F.16 d’Alembert’s Theorem — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.16 (d’Alembert’s Theorem) (SeeligNotebooks).

Folio 0114 (Page 1 of 4): D’Alembert’s Theorem Multi-Color Constructions

[Folio 0114]

Folio 0114 (original scan 0114.jpg) from resources/seelig/16_dalembert.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0114 (0114.jpg, Sheet 1 of 4):

Two constructions with red external tangent cones and green internal tangent cones whose apexes lie on thick black collinearity lines.

\[L_{+++}:\; \{S_{\text{EX}\,AB},\, S_{\text{EX}\,BC},\, S_{\text{EX}\,CA}\}, \qquad L_{+--}:\; \{S_{\text{EX}\,AB},\, S_{\text{IN}\,BC},\, S_{\text{IN}\,CA}\}\]

Editorial & Symbolic Notes

Archival scan 0114.jpg from Fred W. Seelig’s notebook dossier on F.16 d’Alembert’s Theorem (Page 1 of 4). Companion OpenDocument worksheet(s): Centers of Similitude. Classical Euclidean foundations: VI.2, VI.4.

Folio 0115 (Page 2 of 4): The Four Similitude Axes L_{+++}, L_{–+}, L_{-+-}, L_{+–}

[Folio 0115]

Folio 0115 (original scan 0115.jpg) from resources/seelig/16_dalembert.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0115 (0115.jpg, Sheet 2 of 4):

Grid-paper drawing of circles \(A, B, C\) showing all 6 centers of similitude and all 4 Monge–d’Alembert axes.

\[L_{+++}:\; \{S_{\text{EX}\,AB},\, S_{\text{EX}\,BC},\, S_{\text{EX}\,CA}\}, \qquad L_{+--}:\; \{S_{\text{EX}\,AB},\, S_{\text{IN}\,BC},\, S_{\text{IN}\,CA}\}\]

Editorial & Symbolic Notes

Archival scan 0115.jpg from Fred W. Seelig’s notebook dossier on F.16 d’Alembert’s Theorem (Page 2 of 4). Companion OpenDocument worksheet(s): Centers of Similitude. Classical Euclidean foundations: VI.2, VI.4.

Folio 0116 (Page 3 of 4): Connection to the Problem of Apollonius

[Folio 0116]

Folio 0116 (original scan 0116.jpg) from resources/seelig/16_dalembert.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0116 (0116.jpg, Sheet 3 of 4):

Lists the 6 similitude centers and sets up the X/I tangency state table for circles tangent to \(\Gamma_A, \Gamma_B, \Gamma_C\).

\[L_{+++}:\; \{S_{\text{EX}\,AB},\, S_{\text{EX}\,BC},\, S_{\text{EX}\,CA}\}, \qquad L_{+--}:\; \{S_{\text{EX}\,AB},\, S_{\text{IN}\,BC},\, S_{\text{IN}\,CA}\}\]

Editorial & Symbolic Notes

Archival scan 0116.jpg from Fred W. Seelig’s notebook dossier on F.16 d’Alembert’s Theorem (Page 3 of 4). Companion OpenDocument worksheet(s): Centers of Similitude. Classical Euclidean foundations: VI.2, VI.4.

Folio 0117 (Page 4 of 4): Compass Construction of Mutually Tangent Soddy Circles

[Folio 0117]

Folio 0117 (original scan 0117.jpg) from resources/seelig/16_dalembert.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0117 (0117.jpg, Sheet 4 of 4):

Precision compass drawing of three mutually tangent circles enclosed by an outer tangent circle and surrounding a small inner tangent circle.

\[L_{+++}:\; \{S_{\text{EX}\,AB},\, S_{\text{EX}\,BC},\, S_{\text{EX}\,CA}\}, \qquad L_{+--}:\; \{S_{\text{EX}\,AB},\, S_{\text{IN}\,BC},\, S_{\text{IN}\,CA}\}\]

Editorial & Symbolic Notes

Archival scan 0117.jpg from Fred W. Seelig’s notebook dossier on F.16 d’Alembert’s Theorem (Page 4 of 4). Companion OpenDocument worksheet(s): Centers of Similitude. Classical Euclidean foundations: VI.2, VI.4.