Erotusmenetelmässä

Erotusmenetelmässä, often translated as the method of exhaustion, is a mathematical technique used to find the area of a region by inscribing and circumscribing a sequence of polygons, the areas of which approach the area of the region. This method was notably used by Archimedes in the 3rd century BC to calculate the area of a circle, the surface area of a sphere, and the volume of a sphere. The core idea is to divide a complex shape into simpler shapes whose areas can be calculated. As the number of these simpler shapes increases and their size decreases, their combined area gets arbitrarily close to the area of the original shape.

The process involves establishing upper and lower bounds for the area. For instance, when calculating the area of a circle, Archimedes inscribed regular polygons within the circle and circumscribed regular polygons around it. He then proved that the area of the circle was greater than the area of the inscribed polygon and less than the area of the circumscribed polygon. By doubling the number of sides of the polygons repeatedly, he showed that the difference between the areas of the inscribed and circumscribed polygons could be made as small as desired. This convergence of the areas of the polygons to a single value allowed him to determine the area of the circle. The erotusmenetelmässä is considered a precursor to modern integral calculus, as it relies on the concept of limits and the approximation of areas.