Huygens and $π$

TL;DR

The paper reexamines Huygens' De circuli magnitudine inventa to show how he converts circular-segment areas into arc-length bounds via parabolic approximations and barycentric analysis. It reconstructs Huygens' proofs of the Cusa and Snell inequalities, derives novel barycentric inequalities that yield sharp upper and lower bounds on $\pi$ with nine-decimal accuracy using a $60$-gon, and provides modern interpretations through Hofmann's and related proofs. A central contribution is the explicit barycentric equation $\frac{\Sigma}{\delta} = \frac{2}{3}\cdot\frac{2r-a}{r-\xi}$, linking segment area to the barycenter location and enabling precise arc-length estimates. The authors also propose a historical conjecture that Archimedes employed Snell--Cusa convergence-improving ideas, potentially connected to his famed trisection figure, thereby bridging ancient geometry and contemporary analytic techniques.

Abstract

The Dutch scientist Christiaan Huygens refined Archimedes' celebrated geometrical computation of $π$ to its highest point. Yet the rich content of his beautiful treatise \emph{De circuli magnitudine inventa} (1654) has apparently never been presented in modern form. Here we offer a detailed and contemporary development of several of his most striking results. We also make a historical conjecture concerning Archimedes' trisection figure.

Figures (18)

• Figure 1: A circular segment less than a semicircle
• Figure 2: Maximum triangles in circle segments
• Figure 3: Polygon perimeter comparisons
• Figure 4: Tangent and arc length comparison
• Figure 5: Outer and inner triangles of a circle segment

Theorems & Definitions (47)

• Theorem 1.1
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3: Heron--Huygens I: Theorem III of Inventa
• proof
• Theorem 2.4: First arc length inequality: Theorem VII of Inventa
• proof
• Theorem 2.5: Nikolaus of Cusa: Theorem XIII of Inventa
• proof