F.30 Isosceles Triangles — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.30 (Isosceles Triangles) (SeeligNotebooks).

Folio 0210 (Page 1 of 5): Scanner Separator Sheet

[Folio 0210]

Folio 0210 (original scan 0210.jpg) from resources/seelig/30_isosceles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0210 (0210.jpg, Sheet 1 of 5):

Blank dark separator scan preceding the four-page isosceles cubic manuscript.

\[a = b = s - x, \qquad x = \frac{c}{2}, \qquad K = x\sqrt{s(s - 2x)} \implies 2 s x^3 - s^2 x^2 + K^2 = 0\]

Editorial & Symbolic Notes

Archival scan 0210.jpg from Fred W. Seelig’s notebook dossier on F.30 Isosceles Triangles (Page 1 of 5). Companion OpenDocument worksheet(s): Isosceles Triangle, Right Triangle. Classical Euclidean foundations: I.5, I.6.

Folio 0211 (Page 2 of 5): Isosceles Triangle — Area-Perimeter Cubic (Page 1)

[Folio 0211]

Folio 0211 (original scan 0211.jpg) from resources/seelig/30_isosceles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0211 (0211.jpg, Sheet 2 of 5):

Cites Baloglou & Helfgott (2008) and Ivars Peterson; sets \(x = c/2, a = s-x, h = \sqrt{s(s-2x)}\) to derive \(2sx^3 - s^2 x^2 + K^2 = 0\).

\[x = y + \frac{s}{6} \implies y^3 - \frac{s^2}{12} y + \left(\frac{K^2}{2s} - \frac{s^3}{54}\right) = 0, \qquad c_{\text{right}} = s - \frac{K}{s}\]

Editorial & Symbolic Notes

Archival scan 0211.jpg from Fred W. Seelig’s notebook dossier on F.30 Isosceles Triangles (Page 2 of 5). Companion OpenDocument worksheet(s): Isosceles Triangle, Right Triangle. Classical Euclidean foundations: I.5, I.6.

Folio 0212 (Page 3 of 5): Newton’s Right-Triangle Formula & Cubic Setup (Page 2)

[Folio 0212]

Folio 0212 (original scan 0212.jpg) from resources/seelig/30_isosceles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0212 (0212.jpg, Sheet 3 of 5):

Contrasts Newton’s Universal Arithmetick right-triangle identity \(c = s - K/s\) with the isosceles cubic \(x^3 - \frac{s}{2}x^2 + \frac{K^2}{2s} = 0\).

\[a = b = s - x, \qquad x = \frac{c}{2}, \qquad K = x\sqrt{s(s - 2x)} \implies 2 s x^3 - s^2 x^2 + K^2 = 0\]

Editorial & Symbolic Notes

Archival scan 0212.jpg from Fred W. Seelig’s notebook dossier on F.30 Isosceles Triangles (Page 3 of 5). Companion OpenDocument worksheet(s): Isosceles Triangle, Right Triangle. Classical Euclidean foundations: I.5, I.6.

Folio 0213 (Page 4 of 5): Reduction to Depressed Cubic (Page 3)

[Folio 0213]

Folio 0213 (original scan 0213.jpg) from resources/seelig/30_isosceles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0213 (0213.jpg, Sheet 4 of 5):

Substitutes \(x = y - a/3 = y + s/6\) into \(x^3 + ax^2 + bx + c = 0\) to eliminate the quadratic term, obtaining \(p = -s^2/12\) and \(q = -s^3/54 + K^2/(2s)\).

\[x = y + \frac{s}{6} \implies y^3 - \frac{s^2}{12} y + \left(\frac{K^2}{2s} - \frac{s^3}{54}\right) = 0, \qquad c_{\text{right}} = s - \frac{K}{s}\]

Editorial & Symbolic Notes

Archival scan 0213.jpg from Fred W. Seelig’s notebook dossier on F.30 Isosceles Triangles (Page 4 of 5). Companion OpenDocument worksheet(s): Isosceles Triangle, Right Triangle. Classical Euclidean foundations: I.5, I.6.

Folio 0214 (Page 5 of 5): Cardano’s Formula & Casus Irreducibilis (Page 4)

[Folio 0214]

Folio 0214 (original scan 0214.jpg) from resources/seelig/30_isosceles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0214 (0214.jpg, Sheet 5 of 5):

Writes Cardano’s radical solution for \(y^3 + py + q = 0\) and checks the discriminant boundary \(q^2/4 + p^3/27 \le 0\) corresponding to two real isosceles solutions coalescing at the equilateral triangle.

\[a = b = s - x, \qquad x = \frac{c}{2}, \qquad K = x\sqrt{s(s - 2x)} \implies 2 s x^3 - s^2 x^2 + K^2 = 0\]

Editorial & Symbolic Notes

Archival scan 0214.jpg from Fred W. Seelig’s notebook dossier on F.30 Isosceles Triangles (Page 5 of 5). Companion OpenDocument worksheet(s): Isosceles Triangle, Right Triangle. Classical Euclidean foundations: I.5, I.6.