De Zolt’ Postulate: The Geometrical Path

In Chap. 1, we saw that the fundamental problem in the geometrical theory of equivalence is to guarantee that plane polygons can be totally or linearly ordered with respect to their areas. In other words, a crucial task in the rigorous development of this theory is to secure the comparability of (simple) polygons based on an adequate geometrical relation of ‘equality of area’ or equivalence. In the Elements, the first systematic exposition in Greek mathematics of the theory of equivalence, Euclid proved an array of propositions (I.42–I.45) that put forward a method to compare any two given polygonal figures, making operational the comparison of plane areas. The central idea of Euclid’s method was the following well-known result: any polygon can be transformed into an equivalent parallelogram with a given angle and side. Then, any two polygons can always be transformed into two equivalent rectangles of equal altitudes, which can be “juxtaposed” and ordered by comparing their bases. It should be noted that Euclid never explicitly defined when a polygon \(\textbf{P}\) is greater (or lesser) in area than a polygon \(\textbf{Q}\) , but his geometrical practice suggested that he grounded the relation of order of polygonal areas on the relation of inclusion or, better, on the mereological relation of parthood. Accordingly, a polygon \(\textbf{P}\) is said to be greater in area than another polygon \(\textbf{Q}\) if there is a polygon \(\mathbf {P'}\) , properly contained in \(\textbf{P}\) , such that \(\mathbf {P'}\) is equal in area to \(\textbf{Q}\) .