 In two stages, then, Descartes has demonstrated the solution to the general Pappus Problem. In Book One he offers his algebraic analysis of the problem, and in Book Two he claims to provide the synthesis (or demonstration) that the curves that solve the general problem are legitimately geometrical curves which meet his stated standard for geometrical exactness and precision. And with these two stages completed, Descartes claims to Mersenne six months after La Géométrie is published that his treatment of the general Pappus Problem is proof that his new method for geometrical-problem solving is an improvement over the methods of his predecessors: