Space

Euclid's geometry

Tell an axiom from a postulate, and use Euclid's first assumptions to justify a step.

Every proof has to start somewhere. You cannot prove absolutely everything, because each proof leans on something earlier, and sooner or later you run out of earlier things. Euclid's answer, more than two thousand years ago, was to write down exactly what he was assuming and then never assume anything else. Everything after that had to be earned.

Axiom or postulate?

Both are assumed rather than proved. An axiom is a common-sense truth about anything at all — the whole is greater than the part. A postulate is an assumption about geometry in particular — a straight line can be drawn between any two points. Anything Euclid then argued into being is a theorem.

Have a play

Four drawings, and the only difference between them is where they stop. Find out what each one is called.

Tap any part of the picture.

Worked example

Why can two straight lines never cross twice?

  1. Suppose two different lines did cross at two points, P and Q.

    Assuming the opposite and watching it fall apart is the oldest move in geometry.

Try it together

B is a point on segment AC, with AB = 4 units and BC = 3 units. Show that AC is longer than AB.

Two of Euclid's axioms are enough, and neither of them needs a ruler.

    1.AC coincides with AB and BC laid end to end, and things which coincide with one another are equal — so AC = AB + BC. If AB = 4 and BC = 3, what is AC?

    The postulate that would not settle

    Euclid's fifth postulate — the one about co-interior angles adding to less than 180° — is far wordier than the other four, and for centuries people tried to prove it from them. Nobody managed, and in the end that turned out to be the point: assume something different instead and you get a whole other geometry, one that describes curved surfaces.

    Have a go

    Have a go on your own: "All right angles are equal to one another" is one of Euclid's...

    Ready to practise?

    Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.

    Print a worksheet