How did Eudoxus use the method of exhaustion to work with infinite processes?

Eudoxus of Cnidus developed the method of exhaustion as a rigorous logical procedure to calculate the areas and volumes of geometric figures without relying on the concept of infinity or infinitesimal quantities, which the Greeks generally rejected.123

Instead of using modern limits, Eudoxus employed a bisection principle: any given quantity can be made smaller than another by successively halving it a finite number of times. He applied this to measure complex shapes by "exhausting" them with combinations of simpler polygons or polyhedra of known dimensions.142

For example, to determine the volume of a pyramid, Eudoxus demonstrated that:
* Any volume less than the target value could be exceeded by a stack of prisms inside the figure.2
* Any volume greater than the target could be undercut by a stack of prisms containing the figure.2

By showing that all other possibilities were "exhausted," he proved the volume must be exactly $Bh/3$ (where $B$ is the base area and $h$ is the height). This approach provided a foundational, rigorous framework for handling continuous magnitudes that prefigured integral calculus.213