Mathematicians nevertheless succeeded in developing another approach: at the age of 26, Jacques Charles François Sturm (1803–1855) proposed a method for determining the number of real roots between two bounds.
Once Sturm's method has been applied to a polynomial equation, each root can be approximated, for example using algorithmic methods. The first phase of this analysis is called qualitative, and the second quantitative.
When Henri Poincaré adopted his "new goal of qualitative geometry" in his early work on differential equations, he was drawing on and extending this approach. The impossibility of determining the solutions of these equations did not close off the horizon; it opened up another approach—one in which discovering "the properties of differential equations is of the greatest interest."
When his questions concern the relative positions of celestial bodies (stability, moving apart or drawing closer together), the qualitative study of the differential equations modeling their motion is particularly apt.
The study of the relationships that persist when a shape is deformed arbitrarily, without tearing or folding it—a study that would lead to algebraic topology—is another example of his qualitative approach to problems.
