The History of Ancient Greek Mathematics as a guide of mathematical research

Weil, in his reputed paper entitled, "History of mathematics : why and how", stresses that the study of particularly distinguished examples constitutes a powerful tool to promote scientific research.
In the History of Ancient Greek mathematics, we have a prolific source of distinguished examples: i.e.:
The discovery of incommensurability gave birth to the study of irrational numbers from Eudoxus to Dedekind. The quadrature of the circle served as an experiment in order to develop the inherent reasoning and advance the exhaustion's method from Archimedes to the founders of the calculus, as well as the nature of π.Zenon's paradoxes reveal two opposing concepts of space and time ( space and time are infinitely divisible / space and time are made up of indivisible small intervals ) and are the first indication of the difficulties to understand infinite sets. On the other hand Apollonius' Conics offers the theoretical background to Kepler's Laws of planetary motion and to Newton's celestial mechanics.

Without apparent application Ancient Greek mathematics have offered us an excellent example of the know-how of this procedure. It occured however not at the time of its origination but much later , when Ancient Greek mathematics became the rational cause for the advancement of new mathematical theories.