F.26 Harmonic Conjugate Inversion — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.26 (Harmonic Conjugate Inversion) (SeeligNotebooks).

Folio 0168 (Page 1 of 15): Harmonic Conjugation, Inversion & The Important Parallelogram (FWS 11/19/2013)

[Folio 0168]

Folio 0168 (original scan 0168.jpg) from resources/seelig/26_harmonic-conjugate-inversion.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0168 (0168.jpg, Sheet 1 of 15):

Introduces Fig. 1 showing circle \(\Gamma(M, r)\), inverse points \(C, D\) (\(xy = r^2\)), tangent \(DT = T\), and parallelogram \(M D M_T T\) with diagonal \(z\).

\[x y = |MC| \cdot |MD| = r^2, \qquad T^2 = y^2 - r^2 = |AD| \cdot |BD|, \qquad z^2 + T^2 = 2(r^2 + y^2) \implies z^2 = y^2 + 3 r^2\]

Editorial & Symbolic Notes

Archival scan 0168.jpg from Fred W. Seelig’s notebook dossier on F.26 Harmonic Conjugate Inversion (Page 1 of 15). Companion OpenDocument worksheet(s): Harmonic Conjugates, HC Tables. Classical Euclidean foundations: III.36, VI.8.

Folio 0169 (Page 2 of 15): Jakob Steiner’s Power of a Point (Page 2)

[Folio 0169]

Folio 0169 (original scan 0169.jpg) from resources/seelig/26_harmonic-conjugate-inversion.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0169 (0169.jpg, Sheet 2 of 15):

Cites Jakob Steiner (1796–1863) for \(T^2 = y^2 - r^2 = PD\cdot QD = AD\cdot BD\) (Fig. 2).

\[(x,\, t,\, r,\, y,\, T) = \left(\hat{x}^2,\, \hat{x}\hat{t},\, \hat{x}\hat{r},\, \hat{r}^2,\, \hat{r}\hat{t}\right) \quad \text{where } \hat{x}^2 + \hat{t}^2 = \hat{r}^2\]

Editorial & Symbolic Notes

Archival scan 0169.jpg from Fred W. Seelig’s notebook dossier on F.26 Harmonic Conjugate Inversion (Page 2 of 15). Companion OpenDocument worksheet(s): Harmonic Conjugates, HC Tables. Classical Euclidean foundations: III.36, VI.8.

Folio 0170 (Page 3 of 15): Similarity Proof of TD^2 = PD*QD & Example r=5, x=3 (FWS 11/19/2013)

[Folio 0170]

Folio 0170 (original scan 0170.jpg) from resources/seelig/26_harmonic-conjugate-inversion.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0170 (0170.jpg, Sheet 3 of 15):

Proves \(\triangle DTQ \sim \triangle DPT\) and computes \(y = 25/3, AD = 40/3, BD = 10/3, T = 20/3\).

\[x y = |MC| \cdot |MD| = r^2, \qquad T^2 = y^2 - r^2 = |AD| \cdot |BD|, \qquad z^2 + T^2 = 2(r^2 + y^2) \implies z^2 = y^2 + 3 r^2\]

Editorial & Symbolic Notes

Archival scan 0170.jpg from Fred W. Seelig’s notebook dossier on F.26 Harmonic Conjugate Inversion (Page 3 of 15). Companion OpenDocument worksheet(s): Harmonic Conjugates, HC Tables. Classical Euclidean foundations: III.36, VI.8.

