Part I, Chapter IV — Critical & Symbolic Notes

Textual Collation (1895 vs. 1897), Exact Single-Circle Model in geometor.klein.heptadecagon, and Sister Corpora

Textual Variants & Errata Between the 1895 Teubner and 1897 Ginn Witnesses

Collating the 1895 German Editio Princeps (S. 19–32) against Beman and Smith's 1897 English translation (pp. 24–41) reveals three noteworthy editorial refinements:

  1. Notation Renaming ($x, y, z$ vs. $\eta, \eta', \eta''$): In 1895 DE (S. 21–26), Klein and Tägert use Gauss’s primed period notation $\eta_0, \eta_1$ ($8$-periods), $\eta'_0, \eta'_1, (\eta'_0), (\eta'_1)$ ($4$-periods), and $\eta''_0, \eta''_1$ / $C_1, \dots, C_8$ ($2$-periods). In 1897 EN (pp. 26–32), Beman and Smith introduce the cleaner unprimed alphabet $x_1, x_2$, $y_1, y_2, y_3, y_4$ (with $y_4 = (\eta'_0) > 0 > y_3 = (\eta'_1)$), and $z_1, \dots, z_8$, naming the four quadratics $(\xi), (\eta), (\eta'), (\zeta)$. Both systems are preserved side-by-side in our synchronized tracks and in geometor.klein.heptadecagon.

  2. Correction of the S. 29 Sign Misprint in 1895 DE: On S. 24 of the 1895 Teubner edition, the first quadratic equation is derived correctly as $\eta^2 + \eta - 4 = 0$ (since $\eta_0 + \eta_1 = -1$ and $\eta_0 \eta_1 = -4$), and S. 29 rightly connects $+4$ below with $-4$ above ($q/p = -4/(-1) = +4$, $4/p = -4$); however, the summary display at the top of S. 29 printed \eta^2 - \eta + 4 = 0 by typographical slip. The 1897 English edition (p. 36) silently restores the exact equation $(\xi): x^2 + x - 4 = 0$, which we likewise restore in the German track with this apparatus note.

  3. L. Gérard’s 1896 Mascheroni Construction (§ 8 Footnote): In October 1895 (DE S. 27 fn.), Klein remarked: „Eine Construction des 17-Ecks nach Mascheroni nur mit dem Zirkel ist noch nicht versucht, obgleich sie jedenfalls möglich ist.“ Prompted directly by Klein’s Göttingen Festschrift, Louis Gérard published the first pure-compass construction of the regular $17$-gon within months (Mathematische Annalen, Vol. XLVIII, 1896, pp. 390–392), and Beman & Smith updated the 1897 English footnote (p. 34 fn.) accordingly!

Exact Symbolic Realization in geometor.klein.heptadecagon

The function geometor.klein.heptadecagon.construct_gauss_17gon_periods() executes Klein’s §§ 8–9 inside geometor.model.Model (auto_intersect=False) backed by a dedicated exact polynomial-quotient arithmetic engine _K16Engine over $\mathbb{Q}[t]/(M_8(t))$ ($M_8(t) = t^8 - 68t^6 + 1088t^4 - 5440t^2 + 4352$, minimal polynomial of $R_1 = \sqrt{17 + 3\sqrt{17} - \sqrt{34 - 2\sqrt{17}} - 2\sqrt{34 + 2\sqrt{17}}}$):

  • Strict Poncelet–Steiner Purity (`len(model.circles) == 1`): Only the single fixed circle $C_0: x^2 + y(y - 2) = 0$ is ever constructed in model. Even the final vertical chords through $K_m = (\cos\frac{2m\pi}{17}, 1)$ (which Klein § 9 describes simply as “erecting a perpendicular”) are constructed by pure straightedge alone using Steiner’s parallel-tangent harmonic bundle on $C_0$, inscribing all 17 exact vertices V0..``V16`` with zero float or sp.Float coordinates!

  • Sister-Corpus Links (`geometor/pappus`, `geometor/petersen`, `geometor/lemoine`): Klein’s projective inversion $x' = \frac{4}{x}$ and harmonic center $\left(0, \frac{2}{5}\right)$ directly exemplify the cross-ratio and involution machinery of geometor/pappus (Pappus, Book VII), while the structural comparison of straightedge-only, compass-only (Mascheroni, 1797), single-circle (Steiner, 1833), and standard Euclidean constructions bridges directly to Julius Petersen's Methods and Theories (1879) and Émile Lemoine's Géométrographie (1902).