F.44 Homothetic Ratios and Crossed Diagonals — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.44 (Homothetic Ratios and Crossed Diagonals) (SeeligNotebooks).

Folio 0282 (Page 1 of 5): Homothetic Right Triangles & Crossed Diagonals (001)

[Folio 0282]

Folio 0282 (original scan 0282.jpg) from resources/seelig/44_ratios.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0282 (0282.jpg, Sheet 1 of 5):

Sets up uprights \(h_1, h_2\) separated by \(d\) from apex \(O\) (angle \(\phi\)) and derives \(x = d h_1/(h_1+h_2)\) and \(y-x = d h_1/(h_2-h_1)\).

\[x = d\frac{h_1}{h_1 + h_2}, \qquad y - x = d\frac{h_1}{h_2 - h_1}, \qquad h = \frac{h_1 h_2}{h_1 + h_2}\]

Editorial & Symbolic Notes

Archival scan 0282.jpg from Fred W. Seelig’s notebook dossier on F.44 Homothetic Ratios and Crossed Diagonals (Page 1 of 5). Companion OpenDocument worksheet(s): Centers of Similitude, Harmonic Conjugates. Classical Euclidean foundations: VI.2, VI.4.

Folio 0283 (Page 2 of 5): Total Span y & Midpoint z Between Similitude Centers (002)

[Folio 0283]

Folio 0283 (original scan 0283.jpg) from resources/seelig/44_ratios.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0283 (0283.jpg, Sheet 2 of 5):

Combines \(x\) and \(y-x\) to obtain \(y = \frac{2dh_1h_2}{h_2^2-h_1^2}\) and \(z = \frac{dh_1h_2}{h_2^2-h_1^2}\).

\[y = \frac{2 d h_1 h_2}{h_2^2 - h_1^2}, \qquad z = \frac{y}{2} = \frac{d h_1 h_2}{h_2^2 - h_1^2}\]

Editorial & Symbolic Notes

Archival scan 0283.jpg from Fred W. Seelig’s notebook dossier on F.44 Homothetic Ratios and Crossed Diagonals (Page 2 of 5). Companion OpenDocument worksheet(s): Centers of Similitude, Harmonic Conjugates. Classical Euclidean foundations: VI.2, VI.4.

Folio 0284 (Page 3 of 5): Crossed-Ladder Height & Harmonic Mean (003)

[Folio 0284]

Folio 0284 (original scan 0284.jpg) from resources/seelig/44_ratios.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0284 (0284.jpg, Sheet 3 of 5):

Shows that the height of the crossed-diagonal intersection is \(h = \frac{h_1 h_2}{h_1+h_2} = \frac{1}{2}H(h_1,h_2)\), independent of \(d\).

\[x = d\frac{h_1}{h_1 + h_2}, \qquad y - x = d\frac{h_1}{h_2 - h_1}, \qquad h = \frac{h_1 h_2}{h_1 + h_2}\]

Editorial & Symbolic Notes

Archival scan 0284.jpg from Fred W. Seelig’s notebook dossier on F.44 Homothetic Ratios and Crossed Diagonals (Page 3 of 5). Companion OpenDocument worksheet(s): Centers of Similitude, Harmonic Conjugates. Classical Euclidean foundations: VI.2, VI.4.

Folio 0285 (Page 4 of 5): Dimensionless Ratio Parameterization k = h_2 / h_1 (004)

[Folio 0285]

Folio 0285 (original scan 0285.jpg) from resources/seelig/44_ratios.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0285 (0285.jpg, Sheet 4 of 5):

Divides top and bottom by \(h_1^2\) to recover the canonical Apollonian formulas \(x = d/(k+1), y-x = d/(k-1), z = dk/(k^2-1)\).

\[y = \frac{2 d h_1 h_2}{h_2^2 - h_1^2}, \qquad z = \frac{y}{2} = \frac{d h_1 h_2}{h_2^2 - h_1^2}\]

Editorial & Symbolic Notes

Archival scan 0285.jpg from Fred W. Seelig’s notebook dossier on F.44 Homothetic Ratios and Crossed Diagonals (Page 4 of 5). Companion OpenDocument worksheet(s): Centers of Similitude, Harmonic Conjugates. Classical Euclidean foundations: VI.2, VI.4.

Folio 0286 (Page 5 of 5): Numerical & Graphical Verification (005)

[Folio 0286]

Folio 0286 (original scan 0286.jpg) from resources/seelig/44_ratios.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0286 (0286.jpg, Sheet 5 of 5):

Tests the formulas on integer height pairs \((h_1, h_2) = (1,2)\) and \((2,3)\).

\[x = d\frac{h_1}{h_1 + h_2}, \qquad y - x = d\frac{h_1}{h_2 - h_1}, \qquad h = \frac{h_1 h_2}{h_1 + h_2}\]

Editorial & Symbolic Notes

Archival scan 0286.jpg from Fred W. Seelig’s notebook dossier on F.44 Homothetic Ratios and Crossed Diagonals (Page 5 of 5). Companion OpenDocument worksheet(s): Centers of Similitude, Harmonic Conjugates. Classical Euclidean foundations: VI.2, VI.4.