F.8 Chord Intersection — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.8 (Chord Intersection) (SeeligNotebooks).

Folio 0050 (Page 1 of 4): Sagitta of a Chord in a Circle of Diameter 15

[Folio 0050]

Folio 0050 (original scan 0050.jpg) from resources/seelig/08_chord-intersection.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0050 (0050.jpg, Sheet 1 of 4):

Computes the sagitta \(x = \frac{15}{2} - \sqrt{(15/2)^2 - 5^2} = \frac{5(3-\sqrt{5})}{2} \approx 1.91\) for a chord of length 10.

\[|AX| \cdot |BX| = |CX| \cdot |DX| = R^2 - |OX|^2, \qquad x_{\text{sagitta}} = R - \sqrt{R^2 - \left(\frac{c}{2}\right)^2}\]

Editorial & Symbolic Notes

Archival scan 0050.jpg from Fred W. Seelig’s notebook dossier on F.8 Chord Intersection (Page 1 of 4). Companion OpenDocument worksheet(s): Chord Intersection. Classical Euclidean foundations: III.35, III.36.

Folio 0051 (Page 2 of 4): Intersecting Chords Problem (m = 8, n = 10)

[Folio 0051]

Folio 0051 (original scan 0051.jpg) from resources/seelig/08_chord-intersection.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0051 (0051.jpg, Sheet 2 of 4):

Solves \(4\cdot 4 = x(10-x) \implies x = 2, 8\) and sets up isosceles triangles \(\triangle OAB\) and \(\triangle OCD\) to find \(|OX| = r\).

\[|AX| \cdot |BX| = |CX| \cdot |DX| = R^2 - |OX|^2, \qquad x_{\text{sagitta}} = R - \sqrt{R^2 - \left(\frac{c}{2}\right)^2}\]

Editorial & Symbolic Notes

Archival scan 0051.jpg from Fred W. Seelig’s notebook dossier on F.8 Chord Intersection (Page 2 of 4). Companion OpenDocument worksheet(s): Chord Intersection. Classical Euclidean foundations: III.35, III.36.

Folio 0052 (Page 3 of 4): Proof via Similar Triangles & Power of Point X

[Folio 0052]

Folio 0052 (original scan 0052.jpg) from resources/seelig/08_chord-intersection.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0052 (0052.jpg, Sheet 3 of 4):

Proves \(AX\cdot BX = CX\cdot DX\) from \(\triangle AXC \simeq \triangle DXB\) and equates it to \(R^2 - |OX|^2\).

\[|AX| \cdot |BX| = |CX| \cdot |DX| = R^2 - |OX|^2, \qquad x_{\text{sagitta}} = R - \sqrt{R^2 - \left(\frac{c}{2}\right)^2}\]

Editorial & Symbolic Notes

Archival scan 0052.jpg from Fred W. Seelig’s notebook dossier on F.8 Chord Intersection (Page 3 of 4). Companion OpenDocument worksheet(s): Chord Intersection. Classical Euclidean foundations: III.35, III.36.

Folio 0053 (Page 4 of 4): Concentric Circles & Ptolemy’s Formula (CRC, p. 10)

[Folio 0053]

Folio 0053 (original scan 0053.jpg) from resources/seelig/08_chord-intersection.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0053 (0053.jpg, Sheet 4 of 4):

Shows that \(R^2 - |OX|^2 = 16\) with \(2R \ge 10\) leaves one degree of freedom (the angle between chords) and applies Ptolemy’s theorem \(AC\cdot BD + AD\cdot BC = 80\).

\[|AX| \cdot |BX| = |CX| \cdot |DX| = R^2 - |OX|^2, \qquad x_{\text{sagitta}} = R - \sqrt{R^2 - \left(\frac{c}{2}\right)^2}\]

Editorial & Symbolic Notes

Archival scan 0053.jpg from Fred W. Seelig’s notebook dossier on F.8 Chord Intersection (Page 4 of 4). Companion OpenDocument worksheet(s): Chord Intersection. Classical Euclidean foundations: III.35, III.36.