Part II, Chapter II — Analytical Summary

Five Epochs of Circle Measurement from Ahmes to Gordan (§§ 1–5)

Chronological Architecture (Following Rudio 1892)

Drawing on Ferdinand Rudio's historical anthology (1892), Klein organizes $4000$ years of circle-squaring research into five methodological epochs:

Five Historical Stages in the Problem of $\pi$

Stage	Key Figures & Dates	Mathematical Milestone
§ 1. Empirical	Ahmes (c. 2000 B.C.); 1 Kings 7:23	Mensuration rules: Egyptian $\pi \approx (16/9)^2 \approx 3.1605$; Biblical $\pi \approx 3$.
§ 2. Classical Greek	Hippias, Dinostratus, Archimedes, Huygens (1654)	Rigorous polygonal exhaustion ($3\frac{10}{71} < \pi < 3\frac{1}{7}$) and kinematic quadratrix $x = \frac{y}{\tan(\pi y/2)} \to \frac{2}{\pi}$.
§ 3. Analytic Series	Napier (1614), Leibniz, Euler (1670–1770)	Infinite series ($\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \dots$) and Euler’s bridge $e^{ix} = \cos x + i\sin x \implies 1 + e^{i\pi} = 0$.
§ 4. Irrationality	Lambert (1770), Legendre (1794)	Continued-fraction proofs that $\pi \notin \mathbb{Q}$ and $\pi^2 \notin \mathbb{Q}$ (leaving open whether $\pi \in \overline{\mathbb{Q}}$).
§ 5. Transcendence	Hermite (1873), Lindemann (1882), Weierstrass (1885), Hilbert, Hurwitz, Gordan (1893)	Proof that $e \notin \overline{\mathbb{Q}}$ and $\pi \notin \overline{\mathbb{Q}}$, reduced in Spring 1893 to purely elementary polynomial divisibility modulo $p$.