Unit VI Summary: Unifying Spatial Form and Eudoxian Ratio

How Book VI Reconnects Book V Algebra to Planar Geometry

1. Proportionality Without Commensurability: Armed with Book V’s general ratio calculus, Book VI proves Thales’s intercept theorem (VI.2), AAA/SAS similarity (VI.4–VI.7), and quadratic area scaling (VI.19–VI.20) without ever assuming rational side ratios.

2. Two Faces of the Golden Ratio (``II.11`` vs. ``VI.30``): Whereas II.11 formulated the golden cut purely as an area equality ($AH^2 = AB \cdot HB$) because ratio theory did not yet exist, VI.30 expresses it directly as a continuous proportion ($AB : AE \mathrel{∷} AE : EB$) via VI.14 and parabolic/hyperbolic application of areas (VI.29).