Tobias Dantzig on René Descartes

“The great invention... Descartes gave to the world, the analytical diagram, ...gives at a glance a graphical picture of the law governing a phenomenon, or of the correlation which exists between dependent events, or of the changes which a situation undergoes in the course of time. ...the invention of Descartes not only created the important discipline of analytic geometry, but it gave Newton, Leibnitz, Euler, and the Bernoullis that weapon for the lack of which Archimedes and later Fermat had to leave inarticulate their profound and far-reaching thoughts.”

“Descartes implicitly assumed a complete correspondence existed between the real numbers and the points of a fixed axis. ...tacitly, because it seemed so natural as to go without saying, he accepted it as axiomatic that between the points of a plane and the aggregate of all pairs of real numbers there can be established a perfect correspondence. Thus the Dedekind-Cantor axiom, extended to two dimensions, was tacitly incorporated in a discipline which was created two hundred years before Dedekind and Cantor saw the day. This discipline became the [tool and] proving-grounds for all achievements of the following two centuries: the calculus, the theory of functions, mechanics, and physics. Nowhere did this discipline, analytic geometry, strike any contradictions; and such was its power to suggest new problems and forecast the results that wherever applied it would soon become the indispensable tool of investigation.”

“The famous problems of antiquity [doubling the cube, angle trisection, and squaring the circle]... were now disposed of by Descartes in a matter-of-fact statement that any problem which leads to an equation of the first degree is capable of a geometric solution by straight-edge only; that a straight-edge [and] compass construction is equivalent to the solution of a quadratic equation; but that if a problem leads to an irreducible equation of degree higher than the second, its geometrical solution is not possible by means of a ruler and compass only.”