Chapter IV: The Theory of Plane Areas — Summary¶
Structural & Mathematical Overview — Kapitel IV: Die Lehre von den Flächeninhalten in der Ebene
Architectural Summary of Chapter IV (§§ 18–21)¶
Building directly upon Chapter III’s non-Archimedean Streckenrechnung, Chapter IV reconstructs Euclid’s Book I (I.35–45) theory of polygonal area in the plane solely from planar Axioms `I 1–2, II–IV` without Continuity (`V`):
§ 18 — Equidecomposability vs. Equicomplementability (Theorem 24): Introduces the decisive distinction between two polygons being flächengleich (1899 term; renamed zerlegungsgleich in later editions; Townsend: “equal by addition” / equidecomposable into pairwise congruent triangles) versus inhaltsgleich (Townsend: “of equal content” / equicomplementable by adjoining equidecomposable pieces). Proves transitivity of both relations by common subdivision (Theorem 24).
§ 19 — Parallelograms and Triangles (Theorems 25–28): Reproduces Euclid’s I.35–38 trapezoid-subtraction argument to prove that parallelograms and triangles on equal bases with equal altitudes are always inhaltsgleich (Theorems 25–27), but exhibits a dramatic counterexample in $\Omega(t)$ (§ 12): two triangles of base $1$ and altitude $1$ whose apex projection is offset by the non-Archimedean element $t$ are inhaltsgleich yet cannot be dissected into congruent triangles (not flächengleich)! Formulates the converse (Theorem 28, Euclid I.39) and notes that Euclid relied on Common Notion 5 („Καὶ τὸ ὅλον τοῦ μέρους μεῖζόν ἐστιν“) as an unproved area axiom.
§ 20 — Measure of Area (*Flächenmass*, Theorem 29): Defines the area measure $F(\Delta) = \tfrac{1}{2} a h_a = \tfrac{1}{2} b h_b$ in the segment field via Theorem 22, proves invariance of $F(\Delta)$ under arbitrary simplicial subdivision by reducing any triangulation to chains of transversal splits (Theorem 29), and deduces that inhaltsgleich polygons have equal area measure—immediately proving Theorem 28.
§ 21 — Converse Equivalence and De Zolt’s Postulate (Theorems 30–31): Proves conversely that any two polygons with equal segment-valued area measure are inhaltsgleich (Theorem 30), and deduces Antonio De Zolt’s postulate / Euclid C.N. 5 (Theorem 31): if a rectangle is dissected into triangles and even a single triangle is omitted, the remaining triangles can never refill the rectangle!