Unit 4 Summary: Coordinates Without Measure (The Algebra of Throws)

Analytical Synopsis of Beiträge zur Geometrie der Lage, Heft 2 (1857), §§ 19, 20, 29

1. Throw (*Wurf*, § 19 art. 256): An ordered projective equivalence class $(ABCD)$ of four collinear elements; fixing gauge triad $(P_0, P_1, P_\infty)$ identifies each point $X \in \ell \setminus \{P_\infty\}$ with $(P_\infty P_0 P_1 X)$.

2. Straightedge Addition (§ 19 art. 259): Centers $U, V$ collinear with $P_\infty$ and axis $a = [P_\infty R]$ ($R = [U P_0] \cap [V X]$) construct $S = X + Y$ by a parabolic double perspectivity fixing $P_\infty$.

3. Straightedge Multiplication (§ 20 art. 268): Centers $U, V$ collinear with $P_0$ and axis $a = [P_\infty W]$ ($W = [U P_1] \cap [V X]$) construct $P = X \cdot Y$ by a hyperbolic double perspectivity fixing $P_0$ and $P_\infty$.

4. Harmonic Scale & Projective Abscissae (§ 29 arts. 403–404): Iterated sums and quotients construct every rational coordinate $P_{p/q}$ by straightedge alone, coordinatizing $\mathbb{P}^1$ intrinsically without metric distance.