Critical & Symbolic Notes — Unit II: Six Quadrances & The Euler Line (§§ 10–18)

Folio Errata, Stewart’s Theorem (1746) & Exact Symbolic Verification

1. Compositorial Errata on pp. 109–113 of the 1767 Edition

Visual inspection of the 1767 Novi Commentarii pages reveals four minor printing slips in the intermediate derivations, all of which Euler silently corrects in his summary table § 18 on p. 114:

  • p. 109, § 12: (a) Line 1 prints (3cc - bb + aa)/(6c) for (3cc + bb - aa)/(6c), while giving the true difference (bb - aa)/(3c); (b) the intermediate display prints denominator 16 AA for 36 AA (stemming from \((12c\mathcal{A})^2 / (4c^2) = 36\mathcal{A}^2\), corrected six lines below); (c) the expansion of \(a^6 + b^6 + c^6\) prints -3rr at the end of line 2 for +3rr; and (d) the final line prints EF^2 = rr/(4AA) = 4/9(pp - 2q) with = instead of - (corrected in § 18 I).

  • p. 112, § 15, line 2: Prints GF - RG for QF - RG.

  • p. 113, § 17, penult. line: Prints - between (r / 16pAA)(p^4 - 4ppq + 9pr) and r(p^3 - 4pq + 9r)/(16AA) where = is intended.

2. Connection to Matthew Stewart’s Proposition II (1746)

At the close of § 18 (p. 114), Euler deduces from \(FH = \frac{1}{2}EF\) and \(EH = \frac{3}{2}EF\) the identity

\[4GH^2 + 2EG^2 = 3EF^2 + 6FG^2.\]

This is a direct specialization of Matthew Stewart’s theorem (Some General Theorems of Considerable Use in the Higher Parts of Mathematics, Edinburgh, 1746, Prop. II): for collinear points \(E - F - H\) and any point \(G\) in the plane,

\[GH^2 \cdot EF + EG^2 \cdot FH = EH\left(FG^2 + EF \cdot FH\right).\]

Substituting \(FH = \tfrac{1}{2}EF\) and \(EH = \tfrac{3}{2}EF\) gives \(GH^2 + \tfrac{1}{2}EG^2 = \tfrac{3}{2}\left(FG^2 + \tfrac{1}{2}EF^2\right)\), and multiplying by \(4\) yields \(4GH^2 + 2EG^2 = 3EF^2 + 6FG^2\). In geometor.feuerbach.euler.prove_euler_1765_identities(), both each of Formulas I–VI and Stewart’s relation simplify to symbolic 0 over \(\mathbb{Q}(a, b, c)\) via sp.cancel.

3. Bridge Forward to Feuerbach (1822, §§ 49–57)

Formula VI (\(GH^2 = R^2 - 2Rr_{\text{in}}\)) and Formula III (\(EH^2 = 9R^2 - (a^2 + b^2 + c^2)\)) are explicitly cited by Karl Wilhelm Feuerbach in § 49 and § 53 of his 1822 Nürnberg monograph (Unit III: The Nine-Point Circle, Feuerbach’s Theorem & The Euler Line (§§ 49–60, Complete)). Extending Formula VI to the three excenters (\(KS^{(k)2} = R^2 + 2Rr^{(k)}\)) and combining it with Apollonius’s median theorem applied to the midpoint \(L = X(5)\) of segment \(EH\) (renamed \(OK\) in Feuerbach) produces Feuerbach’s Theorem (\(LS = \tfrac{1}{2}R - r_{\text{in}}\)).

4. Rational Quadrances on the (15, 13, 14) Triangle

On our canonical triangle \((a, b, c) = (15, 13, 14)\) with \(p = 42, q = 587, r = 2730, \mathcal{A} = 84\):

\[EF^2 = \frac{265}{144}, \quad EG^2 = \frac{17}{16}, \quad EH^2 = \frac{265}{64}, \quad FG^2 = \frac{1}{9}, \quad FH^2 = \frac{265}{576}, \quad GH^2 = \frac{65}{64},\]

satisfying \(4\left(\frac{65}{64}\right) + 2\left(\frac{17}{16}\right) = \frac{99}{16} = 3\left(\frac{265}{144}\right) + 6\left(\frac{1}{9}\right)\) identically.