F.24 Gothic Arch — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.24 (Gothic Arch) (SeeligNotebooks).

Folio 0135 (Page 1 of 5): Gothic Arch & Oculus Circle System in Square of Side s

[Folio 0135]

Folio 0135 (original scan 0135.jpg) from resources/seelig/24_gothic.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0135 (0135.jpg, Sheet 1 of 5):

Sets up circle \(r^2 = (x-s/2)^2 + (y-Q)^2\), arch \(s^2 = x^2+y^2\), and line \(l_{CF}: y = \sqrt{3}x + s(1-\sqrt{3})\).

\[r^2 = \left(x - \frac{s}{2}\right)^2 + (y - Q)^2, \qquad p s = P_x H_x + P_y H_y \quad (r = s - p), \qquad g^2 = G_x^2 + G_y^2 = 2 s G_x\]

Editorial & Symbolic Notes

Archival scan 0135.jpg from Fred W. Seelig’s notebook dossier on F.24 Gothic Arch (Page 1 of 5). Companion OpenDocument worksheet(s): General Reference. Classical Euclidean foundations: I.1, III.11, III.12.

Folio 0136 (Page 2 of 5): Three-Circle Symmetry Analysis at Intersections H and G

[Folio 0136]

Folio 0136 (original scan 0136.jpg) from resources/seelig/24_gothic.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0136 (0136.jpg, Sheet 2 of 5):

Exploits horizontal symmetry \(H_y = G_y\) and \(G_x + H_x = s\) across the centerline \(x = s/2\).

\[r^2 = \left(x - \frac{s}{2}\right)^2 + (y - Q)^2, \qquad p s = P_x H_x + P_y H_y \quad (r = s - p), \qquad g^2 = G_x^2 + G_y^2 = 2 s G_x\]

Editorial & Symbolic Notes

Archival scan 0136.jpg from Fred W. Seelig’s notebook dossier on F.24 Gothic Arch (Page 2 of 5). Companion OpenDocument worksheet(s): General Reference. Classical Euclidean foundations: I.1, III.11, III.12.

Folio 0137 (Page 3 of 5): Tangency Inner-Product Identity p*s = P_x H_x + P_y H_y

[Folio 0137]

Folio 0137 (original scan 0137.jpg) from resources/seelig/24_gothic.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0137 (0137.jpg, Sheet 3 of 5):

Uses internal tangency \(r = s - p\) (\(p^2 = P_x^2 + P_y^2\)) to derive \(p s = P_x H_x + P_y H_y\).

\[r^2 = \left(x - \frac{s}{2}\right)^2 + (y - Q)^2, \qquad p s = P_x H_x + P_y H_y \quad (r = s - p), \qquad g^2 = G_x^2 + G_y^2 = 2 s G_x\]

Editorial & Symbolic Notes

Archival scan 0137.jpg from Fred W. Seelig’s notebook dossier on F.24 Gothic Arch (Page 3 of 5). Companion OpenDocument worksheet(s): General Reference. Classical Euclidean foundations: I.1, III.11, III.12.

Folio 0138 (Page 4 of 5): Semicircle Geometric Mean Proof of g^2 = 2*s*G_x

[Folio 0138]

Folio 0138 (original scan 0138.jpg) from resources/seelig/24_gothic.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0138 (0138.jpg, Sheet 4 of 5):

Proves \(g^2 = G_x^2 + G_y^2 = 2sG_x\) in a semicircle of diameter \(2s\) centered at \(B\).

\[r^2 = \left(x - \frac{s}{2}\right)^2 + (y - Q)^2, \qquad p s = P_x H_x + P_y H_y \quad (r = s - p), \qquad g^2 = G_x^2 + G_y^2 = 2 s G_x\]

Editorial & Symbolic Notes

Archival scan 0138.jpg from Fred W. Seelig’s notebook dossier on F.24 Gothic Arch (Page 4 of 5). Companion OpenDocument worksheet(s): General Reference. Classical Euclidean foundations: I.1, III.11, III.12.

Folio 0139 (Page 5 of 5): Biquadratic Reduction z^4 - 74z^2 + 928 = 0

[Folio 0139]

Folio 0139 (original scan 0139.jpg) from resources/seelig/24_gothic.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0139 (0139.jpg, Sheet 5 of 5):

Solves the auxiliary circle-intersection system \((P, Q) = (\frac{z^2-40}{2z}, \frac{z^2-16}{2z})\) with \(P^2+Q^2=9\) to get \(z^2 = 58\).

\[r^2 = \left(x - \frac{s}{2}\right)^2 + (y - Q)^2, \qquad p s = P_x H_x + P_y H_y \quad (r = s - p), \qquad g^2 = G_x^2 + G_y^2 = 2 s G_x\]

Editorial & Symbolic Notes

Archival scan 0139.jpg from Fred W. Seelig’s notebook dossier on F.24 Gothic Arch (Page 5 of 5). Companion OpenDocument worksheet(s): General Reference. Classical Euclidean foundations: I.1, III.11, III.12.