Pascal became interested in the problem and communicated it to Fermat. In which the problem was correctly but differently solved by each. In their correspondence, then, Pascal and Fermat laid the foundations of the science of probability. While Desargues and Pascal were opening the new field of projictive geometry, Descartes and Fermat were conceiving ideas of modern analytic geometry . The projective geometry is a 'branch' of geometry whereas the analytic geometry is a 'method' of geometry.
Analytic geometry combines algebra with geomety by introducing coordinates.
The real essence of the subject lies in the transference of a geometric investigation into a corresponding algebraic inverstigation. French mathematician Descartes and Fermat launched modern analytic gometry. They contributed so much to the mathematics in the 17th century. Thus, where to a large extent Descartes began with a locus and then found it equation, Fermat started with the equation and then studied the locus. That is to say, Descartes' analytic geometry is the algebraic geometry but Fermat's is geometrical algebra.