Part II, Chapter II — Critical & Symbolic Notes

Textual Collation of Two Slips in the 1895/1897 Editions and Pappus’s Critique of the Quadratrix

Textual Variants & Corrections Between 1895 DE and 1897 EN

  1. The Leibniz Series Slip in 1895 DE (§ 3, S. 45): In the 1895 Teubner edition (S. 45), Tägert's lecture transcription inadvertently printed the alternating harmonic series $\frac{\pi}{4} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$ (whose true sum is $\ln 2$, not $\frac{\pi}{4}$!). In the 1897 Ginn edition (p. 59), Beman and Smith silently corrected the denominators to the odd integers $\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots$, which we likewise restore in the German track with this critical note.

  2. Huygens’s Date 1654 in 1897 EN (§ 2, p. 56): Where 1895 DE (S. 44) wrote „Huygens (1654) in seinem Werke: De circuli magnitudine inventa“—giving the publication year 1654 of Huygens’s treatise—1897 EN (p. 56) mistranslated the parenthetical year as “Huygens (d. 1654)” (Christiaan Huygens actually lived 1629–1695 and published De circuli magnitudine inventa at Leiden in 1654). Our English track restores 1654 in its proper bibliographic position.

Sister-Corpus Connections (geometor/pappus & geometor/hilbert)

  • Sporus and Pappus on the Quadratrix (`geometor/pappus`): Klein’s remark in § 2 that the quadratrix of Hippias and Dinostratus “amounts simply to a geometric formulation of the problem of rectification” echoes the famous classical critique preserved by Pappus in Collection, Book IV, Props. 25–26: to synchronize the linear velocity of $M$ along radius $OB$ with the angular velocity of $L$ along arc $AB$ already presupposes knowing the ratio $\frac{\pi}{2} : 1$ of quadrant arc to radius, and moreover the terminal intersection $x_0 = \frac{2}{\pi}$ at $y = 0$ can only be found as a limit where the rotating ray and sliding parallel coincide!

  • Spring 1893 in *Mathematische Annalen* Bd. 43: Notice a striking historical coincidence within our dual codex: Volume 43 (1893) of Klein’s Mathematische Annalen simultaneously published the reprint of Klein’s 1872 *Erlangen Program* (pp. 63–100) and the three transcendence memoirs of Hilbert, Hurwitz, and Gordan (pp. 216–224) on which Part II of Famous Problems is built!