Chapter V: Desargues’s Theorem — Summary¶
Structural & Mathematical Overview — Kapitel V: Der Desarguessche Satz
Architectural Summary of Chapter V (§§ 22–30)¶
Chapters V and VI pivot from metric Euclidean geometry to projective-affine foundations, investigating what happens when Congruence (`IV`) is dropped altogether:
§ 22 — Desargues’s Theorem (Theorem 32): Formulates the affine Desargues's Theorem (Theorem 32) for perspective triangles with parallel sides, observing that it follows effortlessly either from 3D spatial incidence (I 1–7, II, III) or from 2D planar congruence (I 1–2, II–IV via Chapter III coordinates).
§ 23 — Hilbert’s Non-Desarguesian Plane (Theorem 33): Constructs a celebrated planar countermodel inside the Euclidean $XY$-plane: straight lines cutting the ellipse $x^2 + 4y^2 = 1$ have their interior chord replaced by the circular arc passing through the two boundary intersections $P, Q$ and the exterior point $F(\tfrac{3}{2}, 0)$ (since $\tfrac{3}{2} > \tfrac{\sqrt{7}}{2}$, no circle meeting the ellipse four times can pass through $F$). This deformed plane satisfies all planar axioms `I 1–2, II, III, IV 1–5, V` while violating Desargues’s Theorem—proving planar Desargues unprovable without either 3D spatial incidence (I 3–7) or triangle congruence (IV 6)!
§§ 24–26 — Congruence-Free Segment Calculus: Fixes two intersecting axes $OE, OE'$ and a unit line $EE'$ in any plane satisfying I 1–2, II, III + Desargues (Theorem 32), defining addition $c = a + b$ and multiplication $c = ab$ purely by parallel projections. Using intricate multi-triangle Desargues configurations (Figures 41–45, drafted with Hans von Schaper), Hilbert proves commutative and associative addition (§ 25), associative multiplication $a(bc) = (ab)c$, and both left and right distributive laws (§ 26)—all without Congruence!
§§ 27–30 — Skew-Field Coordinatization and 3D Embeddability (Theorems 34–35): Proves every line has a left-linear equation $ax + by + c = 0$ (Theorem 34, whereas $xa + yb + c = 0$ need not be a line when multiplication is non-commutative!), shows the segments form an ordered associative division ring (Desarguessches Zahlensystem, rules 1–11, 13–16 of § 13), reconstructs 3D space over any Desarguesian division ring (§ 29), and crowns the chapter with Theorem 35: Desargues’s Theorem is the necessary and sufficient condition for a planar geometry (I 1–2, II, III) to be embeddable as a plane inside a 3D space (I, II, III).