Euclid and his Elements of geometry were considered for more than 2000 years to be the pinnacle and model of mathematical perfection. We now know better but the Elements remain an amazing achievement: so simple and so powerful despite their imperfections.

Euclid started by listing 23 definitions, 5 postulates, and 5 common notions as a foundation for the whole of geometry. Below are some of the definitions and all the postulates. The common notions really do seem ‘common’ and quite obvious, for example that equals added to equals, make equals. The postulates consists of three actions which Euclid assumes can always be carried out, and two assumptions. The first seems natural enough – but Euclid still felt obliged to state it – but the famous 5th postulate is by no means obvious and it was eventually discovered in the nineteenth century that if you deny it and substitute a suitable alternative you can get several perfectly consistent geometries that are non-Euclidean.