Portrait of Descartes by Frans Hals [Public domain], via Wikimedia Commons
The ancient Greeks could do arithmetic – with difficulty, since they didn’t have our system of numbers – but their big contribution to mathematics was Euclid’s geometry (that stuff you learned at school), and the formula for right-angled triangles that is attributed to a weird cult leader called Pythagoras. It wasn’t until the Renaissance and the influence of the Asian and Arab civilisations that a decent number system and the symbolic manipulation system called Algebra that the western world gained tools that made it easy to analyse problems that Euclid’s geometry couldn’t handle.
However, there was no common ground between algebra and geometry until René Descartes showed that equations could be represented by lines on a graph, giving people an insight into their meaning, and the meaning of solutions to these equations. Cartesian Co-ordinates – the representation of equations for given values of their variables on a system of orthogonal (i.e. at right-angles to each other) axes, are named after him.
This means that we can do geometry Euclid’s way – by drawing diagrams – or Descartes’ way – using algebra. However, using algebra allows us to explore things that Euclid would have found impossible to represent, let alone define ways of constructing them. Did you know that there are mathematicians who explore the properties of a 4-dimensional Rubik’s cube?
