F.49 Vesica Piscis and Acute-Obtuse Probability — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.49 (Vesica Piscis and Acute-Obtuse Probability) (SeeligNotebooks).

Folio 0315 (Page 1 of 3): Locus of Longest-Side Triangles in the Vesica Piscis (001)

[Folio 0315]

Folio 0315 (original scan 0315.jpg) from resources/seelig/49_vesica-pisces.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0315 (0315.jpg, Sheet 1 of 3):

Identifies the region where \(|AC| \le c\) and \(|BC| \le c\) with the vesica piscis on \(AB\) and computes the quarter-vesica area \(c^2\frac{4\pi-3\sqrt{3}}{24}\).

\[(O) + (A) = c^2\left(\frac{\pi}{6} - \frac{\sqrt{3}}{8}\right) = c^2\frac{4\pi - 3\sqrt{3}}{24}, \qquad (O) = \frac{1}{4}\pi\left(\frac{c}{2}\right)^2 = c^2\frac{\pi}{16}\]

Editorial & Symbolic Notes

Archival scan 0315.jpg from Fred W. Seelig’s notebook dossier on F.49 Vesica Piscis and Acute-Obtuse Probability (Page 1 of 3). Companion OpenDocument worksheet(s): Monte Carlo, General Reference. Classical Euclidean foundations: I.1, III.31.

Folio 0316 (Page 2 of 3): Thales Semicircle Partition into Obtuse (O) & Acute (A) Regions (002)

[Folio 0316]

Folio 0316 (original scan 0316.jpg) from resources/seelig/49_vesica-pisces.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0316 (0316.jpg, Sheet 2 of 3):

Uses the semicircle of radius \(c/2\) on diameter \(AB\) to separate obtuse triangles \((O) = c^2\pi/16\) from acute triangles \((A) = c^2\frac{5\pi-6\sqrt{3}}{48}\).

\[(A) = c^2\frac{5\pi - 6\sqrt{3}}{48}, \qquad \frac{(O)}{(A)} = \frac{3\pi}{5\pi - 6\sqrt{3}} \approx 1.7731\]

Editorial & Symbolic Notes

Archival scan 0316.jpg from Fred W. Seelig’s notebook dossier on F.49 Vesica Piscis and Acute-Obtuse Probability (Page 2 of 3). Companion OpenDocument worksheet(s): Monte Carlo, General Reference. Classical Euclidean foundations: I.1, III.31.

Folio 0317 (Page 3 of 3): Exact Obtuse-to-Acute Probability Ratio (003)

[Folio 0317]

Folio 0317 (original scan 0317.jpg) from resources/seelig/49_vesica-pisces.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0317 (0317.jpg, Sheet 3 of 3):

Evaluates \((O)/(A) = \frac{3\pi}{5\pi-6\sqrt{3}} \approx 1.773\) and compares with the empirical counts in the Monte Carlo spreadsheet.

\[(O) + (A) = c^2\left(\frac{\pi}{6} - \frac{\sqrt{3}}{8}\right) = c^2\frac{4\pi - 3\sqrt{3}}{24}, \qquad (O) = \frac{1}{4}\pi\left(\frac{c}{2}\right)^2 = c^2\frac{\pi}{16}\]

Editorial & Symbolic Notes

Archival scan 0317.jpg from Fred W. Seelig’s notebook dossier on F.49 Vesica Piscis and Acute-Obtuse Probability (Page 3 of 3). Companion OpenDocument worksheet(s): Monte Carlo, General Reference. Classical Euclidean foundations: I.1, III.31.