Critical & Symbolic Notes — Unit III: Exact SymPy Audit of §§ 49–60 & The § 51 Sign Slip

Why Symbolic Verification Matters in Historical Editions

1. Discovery & Diagnosis of Feuerbach’s § 51 / § 52 / § 60 Sign-Substitution Slip

When encoding every equation of Feuerbach (1822) §§ 49–60 in geometor.feuerbach.euler.prove_euler_1765_identities() over \(\mathbb{Q}(a, b, c)\) in sympy, a striking philological fact emerged on pages 35, 36, and 40–41:

• In § 51 (p. 35), Feuerbach proves directly from coordinates the 100% exact incenter quadrance

  \[\overline{OS}^2 = 2r^2 - 2\rho R.\]

• Immediately below on p. 35, he applies his customary substitution \((a, b, c) \mapsto (-a, b, c)\) (introduced in § 8, p. 6, where he noted r wird zu r' und +R zu -R) to write

  \[\overline{OS^{(k)}}^2 \stackrel{\text{1822 text}}{=} 2r^{(k)2} + 2\rho R \qquad \text{(printed slip!)}.\]

• Why this fails algebraically: By § 24 (p. 18) and \(R = \frac{abc}{4\Delta}\),

  \[\rho = \frac{(-a^2+b^2+c^2)(a^2-b^2+c^2)(a^2+b^2-c^2)}{16abc\Delta}, \qquad \rho R = \frac{(-a^2+b^2+c^2)(a^2-b^2+c^2)(a^2+b^2-c^2)}{64\Delta^2}.\]

  Under \(a \mapsto -a\) (holding \(+\Delta\) positive), both \(R\) and \(\rho\) reverse sign (\(R \mapsto -R\) and \(\rho \mapsto -\rho\)), so their product \(\rho R\)—which depends solely on \(a^2, b^2, c^2\)—is strictly invariant!

Exact Symbolic Comparison on (a, b, c) = (15, 13, 14) (\(R = \tfrac{65}{8}, r = 4, r_a = 14, \rho = \tfrac{99}{52}\))

Locus	1822 Printed Formula	True Exact Formula (sympy)	Value on (15, 13, 14) (\(2\rho R = \frac{495}{16}\))
§ 51 (\(\overline{OS}^2\))	\(2r^2 - 2\rho R\) (exact)	\(2r^2 - 2\rho R\)	\(32 - \frac{495}{16} = \frac{17}{16}\)
§ 51 (\(\overline{OS_a}^2\))	\(2r_a^2 + 2\rho R\) (slip)	\(2r_a^2 - 2\rho R\)	\(392 - \frac{495}{16} = \frac{5777}{16}\) (printed gives \(\frac{6767}{16}\))
§ 52 (\(\sum \overline{OS_i}^2\))	\(4R(4R - \rho)\) (slip)	\(16R(R - \rho)\)	\(\frac{12935}{16}\)
§ 57 (\(LS, LS_k\))	\(\frac{1}{2}R - r,\; \frac{1}{2}R + r_k\) (exact!)	\(\left|\frac{1}{2}R - r\right|,\; \frac{1}{2}R + r_k\)	\(\frac{1}{16},\; \frac{289}{16},\; \frac{233}{16},\; \frac{257}{16}\)
§ 59 (\(\sum \overline{LS_i}^2\))	\(13R^2 - 4\rho R\) (exact!)	\(13R^2 - 4\rho R\)	\(\frac{52945}{64}\)
§ 60 (\(\overline{GS}^2\))	\(\frac{2}{9}R(2R+\rho) - \frac{2}{3}r(2R-r)\) (exact)	\(\frac{2}{9}R(2R+\rho) - \frac{2}{3}r(2R-r)\)	\(\frac{1}{9}\)
§ 60 (\(9\sum \overline{GS_i}^2\))	\(7\sum \overline{OS_i}^2 = 28R(4R-\rho)\) (slip)	\(16R(7R - \rho)\)	\(\frac{116855}{16}\)

2. Why § 57 (Feuerbach’s Theorem) and § 59 Are 100% Unscathed!

Why did the § 51 sign slip fail to corrupt § 57 or § 59? Look at the exact order of operations on page 38 (fb_058.jpg):

1. First, for the incenter \(S\) (where \(\overline{OS}^2 = 2r^2 - 2\rho R\) is 100% exact!), Apollonius’s median theorem \(2\overline{LS}^2 + 2\overline{OL}^2 = \overline{KS}^2 + \overline{OS}^2\) cancels \(-2\rho R\) against \(2\overline{OL}^2 = \frac{1}{2}R^2 - 2\rho R\), leaving \(LS = \frac{1}{2}R - r\) with \(\rho\) eliminated.

2. Second, rather than substituting his § 51 expression for \(\overline{OS'}^2\) into the median theorem (which would have left a spurious \(+4\rho R\)), Feuerbach substituted \(a, b, c \mapsto -a, -b, -c\) directly into \(LS = \frac{1}{2}R - r\), where \(\rho\) was already gone! Thus \(R \mapsto -R\) and \(r \mapsto r^{(k)}\) yielded the completely exact \(LS^{(k)} = \frac{1}{2}R + r^{(k)}\), and squaring those exact expressions in § 59 gave the exact sum \(13R^2 - 4\rho R\) (which also validates \(\sum \overline{OS_i}^2 = 16R(R - \rho)\) via \(2\sum \overline{LS_i}^2 + 8\overline{OL}^2 = \sum \overline{KS_i}^2 + \sum \overline{OS_i}^2\)!).

3. Compositorial Typo in § 55 (p. 37) and Nine-Point History

• p. 37, § 55, line 6: Prints OL^2 = 1/2 R^2 - rho R where § 54 (three lines above on p. 37) established OL^2 = 1/4 R^2 - rho R; the immediately following deduction LP^2 - OL^2 = OP . Kc = rho R uses the true 1/4 R^2 - rho R.

• From Six Points to Nine Points (1821–1842): In §§ 54–56 (and synthetically in 9. Satz), Feuerbach explicitly proves concyclicity of the six points (three altitude feet + three side midpoints), while Brianchon and Poncelet (1821) independently proved concyclicity of all nine points (adding the three Euler midpoints E₁, E₂, E₃ of \(AO, BO, CO\)), and Olry Terquem (1842) coined the name cercle des neuf points (nine-point circle). Notably, Feuerbach already knew \(AO = 2\,Ka\) (§ 47) and \(L = \operatorname{mid}(KO)\) (§ 55), from which \(LE_a = LK_a\)? Rather, \(LE_a = \frac{1}{2}AK = \frac{1}{2}R\) follows in a single line by midline homothety!