Classical surveyors and astronomers, such as Hipparchus, had specified the position of a place on the earth's surface by the two " coordinates " latitude and longitude. Further, historians have detected the idea of a graph as early as the 10th century. But none of these contain the essence of Cartesian geometry. This lies in representing curves or graphs by coordinates connected by a formula.

While the 16th century had made progress in the theory of equations, these had only been (as we should say) in one variable. The idea of a relation of two current variables, of a function, was in effect as new in algebra as was its application in geometry. It was an idea, and an application, the unfolding of the riches of which is still not finished ; and Descartes made only a beginning. He stated, for instance, that if any curve could be mechanically described, it could be represented by an equation. He saw that the axes could be either perpendicular or oblique, but he did not realise, as did Newton later, the immense usefulness of the negative extensions of the axes.

The new method was not widely known until the annotated edition of 1659. Wallis (1616-1703), however, systematised and extended it, and took the step of identifying cones with curves having equations of the second degree. This clinched the classificatory primacy of degree both for equations and for curves, and made it a natural step when Newton went on to examine cubic curves.

Descartes taught that the language of nature is mathematics and that 'nothing exists, except atoms and empty space; everything else is opinions',  His legacy was accepted by Newton, who used, what we now call, Cartesian methods, to define a world of matter in motion.