The turning point in Descartes's intellectual development occurred on November 10, 1619. He had attended the coronation of Ferdinand II in Frankfurt, and was returning to serve in the army of Maximilian of Bavaria. Due to the onset of winter, he holed himself up for a day, alone in a stove-heated room. With nothing else to occupy him, he set about thinking.
He first mused that accomplishments of single individuals are usually more perfect than group efforts. Cities and buildings are more beautiful when they are made according to a single plan than when they are patched together piecemeal. Similarly, laws are better when they come from a single mind than when they evolve gradually over time. Descartes cites God's law as an instance of this perfection. These musings suggest to him that a person is best served by following the guidance of his reason alone, and not letting his judgments be clouded by his appetites and by the opinions of others.
While it would be impossible to resolve the imperfections of a state or a body of sciences by tearing it all down and starting again from scratch, Descartes suggests that such a method is not quite as unreasonable on the individual level. He decided to let go of all his former opinions at once, and re-build them anew according the exacting standards of his own reason.
Descartes is very careful, first of all, to point out that this method is meant only on an individual level, and he strongly opposes those who would try to topple a public institution and rebuild it from the ground up. Second, he reminds us that he only wants to discuss his method with us; he is not telling us to imitate him. In particular, he notes that there are two types of people for whom this method would be unsuited: those who think they know more than they do and who lack the patience for such careful work, and those who are modest enough to think that they are more capable of finding out the truth if they follow a teacher. Descartes would count himself among this second group if he hadn't had such a number of teachers and embarked on so many travels as to realize that the opinions of even learned men vary greatly.
Before abandoning his former opinions entirely, Descartes formulates four laws that will direct his inquiry: First, not to accept anything as true unless it is evident; this will prevent hasty conclusions. Second, to divide any given problem into the greatest possible number of parts to make for a simpler analysis. Third, to start with the simplest of objects and to slowly progress toward increasingly difficult objects of study. Fourth, to be circumspect and constantly review the progress made in order to be sure that nothing has been left out.
An obvious starting place was in the mathematical sciences, where a great deal of progress and certain knowledge had been achieved by means of demonstration. Descartes found his work made considerably easier if, on the one hand, he considered every quantity as a line, and, on the other hand, developed a system of symbols that could express these quantities as concisely as possible. Taking the best elements of algebra and geometry, he had tremendous success in both these fields.