In his 1676 manuscript De quadratura arithmetica circuli, ellipseos et hyperbolae, Leibniz describes the Resection Method for determining areas under curves. This method establishes a relationship between the area under a given curve and the area under another, often simpler, curve, thereby enabling the computation of a wide range of areas. After introducing the Resection Method, we turn to the so-called Fringe Method, which likewise relates areas under different curves. We show that the Fringe Method in fact inverts the Resection Method, a result that constitutes the central contribution of this chapter. Particular emphasis is placed on the geometric interpretation of the Fringe Method, which in turn sheds new light on the Resection Method itself. A series of examples is used to illustrate both approaches. By studying the inverse of Leibniz’s Resection Method, we reveal a largely unexplored potential that merits closer attention.