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If your textbook calls one of Euclid's starting rules an "axiom" and another a "postulate", the split goes back to how Euclid organised Book I of the *Elements*. He set down 23 definitions, five postulates and five common notions, and justified each step of his proofs by one of these or by a proposition already proved (Euclid's Elements, Book I, D. E. Joyce, Clark University).
What is the difference between Euclid's axioms and postulates?
In classical terminology, an axiom was a self-evident assumption common to many branches of science, while the postulates of each particular science were different (Axiom, Wikipedia, "Early Greeks"). Euclid's common notions are the axioms: they talk about equals, wholes and parts, so they apply to any kind of magnitude, whether lengths, angles or areas. His postulates talk about points, lines, circles and angles, so they only make sense inside geometry. Proclus also reports that Geminus wanted Postulate 4 classed as an axiom, since it states a property rather than a construction.
What are Euclid's five common notions?
In Joyce's edition of Book I (Clark University) they read:
Things which equal the same thing also equal one another.
If equals are added to equals, then the wholes are equal.
If equals are subtracted from equals, then the remainders are equal.
Things which coincide with one another equal one another.
The whole is greater than the part.
None mentions a line or an angle; you use the first whenever a = b and b = c gives a = c.
What are Euclid's five postulates?
From the same text:
To draw a straight line from any point to any point.
To produce a finite straight line continuously in a straight line.
To describe a circle with any center and radius.
That all right angles equal one another.
That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
How do axioms and postulates work together in a proof?
Take Euclid's very first proposition, building an equilateral triangle on a line AB (Proposition I.1, Clark University):
Draw circles with centre A, radius AB, and centre B, radius BA: Postulate 3.
From C, where they cross, draw CA and CB: Postulate 1.
AC equals AB and BC equals BA, as radii: the definition of a circle.
So AC equals BC: Common Notion 1.
The postulates draw the figure; the common notion reasons about it.
Try it on these five, then check the answers:
"If two students have equal scores and each gets 5 bonus marks, their totals are equal." Common Notion 2.
"You can draw a straight line through any two points." Postulate 1.
"You can draw a circle with centre O and radius 4 cm." Postulate 3.
"A whole pizza is bigger than one slice." Common Notion 5.
"Every right angle equals every other right angle." Postulate 4.
Why did the fifth postulate stand apart?
The fifth talks about lines "produced indefinitely", which no drawing can check. Euclid does not use it until Proposition I.29, though nearly all of the rest of Book I depends on it (Postulate 5 commentary, Clark University).
For centuries, mathematicians tried to prove it from the other four. Proclus (410-485) gave a false proof of his own, and in 1766 Lambert investigated a geometry that denied it without reaching a contradiction (Non-Euclidean geometry, MacTutor, University of St Andrews). In 1795 Playfair proposed replacing it with an equivalent statement, now called Playfair's axiom: given a line and a point not on it, exactly one line can be drawn through the point parallel to the line.
By 1817 Gauss had become convinced that the fifth postulate was independent of the other four, but he never published his work. Lobachevsky published a non-Euclidean geometry in 1829, and János Bolyai published his own a few years later. In 1868 Beltrami built a model of this geometry inside Euclidean geometry in which the first four postulates hold but the fifth does not, so the new geometry is consistent if Euclid's is (MacTutor). So the fifth postulate is a choice, not a theorem: keep it and you get Euclid's geometry; replace it and you get another consistent geometry.
How should you revise this for an exam?
Write the five common notions and five postulates from memory.
Mark which postulates let you construct something and which state a property, such as Postulate 4.
Redo Proposition I.1, labelling every step with its rule.
Explain in two sentences why the fifth postulate is different.
If a concept still feels shaky, the Fermi Concept Library has its own lesson for every concept in mathematics. If you keep citing the wrong rule in proofs, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
Start with the sorting test: once you can sort all ten of Euclid's starting rules quickly, you have the difference.