Folio 0171 (Page 4 of 15): Stewart’s Theorem on Parallelogram MDM_TT for Diagonal z

[Folio 0171]

Folio 0171 (original scan 0171.jpg) from resources/seelig/26_harmonic-conjugate-inversion.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0171 (0171.jpg, Sheet 4 of 15):

Proves \(z^2 + T^2 = 2(r^2+y^2) \implies z^2 = y^2 + 3r^2\) and derives the reverse \(\Gamma'\) formula in terms of \(x', R\).

\[(x,\, t,\, r,\, y,\, T) = \left(\hat{x}^2,\, \hat{x}\hat{t},\, \hat{x}\hat{r},\, \hat{r}^2,\, \hat{r}\hat{t}\right) \quad \text{where } \hat{x}^2 + \hat{t}^2 = \hat{r}^2\]

Editorial & Symbolic Notes

Archival scan 0171.jpg from Fred W. Seelig’s notebook dossier on F.26 Harmonic Conjugate Inversion (Page 4 of 15). Companion OpenDocument worksheet(s): Harmonic Conjugates, HC Tables. Classical Euclidean foundations: III.36, VI.8.

Folio 0172 (Page 5 of 15): Reverse Formula Verification on Example 1 (FWS 11/19/2013)

[Folio 0172]

Folio 0172 (original scan 0172.jpg) from resources/seelig/26_harmonic-conjugate-inversion.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0172 (0172.jpg, Sheet 5 of 15):

Rewrites \(z^2 = J + K + R^4(1/J + 1/K)\) and confirms \(z^2 = 1300/9\) from both forward and reverse equations.

\[x y = |MC| \cdot |MD| = r^2, \qquad T^2 = y^2 - r^2 = |AD| \cdot |BD|, \qquad z^2 + T^2 = 2(r^2 + y^2) \implies z^2 = y^2 + 3 r^2\]

Editorial & Symbolic Notes

Archival scan 0172.jpg from Fred W. Seelig’s notebook dossier on F.26 Harmonic Conjugate Inversion (Page 5 of 15). Companion OpenDocument worksheet(s): Harmonic Conjugates, HC Tables. Classical Euclidean foundations: III.36, VI.8.

Folio 0173 (Page 6 of 15): Graphic Scaling & Integer Solutions Table

[Folio 0173]

Folio 0173 (original scan 0173.jpg) from resources/seelig/26_harmonic-conjugate-inversion.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0173 (0173.jpg, Sheet 6 of 15):

Identifies the graphing sweet spot \(1/3 < x/r < 2/3\) and tabulates integer 5-tuples \((x, t, r, y, T)\).

\[(x,\, t,\, r,\, y,\, T) = \left(\hat{x}^2,\, \hat{x}\hat{t},\, \hat{x}\hat{r},\, \hat{r}^2,\, \hat{r}\hat{t}\right) \quad \text{where } \hat{x}^2 + \hat{t}^2 = \hat{r}^2\]

Editorial & Symbolic Notes

Archival scan 0173.jpg from Fred W. Seelig’s notebook dossier on F.26 Harmonic Conjugate Inversion (Page 6 of 15). Companion OpenDocument worksheet(s): Harmonic Conjugates, HC Tables. Classical Euclidean foundations: III.36, VI.8.

Folio 0174 (Page 7 of 15): Generating Integer 5-Tuples from Pythagorean Triples

[Folio 0174]

Folio 0174 (original scan 0174.jpg) from resources/seelig/26_harmonic-conjugate-inversion.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0174 (0174.jpg, Sheet 7 of 15):

Proves that scaling \((\hat{x}, \hat{t}, \hat{r})\) by \(\hat{x}\) and \(\hat{r}\) yields integer \((x, t, r, y, T)\).

\[x y = |MC| \cdot |MD| = r^2, \qquad T^2 = y^2 - r^2 = |AD| \cdot |BD|, \qquad z^2 + T^2 = 2(r^2 + y^2) \implies z^2 = y^2 + 3 r^2\]

Editorial & Symbolic Notes

Archival scan 0174.jpg from Fred W. Seelig’s notebook dossier on F.26 Harmonic Conjugate Inversion (Page 7 of 15). Companion OpenDocument worksheet(s): Harmonic Conjugates, HC Tables. Classical Euclidean foundations: III.36, VI.8.

