Välistamismeetod

Välistamismeetod, known in English as the method of exhaustion, is a mathematical procedure for determining the area of a region by inscribing and circumscribing it with polygons whose areas are known and which approach the area of the region as the number of sides increases. This technique was extensively used by ancient Greek mathematicians, most notably Archimedes, in his work on calculating the areas of circles, parabolic segments, and volumes of spheres and cylinders.

The core idea of the välistamismeetod is to establish upper and lower bounds for the area in question. For instance, to find the area of a circle, one could inscribe a regular polygon within the circle and circumscribe a regular polygon around it. As the number of sides of these polygons increases, their areas progressively converge towards the area of the circle. By demonstrating that the difference between the areas of the inscribed and circumscribed polygons can be made arbitrarily small, Archimedes was able to rigorously determine the area of the circle, which he showed to be proportional to the square of its radius. This method is a precursor to modern integral calculus, laying the groundwork for the concept of limits and the summation of infinitesimally small quantities. It represents a powerful geometric approach to solving problems that would later be tackled with more abstract analytical tools.