F.1 Albert’s Parabola — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.1 (Albert’s Parabola) (SeeligNotebooks).

Folio 0001 (Page 1 of 2): Albert’s Parabola Problem (Attempt #2, FWS Apr. 2015)

[Folio 0001]

Folio 0001 (original scan 0001.jpg) from resources/seelig/01_alberts-parabola.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0001 (0001.jpg, Sheet 1 of 2):

Derives slope-intercept and double-intercept equations for tangent \(T\) and normal \(N\) to \(y = x^2\), records C.a.R. expressions, and reduces the normal intercept condition at \((M,0)\) (tested at \(M=4\)) to the depressed cubic \(P_x^3 + \frac{1}{2}P_x - 2 = 0\).

\[T:\; y = 2 P_x x - P_x^2, \qquad N:\; y = -\frac{x}{2 P_x} + \left(P_x^2 + \frac{1}{2}\right)\]

Editorial & Symbolic Notes

Archival scan 0001.jpg from Fred W. Seelig’s notebook dossier on F.1 Albert’s Parabola (Page 1 of 2). Companion OpenDocument worksheet(s): General Reference. Classical Euclidean foundations: I.11, I.12.

Folio 0002 (Page 2 of 2): Albert’s Parabola Problem — Attempt #3 (Directrix Frame)

[Folio 0002]

Folio 0002 (original scan 0002.jpg) from resources/seelig/01_alberts-parabola.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0002 (0002.jpg, Sheet 2 of 2):

Shifts the parabola to \(y = x^2 + 1\) with directrix \(y = 0\), vertex \(V(0,1)\), and focus \(F(0,2)\), and formulates the 3-step calculus minimization of \(d^2 = (P_x - Q_x)^2 + (P_y - Q_y)^2\).

\[P_x^3 + \frac{1}{2} P_x - \frac{M}{2} = 0, \qquad r = \frac{P_x + 4 P_x^3}{4}\]

Editorial & Symbolic Notes

Archival scan 0002.jpg from Fred W. Seelig’s notebook dossier on F.1 Albert’s Parabola (Page 2 of 2). Companion OpenDocument worksheet(s): General Reference. Classical Euclidean foundations: I.11, I.12.