Chapter III: The Theory of Proportion — Summary¶
Structural & Mathematical Overview — Kapitel III: Die Lehre von den Proportionen
Architectural Summary of Chapter III (§§ 13–17)¶
Chapter III resolves one of the deepest structural problems in the history of geometry: reconstructing Euclid’s Book V–VI theory of proportion and similar triangles purely in the plane (`I 1–2, II–IV`) without the Axiom of Archimedes (`V`):
§ 13 — Complex Number Systems: Catalogues 17 algebraic axioms (1–12 field laws, 13–16 ordered ring laws, 17 Archimedean property), framing the central question of geometric algebra: which geometric axioms correspond to which algebraic field laws?
§ 14 — Proof of Pascal’s (Pappus’s) Theorem (Theorem 21): States the affine two-line specialization of Pascal’s hexagram—historically Pappus’s Hexagon Theorem (Synagoge VII.138–139): if $A, B, C$ and $A', B', C'$ lie on two intersecting lines through $O$ with $CB' \parallel BC'$ and $CA' \parallel AC'$, then $BA' \parallel AB'$. Using right-triangle projection operators $a = \alpha c$ and inscribed cyclic quadrilaterals ($ABCD$ on a diameter circle), Hilbert proves operator commutativity $\alpha\beta c \equiv \beta\alpha c$ and deduces Theorem 21 (adding a second direct four-circle proof).
§ 15 — Geometric Calculus of Segments (*Streckenrechnung*): Defines segment addition $c = a + b$ by collinear juxtaposition and segment multiplication $c = ab$ on perpendicular rays from $O$ via a fixed unit segment $1$ and parallel lines. Crucially, Theorem 21 (Pascalscher Satz) immediately yields commutativity $ab = ba$ and associativity $a(bc) = (ab)c$, while congruent right triangles establish distributivity $a(b + c) = ab + ac$.
§ 16 — Proportion and Similitude (Theorems 22–23): Defines $a : b = a' : b'$ algebraically as $ab' = ba'$, proves proportionality of corresponding sides in similar triangles (Theorem 22, first for right triangles and then generally via the incenter $S$ of the three angle bisectors!), and deduces the Fundamental Theorem of Proportion (Theorem 23, Thales’s intercept theorem Euclid VI.2).
§ 17 — Linear Equations of Lines and Planes: Extends positive segments to signed segments ($+, 0, -$) forming an ordered commutative field, establishes rectangular coordinates $(x, y)$, proves every straight line has a linear equation $bx - ay - bc = 0$, and shows how adjoining Archimedes (V) embeds the coordinate domain into $\mathbb{R}^3$.