The implications of Descartes' analytic reformulation of geometry are obvious. Not only did the new method make possible a systematic investigation of known curves, but, what is of infinitely greater significance, it potentially created a whole universe of geometric forms beyond conception by the synthetic method. Descartes also saw that his method applies equally as well to surfaces... but he did not develop this. With the extension to surfaces, there was no reason why geometry should stop with equations in three variables; and the generalization to systems of equations in any finite number of variables was readily made in the nineteenth century. Finally, in the twentieth century, the farthest extension possible in this direction led to spaces of a non-denumerable infinity of dimensions. ...The path from Descartes to the creators of higher space is straight and clear; the remarkable thing is that it was not traveled earlier than it was.