Folio 0175 (Page 8 of 15): Collinearity Warning & Power of a Point Sketch

[Folio 0175]

Folio 0175 (original scan 0175.jpg) from resources/seelig/26_harmonic-conjugate-inversion.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0175 (0175.jpg, Sheet 8 of 15):

Illustrates non-collinear tangent points and the secant power relation \(x^2 - r^2\).

\[(x,\, t,\, r,\, y,\, T) = \left(\hat{x}^2,\, \hat{x}\hat{t},\, \hat{x}\hat{r},\, \hat{r}^2,\, \hat{r}\hat{t}\right) \quad \text{where } \hat{x}^2 + \hat{t}^2 = \hat{r}^2\]

Editorial & Symbolic Notes

Archival scan 0175.jpg from Fred W. Seelig’s notebook dossier on F.26 Harmonic Conjugate Inversion (Page 8 of 15). Companion OpenDocument worksheet(s): Harmonic Conjugates, HC Tables. Classical Euclidean foundations: III.36, VI.8.

Folio 0176 (Page 9 of 15): Tangent-Chord Proof of TD^2 = DE*DF

[Folio 0176]

Folio 0176 (original scan 0176.jpg) from resources/seelig/26_harmonic-conjugate-inversion.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0176 (0176.jpg, Sheet 9 of 15):

Detailed geometric proof using inscribed angles that \(\triangle DTF \sim \triangle DET\).

\[x y = |MC| \cdot |MD| = r^2, \qquad T^2 = y^2 - r^2 = |AD| \cdot |BD|, \qquad z^2 + T^2 = 2(r^2 + y^2) \implies z^2 = y^2 + 3 r^2\]

Editorial & Symbolic Notes

Archival scan 0176.jpg from Fred W. Seelig’s notebook dossier on F.26 Harmonic Conjugate Inversion (Page 9 of 15). Companion OpenDocument worksheet(s): Harmonic Conjugates, HC Tables. Classical Euclidean foundations: III.36, VI.8.

Folio 0177 (Page 10 of 15): Diophantine Problem Statement for (r, T, y) (FWS 11/19/2013)

[Folio 0177]

Folio 0177 (original scan 0177.jpg) from resources/seelig/26_harmonic-conjugate-inversion.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0177 (0177.jpg, Sheet 10 of 15):

Formulates the number-theoretic problem of finding all-integer harmonic inversion configurations.

\[(x,\, t,\, r,\, y,\, T) = \left(\hat{x}^2,\, \hat{x}\hat{t},\, \hat{x}\hat{r},\, \hat{r}^2,\, \hat{r}\hat{t}\right) \quad \text{where } \hat{x}^2 + \hat{t}^2 = \hat{r}^2\]

Editorial & Symbolic Notes

Archival scan 0177.jpg from Fred W. Seelig’s notebook dossier on F.26 Harmonic Conjugate Inversion (Page 10 of 15). Companion OpenDocument worksheet(s): Harmonic Conjugates, HC Tables. Classical Euclidean foundations: III.36, VI.8.

Folio 0178 (Page 11 of 15): Master Landscape Table for kappa = 3/5 (FWS 11/20/2013)

[Folio 0178]

Folio 0178 (original scan 0178.jpg) from resources/seelig/26_harmonic-conjugate-inversion.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0178 (0178.jpg, Sheet 11 of 15):

Complete table of graphical and exact integer coordinates (\(x=9, t=12, r=15, y=25, T=20, R=8, k=4\)) with scale diagram.

\[x y = |MC| \cdot |MD| = r^2, \qquad T^2 = y^2 - r^2 = |AD| \cdot |BD|, \qquad z^2 + T^2 = 2(r^2 + y^2) \implies z^2 = y^2 + 3 r^2\]

Editorial & Symbolic Notes

Archival scan 0178.jpg from Fred W. Seelig’s notebook dossier on F.26 Harmonic Conjugate Inversion (Page 11 of 15). Companion OpenDocument worksheet(s): Harmonic Conjugates, HC Tables. Classical Euclidean foundations: III.36, VI.8.

