Method of exhaustion

X.1 gives a reduction principle: for magnitudes satisfying the required multiple-comparison premise, repeatedly removing more than half the remainder eventually makes it smaller than a given smaller magnitude.

The remainder need not become zero at a particular step. It can stay positive while eventually falling below any specified positive area. This permits a finite number of subdivisions to confront a particular assumed discrepancy.

References: [1]

XII.2 proves that circle areas are in the ratio of squared diameters. Repeated arc bisection adds sides to inscribed polygons and reduces the uncovered circular segments.

A hypothetical smaller comparison area is contradicted by choosing subdivisions fine enough for the inscribed polygon to exceed it, while corresponding polygon ratios demand the opposite. Excluding both directions of discrepancy establishes the exact ratio.

References: [2]

XII.5 subdivides equal-height pyramids with triangular bases into prisms and smaller residual pyramids to establish their volume ratio. Remainder control therefore extends beyond curved plane regions.

Numerical subdivision may estimate a magnitude, but several accurate digits are not the same as proof of an exact relationship. The argument must justify the required error bound and the contradiction, rather than relying on a sufficiently dense-looking drawing.

References: [3]