# Visualizing algebra before Descartes

The Cartesian coordinate system is what I have been told provided the first link between algebra and geometry. However, I also have learned that, for instance, Omar Khayyam solved cubic equations as the intersection between parabolas.

So, how was this visualization done before Descartes? Also, what is so fundamentally different between the Cartesian coordinate system and the older methods?

The use of geometry to solve (what we would call) polynomial equations is also ancient. Omar Khayyam was following in the footsteps of Menaechmus, Archimedes and Diocles, the first of whom lived before Euclid. Of course, ancient geometers did not start with equations, usually the initial problem was itself geometric, e.g. Menaechmus used intersection of curves to duplicate the cube, which in algebraic terms translates into solving $x^3=2$. Once Islamic mathematicians developed algebraic notation however, starting with al-Khwarismi in the 9th century AD, they started converting geometric problems into equations and back, and using ancient geometric methods to solve them. See How was geometry historically used to solve polynomial equations?.