Folio 0179 (Page 12 of 15): Worked Example r = 3, R = 4, kappa = 1/3

[Folio 0179]

Folio 0179 (original scan 0179.jpg) from resources/seelig/26_harmonic-conjugate-inversion.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0179 (0179.jpg, Sheet 12 of 15):

Verifies \((W+w)/2 = (AD+BC)/2 = r+R = 7\) for \(AB=6, CD=8\).

\[(x,\, t,\, r,\, y,\, T) = \left(\hat{x}^2,\, \hat{x}\hat{t},\, \hat{x}\hat{r},\, \hat{r}^2,\, \hat{r}\hat{t}\right) \quad \text{where } \hat{x}^2 + \hat{t}^2 = \hat{r}^2\]

Editorial & Symbolic Notes

Archival scan 0179.jpg from Fred W. Seelig’s notebook dossier on F.26 Harmonic Conjugate Inversion (Page 12 of 15). Companion OpenDocument worksheet(s): Harmonic Conjugates, HC Tables. Classical Euclidean foundations: III.36, VI.8.

Folio 0180 (Page 13 of 15): Harmonic Conjugate Kite Sample Problem (r = 5, x = 3)

[Folio 0180]

Folio 0180 (original scan 0180.jpg) from resources/seelig/26_harmonic-conjugate-inversion.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0180 (0180.jpg, Sheet 13 of 15):

Computes all segments and ratios for \(\kappa = 3/5, k = 4, k' = 5/3\).

\[x y = |MC| \cdot |MD| = r^2, \qquad T^2 = y^2 - r^2 = |AD| \cdot |BD|, \qquad z^2 + T^2 = 2(r^2 + y^2) \implies z^2 = y^2 + 3 r^2\]

Editorial & Symbolic Notes

Archival scan 0180.jpg from Fred W. Seelig’s notebook dossier on F.26 Harmonic Conjugate Inversion (Page 13 of 15). Companion OpenDocument worksheet(s): Harmonic Conjugates, HC Tables. Classical Euclidean foundations: III.36, VI.8.

Folio 0181 (Page 14 of 15): Center Distance d and Radius R in Terms of kappa

[Folio 0181]

Folio 0181 (original scan 0181.jpg) from resources/seelig/26_harmonic-conjugate-inversion.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0181 (0181.jpg, Sheet 14 of 15):

Tabulates \(\kappa, 1/\kappa - \kappa, d/r\) and remarks ‘(Euclid and Apollonius would approve their use, I’m sure)’.

\[(x,\, t,\, r,\, y,\, T) = \left(\hat{x}^2,\, \hat{x}\hat{t},\, \hat{x}\hat{r},\, \hat{r}^2,\, \hat{r}\hat{t}\right) \quad \text{where } \hat{x}^2 + \hat{t}^2 = \hat{r}^2\]

Editorial & Symbolic Notes

Archival scan 0181.jpg from Fred W. Seelig’s notebook dossier on F.26 Harmonic Conjugate Inversion (Page 14 of 15). Companion OpenDocument worksheet(s): Harmonic Conjugates, HC Tables. Classical Euclidean foundations: III.36, VI.8.

Folio 0182 (Page 15 of 15): Fibonacci & Lucas Prime Factorization Table (F_1 to F_25)

[Folio 0182]

Folio 0182 (original scan 0182.jpg) from resources/seelig/26_harmonic-conjugate-inversion.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0182 (0182.jpg, Sheet 15 of 15):

Reverse side of Folio 14 listing prime factorizations of Fibonacci and Lucas numbers.

\[x y = |MC| \cdot |MD| = r^2, \qquad T^2 = y^2 - r^2 = |AD| \cdot |BD|, \qquad z^2 + T^2 = 2(r^2 + y^2) \implies z^2 = y^2 + 3 r^2\]

Editorial & Symbolic Notes

Archival scan 0182.jpg from Fred W. Seelig’s notebook dossier on F.26 Harmonic Conjugate Inversion (Page 15 of 15). Companion OpenDocument worksheet(s): Harmonic Conjugates, HC Tables. Classical Euclidean foundations: III.36, VI.8.