Shape and space

c. 300 BC

Euclid’s Elements

Euclid of Alexandria

If something seems obviously true, how can you be sure it is always true?

In Alexandria, Euclid gathered Greek geometry into thirteen books. He started from a handful of simple assumptions and proved everything else step by step. It was used to teach maths for over two thousand years.

How they figured it out

Euclid opened the Elements with a short list of definitions, five postulates and five “common notions”, then built more than 450 propositions, each proved only from what came before. Nothing is allowed in just because it looks right in a drawing.

His fifth postulate, about parallel lines, is longer and clumsier than the others, and Euclid avoids using it for his first 28 propositions, almost as if he were uneasy. For two thousand years mathematicians tried to prove it from the other four. They failed, and that failure turned out to be one of the most productive in history: it led to curved geometries, and eventually to Einstein.

The story goes that a king asked Euclid for a quicker way to learn geometry, and he replied that there is no royal road. It is probably a legend, but a good one.

The equation

axioms  ⇒  theorems\text{axioms} \;\Rightarrow\; \text{theorems}

Start from a few things everyone agrees on, and prove everything else from them.

Everyday echoEvery proof you write in a maths exam follows Euclid’s pattern.

See it in real life

Sudoku is Euclid’s game

In a good Sudoku you never guess. Each number is forced by the numbers already there, and you can always say why. That is exactly how a proof works: start from what is given, and make every step follow.

Where engineers use it

  • Proof and verification

    Software that checks chip designs and aircraft control code works Euclid’s way: from axioms, step by step.

  • Engineering drawing

    Constructions with compass and straight edge underlie technical drawing and CAD constraints.

Where it came from

What it led to