Chapter VII: Geometrical Constructions Based Upon Axioms I–V — Summary¶
Structural & Mathematical Overview — Kapitel VII: Die geometrischen Konstruktionen auf Grund der Axiome I–V
Architectural Summary of Chapter VII (§§ 36–39)¶
In the final chapter, Hilbert returns to Euclid’s constructive postulates (Postulates 1–3), asking precisely which geometric construction problems solvable by straightedge and compass can already be executed using only Axioms I–V—implemented physically by a straightedge (Lineal) and a transferer of segments (Streckenübertrager, specialized in later editions to the fixed-unit gauge Eichmass):
§ 36 — Elementary Constructions Without Full Circles (Theorem 40): Reduces the five fundamental planar constructions—joining points/intersecting lines (Problem 1), drawing parallels (Problem 2), transferring segments (Problem 3), transferring angles (Problem 4), and dropping perpendiculars (Problem 5, solved via semicircle chords and the triangle orthocenter!)—to Problems 1 and 3 alone (Theorem 40).
§ 37 — Pythagorean Radical Criterion (Theorem 41): Proves that a construction problem is solvable by straightedge and segment-transferer if and only if the coordinates of the sought points belong to the domain $\Omega(R)$ generated from the given coordinates $R$ by $+, -, \times, \div$ and $\sqrt{\alpha^2 + \beta^2}$ (Theorem 41). Exhibits an explicit compass-constructible right triangle (hypotenuse $1$, leg $|\sqrt{2}| - 1$, second leg $\sqrt{2|\sqrt{2}| - 2}$) impossible with straightedge and segment-transferer because its conjugate $\sqrt{-2|\sqrt{2}| - 2}$ is imaginary, whereas every number in $\Omega$ is totally real!
§ 38 — Totally Positive Numbers and Sums of Squares (Theorems 42–43): Generalizes Lagrange’s / Fermat’s four-square theorem to algebraic number fields via relative quadratic field theory: every totally positive element of a number field $k$ is a sum of four squares in $k$ (Theorem 42), and every positive-definite univariate polynomial $f(x) \in \mathbb{Q}[x]$ is a quotient of two sums of squares of polynomials in $\mathbb{Q}[x]$ (Theorem 43—the direct progenitor of Hilbert’s Seventeenth Problem, 1900!).
§ 39 — Geometric Multiplicity Criterion (Theorem 44): Combines Theorems 41–43 to establish a purely geometric necessary and sufficient criterion (Theorem 44, proved for 1-parameter problems): a ruler-and-compass problem requiring $n$ minimal square roots is solvable by straightedge and segment-transferer if and only if it possesses exactly $2^n$ real solutions for all real parameter positions of the given points! Concludes that all Gauss-constructible regular polygons and Malfatti’s problem are solvable with straightedge and segment-transferer, whereas Apollonius’s circle-tangency problem is not.