Part I, Chapter V — Analytical Summary

Beyond Compass and Straightedge: Origami, Conics, Cissoid, Conchoid, and Linkages (§§ 1–5)

Hierarchy of Constructive Auxiliaries

Having completed the positive and negative theory of quadratic radical towers (Chapters I–IV), Klein concludes Part I by surveying algebraic means that transcend degree $2^h$:

  1. Paper Folding (§ 1): Hermann Wiener (polyhedral nets) and T. Sundara Row (1893) introduce crease-based point and tangent constructions.

  2. Conic Intersections (§ 2): Substituting $y = x^2$ reduces any cubic $x^3 + ax^2 + bx + c = 0$ or quartic $x^4 + ax^3 + bx^2 + cx + d = 0$ to intersecting the fixed parabola $y = x^2$ with a hyperbola ($3$ finite intersections + $1$ point at infinity for cubics; $4$ finite intersections for quartics).

  3. Cissoid of Diocles (§ 3): The rational cuspidal cubic $(x^2 + y^2)x - y^2 = 0$ ($r = \frac{\sin^2\theta}{\cos\theta}$) satisfies $\left(\frac{y}{x}\right)^3 = \frac{y}{1 - x}$, converting the intercept $2$ on $x = 0$ directly into the cube-duplication ordinate $\sqrt[3]{2}$ on $x = 1$.

  4. Conchoid of Nicomedes & Bézout Accounting (§ 4): The circular quartic $(x^2 + y^2)(x - a)^2 - b^2 x^2 = 0$ trisects $\angle MOY = \varphi$. Because the trisection cubic is irreducible, any algebraic curve intersection must yield all $3$ roots simultaneously: out of $4 \times 2 = 8$ Bézout intersections with the circle, $2$ lie at the origin node $O$, $2$ at the circular points at infinity, and $1$ at the rational point $B$ on ray $OM$, leaving exact $3$ trisecting roots $\frac{\varphi + 2k\pi}{3}$ ($k = 0, 1, 2$).

  5. Continuous-Motion Trammels (§ 5): Theoretical exactness with higher curves requires continuous mechanical generation (such as Nicomedes’ conchoid trammel), not discrete point-plotting.