Chapter IV: The Theory of Plane Areas — Notes¶
Historical, Philological & Cross-Repository Commentary
GEOMETOR Commentary & Cross-Repository Notes¶
1. Heath on Euclid I.35 and Hilbert’s Edition Shift (flächengleich → zerlegungsgleich)¶
In geometor/euclid (volume_I/book_i/propositions/35/commentary.rst, folio 0342, Heath 1908, Vol. I, p. 328), T. L. Heath devotes a special section to Hilbert’s Chapter IV distinctions. Readers comparing the 1899 German Festschrift, Townsend’s 1902 English, and Heath’s 1908 commentary should note an essential terminological evolution:
Mathematical Concept |
1899 Teubner (1st ed.) |
1902 Townsend (EN) |
Later German eds. & Heath (1908) |
|---|---|---|---|
Dissectible into finitely many pairwise congruent triangles |
equal area / equal by addition |
zerlegungsgleich (equidecomposable) |
|
Obtained by subtracting flächengleich pieces from flächengleich wholes |
equal content (of equal content) |
inhaltsgleich / ergänzungsgleich (equicomplementable) |
Why does Euclid I.35 (“Parallelograms which are on the same base and in the same parallels are equal to one another”) work without Archimedes’s Axiom only for inhaltsgleich and NOT for flächengleich? Look at fig_polygon_equivalence.svg (§ 19): when parallelogram $AB'C'D$ is sheared arbitrarily far to the right—say by the non-Archimedean distance $AF = t$ in $\Omega(t)$ while $AD = 1$—one can still add the single triangle $AED$ and subtract congruent trapezoids in one step (inhaltsgleich), whereas dissecting one triangle into congruent pieces to tile the sheared triangle would require $\ge t$ steps, which no finite integer $n$ can reach! With Archimedes’s Axiom restored, Farkas Bolyai and Paul Gerwien (1833) proved that inhaltsgleich and zerlegungsgleich coincide in the plane.
2. Proving Euclid’s Common Notion 5 (De Zolt’s Postulate, §§ 19–21)¶
In § 19, Hilbert quotes Euclid’s Greek text of Common Notion 5 verbatim:
„Καὶ τὸ ὅλον τοῦ μέρους μεῖζόν ἐστιν“ (“And the whole is greater than the part”)
pointing out that when Euclid invokes C.N. 5 in I.39 to prove equal-area triangles on the same base have equal altitude (Theorem 28), it is not a purely logical truism—without a proof that polygonal area cannot remain unchanged when a piece is removed (Antonio De Zolt’s 1881 postulate, De Zolt 1881), every polygon could trivially be inhaltsgleich to every other! By constructing the segment-valued area functional $F(P) \in \Omega_{\text{seg}}$ in § 20 via Chapter III’s Pappus-based Streckenrechnung and proving triangulation invariance (Theorem 29), Hilbert converts Euclid’s Common Notion 5 into a rigorous theorem (`Theorem 31`)!
Cross-Repository Concordance Links¶
(See the full Euclid–Hoenen–Hilbert–Pappus Master Concordance for complete tables.)
§ 18, § 19, § 20, § 21 (§ 18, § 19, § 20, § 21 (chapter-4)): Reconstructs Euclid I.35–45 without Continuity by separating equidecomposability (flächengleich / zerlegungsgleich) from equicomplementability (inhaltsgleich), exhibits a non-Archimedean triangle pair in Ω(t) (§ 19) that is inhaltsgleich without being flächengleich, and defines additive triangle area via Chapter III’s segment calculus to prove Common Notion 5 / De Zolt’s Theorem (Theorem 31). - Euclid — C.N. 2, C.N. 3, C.N. 5; Book I, Props. 35–45: Heath (Vol. I, p. 328 on I.35) directly cites Hilbert’s distinction between polygons equal by decomposition (1899 flächengleich / later zerlegungsgleich) and polygons of equal content by subtraction of congruent pieces (inhaltsgleich / ergänzungsgleich). Moreover, Hilbert proves Euclid’s Common Notion 5 (‘The whole is greater than the part’ for polygon area, De Zolt’s postulate) as Theorem 31 using segment-algebra area measures! - : Clarifies the axiomatic vs. intuitive basis of ‘The whole is greater than the part’ (C.N. 5) in polygonal dissection.