mistake. A picture is not a proof. Overlapping marks on a diagram can hide an unjustified coincidence. Name the postulate or prior proposition that licenses each step.