From Euclidean to Hilbertian Practice: The Theory of Plane Area

The theory of plane area played a central role in Euclid’s development of plane geometry in the Elements. In Books I–VI, Euclid presented the first systematic treatment of the concept of area of a plane rectilinear figure in Greek geometry. The propositions about plane areas were also crucial for other parts of his geometrical theory; for instance, they constituted the cornerstone of the theory of similar figures in Book VI. A central feature of the strategy deployed by Euclid for the study of areas was the complete absence of metrical considerations, especially the concept of area measure. In the early books of the Elements, Euclid developed a theory of the comparison of plane areas, not a theory of measure of area in the modern sense, i.e., as numerical functions that assign (positive) real numbers to every rectilinear figure. His approach established the equality of area or “content” of polygonal figures based on the possibility of decomposing and (re)composing them into polygonal parts, congruent in pairs, respectively. Thus, this strictly geometrical method to study plane areas, labeled in modern times the “geometrical theory of equivalence,” was fundamentally grounded on the relation of geometrical congruence.